A Study of Spatio-Temporal Spread of
Infectious Disease:SARS
Afia Naheed
A thesis submitted in fulfilment of requirements for the Degree of
Doctor of Philosophy
Mathematics Department
Faculty of Science Engineering and Technology
Swinburne University of Technology
Australia
2015
Dedicated
To my loving parents and husband, my source of strength and inspiration
i
Abstract
This thesis is based on using three different types of deterministic compart-
mental epidemic models to investigate the transmission dynamics of Severe Acute
Respiratory Syndrome (SARS). These models are represented by ordinary and
partial differential equations, with the inclusion of a reaction-diffusion system. The
first model assumes Susceptible-Exposed-Infected-Diagnosed-Recovered (SEIJR)
populations, whereas the second and third models are extensions of the first model,
with treatment and quarantine compartments added. For different initial popula-
tion distributions the system of differential equations, representing different com-
partments, has been solved in the presence of diffusion in the first three compart-
ments i.e susceptible, exposed and infected. In this study the effects of diffusion on
SARS transmission are investigated as are the effects of some intervention strate-
gies. It is shown that diffusion and initial population distribution play crucial roles
in disease transmission. Then, the same system is solved numerically with diffusion
in the susceptible, exposed and infected compartments and with cross-diffusion in
the susceptible and exposed compartments for different cases. Using clinical and
demographic information for SARS, the SEIJR model is further extended to
a Susceptible-Exposed-Infected-Diagnosed-Treated-Recovered (SEIJTR) model.
The SEIJTR model’s parameters are estimated, again using available field data
on the 2003 SARS epidemic in Hong Kong. After that, model parameters are
analysed for sensitivity and uncertainty. Three different techniques are used to per-
form the sensitivity analysis. The effect of the treatment compartment on SARS
transmission is then numerically studied. Stability analyses of steady state and
treatment-reduced basic reproduction number are performed. Studies show that
availability of treatment can reduce infection significantly. Finally, a quarantine
compartment is added to the SEIJTR model in order to study the effects of quar-
antine on SARS transmission. The results for this extended SEQIJTR model are
compared with those for the SEIJTR model in which only isolation and treatment
but no quarantine measures, are used as intervention. The investigations show that
the presence of quarantine measures effectively reduces disease transmission.
ii
Acknowledgements
I would like to express the deepest appreciation to my coordinating supervisor
Dr. Manmohan Singh for his extreme kindness, extraordinary guidance, intellec-
tual and moral support through out all stages of my doctorate. I am very grateful
to him for introducing me to exciting subject of Mathematical Biology. Without
his thoughtful attention, vast knowledge and persistent help this thesis would not
have been possible.
I’m also grateful to my associate supervisors Dr. David Lucy and Mr. David
Richards for their thoughtful attention, guidance and invaluable suggestions. Their
constructive criticism, valuable comments and constant encouragement have re-
markably improved the quality of my thesis. I have become a stronger writer with
their kind help in editing my written pieces. My sincere thanks goes to Dr. Md
Samsuzzoha for his help and fruitful discussions during the completion of this work.
My special gratitude to Professor Geoffrey Brooks, Professor Billy Todd, and other
staff from Mathematics Department, Faculty of Science, Engineering and Technol-
ogy, for their kind help and friendly attitude during the preparation and completion
of the thesis.
I would like to thank Swinburne University of Technology for providing me Swin-
burne University Postgraduate Research Award (SUPRA) to carry out PhD. It
has been challenging, enjoyable and worthwhile time. I am deeply indebted to the
Faculty of Science Engineering and Technology for their technological support.
My sincere thanks to my amazing friends and coleagues for their love, care, advice,
and support through out my doctorate. I would like to share a special thanks to
Afsana Ahmed, Fatemeh Mekanic, Hou Wen, Qudsia Arooj, Rashida Bashir, Shab-
nam Sabah and Wajeehah Aayeshah for always being available for me and offering
the positive words and advice to keep me afloat .
Finally and most importantly, I would like to extend my gratitude to my wonder-
ful family (regular and new). Thank you mama and papa for your love, unlimited
support and inspiration of all my goals in life. You taught me to love in silence
and instilled in me the strength and confidence to continue in the way of success
to make my dreams come true. A big thanks to my siblings who have always been
iii
the best friends to me and my inlaws who gave me a home away from home. I have
no words to thanks my dear husband for his love, support and encouragement in
the journey of making this work possible.
iv
Declaration
The candidate hereby declares that the work in this thesis, presented for the Degree
of Doctorate in Mathematics submitted in the faculty of Science, Engineering and
Technology, Swinburne University of Technology:
1. is that the candidate alone and has not been submitted previously, in whole
or in part, in respect of any other academic award and has not been published
in any form by other person except where due references are given, and
2. has been carried out during the period from March 2011 to February 2015
under the supervision of Dr. Manmohan Singh, Dr. David Lucy and Mr.
David Richards.
—————————
Afia Naheed Date: February, 2015
v
Contents
1 Summary of the thesis 1
1.1 Research overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Literature Survey 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Mathematical epidemiology a brief look at history . . . . . . . . . . 9
2.3 The case of severe acute respiratory syndrome . . . . . . . . . . . . 17
2.3.1 The outbreak . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.2 SARS virus . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.3 Symptoms of SARS . . . . . . . . . . . . . . . . . . . . . . 22
2.3.4 Transmission route . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.5 Mathematical modeling of SARS . . . . . . . . . . . . . . . 24
3 Numerical Study of SARS Model with the Inclusion of Diffusion
in the System 28
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 The SEIJR epidemic model . . . . . . . . . . . . . . . . . . . . . . 29
3.2.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.2 Initial and boundary conditions . . . . . . . . . . . . . . . . 30
3.3 Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.1 Reproduction number and disease-free equilibrium (DFE) . 34
3.4.2 Disease-free equilibrium and stability analysis . . . . . . . . 36
3.4.3 Stability of endemic equilibrium without diffusion . . . . . . 37
vi
3.4.4 Stability of endemic equilibrium with diffusion . . . . . . . . 38
3.4.5 Excited mode and bifurcation value . . . . . . . . . . . . . . 40
3.5 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5.1 Solutions of SEIJR model without diffusion (Case 1) . . . . 42
3.5.2 Solutions of SEIJR model with diffusion (Case 1) . . . . . 47
3.5.3 Other cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Numerical Simulation of Cross Diffusion on Transmission Dynam-
ics of SARS 57
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 The SEIJR epidemic model . . . . . . . . . . . . . . . . . . . . . . 59
4.2.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.2 Initial and boundary conditions . . . . . . . . . . . . . . . . 61
4.3 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.1 Reproduction number and disease-free equilibrium (DFE) . 62
4.3.2 Stability of endemic equilibrium with cross-diffusion . . . . . 63
4.3.3 Reproduction number with diffusion . . . . . . . . . . . . . 65
4.3.4 Excited mode and bifurcation value . . . . . . . . . . . . . . 66
4.4 Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.5 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5.1 Numerical solution for initial condition (i) . . . . . . . . . . 69
4.5.2 Numerical solution for initial condition (ii) . . . . . . . . . . 72
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5 Parameter Estimation with Uncertainty and Sensitivity Analysis
for the SARS Outbreak in Hong Kong 86
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2.1 Reproduction number RIT for SARS . . . . . . . . . . . . . 92
5.2.2 Epidemiological data . . . . . . . . . . . . . . . . . . . . . . 92
5.2.3 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . 92
vii
5.2.4 Validation statistics . . . . . . . . . . . . . . . . . . . . . . . 94
5.3 Uncertainty and sensitivity analysis . . . . . . . . . . . . . . . . . . 97
5.3.1 Sensitivity indices of RIT . . . . . . . . . . . . . . . . . . . 99
5.3.2 Partial rank correlation coefficient (PRCC) . . . . . . . . . 101
5.3.3 Factor prioritization by reduction of variance . . . . . . . . . 107
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6 Numerical Study of SARS Model with Treatment (SEIJTR) and
Diffusion in the System 112
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.2 SEIJTR epidemic model . . . . . . . . . . . . . . . . . . . . . . . 113
6.2.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2.2 Initial and boundary conditions . . . . . . . . . . . . . . . . 114
6.3 Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.4 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.4.1 Disease-free equilibrium (DFE) . . . . . . . . . . . . . . . . 121
6.4.2 Endemic equilibrium without diffusion . . . . . . . . . . . . 122
6.4.3 Endemic equilibrium with diffusion . . . . . . . . . . . . . . 124
6.4.4 Excited mode and bifurcation value . . . . . . . . . . . . . . 126
6.5 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.5.1 Solutions of SEIJTR model in the absence of diffusion (Case
1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.5.2 Solutions of SEIJTR model in the presence of diffusion
(Case 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.5.3 Other cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7 Simulating the Effect of Quarantine on Isolation Treatment Model
for SARS Epidemic 144
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.2 The SEQIJTR epidemic model . . . . . . . . . . . . . . . . . . . . 147
7.2.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
viii
7.2.2 Initial and boundary conditions . . . . . . . . . . . . . . . . 151
7.2.3 Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . 152
7.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.3.1 Reproduction number without diffusion . . . . . . . . . . . . 154
7.3.2 Disease-free equilibrium (DFE) . . . . . . . . . . . . . . . 155
7.3.3 Endemic equilibrium without diffusion . . . . . . . . . . . . 157
7.3.4 Endemic equilibrium with diffusion . . . . . . . . . . . . . . 159
7.3.5 Reproduction number with diffusion . . . . . . . . . . . . . 161
7.3.6 Excited mode and bifurcation value . . . . . . . . . . . . . . 164
7.4 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.4.1 Numerical solution without diffusion . . . . . . . . . . . . . 166
7.4.2 Numerical solution with diffusion . . . . . . . . . . . . . . . 170
7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8 Conclusions 177
Appendix i
List of Publications and Conference Presentations xxxiv
ix
List of Figures
2.1 Emerging and re-emerging infectious diseases in the world. Red de-
notes newly emerging diseases, blue, re-emerging/resurging diseases,
black, a deliberately emerging disease [69]. . . . . . . . . . . . . . . 7
2.2 Annual deaths worldwide due to infectious diseases [236]. . . . . . 8
2.3 Sever acute respiratory syndrome (SARS) a deadly threat [124]. . . 19
2.4 Recent patient of SARS-like disease [123]. . . . . . . . . . . . . . . 20
2.5 SARS corona-virus under the microscope [122]. . . . . . . . . . . . 21
2.6 Symptoms of SARS [117]. . . . . . . . . . . . . . . . . . . . . . . . 22
2.7 Transmission route of SARS virus [119]. . . . . . . . . . . . . . . . 23
3.1 Initial conditions (i)− (iv). . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Determination of first excited mode with β as an unknown parameter. 41
3.3 Solutions with initial condition (i) and without diffusion. . . . . . . 43
3.4 Solutions with initial condition (ii) and without diffusion. . . . . . . 44
3.5 Solutions with initial condition (iii) and without diffusion. . . . . . 45
3.6 Solutions with initial condition (iv) and without diffusion. . . . . . 46
3.7 Solutions with initial condition (i) and with diffusion. . . . . . . . . 47
3.8 Solutions with initial condition (ii) and with diffusion. . . . . . . . 48
3.9 Solutions with initial condition (iii) and with diffusion. . . . . . . . 49
3.10 Solutions with initial condition (iv) and with diffusion. . . . . . . . 50
4.1 Initial Conditions (i) and (ii). . . . . . . . . . . . . . . . . . . . . . 62
4.2 Determination of first excited mode with β as an unknown parameter. 67
4.3 Solutions with initial condition (i) for Case (a). . . . . . . . . . . . 72
4.4 Solutions with initial condition (i) for Case (b). . . . . . . . . . . . 73
x
4.5 Solutions with initial condition (i) for Case (c). . . . . . . . . . . . 74
4.6 Solutions with initial condition (i) for Case (d). . . . . . . . . . . . 75
4.7 Solutions with initial condition (ii) for Case (a). . . . . . . . . . . . 78
4.8 Solutions with initial condition (ii) for Case (b). . . . . . . . . . . . 79
4.9 Solutions with initial condition (ii) for Case (c). . . . . . . . . . . . 80
4.10 Solutions with initial condition (ii) for Case (d). . . . . . . . . . . . 81
5.1 SARS infected incidence data, Hong Kong 2003. . . . . . . . . . . . 93
5.2 Predicted model of SARS. . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 The Model fitted to the data for the infected individuals : Initial (a)
and final fit (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.4 Residual (a) and residual correlation (b). . . . . . . . . . . . . . . . 97
5.5 Plots of probability distributions for all parameters generated with
10, 000 sample size. . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.6 Scatter plots for the basic reproduction number and eight sampled
input parameters values with 10, 000 random samples. . . . . . . . . 103
5.7 Scatter plots for the basic reproduction number and eight sampled
input parameters values with 10, 000 LHS samples. . . . . . . . . . 104
5.8 PRCCs for the full range of parameters from Table 5.6 for LHSb1 =
10, 000 (a) and RSa1 = 10, 000 (b). . . . . . . . . . . . . . . . . . . 105
5.9 PRCCs for the full range of parameters from Table 5.6 LHSb2 =
20, 000 (a) and RSa2 = 20, 000 (b). . . . . . . . . . . . . . . . . . . 106
5.10 Pie chart of factor prioritization sensitivity indices LHSb1 and RSa1. 108
5.11 Pie chart of factor prioritization sensitivity indices LHSb2 and RSa2. 109
6.1 Initial conditions (i) and (ii). . . . . . . . . . . . . . . . . . . . . . 118
6.2 Initial condition (iii). . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.3 Determination of first excited mode with β as an unknown parameter.127
6.4 Solutions for initial condition (i) without diffusion. . . . . . . . . . 130
6.5 Solutions for initial condition (ii) without diffusion. . . . . . . . . . 132
6.6 Solutions for initial condition (iii) without diffusion. . . . . . . . . . 133
6.7 Solutions for initial condition (i) with diffusion. . . . . . . . . . . . 135
xi
6.8 Solutions for initial condition (ii) with diffusion. . . . . . . . . . . . 136
6.9 Solutions for initial condition (iii) with diffusion. . . . . . . . . . . 138
7.1 Initial conditions (i) and (ii). . . . . . . . . . . . . . . . . . . . . . 152
7.2 Determination of first excited mode with β as an unknown parameter.165
7.3 Solutions for initial condition (i) without diffusion. . . . . . . . . . 168
7.4 Solutions for initial condition (ii) without diffusion. . . . . . . . . . 169
7.5 Solutions for initial condition (i) with diffusion. . . . . . . . . . . . 172
7.6 Solutions for initial condition (ii) with diffusion. . . . . . . . . . . . 173
1 Bifurcation diagram for β without diffusion for Case(1)-(4) . . . . . x
2 Bifurcation diagram for β with diffusion Case(1)-(4) . . . . . . . . . xi
3 Bifurcation diagram for γ1 without diffusion Case(1)-(4) . . . . . . xi
4 Bifurcation diagram for γ1 with diffusion Case(1)-(4) . . . . . . . . xii
5 Bifurcation diagram for γ2 without diffusion Case(1)-(4) . . . . . . xii
6 Bifurcation diagram for γ2 with diffusion Case(1)-(4) . . . . . . . . xiii
7 Bifurcation diagram for β for Case(a)-(d) . . . . . . . . . . . . . . . xv
8 Bifurcation diagram for γ1 for Case(a)-(d) . . . . . . . . . . . . . . xvi
9 Bifurcation diagram for γ2 for Case(a)-(d) . . . . . . . . . . . . . . xvi
10 Bifurcation diagrams for without diffusion . . . . . . . . . . . . . . xxvi
11 Bifurcation diagrams with diffusion . . . . . . . . . . . . . . . . . . xxvii
12 Bifurcation diagrams without diffusion . . . . . . . . . . . . . . . . xxxii
13 Bifurcation diagrams with diffusion . . . . . . . . . . . . . . . . . . xxxiii
xii
List of Tables
3.1 Interpretation of parameters (per day) . . . . . . . . . . . . . . . . 30
3.2 Values of LHS Routh-Hurwitz criterion of equilibrium without dif-
fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Values of LHS Routh-Hurwitz criterion of equilibrium with diffusion 40
3.4 Bifurcation value of β . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.5 Bifurcation values of γ1 andγ2 . . . . . . . . . . . . . . . . . . . . . 41
3.6 Four cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.7 Peak values of susceptible (S) and exposed (E) (without diffusion) 52
3.8 Peak values of infected (I) and recovered (R) (without diffusion) . . 52
3.9 Peak values of susceptible (S) and exposed (E) (with diffusion) . . 53
3.10 Peak values of infected (I) and recovered (R) (with diffusion) . . . 53
3.11 Peak values of infected at t = 20 . . . . . . . . . . . . . . . . . . . 54
4.1 Interpretation of parameters (per day) . . . . . . . . . . . . . . . . 60
4.2 Cases for cross-diffusion . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3 Routh-Hurwitz criteria with and without cross-diffusion . . . . . . . 65
4.4 Reproduction number . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5 Bifurcation value of β, γ1 and γ2 with cross-diffusion . . . . . . . . 67
5.1 Biological definition of parameters and state variables . . . . . . . . 91
5.2 Estimated parameters value for model . . . . . . . . . . . . . . . . 95
5.3 Covariance relations among parameters of SEIJTR model . . . . . 99
5.4 Parameters’ sensitivity analysis . . . . . . . . . . . . . . . . . . . . 100
5.5 Probability distribution functions (PDF ) for parameters . . . . . . 102
5.6 Estimates of partial rank correlation coefficients . . . . . . . . . . . 105
xiii
5.7 Percentage values of sensitivity index based on reduction of variance 108
6.1 Biological definition of parameters . . . . . . . . . . . . . . . . . . . 115
6.2 Parameters’ value for model . . . . . . . . . . . . . . . . . . . . . . 116
6.3 Values of LHS of Routh-Hurwitz criteria of equilibrium without dif-
fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.4 Values of LHS of Routh-hurwitz criteria of equilibrium with diffusion126
6.5 Bifurcation values of β and α. . . . . . . . . . . . . . . . . . . . . . 127
6.6 Peak values of susceptible(S), exposed(E), infective(I) and recovered(R)
(without diffusion) . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.7 Peak values of susceptible(S), exposed(E), infective(I) and recovered(R)
(with diffusion) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.8 Basic reproduction number RIT . . . . . . . . . . . . . . . . . . . . 142
7.1 Biological definition of parameters . . . . . . . . . . . . . . . . . . 149
7.2 Parameters values for model . . . . . . . . . . . . . . . . . . . . . . 150
7.3 Values for Routh-Hurwitz criteria of equilibrium . . . . . . . . . . . 159
7.4 Value of reproduction number (without diffusion) . . . . . . . . . . 164
7.5 Bifurcation value of influential parameters . . . . . . . . . . . . . . 165
7.6 Peak values for initial condition (ii) without diffusion . . . . . . . . 167
7.7 Peak values for initial condition (ii) without diffusion . . . . . . . . 171
xiv
Chapter 1
Summary of the thesis
1.1 Research overview
The main aim of this thesis is to understand the spatio-temporal spread of Severe
Acute Respiratory Syndrome (SARS) through the numerical study of the factors
related to the transmission dynamics of the disease in order to develop effective
control measures and strategies for its better control. SARS is an infectious disease
caused by a corona virus, called SARS associated corona virus (SARS − Cov).
At first it was thought to be transmitted by close person-to-person contact most
readily by droplet and airborne spread [210], but later many studies showed that
it can also spread through faecal-oral transmission [191, 226]. It was first reported
in Asia in February 2003 and with-in the next few months, the illness spread to
more than two dozen countries in North America, South America, Europe and
Asia. According to the World Health Organization (WHO) [238], a total of 8, 450
people worldwide became sick with SARS during the 2003 outbreak and of these,
774 died. In 2004 and 2005 many separate outbreaks emerged in Taiwan, China
and Singapore due to the accidental release of virus from laboratories or infected
animals. Recently, in 2012, a SARS−like disease named afterwards as MERS or
Middle East Respiratory Syndrome was reported [248]. All this shows the danger
of appearance of such deadly disease anytime in future. There is no specific treat-
ment or vaccine available for SARS till now. So to control any future outbreak
of such infectious disease, it is crucial to understand its transmission dynamics
1
and factors involved in aggravating its transmission. Mathematical modelling of
SARS has played a crucial role in understanding and predicting the transmission
of SARS for different parts of world. This has also helped in applying medical and
non medical intervention in order to control not only SARS, but also any deadly
new emerging infectious disease [31, 39, 40, 67, 86, 102, 153, 191, 101, 212, 233,
231, 241, 243, 249, 239, 233, 240, 245, 241, 239, 240, 245].
The focus of this thesis is to study the transmission dynamics of SARS epidemic
as well as the efficiency of different control strategies. Different initial conditions
are used to study the disease transmission with different population distributions.
The operator splitting method has been used to solve the differential equations
governing the system under investigation. The bifurcation effects for the most sen-
sitive parameters in all models have been investigated. In order to investigate the
effects of cross and diffusion on transmission dynamics, the differential equations
have been solved with and without diffusion in the system. Different cases have
been constructed in order to investigate the medical (treatment) and non medical
(quarantine, diagnoses and isolation) interventions for the disease. The nature of
the periodic solutions in case of cross diffusion have been studied. In order to study
the effect of quarantine, isolation and treatment, three different models with the
addition of three different classes or compartment J (diagnosed and isolated), T
(treatment) and Q (quarantined) in the basic SEIR model have been formulated.
The main threshold parameter, the reproduction number, R0 is calculated for all
models. R0 is calculated both with self and cross diffusion for SEIJR model. In
order to re-estimate the parameters for SEIJTR model, field data from the Hong
Kong SARS epidemic 2003, has been used.
Main aims of this thesis are to:
• investigate the effect of diagnosis and isolation on the spread of SARS,
• study the effects of self and cross diffusion on the transmission dynamics of
the SARS outbreak 2003,
• estimate the parameters for the SARS epidemic in Hong Kong based on field
data,
2
• determine the sensitivity and uncertainty in the model parameters in order
to examine the influence of parameters on the spread of disease,
• study the effects of treatment in controlling SARS disease,
• investigate the effects of quarantine in controlling SARS disease.
MATHEMATICA, MATLAB and Sampling and Sensitivity Analysis Tools (SaSAT )
are the softwares that have been used for the numerical calculations. LATEX and
WinEdt are used for the preparation of this thesis.
1.2 Thesis outline
This thesis is divided into eight chapters and an appendix.
Main aims of this thesis are:
• Chapter 1. Summary of the Thesis
This chapter contains an introduction to the work done in this thesis.
• Chapter 2. Literature Survey
This chapter reviews the history of mathematical biology with mathemati-
cal modelling of infectious diseases. A review of Severe Acute Respiratory
Syndrome (SARS) epidemic and mathematical models that represent the
transmission dynamics of SARS is given here.
• Chapter 3. Numerical Study of SARS Epidemic Model with the
Inclusion of Diffusion in the System
This chapter discusses the numerical study of population model based on
the epidemics of Severe Acute Respiratory Syndrome (SARS). The SEIJR
(Susceptible, Exposed, Infected, Diagnosed, Recovered) model of SARS epi-
demic is considered with net inflow of individuals into a region. Transmission
of the disease is analyzed by solving the system of differential equations using
numerical methods with different initial population distributions. The effect
of diffusion on the spread of disease is examined. Stability is established for
3
the numerical solutions. The reproduction numbers for all possible cases with
and without diffusion are estimated and compared. Effects of interventions
(medical and non medical) are also analyzed.
• Chapter 4. Numerical Simulation of Cross-Diffusion on Transmission
Dynamics of SARS Epidemic
This chapter describes the numerical study of spatial distribution dynam-
ics of SARS epidemic under the influence of self and cross-diffusion using
an SEIJR (Susceptible, Exposed, Infected, Diagnosed, Recovered) compart-
mental model. For both self-diffusion and cross-diffusion, the nonlinear par-
tial differential system is solved using the operator splitting and forward
difference technique. Criteria for local stability are obtained and the effect of
cross-diffusion on stability is also analysed.
• Chapter 5. Parameter Estimation with Uncertainty and Sensitivity
Analysis for the SARS Outbreak in Hong Kong
Parameter estimation is a key issue in systems biology, as it represents the
crucial step to obtain predictions from computational models of biological
systems. Based on the SARS epidemic of 2003 the parameters are esti-
mated using Runge-Kutta (Dormand-Prince pairs) and Least squares meth-
ods. Graphical and numerical techniques are used to validate the estimates.
The effect of the model parameters on the dynamics of the disease is examined
using sensitivity and uncertainty analysis.
• Chapter 6. Numerical Study of SARS Epidemic Model with Treat-
ment (SEIJTR) and Diffusion in the System
A numerical study of Severe Acute Respiratory Syndrome (SARS) using an
open population model consisting of six compartments i.e SEIJTR (Sus-
ceptible, Exposed, Infected, Diagnosed, Treated, Recovered) has been done
in this chapter. Different population distributions are used to generate the
numerical simulations for the disease in different situations. Stability of the
system is analysed. In order to observe the effect on disease transmission,
different cases are studied in the absence and presence of diffusion in the
4
system. The impact of treatment on the transmission of the disease is also
analysed
• Chapter 7. Simulating the Effect of Quarantine on Isolation Treat-
ment Model for SARS Epidemic
This chapter describes a compartmental model to predict the geographic
spread of infectious diseases in the presence of quarantine measures. The
model described in Chapter 6 with the inclusion of quarantine compartment is
numerically studied, in this chapter. Stability of the disease free and endemic
equilibria are analysed with the help of the Routh-Hurwitz criterion. The
reproduction number in the presence and absence of diffusion and quarantine
is estimated and analysed. Finally, the numerical simulations of the present
model are compared with the SEIJTR model discussed in Chapter 6.
• Chapter 8. Conclusions
This chapter concludes with the crucial findings and suggests opportunities
for future work.
5
Chapter 2
Literature Survey
2.1 Introduction
Through out human history, infectious diseases have been a serious cause of mor-
bidity and mortality. According to historians with-in 50 years of 1500, more than
half of the population of the American region reduced as a result of the spread
of infectious diseases [209]. Although the presence of infectious diseases is not ig-
norable at all times in human population but the epidemics are more noticeable
and striking. One of such major epidemic in history was Black Death, that caused
25 million causalities in Europe in 14th century [44]. According to Meltzer [168]
“It is often said that in the centuries after Columbus landed in the New World on
12 October 1492, more native Americans died each year from infectious diseases
brought by the European settlers than were born.” Small pox was the first disease
to appear in America in 1518 in Hispaniola, from where the disease spread rapidly
to Mexico in 1520 and then to the whole world. In the years of 1525 − 26, this
disease killed half of the population in Aztecs, Guatemala and the territories of
Incas. A few years later another infectious disease, influenza attracted the world’s
attention through a massive epidemic in 1558 − 59. With-in one year in 1919, 20
million people died due to influenza world wide [8]. Other than these epidemics,
typhus caused half a million deaths from 1918 to 1921 in Russia.
Although in the beginning of 21st century with the appearance of different med-
ical and non-medical interventions like vaccines, antibiotics, early quarantine and
6
Figure 2.1: Emerging and re-emerging infectious diseases in the world. Red denotes
newly emerging diseases, blue, re-emerging/resurging diseases, black, a deliberately
emerging disease [69].
isolation of infected, reduced the danger of infectious diseases and the world’s atten-
tion diverted to chronic diseases like diabetes, cancer, arthritis and cardiovascular
disease, in developed countries. On the other hand, infectious diseases not only con-
tinued to cause sufferings in developing countries but also their agents developed
and re developed, as a result many infectious diseases emerged, re-emerged and de-
liberately emerged as shown in Fig. 2.1). In a short period of nineteen years from
1975 to 1993, the world came across many deadly new infectious disease like Lyme
(1975), Legionnaires (1976) toxic-shock syndrome (1978), the sexually transmitted
disease Acquired Immunodeficiency Syndrome (AIDS) (1981), hepatitis C (1989),
hepatitis E (1990) and hantavirus (1993). Many diseases such as gonorrhea (sexu-
ally transmitted diseases), tuberculosis, pneumonia, malaria, dengue, yellow fever
reappeared while the epidemics of cholera, hemorrhagic fevers (Bolivian, Ebola,
Lassa, Marburg, etc.) [108] and SARS emerged as new infections. After bacte-
ria, helminths (worms), viruses and protozoa, a new agent called prions has joined
the group. Prions are considered to be the cause of spongiform encephalopathies
such as bovine Creutzfeldt-Jakob Disease (CJD), scabies in sheep, bovine spongi-
7
Figure 2.2: Annual deaths worldwide due to infectious diseases [236].
form encephalopathy (BSE, mad cow disease) and kuru [182]. The pandemics of
influenza in the years of 1918, 1957, 1968, 1977 and 2009 and Middle East respira-
tory syndrome (MERS) are a few of the infectious diseases that still threaten the
world.
The rate of mortality due to infectious diseases is higher in developing countries
[94], specifically as a major cause of child death in developing areas of the world. A
rough estimate of a study shows that three million children die every year because of
diarrhoeal diseases and malaria alone [236]. According to Morens [172] “ About 15
million (> 25%) of 57 million annual deaths worldwide are estimated to be related
directly to infectious diseases; this figure does not include the additional millions
of deaths that occur as a consequence of past infections (for example, streptococcal
rheumatic heart disease), or because of complications associated with chronic in-
fections, such as liver failure and hepatocellular carcinoma in people infected with
hepatitis B or C viruses.” Fig 2.2 gives a brief overview of worldwide causes of
deaths where infectious disease contribute to a major number of deaths. In this
situation, in addition to medicines epidemiological models are the most helpful
tools to improve the efficiency of control strategies by understanding the under
lying mechanism of infectious diseases and finally, can wipe out the infection from
population. The present chapter outlines the history of infectious diseases and his-
tory of mathematical modelling for disease dynamics. It also describes the story of
Sever Acute Respiratory Syndrome (SARS), the fatal epidemic which appeared in
2003 in the context of mathematical modelling.
8
2.2 Mathematical epidemiology a brief look at
history
MacMahon and Pugh [157] defined epidemiology as “ The study of the distribution
and determinants of disease frequency in man.” Epidemiology is a blend of four
objectives given as follows [109]:
• First objective is to define dispersion of disease in real world.
• Second objective is to determine the causes, risk factors and complexities
that effect the disease dynamics.
• Third objective is to develop and test theories.
• Fourth objective is to develop, maintain and apply control measures.
An infection in a community can be studied in the best possible way with the
help of its epidemiological information using mathematical models. According to
Nokes [179] “A major goal of theoretical or mathematical study in epidemiology
is to develop understanding of the interplay between the variables that determine
the course of infection within an individual, and the variables that control the pat-
tern of infections within communities of people. In view of the successes achieved
by combining empirical and theoretical work in the physical sciences, it is sur-
prising that many people still question the potential usefulness of mathematical
models in epidemiology.” If we look at the literature mathematical epidemiology
can be traced back more than two hundred years. the Great Plague of London
(1665− 1666) is said to be the first epidemic, that was investigated through mod-
elling techniques while in the 18th century a small pox model to measure the effect
of variolation for common public health was constructed and estimated by Daniel
Bernoulli [22] in 1760. This was followed by a long gap till the middle of 19th
century when the popular dynamical systems techniques were used and the deter-
ministic epidemiology era started. In 1840, Dr. William Farr [68] fitted a normal
curve model to small pox induced mortality data (1837-1839) for Wales and Eng-
land. The work done by Enko [55] between 1873 and 1894 is considered as the first
9
step to develop modern mathematical epidemiology. Dr John Brownlee continued
Farr’s work and developed theories for epidemic systems. He published excellent
work on epidemics, where in the beginning he relied on normal curves but after
analysing many epidemics he found that Pearson Frequency Distribution fits bet-
ter for epidemics. According to Brownlee “An epidemic is an organic phenomenon,
the course of which seems to depend on the acquisition by an organism of a high
grade of infectivity at the point where the epidemic starts, this infectivity being
lost from that period till the end of the epidemic at a rate approaching to the terms
of a geometrical progression.” His work is considered as an important contribution
to the statistical approach in mathematical modelling [28, 29, 30]. Mathematical
epidemiology is used to study the mechanism of transmission of infectious diseases.
There are several ways of transfer of infections for these diseases. Some of them
are transferred by viral agents such as influenza, ebola, measles, severe acute res-
piratory syndrome, rubella (German measles), rabies and chicken pox. Some are
transmitted through bacteria, for example tuberculosis, pneumonia, meningitis,
and gonorrhea while some are transferred through vectors such as West Nile virus
malaria, dengue, chagas disease, lyme disease and yellow fever etc.
According to O’Neill [181] “The models for infectious disease transmission essen-
tially subdivide into two categories, namely deterministic models and stochastic
models. The former are frequently defined via a system of ordinary or partial dif-
ferential equations, which describe how the numbers or proportions of individuals
in different states (susceptible, infective, etc.) evolve through time. An attractive
feature of deterministic models is that it is usually fairly straight forward to obtain
numerical solutions for a given set of parameter values. They are generally most
effective as descriptions of reality in large populations, where, roughly speaking,
laws of large numbers act to reduce the order of stochastic effects. Stochastic mod-
els are usually thought of as more realistic, although their mathematical analysis
is often much harder. They can capture the stochasticity seen in real-life disease
outbreaks, for example, the phenomenon of fade-out in endemic diseases. They
are generally defined at the level of individuals, for instance, specifying probability
distributions that describe the latent or infectious periods.” The most used branch
10
of deterministic modelling is compartmental models, where a population is divided
in number of compartments based on their epidemiological characteristics. Accord-
ing to Sattenspiel [206] “The compartmental models are mathematical models for
the spread of disease. The process of building a mathematical model begins with
a series of assumptions about how the disease process works and the development
of a simplified model to describe the process. As a consequence of the analysis
of the initial model, comparison with actual disease data, and evaluation of the
assumptions, the model is reformulated in a more realistic manner. In general, the
process of mathematical modeling proceeds from simple to complex.”
Deterministic epidemiological modelling based on compartmental models flourished
immensely between 1900 and 1935 with the research work of Sir Ross, Hamer, McK-
endrick and Kermack. Hamer [100] published his work on diseases transmission.
He formulated a simple mathematical model to study the transmission dynamics
of measles. He introduced the idea of “mass action principal” that is considered
as basic concept for epidemic models nowadays. Inspired by Hamer, Dr Ronald
Ross became interested in the prevalence and control of malaria and he started
working, in 1899, on (bird) malaria where he extended Hammer’s discrete time-
model to continuous-time model in order to study transmission of disease. In 1904,
Ross [194] published his paper about the irregular movements of mosquitoes. Ross
published many papers describing malaria transmission dynamics [195, 196]. In
these papers he developed his “theory of happenings” that is considered as the
basis of modern epidemic theory. Numerous epidemiological models were then de-
veloped by Ross and Hudson, Martini and Lotka [12, 53]. Based on the work of
Hamer and Ross, Kermack and McKendrick developed a theory for the spread of
infection through population and published three articles in 1927, 1932, and 1933
respectively [134, 135, 136] based on simple a compartmental model to study the
progress of an epidemic in homogenous close and open population and developed
the threshold theory. They explained that threshold value of a particular epidemic
depends on infectivity, recovery and death rates for that epidemic and also the
population density should exceed a certain threshold value for a diseases to spread
and become epidemic. This theory gave rise to the vaccination with an impact on
11
immunity, proving that in order to eliminate an infection, it is not necessary to
vaccinate the whole population. This was proved in the case of small pox in 1970,
where eradication of the virus was achieved with only 80% vaccine coverage world-
wide. They also explained that small change in the rate of infection might cause
a huge epidemic. Following the work of Hamer, Soper [215] published his work in
1929 on the periodicity of epidemics. He explained that the reemergence of measles
in any certain region depends on two major factors. First, the dreadful virus of the
disease and secondly, the number of susceptible population for that disease. All
these mathematical models were based on Hamer’s “mass action principal”.
Models on an alternative technique of this principal was developed by Lowell Reed
and Wade Hampton Frost [2, 159] in 1920 and was published in 1950. According
to Sattenspiel [206] “In this model, the transmission of infection is defined in terms
of a probability of effective contact rather than a proportion of contacts that result
in transmission. Effective contact is the type of contact necessary for transmission
of infection, not necessarily just casual contact.” Starting with Reed Frost [2, 159]
the importance of stochastic models for the transmission dynamics of infectious
diseases has been analysed in many ways while Bailey [12] and Becker [18] pro-
vided many examples of stochastic versions of SIR models.
Epidemiological modelling grew significantly after 20th century tremendously. The
first edition of Bailey’s book in 1957 [12] is considered as a significant milestone in
this regard. His scholarly review “The Mathematical Theory of Infectious Diseases”
documented 539 research papers on mathematical epidemiology in its bibliography
written between 1900 and 1973 and 336 of these articles were published in 1964-
1973. Anderson and May [8] made major contributions to the understanding of
population biology in infectious diseases. They investigated the relations between
transmission parameters of their models. They explained the direct and indirect
transmission of microparasites (viruses, bacteria and protozoa) and macroparasites
(helminths and arthropods) through intermediate hosts. A huge variety of mathe-
matical models for infectious diseases were developed in the 21st century. Most of
these models explained the concept of steady loss of vaccination, disease vectors,
mixing of population, age structure, spacial spread, self and cross-diffusion, pas-
12
sive immunity, disease acquired immunity, quarantine, isolation, stage of infection
chemotherapy and vector transmission [17, 32]. According to Hethcote [109] “An
advantage of mathematical modeling of infectious diseases is the economy, clarity
and precision of a mathematical formulation. A model using difference, differential,
integral or functional differential equations is not ambiguous or vague. Of course,
the parameters must be defined precisely and each term in the equations must be
explained in terms of mechanisms, but the resulting model is a definitive statement
of the basic principles involved. Once the mathematical formulation is complete,
there are many mathematical techniques available for determining the threshold,
equilibrium, periodic solutions, and their local and global stability. Thus the full
power of mathematics is available for the analysis of the equations. Moreover,
information about the model can also be obtained by numerical simulation on dig-
ital computers of the equations describing the model. The mathematical analyses
and computer simulations can identify important combinations of parameters and
essential aspects or variables in the model. In order to choose and use epidemio-
logical modeling effectively on specific diseases, one must understand the behavior
of the available formulations and the implications of choosing a particular formula-
tion. Thus mathematical epidemiology provides a foundation for the applications
[106, 107].”
Since 1983 HIV and AIDS are the two main sexually transmitted diseases. Ac-
cording to estimates in 2002, 45 million population got infected from AIDS and
HIV epidemics. Each year, there are almost five million infected and three mil-
lion deaths are recorded in Africa. In the United Kingdom syphilis, chlamydia
and gonorrhea have been reported in increasing rates in the last ten years [133].
This alarming situation drew the attention of epidemiologist to model the trans-
mission dynamics of these sexually transmitted diseases in order to find effective
control measures. On the other hand due to the lack of efficient computers in
early times mathematical methods developed were based on the idea of random
mixing of population that make the models simple enough to generate meaningful
results. However in reality, populations do not mix up randomly. The development
of non-random or heterogenous mixed population models started in mid 1970 and
13
refinement of compartmental models came into light with the paper by Cooke and
Yorke [49]. The heterogenous behavior was initially included in the risk structure
models of sexually transmitted diseases.
Structured models were used to study the transmission dynamics of various in-
fectious diseases like measles, influenza, smallpox and hepatitis. Haggett [98, 99]
developed structured model to investigate the spatial patterns in the spread of
measles outbreak and determined the importance of different spatial processes at
different stages for the development of the epidemic. Cliff et al. [47], developed
the static aspects of spatial structure, regional dynamics, spatial autocorrelation
and spatial forecasting for the mathematical models. Some structured models were
developed for Influenza by Bayroyan et al. [16], Rvachev and Longini [193], Longini
[156]. Sattenspiel [203], Sattenspiel and Simon [205] formulated structured models
for the transmission dynamics of hepatitis. Travis and Lenhart [225] established
structured model for infectious diseases in heterogenous populations for small pox.
They developed the essential conditions to eradicate the diseases with the help of
vaccination. Furthermore Andreasen and Christiansen [9] analysed effects of popu-
lation structure on infectious diseases. All these models were developed on the idea
that, human population is structured essentially in some manner. Thus random
mixing of population in a model is inappropriate for any disease. Another kind of
demographic model also appeared as age-structured models either with continuous
age models or models with age groups. These models were developed on the con-
cept of more interaction of people from the same age group as compared to different
age groups. So the difference between risk-structure (sexually transmitted disease)
and age-structure models is the inclusion of individuals age. The work done by
Anderson and May [6], Castillo-Chavez et al. [32], Dietz, [5], Dietz and Schenzle
[54] and Hethcote [107] contributed as foundation of age structured epidemiological
modeling. According to Hethcote “Indeed, some of the early epidemiology models
incorporated continuous age structure [22, 134]. Modern mathematical analysis
of age-structured models appears to have started with Hoppensteadt [115], who
formulated epidemiological models with both continuous chronological age and in-
fection class age (time since infection), showed that they were well posed, and found
14
threshold conditions for endemicity.”
The new term “Community epidemiology” has recently appeared in mathematical
epidemiology. It is based on the theoretical expansion of simple one-pathogene
one-host model where one pathogene is transferred to multiple host species. There
are two kind of multiple host models. Vector borne diseases such as malaria,
dengue fever and leishmania are among a few of the most dangerous and challeng-
ing multiple host diseases. Also more and more models are required to optimize the
control for macro-parasitic infectious diseases as avian influenza, hantavirus, Lyme
disease, bubonic plague, Q-fever, SARS, rabies, West Nile virus, toxoplasmosis,
trypanosomiasis etc. Ferguson et al. [73], in 1999, developed multiple host model
to investigate effect of antibody-dependent improvement in the multi pathogene
transmission dynamics. They predicted the chaotic behaviour of the epidemic with
the frequent antibody-dependent improvement. In 2002 Gog and Grenfell [93] de-
veloped a model to answer the fundamental question in strain dynamics where they
used specific example of Influenza while in 2003 Ferguson et al. [74], studied dif-
ferent strains of Influenza virus to investigate the factors effecting the variational
epidemiological dynamics patterns with the help of a multi-strain mathematical
model. According to Keeling [133] “ Models incorporating multiple pathogenes
allow us to investigate questions of disease evolution, from theoretical questions
such as understanding current disease behavior in terms of an optimal strategy for
transmission to more applied issues such as predicting the influenza strains for the
coming year or understanding the effects of strain specific control. Finally multi-
strain models offer insights into the increasing prevalence of drug resistant bacteria
and how to limit their control.”
Nowadays contact network models are also very popular not only in epidemiology
but also in various other branches of science. Networks have been used in modelling
to study the spread of infectious disease for the last fifty years [12, 53]. With the
availability of huge and accurate data, computational efficiency and improvement
in methodologies in the last twenty years, the theory of network modelling has been
widely used for the investigation of human diseases. Eubank et al. [65], Meyers et
al. [170], Bansal et al. [14], have successfully developed epidemiological models to
15
study several infectious diseases . According to Welch et al. [232], “The network-
based studies to date have largely focused on the impact of network structure on
disease dynamics and the effect of control strategies. Network structure has largely
been determined by collecting host data to inform probabilistic models of host in-
teractions, which are then used to generate simulated networks over which disease
spread can be studied. Network-based models provide an elegant alternative to
homogeneous-mixing models by intuitively capturing diversity in the underlying
patterns of interaction in a population.”
The effect of diffusion is extensively studied in epidemic of infectious diseases.
Spatial diffusion for measles was first studied by Soper [215] in 1929. His work
contributed to the foundation of the study of spatial and temporal aspects of epi-
demics. In 1975, Bailey [12], summarized some of such spatial diffusion complex
but highly idealized models. The book written by Cliff et al. [48], in the year 1981,
is considered as the most important addition in the study of spatial diffusion of
infectious diseases. Liu et al. [154], states that as “we do not know what kind of
epidemic outbreaks, when it outbreaks, and how it diffuses. Generally, after an epi-
demic outbreaks, public officials are faced with many critical and complex issues,
the most important of which is to make certain how the epidemic diffuses so that
the rescue operation efficiency is maximized.” There is a large number of diffusive
compartmental models [171, 218, 151, 250, 255]. Most recently Kim et al. [138],
examined a diffusive model to study the transmission of avian influenza among
human and birds. Their study suggests that by eradicating the infected birds and
by reducing the contact rate of susceptible and infected humans the danger of pan-
demic of influenza can be reduced. Samsuzzoha et al. [198], formulated an SEIR
diffusive compartmental model and numerically studied the transmission dynamics
of the 1918 Influenza pandemic with different initial population distributions. Also
Samsuzzoha et al. [199], developed SV EIR model in the presence of diffusion to
study the effect of vaccination on transmission dynamics of influenza. Very recently
Lui and Xiao [154] developed a susceptible-infected-susceptible epidemic diffusive
model to study the effect of population migration between two cities on the spread
of an epidemic. They showed that to control the diffusion of an epidemic in both
16
cities the migration of individuals in both cities needs to be controlled in same
manner.
According to Kramer et al. [142], “Preventing and reducing the spread of infectious
disease among humans is an essential function of public health. Epidemiology is
often called the core science of public health, which studies the distribution and de-
terminants of disease risk in human populations. Starting in the middle of the 19th
century, infectious disease epidemiology applies the fundamentals of epidemiology
to study infectious diseases and deals with questions about conditions for disease
emergence, spread and persistence. It describes the prevalence and incidence of
infectious diseases through which the epidemiological trends can be characterized
for different world regions.”
2.3 The case of severe acute respiratory syndrome
Severe acute respiratory syndrome also called SARS is a viral respiratory illness
caused by a coronavirus termed as SARS-associated coronavirus (SARS − CoV )
[131]. This epidemic is different from the conventional atypical pneumonia. The
SARS outbreak which happened in 2003, had no prior history like bird flu, human
enterovirus and Nipah virus. The person infected with SARS experience a severe
lack of oxygen and need medical aid for breathing. In addition the infection is
communicable and can infect massive number of individuals in some situations.
That is why world health organization (WHO) named it as severe acute respira-
tory syndrome. It is refereed as most unrivaled disease in the history of WHO
records. The SARS epidemic was hard to handle due to three main reasons [149]:
• Firstly, The symptoms of SARS are similar to common flu. So its usually
not possible to distinguish between SARS and flu patient. That’s why many
SARS patient were treated with antibiotics for common flu and sent back
home, spreading the disease more.
• Secondly, SARS virus can survive outside the human body for a few hours
which causes rapid transfer of the infection with close contact. That is why at
17
the beginning of SARS outbreak mostly family members, friends and health
workers of the infected individual were also infected.
• Thirdly, the incubation period of SARS is less than ten days, meaning it
not only spreads fast, it also kills faster. Also it is more dangerous for older
people as they have weaker immunity and also they suffer already with health
problems like high blood pressure, diabetes, and heart diseases.
2.3.1 The outbreak
It was in November 2002 when first case of SARS was actually diagnosed in
Guangzhou, the capital of a Chinese province Guangdong. Afterwards, many new
cases of SARS were diagnosed in Heyuan, Jianmen, Shengzhen, Zhaoqing and
Zhongshan between November 2002 and January 2003; the locations of these cities
following a major cluster in Guangzhou [255]. But Chinese government kept it
secret to maintain public confidence, and informed the World Health Organization
(WHO) about the SARS outbreak only in late February 2003, when SARS epi-
demic was beyond their control. Till 9 February 2003, 792 probable cases of SARS
were diagnosed with 31 deaths in China. At the end of the SARS outbreak in
China alone the total number of SARS probable cases reached the figure of 5, 327
with 343 deaths. China is one of the countries where the SARS outbreak hit the
hardest [254]. According to the assessment of a WHO report [237] “If SARS is not
brought under control in China, there will be no chance of controlling the global
threat of SARS. Achieving control of SARS is a major challenge especially in a
country as large and diverse as China.” According to Leung [149] “The disease
soon found its way to Hong Kong, and subsequently, to more than 20 countries
in the rest of the world. According to investigations carried out by Hong Kong
authorities, the territory’s first “index case” was a 64 year old doctor from China
who had treated SARS patients in Guangzhou. The doctor, who stayed at the
Metropole Hotel in Monk Kok on 21st February 2003, was admitted to hospital
with SARS symptoms on the next day, where he died on 4, March 2003. It was
later discovered that five other guests of the hotel who had stayed on the same floor
as the doctor (the ninth floor) also contracted the disease. Three of these guests
18
Figure 2.3: Sever acute respiratory syndrome (SARS) a deadly threat [124].
were female tourists from Singapore while the other two were Canadians. Experts
think that it is highly probable that five of the guests from the hotel contracted
the disease from the doctor, who brought the virus into Hong Kong from mainland
China.”
There is no doubt that SARS took real advantage of international air travel.
Modern fast transport system made the spread of SARS virus very easy and rapid.
Hitoshi Oshitani, [237] the WHO coordinator for SARS, calls this “the most sig-
nificant outbreak that has been spread through air travel in history”. According
to Leung [149] “In Hanoi, Vietnam, a 48 year old businessman from the US fell ill
and was admitted to the French Hospital on 26 February 2003, He had traveled to
Hong Kong and China before arriving in Vietnam, shortly after that, an outbreak
occurred in Vietnam. The outbreak in Singapore is believed to have been caused
by the three women who were infected in the Metropole Hotel in Hong Kong. After
returning to their home country, all three of them fell ill, were hospitalized and were
found to have contracted SARS. The SARS cases in Canada are linked to the
two Canadian tourists who stayed at the Metropole Hotel in Hong Kong. Similar
to what happened in Singapore, the disease was brought back to their home coun-
tries when they returned. Ontario was most badly affected, while British Columbia
saw a couple of cases. By the end of March 2003, the former had more than 40
cases of infection while the latter witnessed 2 cases. Both of the source patients
later succumbed to the illness and died. By mid-March 2003, numerous cases of
19
Figure 2.4: Recent patient of SARS-like disease [123].
SARS were reported in many other countries such as US, Taiwan, Thailand and
several European countries including France, Germany, Italy, Republic of Ireland,
Spain, Switzerland and the United Kingdom. Initial investigations revealed that
all the outbreaks had origins in Asia. The viral carrier, when that could be traced,
was almost always someone who had made a recent trip to China or Hong Kong.”
According to the estimates of WHO [238] between November 2002 and June 2003,
8, 450 people were infected. Among those infected, 21% were health care workers.
There were 774 deaths in 33 countries over the five continents. Outside Asia the
largest outbreak happened in Canada. After June 2003 and again in the start of
2004 many cases of SARS were reported either by the individuals who were in con-
tact with the SARS viruses in laboratories or with the animals carring SARS-like
virus.
Recently a SARS-like virus was diagnosed in humans. According to the director-
general of the United Nations this virus is “a threat to the entire world”. At
first the virus was named as novel coronavirus showing the same symptoms as
SARS virus [123]. According to Chris et al. [43], “The emergence in 2012 of a
new disease-causing coronavirus has generated substantial concern. As of June 26,
2013, middle east respiratory syndrome coronavirus (MERS-CoV ) had caused 77
laboratory-confirmed cases and 40 deaths. The virus is related to the Severe Acute
Respiratory Syndrome coronavirus (SARS-CoV ) that emerged in 2002-03 and as
SARS-CoV had during its prepandemic stage, MERS-CoV has probably been
20
Figure 2.5: SARS corona-virus under the microscope [122].
transmitted from an unknown animal host to human beings repeatedly in the past
year. Cases of human-to-human transmission have also been documented in several
countries.”
2.3.2 SARS virus
Corona-virus belongs to a large class of viruses that cause common cold or minor
respiratory problems. In some severe cases they can also cause pneumonia or acute
respiratory distress syndrome (ARDS) [1]. The exact origin of corona-virus is still
not known, but it is believed that transmission of corona-virus to humans may
happened through a cat-like mammal called civets. SARS first appeared in the
Chinese province of Guangdong and this region is rich in its number of civets. A
similar SARS-like virus is found in Horseshoe bats. There is a possibility that these
bats transferred the virus to civets that infected the humans with SARS disease.
As a result of this discovery civets were banned and slaughtered in Guangdong
province. According to Leung [149] “Coronaviruses, so called because of their
spiky crown of protein globules, are generally not mortally harmful. They are a
pest to livestock, and in humans are responsible for more than one-third of common
cold cases. But in this case, researchers believe that the bugs have mutated into
something far deadlier as a rogue virus that triggers a killer pneumonia, now widely
known as SARS. The new coronavirus was isolated in Vero E6 cells from nasal and
throat swab specimens of two patients in Thailand and Hong Kong with suspected
21
Figure 2.6: Symptoms of SARS [117].
SARS. The isolate was identified initially as a coronavirus by electron microscopy
(EM). The little hooks sticking out of the viral body are the telltale characteristics
that help.”
2.3.3 Symptoms of SARS
Prompt identification of the features of SARS − CoV , is not yet available, so the
diagnosis of SARS disease is based on the existence of clinical symptoms and the
evidence of exposure to an infected SARS patient. If a person meets these two cri-
teria then he/she have been termed as ’probable’ case of SARS. Initially flu, cough,
fever, chills, loss of appetite, muscle aches and sore throat can be the symptoms of
SARS. According to the report of Center for Disease Control (CDC) [210] “In gen-
eral, SARS begins with a high fever (temperature greater than 100.4F [> 38.0C]).
Other symptoms may include headache, an overall feeling of discomfort, and body
aches. Some people also have mild respiratory symptoms at the outset. About 10
percent to 20 percent of patients have diarrhea. After 2 to 7 days, SARS patients
may develop a dry cough. Most patients develop pneumonia.” The usual changes
in chest X-ray were irregular fortification leading to bilateral bronchopneumonia
for five to ten days. In most cases symptoms of the disease appear after 3 to 17
22
Figure 2.7: Transmission route of SARS virus [119].
days of latent period, and most of the patients recover within two weeks. The rate
of mortality for SARS was over all 15% where as among the elderly this rate was
higher 50% [95].
2.3.4 Transmission route
The transmission route of severe acute respiratory syndrome (SARS) is quite un-
certain. In some mathematical models it is discovered to be transmitted by close
person-to-person contact similar to Influenza usually where droplets of respiratory
secretions transfer the virus from one person to another. When an infected person
coughs or sneezes these droplets can travel over 3 feet through air and fall on the
mucous membrane of nose, eyes or mouth of surrounding individuals. This trans-
mission is also possible if a person touches his nose or mouth or eye after touching
a contaminated surface. The report of the Center for Disease Control (CDC) [210]
defines the close person-to-person contact as “In the context of SARS, close con-
tact means having cared for or lived with someone with SARS or having direct
contact with respiratory secretions or body fluids of a patient with SARS. Exam-
ples of close contact include kissing or hugging, sharing eating or drinking utensils,
talking to someone within 3 feet and touching someone directly. Close contact does
not include activities like walking by a person or briefly sitting across a waiting
room or office.” On the other hand in some cases the airborne spread of SARS
virus is also identified [126]. Some studies even suspect the fecal-oral transmission
23
for the SARS virus [191, 226].
2.3.5 Mathematical modeling of SARS
When the SARS outbreak happened in 2003, there were no vaccines or specific
treatment available, this situation still holds true today. Quarantine and isolation
were the only means to restrict or control and predicting the disease transmission.
Also many basic questions regarding the number of new infections, its peak, the
time of peak arrival ,and the intensity and length of its peak needed answers to
develop the control policies [191]. In this situation mathematical models facili-
tated the researchers with the tools to provide best answers to these important
questions. Various mathematical models were formulated and solved to study the
SARS transmission dynamics in order to predict its behavior and to plan control
policies. The mathematical models provide the precise quantitative details of the
effect of control strategies. As quarantine and isolation were the first non-medial
interventions used for the prevention of SARS and thus they appeared quite early
in the mathematical modelling of SARS. SARS models of quarantine measures
showed many surprising results. Lipsitch et al. [152], established deterministic
and stochastic mixed homogenous compartmental models to study the effect of
isolation and quarantine on the transmission dynamics of SARS in Hong Kong
and Singapore. They showed that more than one control measure is needed to
control a SARS outbreak that also effect the reproduction number and reduce the
chances of developing outbreak. They also found that the average time of quar-
antine can be reduced by an quarantining a large number of individuals around
infected person under a certain threshold. Lloyd-Smith et al. [155], formulated
a stochastic compartmental model to study the effect of different control mea-
sures. Their studies shows that the increase in the reproduction number of SARS
strongly depends on efficient quarantine and the process is nonlinear. They also
found that applying quarantine strategy to people in hospital is much more effec-
tive than applying it to diagnosed people. In 2003 Riley et al. [191], formulated
a stochastic matapopulation compartmental model to estimate the reproduction
number for SARS for Hong Kong without superspreading events (SSEs). They
24
concluded that implementation of control measures and restricting the contact and
movement of individuals in a region can lead to a slow down of the transmission of
the disease.
Several models of SARS studies show that if the duration between onset of symp-
toms and isolation of diagnosed is reduced some how, it can greatly effect the
transmission of SARS and can be helpful in controlling the outbreak [191, 152, 45].
Many models were used to study the nosocomial outbreaks of SARS disease. One
of such studies was done by Hsieh et al. [116], with the help of a deterministic com-
partmental modal. The results of studies shows that time during the admission of
suspected SARS cases and their classification as probable SARS cases is very im-
portant to increase or decrease the nosocomial transmission of disease. According
to Chris et al [43], “A recurring theme in many SARS models is the importance of
timely application of control measures [95, 152, 155, 177, 230, 231]. These models
show that quarantine and isolation have a disproportionate impact on epidemic
control if applied early in the outbreak. Conversely, delays in imposing control
can lead to large case burdens or even failure of potentially successful containment
measures. The dependence of outbreak size on the time of application is nonlin-
ear: there are crucial periods early in the outbreak beyond which the effectiveness
of control measures is severely degraded.” In the beginning the SARS epidemic
was studied based on homogenous populations but soon different mathematical
models illustrated heterogeneity for SARS transmission in space. Several math-
ematical studies significantly demonstrated the difference in parameter estimates
and prediction about epidemics [45, 191]. Although much difference of infectious-
ness is observed in different age groups of population for SARS [45, 158, 192]. The
concepts of possible treatment, spatial and social structures and susceptibility vari-
ations were also incorporated to study and understand the transmission of SARS
epidemic [152, 155, 163, 170].
Occurrence of super spreading events (SSEs) was one of the captivating features
of the 2003 epidemic of severe acute respiratory syndrome (SARS). They have a
large influence on the early spread of SARS [152, 191, 57]. The major Hong Kong
and Singapore SARS epidemics were the result of two different SSEs in Amoy
25
Gardens estate and hospitals [51, 223, 235] and five sperate SSEs in Singapore
outbreak [148]. So to prevent such SARS epidemics in future, it is essential to un-
derstand the factors of SSEs. According to Fang et al. [66], “SSEs are those rare
events where, in a particular setting, an individual may generate many more than
the average number of secondary cases.” He also states that “The understanding
of SSEs is critical to the containment of SARS. From the mathematical modeling
it is found that SSEs can happen even when the virulences are equal for all the
infective individuals. The long latent periods play a critical role in the appearance
of SSEs. Early awareness of the epidemic, which is also effected by the long latent
periods, is vital for the reduction of the possibility of SSEs and the containment
of SARS.”
China was the place where SARS epidemic hit the hardest. After the appear-
ance of SARS, Chinese mathematical modelling in epidemiology experienced a
new age of modelling. Mathematical models were formulated to study the trans-
mission dynamics of SARS disease with real−time data. From 1994 − 2006 the
total number of mathematical modelling papers published on infectious disease in
Chinese journals were 375. Among those 64 papers were only on SARS model-
ing. In 2003 there was further increase in the number [102]. According to Han et
al. [102], “Clearly, the SARS epidemic has resulted in an enormous boost to this
scientific discipline, leading to disproportional large numbers of papers on SARS
models equalling the annual production of modelling papers in the years before
the outbreak. Perhaps surprisingly, this overall extension of the research field
has not lead to a decrease in the number of modelling papers on other infectious
diseases.” He also states that “For the SARS modelling, relatively often more
complex techniques were used, e.g. individual-based modelling, spatio-temporal
models and other types (Autoregressive Integrated Moving Average (ARIMA)
modelling and small world network models). Spatio-temporal models were largely
confined to application on SARS.” Out of the 64 published papers on the SARS
epidemic in Chinese journals six consist of deterministic compartmental models,
eight consist of stochastic compartmental models, one based on a small-world net-
work model, and two focused on spatiotemporal models, while the remaining 43
26
models were based on simple to complex curve fitting techniques like ARIMA
[56] that described SARS outbreak patterns and explained the intervention ap-
plied [102]. The SARS outbreaks in specific regions were focused and modeled
successfully. Numerous mathematical models were formulated for the epidemic in
Beijing [31, 39, 67, 86, 101, 212, 233, 231, 241, 243, 249, 239], Guangdong province
[40, 233, 240, 245, 241], Inner Mongolia [241], Hong Kong [239, 240, 245, 153]. Some
of the Chinese modelers used data from outside China due to the unavailability of
data [153, 211]. These models also provided the estimates of reproduction number
R0 in the range 3.5 − 4.5. [233, 40, 153]. The value of reproduction number was
verified by other researchers Lipsitch et al. [152] and Riley et al. [191]. According
to Han et al. [102], “It is clear that the Chinese modelling initiatives have had lim-
ited implications for policy advice about the SARS epidemic itself, simply because
the reports were only available in the scientific literature after the epidemic. Also,
as far as we can conclude from the papers, no modelling initiative had a direct
link with decision makers before publication. However, from these papers policy
makers may have learned how models can support their decision making, so that
there may be more interaction in case of re-emergence of SARS or an outbreak of
another infectious disease. In fact, these and other modelling initiatives (including
most of the curve-fitting studies) purely focused on mathematical issues and did
not really consider practical implications with public health relevance. Perhaps the
most important benefit of the Chinese SARS modelling efforts is not preparation
for possible re-emergence of SARS, but rather preparation any future outbreak of
infectious diseases.”
27
Chapter 3
Numerical Study of SARS Model
with the Inclusion of Diffusion in
the System
3.1 Introduction
Epidemiological models are considered as one of the most powerful tools to analyze
and understand the spread and control of infectious diseases. Analysis of trans-
mission dynamics of infectious diseases can lead to the better methodologies to
slow their transmission. These models ranges from simple curve fitting models to
standard compartmental models [8] (MSEIR, MSEIRS, SEIJR, SIR, SIRS,
SEIR, SEIS, SI, and SIS etc.) to complex stochastic models. Fast computer
systems and the availability of huge data bases has made it possible to use complex
mathematical models to analyze the data.
In the beginning, epidemiologist, tried to find out the reasons for the cause and
spread of SARS. They tried to find measures to control it, but the main emphasis
was on research work concerned with the biological properties of the corona virus
[202]. Some work was done to investigate the transmission dynamics and the effect
of various control measures. G. Chowell et al. [45] fitted an SEIJR model for
SARS epidemic for the data from Toronto, Hong Kong and Singapore. Chowell
et al. predicted the behavior of the disease and the role of diagnosis and isola-
28
tion as a control mechanism in these regions showing the difference between the
epidemic dynamics occurred in these three cities. Yang et al. [245] established a
compartmental model to describe the SARS epidemic in spatial-temporal dimen-
sions determining whether people traveling in buses and trains infect one another
or not. They concluded that SARS can spread through people traveling in buses
and trains. In their SEIR models based on data from Beijing and Hong Kong, Wu
et al. [233] and Chen et al. [40] estimated the source of super-spreading events of
SARS with the calculation of the reproductive rate of the disease based on data
from Beijing and Hong Kong.
In this chapter, a SARS model (Chowell et al. [45]) is considered with the in-
clusion of diffusion in the system. Diffusion is introduced in the system to study
the spatial spread of disease. Different initial population distributions are chosen
to investigate the effect of diffusion on the spread of SARS. Also intervention
strategies have been proposed to investigate the effect on spread of disease.
3.2 The SEIJR epidemic model
3.2.1 Equations
This model is based on the SEIJR model (G Chowell et al. [45]) with the inclusion
of diffusion in the equations governing the system. Total population is supposed
to be N where N = S + E + I + J +R.
∂S
∂t= −β
(I + qE + lJ)
NS − µS +Π+ d1
∂2S
∂x2(3.1)
∂E
∂t= β
(I + qE + lJ)
NS − (µ+ κ)E + d2
∂2E
∂x2(3.2)
∂I
∂t= κE − (µ+ α + γ1 + δ)I + d3
∂2I
∂x2(3.3)
∂J
∂t= αI − (µ+ γ2 + δ)J + d4
∂2J
∂x2(3.4)
∂R
∂t= γ1I + γ2J − µR + d5
∂2R
∂x2(3.5)
where the variables S, E, I, J and R denote the proportion of susceptible, exposed,
infected, diagnosed and recovered individuals respectively. d1, d2, d3, d4 and d5 are
29
Table 3.1: Interpretation of parameters (per day)
Parameter Description Values
Π Rate of inflow of susceptible individuals into region 3.3× 10−5b
β Transmission rate 0.75a
µ Rate of natural mortality 3.4× 10−5b
l Relative measure of reduced risk among diagnosed 0.38a
κ Rate of progression from exposed to the infected 0.33a
q Relative measure of infectiousness for exposed individuals 0.1a
α Rate of progression from infected to diagnosed 0.33a
γ1 recovery rate of infected individuals 0.125a
γ2 recovery rate of diagnosed individuals 0.2a
δ SARS induced mortality rate .006a
a(Chowell G. et. al. [45]), b (Gummel A. B. et.al. [95] )
the diffusivity constants. Table 3.1 provides the description and the values of the
parameters.
3.2.2 Initial and boundary conditions
The domain of all the calculations is considered as [−2, 2]. Boundary and initial
conditions are chosen as follows:
∂S(−2, t)
∂x=
∂E(−2, t)
∂x=
∂I(−2, t)
∂x=
∂J(−2, t)
∂x=
∂R(−2, t)
∂x= 0 (3.6)
∂S(2, t)
∂x=
∂E(2, t)
∂x=
∂I(2, t)
∂x=
∂J(2, t)
∂x=
∂R(2, t)
∂x= 0 (3.7)
30
(i)
S0 = 0.98Sech(5x− 1), − 2 ≤ x ≤ 2.
E0 = 0, − 2 ≤ x ≤ 2.
I0 = 0.02Sech(5x− 1), − 2 ≤ x ≤ 2.
J0 = 0, − 2 ≤ x ≤ 2.
R0 = 0, − 2 ≤ x ≤ 2.
(ii)
S0 = 0.98 exp(−5x2)3, − 2 ≤ x ≤ 2.
E0 = 0, − 2 ≤ x ≤ 2.
I0 =
0, − 2 ≤ x < −0.4,
0.02, − 0.4 ≤ x ≤ 0.4,
0, 0.4 < x ≤ 2.
J0 = 0, − 2 ≤ x ≤ 2.
R0 = 0, − 2 ≤ x ≤ 2.
(iii)
S0 = 0.97 exp(−5(x− 1)2), − 2 ≤ x ≤ 2.
E0 = 0, − 2 ≤ x ≤ 2.
I0 = 0.03 exp, (−5(x+ 1)2), − 2 ≤ x ≤ 2.
J0 = 0, − 2 ≤ x ≤ 2.
R0 = 0, − 2 ≤ x ≤ 2.
31
t = 0
t = 0(i)
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0S,I
t = 0
(ii)
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0S,I
t = 0
(iii)
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0S,I
t = 0
(iv)
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0
S,I
Figure 3.1: Initial conditions (i)− (iv).
(iv)
S0 ={
0.96Sech(15x), − 2 ≤ x ≤ 2.
E0 = 0, − 2 ≤ x ≤ 2.
I0 =
0, − 2 ≤ x < −.6,
0.04, − .6 ≤ x ≤ .6,
0, .6 < x ≤ 2.
J0 = 0, − 2 ≤ x ≤ 2.
R0 = 0, − 2 ≤ x ≤ 2.
The initial conditions are shown in Fig. 3.1. In initial condition (i) a large pro-
portion of susceptible and infected populations is concentrated towards the right
half of the main domain. Initial condition (ii) shows both S and I concentrated
around the middle of the main domain. In initial condition (iii), I has high con-
centration in the left half of the domain [−2, 2] and population S has concentration
on the right half of the domain [−2, 2]. Initial condition (iv) shows susceptible S
around the middle of domain [−2, 2] and infectious individuals around the middle
but beyond the domain of S.
32
3.3 Numerical scheme
In this section the operator splitting technique [244] has been used to solve the
SEIJRmodel equations. The equations are divided in two groups of sub equations.
The first group comprises the nonlinear reaction equations to be used for the first
half-time step as given:
1
2
∂S
∂t= −β
(I + qE + lJ)
NS − µS +Π (3.8)
1
2
∂E
∂t= β
(I + qE + lJ)
NS − (µ+ κ)E (3.9)
1
2
∂I
∂t= κE − (µ+ α + γ1 + δ)I (3.10)
1
2
∂J
∂t= αI − (µ+ γ2 + δ)J (3.11)
1
2
∂R
∂t= γ1I + γ2J − µR (3.12)
The second group consists of the linear diffusion equations, to be used for the
second half-time step as follows:
1
2
∂S
∂t= d1
∂2S
∂x2(3.13)
1
2
∂E
∂t= d2
∂2E
∂x2(3.14)
1
2
∂I
∂t= d3
∂2I
∂x2(3.15)
1
2
∂J
∂t= d4
∂2J
∂x2(3.16)
1
2
∂R
∂t= d5
∂2R
∂x2(3.17)
By the forward Euler scheme the above equations transform to
Sj+ 1
2i = Sj
i +∆t(−β(Iji + qEj
i + lJ ji )
N ji
Sji − µSj
i +Π) (3.18)
Ej+ 1
2i = Ej
i +∆t(β(Iji + qEj
i + lJ ji )
N ji
Sji − (µ+ κ)Ej
i ) (3.19)
Ij+ 1
2i = Iji +∆t(κEj
i − (µ+ α + γ1 + δ)Iji ) (3.20)
Jj+ 1
2i = J j
i +∆t(αIji − (µ+ γ2 + δ)J ji ) (3.21)
33
Rj+ 1
2i = Rj
i +∆t(γ1Iji + γ2J
ji − µRj
i ) (3.22)
where Sji , E
ji , I
ji , J
ji and Rj
i are the approximated values of S, E, I, J and R at
position −2 + i∆x, for i = 0, 1, . . . and time j∆t, j = 0, 1, . . . and Sj+ 1
2i , E
j+ 12
i ,
Ij+ 1
2i , J
j+ 12
i and Rj+ 1
2i denote their values at the first half-time step. Similarly, for
the second half-time step,
Sj+1i = S
j+ 12
i + d1∆t
(∆x)2(S
j+ 12
i−1 − 2Sj+ 1
2i + S
j+ 12
i+1 ) (3.23)
Ej+1i = E
j+ 12
i + d2∆t
(∆x)2(E
j+ 12
i−1 − 2Ej+ 1
2i + E
j+ 12
i+1 ) (3.24)
Ij+1i = I
j+ 12
i + d3∆t
(∆x)2(I
j+ 12
i−1 − 2Ij+ 1
2i + I
j+ 12
i+1 ) (3.25)
J j+1i = J
j+ 12
i + d4∆t
(∆x)2(J
j+ 12
i−1 − 2Jj+ 1
2i + J
j+ 12
i+1 ) (3.26)
Rj+1i = R
j+ 12
i + d5∆t
(∆x)2(R
j+ 12
i−1 − 2Rj+ 1
2i +R
j+ 12
i+1 ) (3.27)
The stability condition satisfied by the above described numerical method is given
as:dn∆t
(∆x)2≤ 0.5, n = 1, 2, 3, 4, 5. (3.28)
In each case, ∆x = 0.1, d1 = 0.025, d2 = 0.01, d3 = 0.001, d4 = 0.0, d5 = 0.0 and
∆t = 0.03 are used.
3.4 Stability analysis
3.4.1 Reproduction number and disease-free equilibrium
(DFE)
The threshold parameter for any DFE is R0, referred to as the basic reproduction
number. It is defined as the expected number of secondary cases produced, in a
completely susceptible population, by a typical infected individual [52]. To deter-
mine the expression for the basic reproduction number represented as RI for the
system (3.1) - (3.5), the matrices F and W [60] are chosen as follows:
34
F =
β (I+qE+lJ)N
S
0
0
0
0
and W =
(µ+ κ)E
−κE + (µ+ α + γ1 + δ)I
−αI + (µ+ γ2 + δ)J
−γ1I − γ2J + µR
β (I+qE+lJ)N
S + µS − Π
where F describe the transmission route for infection and W denotes the remaining
dynamics of compartments E, I, J , R and S. For the model represented by the
system of Equations (3.1) - (3.5), E, I and J represent the infected compartments.
Therefore the following matrices F represents the paths to infection and W repre-
sents the remaining dynamics corresponding to the compartments E, I and J . Thus
F =
Coefficient of ES
NCoefficient of IS
NCoefficient of JS
N
Coefficient of ESN
Coefficient of ISN
Coefficient of JSN
Coefficient of ESN
Coefficient of ISN
Coefficient of JSN
=
qβ β lβ
0 0 0
0 0 0
and
W =
Coefficient of E Coefficient of I Coefficient of J
Coefficient of E Coefficient of I Coefficient of J
Coefficient of E Coefficient of I Coefficient of J
=
(µ+ κ) 0 0
−κ (µ+ α + γ1 + δ) 0
0 −α (µ+ γ2 + δ)
This gives
W−1 =
[1
µ+κ0 0
κ(µ+α+γ1+δ)(µ+κ)
1µ+α+γ1+δ
0
ακ(µ+α+γ1+δ)(µ+γ2+δ)(µ+κ)
ακ+µα(µ+α+γ1+δ)(µ+γ2+δ)(µ+κ)
1µ+γ2+δ
]and
F .W−1 =
[qβ
(µ+κ)+ βκ
(µ+α+γ1+δ)(µ+κ)+ lβκα
(µ+α+γ1+δ)(µ+γ2+δ)(µ+κ)β
(µ+α+γ1+δ)+
lβ(µα+κα)(µ+α+γ1+δ)(µ+γ2+δ)(µ+κ)
lβµ+γ2+δ
0 0 0
0 0 0
]
The eigenvalues of F .W−1 are
λ1 = 0,
λ2 = 0,
35
λ3 =qβµ+κ
+ βκ(µ+α+γ1+δ)(µ+κ)
+ lβκα(µ+γ1+α+δ)(µ+γ2+δ)(µ+κ)
.
The dominant eigenvalue i.e, basic reproduction number is given as:
RI =qβ(µ+α+γ1+δ)(µ+γ2+δ)+κ(µ+γ2+δ)+lακ
(µ+α+γ1+δ)(µ+γ2+δ)(κ+µ).
3.4.2 Disease-free equilibrium and stability analysis
The variational matrix of the (3.1) - (3.5) at the disease-free equilibrium P0 =
(1, 0, 0, 0, 0), can be given as:
V0 =
A11 A12
A21 A22
where
A11 =
−µ −qβ
0 (qβ − (κ+ µ)
, A12 =
−β −lβ 0
β lβ 0
,
A21 =
0 κ
0 0
0 0
and A22 =
−(α + γ1 + δ + µ) 0 0
α −(µ+ γ2 + δ) 0
γ1 γ2 −µ
.
The stability of the point of equilibrium, P0(1, 0, 0, 0, 0) depends on the char-
acteristic of the eigenvalues of the matrices A11 and A22. The eigenvalues of the
matrix A11 and A22 are qβ−(κ+µ), −µ and −(α+γ1+δ+µ), −(γ2+δ+µ) and −µ
respectively. All eigenvalues of A22 are clearly negative and real. One eigenvalue
of A11 is negative for other eigenvalue it is shown that:
If RI < 1 then βq(µ+ α+ γ1 + δ)(µ+ γ2 + δ) + βκ(µ+ γ2 + δ) + lβακ
< (µ+ α + γ1 + δ)(µ+ γ2 + δ)(κ+ µ)
⇒ βq < (κ+ µ)
⇒ βq − (κ+ µ) < 0
This implies that P0(1, 0, 0, 0, 0) is stable for ℜ0 < 1.
36
3.4.3 Stability of endemic equilibrium without diffusion
The variational matrix of the system of equations (3.1) - (3.5) at P ∗(S∗, E∗, I∗, J∗R∗),
is given by
V ∗ =
a11 a12 a13 a14 a15
a21 a22 a23 a24 a25
a31 a32 a33 a34 a35
a41 a42 a43 a44 a45
a51 a52 a53 a54 a55
where
a11 = β (I∗+qE∗+lJ∗)S∗
(S∗+E∗+I∗+J∗+R∗)2− β (I∗+qE∗+lJ∗)
(S∗+E∗+I∗+J∗+R∗)− µ,
a12 = β (I∗+qE∗+lJ∗)S∗
(S∗+E∗+I∗+J∗+R∗)2− β qS∗
(S∗+E∗+I∗+J∗+R∗),
a13 = β (I∗+qE∗+lJ∗)S∗
(S∗+E∗+I∗+J∗+R∗)2− β S∗
(S∗+E∗+I∗+J∗+R∗),
a14 = β (I∗+qE∗+lJ∗)S∗
(S∗+E∗+I∗+J∗+R∗)2− β lS∗
(S∗+E∗+I∗+J∗+R∗),
a21 = β 1(S∗+E∗+I∗+J∗+R∗)
− β (I∗+qE∗+lJ∗)S∗
(S∗+E∗+I∗+J∗+R∗)2,
a22 = β qS∗
(S∗+E∗+I∗+J∗+R∗)− β (I∗+qE∗+lJ∗)S∗
(S∗+E∗+I∗+J∗+R∗)2− κ− µ,
a23 = β S∗
(S∗+E∗+I∗+J∗+R∗)− β (I∗+qE∗+lJ∗)S∗
(S∗+E∗+I∗+J∗+R∗)2,
a24 = β lS∗
(S∗+E∗+I∗+J∗+R∗)− β (I∗+qE∗+lJ∗)S∗
(S∗+E∗+I∗+J∗+R∗)2,
a15 = β (I∗+qE∗+lJ∗)S∗
(S∗+E∗+I∗+J∗+R∗)2, a25 = −β (I∗+qE∗+lJ∗)S∗
(S∗+E∗+I∗+J∗+R∗)2,
a32 = κ, a33 = −α− γ1 − δ − µ, a43 = α, a44 = −γ2 − δ − µ, a53 = γ1,
a54 = γ2, a55 = −µ, and a31 = a34 = a35 = a41 = a42 = a45 = a51 = a52 = 0. The
characteristic equation for P ∗(S∗, E∗, I∗, J∗, R∗) can be written as
λ5 + p1λ4 + p2λ
3 + p3λ2 + p4λ+ p5 = 0 (3.29)
Where p1, p2, p3, p4 and p5 are calculated as in [198] and are given in appendix
A.3. The Routh-Hurwitz criterion for the stability is given as in [200]:
C1 : p1 > 0, C2 : p5 > 0,
C3 : p1p2 − p3 > 0,
C4 : p1p2p3 + p1p5 − (p23 + p21p4) > 0,
C5 : (p1p4 − p5)(p1p2p3 − p23 − p21p4) + p21p4p5 − (p5(p1p2 − p3)2 + p1p
25) > 0.
and P1, P2, P3 and P4 are points of equilibrium.
P1 = (.331581, 0.000064, 0.000046, 0.000074, .607887),
P2 = (.357187, .000062, .000044, .000058, .585632),
37
Table 3.2: Values of LHS Routh-Hurwitz criterion of equilibrium without diffusion
Case Equil. Pt. C1 C2 C3 C4 C5 Stability
1 P1 0.97677 6.902× 10−7 .20805 5.343× 10−6 7.613× 10−12 Stable
2 P2 1.02482 7.641× 10−11 .24969 6.903× 10−6 1.114× 10−11 Stable
3 P3 1.02416 6.560× 10−11 .23359 6.059× 10−6 8.399× 10−12 Stable
4 P4 0.97676 6.199× 10−11 .20804 5.109× 10−6 6.421× 10−12 Stable
P3 = (.365393, .000061, .000039, .000064, .577199),
P4 = (.355572, .000062, .000044, .000072, .584715).
3.4.4 Stability of endemic equilibrium with diffusion
To calculate the small perturbations S1(x, t), E1(x, t), I1(x, t),J1(x, t) and R1(x, t),
the equations are linearized about the point of equilibrium P ∗(S∗, E∗, I∗, J∗, R∗)
as described in [35, 201].
∂S1
∂t= a11S1 + a12E1 + a13I1 + a14J1 + a15R1 + d1
∂2S1
∂x2(3.30)
∂E1
∂t= a21S1 + a22E1 + a23I1 + a24J1 + a25R1 + d2
∂2E1
∂x2(3.31)
∂I1∂t
= a31S1 + a32E1 + a33I1 + a34J1 + a35R1 + d3∂2I1∂x2
(3.32)
∂J1∂t
= a41S1 + a42E1 + a43I1 + a44J1 + a45R1 + d4∂2J1∂x2
(3.33)
∂R1
∂t= a51S1 + a52E1 + a53I1 + a54J1 + a55R1 + d5
∂2R1
∂x2(3.34)
where a11, a12, a13 etc are the elements of the variational matrix V ∗ calculated using
the same method as described in [198]. Assume a Fourier series solution exists of
equations (3.30) - (3.34) of the form:
S1(x, t) =∑k
Skeλt cos(kx) (3.35)
E1(x, t) =∑k
Ekeλt cos(kx) (3.36)
I1(x, t) =∑k
Ikeλt cos(kx) (3.37)
38
J1(x, t) =∑k
Jkeλt cos(kx) (3.38)
R1(x, t) =∑k
Rkeλt cos(kx) (3.39)
where k = nπ2, (n = 1, 2, 3, · · · · · · ) is the wave number for the node n. Substituting
the value of S1, E1, I1, R1 in the equations (3.30) - (3.34), the equations are
transformed into∑k
(a11 − d1k2 − λ)Sk +
∑k
a12Ek +∑k
a13Ik +∑k
a14Jk +∑k
a15Rk = 0 (3.40)
∑k
a21Sk +∑k
(a22 − d2k2 − λ)Ek +
∑k
a23Ik +∑k
a24Jk +∑k
a25Rk = 0 (3.41)
∑k
a32Ek +∑k
(a33 − d3k2 − λ)Ik = 0 (3.42)
∑k
a43Ik +∑k
(a44 − d4k2 − λ)Jk = 0 (3.43)
∑k
a53Ik +∑k
a54Jk +∑k
(a55 − d5k2 − λ)Rk = 0 (3.44)
The Variational matrix V for the equations (3.40) - (3.44)
V =
a11 − d1k2 a12 a13 a14 a15
a21 a22 − d2k2 a23 a24 a25
0 a32 a33 − d3k2 0 0
0 0 a43 a44 − d4k2 0
0 0 a53 a54 a55 − d5k2
The characteristic equation for the variational matrix V is given as
λ5 + q1λ4 + q2λ
3 + q3λ2 + q4λ+ q5 = 0 (3.45)
where q1, q2, q3, q4 and q5 are calculated with the same technique as used in [198]
and are given in appendix A.3.
Routh-Hurwitz Conditions are given as:
C1 : q1 > 0, C2 : q5 > 0,
C3 : q1q2 − p3 > 0,
C4 : q1q2q3 + q1q5 − (q23 + q21q4) > 0,
C5 : (q1q4 − q5)(q1q2q3 − q23 − q21q4) + q21q4q5 − (q5(q1q2 − q3)2q1q
25) > 0.
39
Table 3.3: Values of LHS Routh-Hurwitz criterion of equilibrium with diffusion
Case Equil. Pt. C1 C2 C3 C4 C5 Stability
1 P1 1.0656 4.859× 10−8 .29522 .00478 8.061× 10−7 Stable
2 P2 1.11365 5.820× 10−8 .345648 .006458 1.411× 10−6 Stable
3 P3 1.1109 4.696× 10−8 .322143 .004815 9.623× 10−8 Stable
4 P4 1.06559 4.70× 10−8 .29521 .004779 8.049× 10−7 Stable
3.4.5 Excited mode and bifurcation value
The first excited mode of the oscillation n is calculated by the same technique as
used in [35]. According to the definition of mode of excitation the curve
f(β) = (q1q4 − q5)(q1q2q3 − q23 − q21q4) + q21q4q5 − (q5(q1q2 − q3)2q1q
25). (3.46)
for n = 1 represents the first mode of excitation as being closest to the β-axis
as shown in Fig 2. Similarly, n = 1 is first mode of excitation for Cases 2 − 4.
Bifurcation values of the transmission coefficient β are given in Table 3.4. It is
observed that the bifurcation value of transmission coefficient with diffusion is
greater than the value of transmission coefficient without diffusion. Bifurcation
values of the recovery coefficients γ1 and γ2 for which the point of equilibrium
remains stable [36] are given in Table 3.5. Here the bifurcation value of recovery
coefficients with diffusion are smaller than without diffusion. The corresponding
bifurcation diagrams and calculation of bifurcation values of β, γ1 and γ2 are given
in appendix A.3.
3.5 Numerical solutions
Four cases with the variation of β, the transmission coefficient, γ1, the recovery
coefficient in the infectious class and γ2, the recovery coefficient in the diagnosed
class have been chosen as given in Table 3.6. Numerical solutions are obtained
both with and without diffusion for all cases specified in Table 3.6.
40
n=1
0.5 1.0 1.5 2.0 2.5 3.0Β
0.00005
0.00010
0.00015
0.00020
0.00025
f HΒL
n=2
0.5 1.0 1.5 2.0 2.5 3.0Β
0.005
0.010
0.015
0.020
f HΒL
n=3
0.5 1.0 1.5 2.0 2.5 3.0Β
0.1
0.2
0.3
0.4
f HΒL
Figure 3.2: Determination of first excited mode with β as an unknown parameter.
Table 3.4: Bifurcation value of β
Cases Value of β Considered Bifurcation Value
Without Diffusion With Diffusion
1 0.75 0.750435 0.810383
2 0.75 0.750367 0.810239
3 0.75 0.750387 0.809841
4 0.70 0.700381 0.756276
Table 3.5: Bifurcation values of γ1 andγ2
Cases γ1 Without Diffusion With Diffusion γ2 Without Diffusion With Diffusion
1 0.125 0.124708 0.087493 0.2 0.199660 0.161489
2 0.125 0.124752 0.0873489 0.25 0.249586 0.196281
3 0.175 0.17471 0.133485 0.2 0.199694 0.161499
4 0.125 0.124726 0.0875375 0.2 0.199680 0.161530
41
Table 3.6: Four cases
Case Transmission Coefficient(β) Recovery Coefficient(γ1) Recovery Coefficient (γ2)
1 0.75 0.125 0.2
2 0.75 0.125 0.25
3 0.75 0.175 0.2
4 0.7 0.125 0.2
3.5.1 Solutions of SEIJR model without diffusion (Case 1)
Fig. 3.3, shows the output with initial condition (i) and without diffusion. Here
the susceptible population decreases slowly in the first five days of disease but after
that there is a rapid decrease till t = 20 days. There is a rapid increase in popula-
tion exposed to disease in the first five days and this increase continues till t = 10.
It slows down however after five days. After ten days there is a quick decrease
in exposed individuals. Only a few individuals are in the exposed compartment
after fifteen days. There is a slow increase in infected individuals till the fifth day.
But a sudden increase in infected is observed in the next five days. After reaching
maximum level, a decline in infected is observed till t = 15. With the increase of
infected individuals, the number of diagnosed has also increased rapidly but after
attaining the maximum in the first ten days of disease, there is a slow decrease till
t = 15. After that a quick decline is observed at t = 20. Recovered individuals
increase slowly in first ten days of disease but after that a rapid increase in recovery
is observed.
Fig. 3.4, shows the output with initial condition (ii) without diffusion. Here
the proportion of the susceptible population decreases rapidly between 5-10 days
and after that there is very low level of susceptible population. The population
becomes exposed very quickly during first five days. After ten days, there is a
sudden decrease which continues till t = 20 days. Infected individuals increase for
the first ten days with rapid increase between 5-10 days. After that there is a rapid
decrease till t = 15. Then a decrease occurs slowly till t = 20. The proportion of
diagnosed shows an increase till t = 10 and after that diagnosed individuals reduce
42
t = 0
t = 5
t = 10 t = 15t = 20
t=20
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0
S
t = 5
t = 10
t = 15
t = 20t=0
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25E
t = 0
t=10
t=5
t=15t=0
t=20
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25I
t = 20
t=10
t=15
t=0
t=5
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25J
t = 0t = 5
t = 10
t = 15
t = 20
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0R
Figure 3.3: Solutions with initial condition (i) and without diffusion.
43
t = 0
t = 5
t = 10t = 15 t = 20-2 -1 0 1 2
x
0.2
0.4
0.6
0.8
1.0
S
t = 5
* * * * * * * * * * * * * * * *
*
*
*
*
*
*
*
*
*
* * * * * * * * * * * * * * * *
t = 10
t = 15
t=0t=20
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25E
t = 0
* * * * * * * * * * * * * * * *
*
*
*
*
*
*
*
*
*
* * * * * * * * * * * * * * * *
t = 20
t=10
t=5
t=15
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
I
t = 0
t = 15
t = 20
t=10
t=5
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25J
t = 0t = 5
t = 10
t = 15
t = 20
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0R
Figure 3.4: Solutions with initial condition (ii) and without diffusion.
44
t = 0t=5
t=10t=15
t=20
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0
S
t = 10
t = 15
t=20
t=5t=0-2 -1 0 1 2
x
0.05
0.10
0.15
0.20
0.25
0.30E
t = 0
t=5
t=15
t=20
t=10
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
0.10
0.12I
t = 0
t=20
t=5t=10
t=15
-2 -1 0 1 2x
0.01
0.02
0.03
0.04
0.05
0.06
0.07
J
t = 0
t = 10
t=5
t=20
t=15
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
R
Figure 3.5: Solutions with initial condition (iii) and without diffusion.
with a quick fall between t = 15 and 20 days. Recovery is slow initially but after
t = 10, it is fairly quick.
Fig. 3.5, shows the output with initial condition (iii) and without diffusion. The
behavior of the susceptible is quite different as compared to the initial conditions
(i) and (ii). Susceptible move to the right of the initial domain of concentration
slowly and slowly without much change in proportion of susceptible population.
Initially the main concentration of the susceptible population is in the interval
[0, 2] but at t = 20 this shifts to [0.6, 2]. More and more of the population become
exposed during t=10 to 20 days of onset of the disease. During the first five days
of the onset of SARS, the proportion of infected people goes down in its domain
[−2, 0] and after that starts moving to domain [0, 2] with gradual increase. A sharp
increase is observed between t = 10 and 20 days [0, 1]. The number of diagnosed
45
t = 0
t = 5
t = 10t = 15 t = 20-2 -1 0 1 2
x
0.2
0.4
0.6
0.8
1.0S
t = 5
t = 10
t = 15 t = 20
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25
0.30E
t = 0
t=10
t=5
t=15t=20
-2 -1 0 1 2x
0.05
0.10
0.15
0.20I
t = 0
t = 15
t=10
t=20t=5
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25
J
t = 0t = 5
t = 10
t = 15
t=20
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0R
Figure 3.6: Solutions with initial condition (iv) and without diffusion.
individuals increases during the first five days in the domain [−2, 0]. After that
diagnosed individuals decrease with a slow pace. Also the concentration of the
diagnosed moves to the domain [0, 2]. A rapid increase of diagnosed can be seen
between t = 15 and 20 days. Till t = 10 recovery increases in the domain [−2, 0]
and slowly moves to domain [0, 1]. From t = 15 to t = 20 recovery attains maxi-
mum values in the domain [0, 1].
Fig 3.6, shows the output with the initial condition (iv) and without diffusion.
Susceptible individuals decrease rapidly after t = 5. At t = 15, the susceptible re-
duce to a very low level of concentration. Exposed individuals reach the maximum
level in first five days and after that start reducing. Infected individuals increase
till t = 10 and after that there is a sudden fall till t = 20. Diagnosed individuals
increase in the first ten days and after that there is a gradual decrease. There is
46
t = 0
t = 5
t = 10 t = 15 t = 20
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0
S
t = 5
t = 10
t = 20t=15
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
0.10
0.12E
t = 0t = 20
t=15
t=5
t=10
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
I
t = 0
t = 20
t=5
t=15
t=10
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
0.10
0.12J
t = 0
t = 5
t = 10
t = 15t = 20
t=0-2 -1 0 1 2
x
0.1
0.2
0.3
0.4R
Figure 3.7: Solutions with initial condition (i) and with diffusion.
gradual increase in recovered individuals as shown in Fig 3.6.
3.5.2 Solutions of SEIJR model with diffusion (Case 1)
Fig 3.7, shows the output with initial condition (i) and with diffusion. With
the inclusion of diffusion in the system, susceptible spread in the entire region
at t = 5 with peak value 0.2228567. Initially exposed are mainly confined in do-
main [−1, 1.5]. At t = 5 exposed spread in the domain [−1, 1.5] with peak value
0.118055. At t = 10, exposed spread in the whole domain with peak value 0.068133.
After this a rapid decrease in the exposed individuals occurs. Infected individuals
spread in the whole domain [−2, 2] with the passage of time and at t = 10, infected
individual attains its maximum with peak value 0.077894. After this there is a
47
t = 0
t = 5
t = 10t = 15 t = 20-2 -1 0 1 2
x
0.2
0.4
0.6
0.8
1.0
S
t = 5
t = 10
t = 20t=15 t=0
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
0.10
0.12
0.14
E
t = 0
t = 20
t=5
t=10
t=15
-2 -1 0 1 2x
0.01
0.02
0.03
0.04
0.05
0.06I
t = 0
t = 15
t = 20
t=10
t=5
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
0.10J
t = 0
t = 5
t = 10
t = 20
t=10
t=15
-2 -1 0 1 2x
0.1
0.2
0.3
0.4R
Figure 3.8: Solutions with initial condition (ii) and with diffusion.
fall in the infected individuals and at t = 20, there is very low level of infected
individuals. A steady spread of diagnosed is observed in the main domain [−2, 2].
At t = 10 diagnosed are observed in the entire domain with peak value 0.105364.
After t = 10, there starts a decrease in the diagnosed population. Recovery starts
spreading in the domain and at t = 20 days, it completely spreads over the whole
domain with peak value 0.383669.
Fig 3.8, represents the results with the initial condition (ii) and with diffu-
sion. Susceptible quickly spreads in the whole domain [−2, 2] with low peak value
0.129482 at t = 5. In the first five days exposed spread in the domain [−1, 1]
with peak value 0.117691. In first ten days, exposed spread to whole domain with
peak value 0.040644. After ten days the maximum proportion of population gets
infected and spread in the main domain [−2, 2]. Though the population in the
48
t = 0
t = 5
t = 10
t = 15
t=20
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0
S
t = 10
t=5
t=20
t=15
t=0
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
0.10
0.12
0.14
E
t = 0
t=10t=20
t=15
t=5
-2 -1 0 1 2x
0.02
0.04
0.06
0.08I
t = 0
t=20
t=15
t=5
t=10
-2 -1 0 1 2x
0.02
0.04
0.06
0.08J
t = 0t = 5
t = 10
t = 15
t = 20
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25R
Figure 3.9: Solutions with initial condition (iii) and with diffusion.
whole domain [−2, 2] is infected but main concentration of infected lies in domain
[−1, 1]. At time t = 10, diagnosed spread over the whole domain with peak value
0.085846. Diagnosed thereafter start reducing and at time t = 15 days reduce
to peak value 0.051532. Recovery starts slowly and spread to domain [−.6, .6] at
t = 5. At t = 20, recovered are spread over the whole domain with peak value
0.311524.
Fig 3.9, shows the results with the initial condition (iii) and with diffusion. Suscep-
tible spread from the initial domain of concentration [0, 2] to the domain [−1.5, 2]
at t = 5. After 5 days, susceptible start moving back. Susceptible are confined to
the domain [0.5, 2] at t = 15. Exposed individuals also shifts from domain [−2, 0]
to [0, 2] with the passage of time. Initially infected are confined to domain [−2, 0].
49
t = 0
t = 5 t = 10 t = 15 t = 20
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0
S
t = 20t=15
t=10
t=5
-2 -1 0 1 2x
0.01
0.02
0.03
0.04
0.05E
t = 20
t=0
t=5
t=10
t=15
-2 -1 0 1 2x
0.01
0.02
0.03
0.04
I
t = 0
t = 20
t=5
t=10
t=15
-2 -1 0 1 2x
0.005
0.010
0.015
0.020
0.025
0.030
0.035J
t = 0t = 10
t=5
t=10
t=20
t=15
-2 -1 0 1 2x
0.05
0.10
0.15
R
Figure 3.10: Solutions with initial condition (iv) and with diffusion.
Infected spread at very slow pace and a small pulse with peak value 0.004476 can
be observed at t = 5. After ten days, the domain of concentration of infected
people moves to [−1, 1]. At t = 15, infection spread in the whole domain [−2, 2].
At t = 5, a small proportion of diagnosed remain in the domain [−2, 0]. There
is, however, a rapid increase in the number of diagnosed with peak value 0.074575
at t = 20. At t = 5, recovery is restricted to the domain [−2, 0]. After 20 days,
recovered spread in the domain [−2, 2], with peak value 0.206879.
Fig 3.10, shows the results with initial condition (iv) and with diffusion. A sudden
fall in the susceptible is observed at t = 5 with peak value 0.005466. During the
same time, exposed spread to domain [−1.5, 1.5] with peak value 0.043322. At
t = 10, infected spread in the domain [−1.8, 1.8] with peak value 0.010391. At
50
t = 5, diagnosed spread to domain[−1, 1] with peak value 0.033966. Diagnosed
spread further in the domain [−1.8, 1.8] at t = 15 with peak value 0.01385. Recov-
ery initially occurs in the domain [−1, 1] at t = 5 with peak value 0.045636. There
is a further increase of recovered at t = 10 with peak value 0.106593. At t = 20
there is maximum recovery with peak value 0.147773.
3.5.3 Other cases
Graphs of numerical solutions of Cases 2-4, obtained both with and without diffu-
sion, for all cases specified in Table 3.6 are quite similar to Case 1. Thus graphs for
Cases 2-4 are not reproduced here. Summarized results for Cases 2-4 are shown in
Tables 3.7, 3.8, 3.9 and 3.10. Here Sj, Ej, Ij and Rj for j = (i), (ii), (iii) and (iv)
represent the proportion of susceptible, exposed, infected and recovered population
at critical points in the domain [−2, 2] without and with diffusion, for the initial
condition (i), (ii), (iii) and (iv) respectively. The following description is based on
the information provided in Tables 3.7 and 3.10.
In Case 2, there is an increase in the recovery coefficient of diagnosed individual,
γ2 = 0.25 while keeping values of transmission coefficient, β and recovery rate of
infected, γ1 the same as in Case 1. There is a slow decrease in susceptible popu-
lation as compared to Case 1, for all initial conditions, with and without diffusion
in the system. Fewer individuals seem to be exposed and infected to disease in the
first five days with conditions (i), (ii) and (iv) and after five days their proportion
is greater in comparison to Case 1. But with condition (iii), there is a smaller
proportion of exposed all the time. There is higher proportion of recovered in Case
2 as compared to Case 1, both with and without diffusion.
In Case 3, there is increase in recovery rate of infected individuals, γ1 = 0.175 as
compared to Case 1 and Case 2. There is slow decrease in susceptible population
during the first five days for condition (i) − (iv), with and without diffusion as
compared to Cases 1 and 2. Initially exposed are small in proportion as compared
to Case 1 but after five days the proportion of exposed is more than in Case 1
with initial condition (i), (ii) and (iv). On the other hand with initial condition
(iii), there is a decrease in exposed individuals. Population of infected reduces
51
Table 3.7: Peak values of susceptible (S) and exposed (E) (without diffusion)
Case t S(i) S(ii) S(iii) S(iv) E(i) E(ii) E(iii) E(iv)
1 00 98.0 98.0 97.0 96.0 0.00 0.00 0.00 0.00
05 66.4 66.4 96.9 46.3 19.5 19.5 .171 29.0
10 .073 2.34 96.8 .039 21.3 21.3 1.25 .039
15 .001 2.36 96.7 .059 2.94 2.94 7.87 2.05
20 .002 2.38 96.6 .078 .410 .410 30.1 0.288
2 00 98.0 98.0 97.0 96.0 0.00 0.00 0.00 0.00
05 68.2 68.2 96.9 48.6 18.3 18.3 .161 27.57
10 .156 2.34 96.9 .039 22.6 22.6 1.12 15.7
15 .001 2.36 96.8 .059 3.12 3.12 7.38 2.16
20 .003 2.38 96.7 .078 .434 .434 26.3 .303
3 00 98.0 98.0 97.0 96.0 0.00 0.00 0.00 0.00
05 68.7 68.7 96.9 49.3 18.0 18.0 .159 27.3
10 .151 2.34 96.8 .039 22.8 22.8 1.11 15.9
15 .001 2.36 96.8 .059 3.15 3.15 7.36 2.18
20 .002 2.38 96.7 .078 .438 .438 26.2 .306
4 00 98.0 98.0 97.0 96.0 0.00 0.00 0.00 0.00
05 69.7 68.2 96.9 50.5 17.4 18.3 .155 26.6
10 .181 2.34 96.8 .039 23.4 22.6 1.04 16.3
15 .001 2.36 96.8 .059 3.24 3.12 6.92 2.23
20 .002 2.38 96.7 .078 .451 .434 22.7 .313
Table 3.8: Peak values of infected (I) and recovered (R) (without diffusion)
Case t I(i) I(ii) I(iii) I(iv) R(i) R(ii) R(iii) R(iv)
1 002.00 2.00 3.00 4.00 0.00 0.00 0.00 0.00
05 6.91 6.91 .189 11.4 3.53 3.53 1.86 6.61
10 21.4 21.4 .416 17.7 32.4 32.4 2.59 41.1
15 5.37 5.37 2.98 3.91 69.9 69.9 2.833 74.8
20 .891 .891 11.9 .632 87.6 87.6 8.47 89.4
2 00 2.00 2.00 3.00 4.00 0.00 0.00 0.00 0.00
05 6.55 6.55 .189 10.9 3.71 3.71 2.01 6.98
10 21.7 21.7 .403 18.1 33.7 33.7 2.71 43.2
15 5.64 5.64 2.62 4.09 73.4 73.4 2.89 78.3
20 .942 .942 11.6 .665 90.3 90.3 8.39 91.9
3 00 2.00 2.00 3.00 4.00 0.00 0.00 0.00 0.00
05 6.08 6.08 .141 10.1 3.92 3.92 2.01 7.35
10 19.8 19.8 .377 16.2 34.8 34.8 2.66 43.9
15 4.72 4.72 2.48 3.38 72.5 72.5 2.86 77.0
20 .742 .742 10.9 .521 88.9 88.9 8.82 90.5
4 00 2.00 2.00 3.00 4.00 0.00 0.00 0.00 0.00
05 6.20 6.55 .189 10.4 3.32 3.71 1.86 6.27
10 21.9 21.7 .394 18.4 30.5 33.7 2.60 39.4
15 5.82 5.64 2.35 4.23 68.7 73.4 2.83 73.8
20 .975 .942 11.2 .687 87.1 90.3 7.26 89.0
52
Table 3.9: Peak values of susceptible (S) and exposed (E) (with diffusion)
Case t S(i) S(ii) S(iii) S(iv) E(i) E(ii) E(iii) E(iv)
1 00 98.0 98.0 97.0 96.0 0.00 0.00 0.00 0.00
05 22.9 12.9 48.4 .547 11.8 11.8 1.18 4.33
10 .595 .859 34.9 .021 6.81 4.06 8.39 .584
15 .001 .002 15.1 .002 .886 .531 13.2 .077
20 .002 .003 .004 .005 .123 .077 3.05 .017
2 00 98.0 98.0 97.0 96.0 0.00 0.00 0.00 0.00
05 23.8 13.7 48.5 .669 .111 11.2 1.09 4.30
10 .728 .917 34.9 .024 .072 4.25 7.99 .603
15 .002 .002 15.7 .003 .009 .556 13.1 .079
20 .003 .004 .005 .007 .001 .079 3.24 .016
3 00 98.0 98.0 97.0 96.0 0.00 0.00 0.00 0.00
05 23.9 13.9 48.5 .683 11.0 11.1 1.10 4.33
10 .760 .949 35.0 .026 7.21 4.29 7.97 .609
15 .001 .002 15.9 .003 .942 .561 13.2 .079
20 .003 .003 .005 .005 .130 .081 3.28 .017
4 00 98.0 98.0 97.0 96.0 0.00 0.00 0.00 0.00
05 24.4 13.7 48.5 .700 10.7 11.2 1.10 4.35
10 .876 .917 7.78 .029 7.39 4.25 7.78 .615
15 .001 .002 16.7 .003 .966 .556 13.0 .080
20 .002 .004 .005 .005 .134 .079 3.51 .017
Table 3.10: Peak values of infected (I) and recovered (R) (with diffusion)
Case t I(i) I(ii) I(iii) I(iv) R(i) R(ii) R(iii) R(iv)
1 00 2.00 2.00 3.00 4.00 0.00 0.00 0.00 0.00
05 4.63 5.03 .448 3.30 2.94 3.21 1.84 4.56
10 7.79 5.27 3.61 1.04 17.4 16.0 2.78 10.7
15 1.72 1.07 7.73 .166 32.0 26.7 10.5 13.7
20 .272 .167 3.98 .028 38.4 31.2 20.7 14.8
2 00 2.00 2.00 3.00 4.00 0.00 0.00 0.00 0.00
05 3.10 3.40 1.99 4.89 3.10 3.40 1.99 4.89
10 18.1 16.8 2.91 11.2 18.1 16.8 2.91 11.2
15 33.3 27.8 10.9 14.1 33.3 27.8 10.9 14.1
20 39.2 31.7 21.6 14.9 39.2 31.7 21.6 14.9
3 00 2.00 2.00 3.00 4.00 0.00 0.00 0.00 0.00
05 4.08 4.46 .394 2.95 3.25 3.57 1.99 5.03
10 7.11 4.77 3.20 .895 18.4 16.9 2.89 11.1
15 1.48 .905 7.05 .135 32.8 27.3 11.1 13.9
20 .223 .135 3.65 .022 38.6 31.3 21.3 14.8
4 00 2.00 2.00 3.00 4.00 0.00 0.00 0.00 0.00
05 4.21 4.82 .422 3.21 2.79 3.39 35.1 4.45
10 7.99 5.36 3.32 1.07 16.5 16.8 2.78 10.5
15 1.84 1.11 7.53 .174 31.3 27.8 9.98 13.5
20 .294 .174 4.27 .029 37.8 31.7 19.9 14.6
53
Table 3.11: Peak values of infected at t = 20
Cases Peak values for initial conditions Reproductive Number
(i) (ii) (iii) (iv)
1 0.891 0.891 11.90 0.632 2.83
2 0.942 0.942 11.60 0.665 2.64
3 0.742 0.742 10.90 0.521 2.58
4 0.975 0.942 11.20 0.687 2.64
remarkably in Case 3. Diagnosed class also has a decrease in individuals. More
individuals recover in Case 3 as compared to Case 1. But the recovered population
in Case 3 is less than that in Case 2.
In Case 4, there is a reduced value of the transmission coefficient, β = 0.7 as com-
pared to Cases 1, 2, 3. This causes a slow decrease in the susceptible population as
compared to Cases 1, 2 and 3 till t = 10. Exposed behave similarly as in Cases 2
and 3 with the number of individuals first decreasing and then increasing as com-
pared to Case 1. Population of infected is less than Cases 1 at t=5. After that, till
t=20 days , more infected individuals are observed in Case 4 as compared to Cases
1. Recovered individuals slow down here and the number of recovered population
is less in this case as compared to other cases.
3.6 Discussion
An SEIJR Model for SARS (G. Chowell et al. [45]) is considered with the inclu-
sion of diffusion in the system. Four different initial conditions are taken for the
population distribution. The equation governing the system are solved numerically
using operator splitting method. The reproduction number RI is calculated for the
disease. It is shown that disease dies out for RI < 1, in disease-free equilibrium.
It however prevails for endemic equilibrium, where RI > 1 as shown in Table 3.11.
Stability of solutions with and without diffusion is established using Routh-Hurwitz
conditions. The value of the reproduction number RI depends on ten parameters.
54
The parameters transmission coefficient β, recovery rate in infectious class γ1 and
recovery coefficient in diagnosed γ2 have been varied to observe the effects on the
spread of disease. Hence four cases are produced to see the effect on the spread of
disease. Bifurcation values of transmission coefficient β and recovery coefficients
γ1 and γ2 are calculated. It is observed that diffusion causes an increase in the
bifurcation value of β and a decrease in the value of recovery coefficients. This
shows that the system can be stable for larger value of β and smaller values of
recovery rates γ1 and γ2 in the presence of diffusion.
Numerical solution with initial condition (i), as shown in Fig 3.3 and Fig 3.7, that
in the absence of diffusion, only the population in the domain [−1, 1.5] becomes sus-
ceptible, but when diffusion is introduced susceptible spread over the whole domain
in first five days. Similarly the exposed remain confined in the interval [−1, 1.5] in
the absence of diffusion. But with diffusion population outside the domain [−1, 1.5]
also become exposed to infection and after ten days exposed population spread to
whole domain. With and without diffusion in the system, infection reached its peak
in the first ten days. With the inclusion of diffusion, however, infection spreads
over the whole domain [−2, 2]. Recovered also follows the same pattern. Numerical
solution of initial condition (ii), as shown in Fig 3.4 and Fig 3.8, follows the same
pattern as in condition (i) with only difference in concentration of population in
different domain. In the absence of diffusion infected population fluctuate inside
the domain [−.5, .5], but with diffusion in the system fluctuations follows with the
spread in the whole domain after ten days.
Numerical solution with initial condition (iii), as shown in Fig. 3.5 and Fig. 3.9,
shows the main concentration of susceptible shifts slightly to right of domain [0, 2]
in 20 days. Infected move to domain [0, 2] from domain [−2, 0] and with that di-
agnosed and recovered also follow the same pattern. With diffusion in the system
Susceptible start spreading to the left of the domain [−2, 2] but are mainly con-
fined in [−0.8, 2]. Exposed grows in the smaller domain [−1, 0.5] first and then
with passage of time spread in the domain [−1.5, 2]. Infected shifts their domain
from [−2, 0] to [−1, 2] followed with decrease, increase and then again decrease in
proportion. Thus with diffusion more individuals get infected within a short time
55
and infection spreads quickly and reaches its maximum in 15 days covering almost
the whole domain. In the absence of diffusion the maximum number of infected
are observed after twenty days in domain [0, 1], reflecting the intensity of infection
more than that with diffusion during the same time. A large proportion of popu-
lation is recovered with diffusion in the system during the same time as compared
to without diffusion.
Numerical solution with initial condition (iv), as shown in Fig. 3.6 and Fig. 3.10,
demonstrate that diffusion causes the infection to spread out from domain [−.7, .7]
to [−2, 2]. The intensity of the infection also becomes less than the initial intensity
as it spreads to [−1.5, 1.5]. It has been observed in Tables 3.7 and 3.10 that when
recovery is improved in diagnosed class with an increased value of γ2 as in Case
2, a smaller proportion of the population becomes infected and the proportion of
the recovered increases. Even better result is obtained with the greater recovery of
infected with an increased value of γ1 as shown in Case 3, where the proportion of
recovered is higher than previous Case 2 even with a lower value of diagnosed re-
covery, γ2. With recovery coefficients γ1 the same and decreasing the transmission
coefficient β as in Case 4, recovery is observed to be slower than the original Case 1.
In the next chapter the SEIJR model is further examined for the impact of
cross diffusion on the transmission dynamics of SARS. Different cases of positive
and negative cross diffusion are considered to study the effects of cross-diffusion.
56
Chapter 4
Numerical Simulation of Cross
Diffusion on Transmission
Dynamics of SARS
4.1 Introduction
Recently, ecological and epidemiological modelling are increasingly focused on spa-
tially structured models and emergent heterogeneity. The wave of infection is often
caused by the diffusion within the populations in a given spatial region, that gener-
ates periodic infection [219]. Spatial epidemiology can be helpful in this situation
to develop strategies to control the transmission of disease. As a result, spatial
epidemiology with self-diffusion and cross-diffusion has arisen as the principal sci-
entific discipline devoted to understand the causes and consequences of spatial
heterogeneity in infectious diseases, particularly in zoonoses diseases such as in-
fluenza, SARS and MERS [219] that are transmitted to humans from non-human
vertebrate reservoirs. The most important and difficult aims are now to incorpo-
rate spatial effects and specify the dispersion of individuals. Reaction−diffusion
systems have gained much importance in recent years in epidemiological modelling
to target these spacial effects. The investigation of dynamical systems with diffu-
sion in living organism or species has become a fascinating phenomenon in system
biology. Various terms such as self-diffusion, tracer diffusion, mutual diffusion, up-
57
hill diffusion, inter diffusion and cross-diffusion are used to describe diffusion of
objects/species [62, 224]. The term diffusion was first analysed experimentally for
binary liquid mixtures in 1850 [87] and later, in 1855, Fick [75] developed theories
for it. However it was not till 1955 when the existence of cross-diffusion was ex-
perimentally verified by Baldwin et al. [13].
Self-diffusion is termed as passive diffusion where the diffusing object moves along
its intensification. On the other hand, cross-diffusion involves reverse-mobility. So
in terms of population compartments the phenomenon of cross-diffusion can be
defined as the tendency of the susceptible to keep away from the infected as the
susceptible individual has the ability to recognize the infected and move away from
them [213]. This fact has been generally overlooked despite of its potential eco-
logical reality and intrinsic theoretical interest. The value of the cross-diffusion
coefficient can possibly be positive, negative or zero. Positive cross-diffusion refers
to the movement of these susceptible towards lower density of infected while nega-
tive cross-diffusion means that susceptible tend to diffuse towards regions of higher
densities of the infected [219].
Cross-diffusion with reaction diffusion systems occur in living, social and physico-
chemical contexts etc [229]. A number of ecologists and mathematicians have made
contributions to investigate the stability behavior of a system of interacting popu-
lations by taking into account the effect of self-diffusion as well as cross-diffusion.
Kerner [137] found that cross-diffusion can induce pattern-forming instability in an
ecological situation. Gurtin [97] developed mathematical models to investigate the
effects of cross-diffusion and self diffusion on the population dynamic and showed
that cross-diffusion could give rise to the segregation of two species. In the litera-
ture, Jorne [128, 129], Freedman and Shukla [79], Gatto and Rinaldi [82], Hastings
[103], Okubo [180] and Chattopadhyay et al. [38], developed models for interactive
populations to study the effect of cross-diffusion. Shukla and Verma [214] showed
that the cross-diffusion of species may lead to stability, depending upon the na-
ture and the magnitudes of the self and cross-diffusion coefficients. Kuznetsov
et al. [145] developed a mathematical model of cross-diffusion with two interacting
components qualitatively describing the spatial-temporal dynamics of a mixed-age
58
monospecies forest. Zhang and Fu [253] analysed the stability of the nonnegative
constant steady states for a realistic and mathematically complex cross-diffusive
prey-predator system. They showed that in the presence of cross-diffusion, the
unique positive constant steady state is asymptotically stable, also non-constant
positive steady solutions can exist. Chattopadhyay and Tapaswi [37] showed the
crucial role of negative cross-diffusion in the structure of tumour growth. Sun et al.
[219] investigated the spatial model with cross-diffusion in the susceptible. In re-
cent years several researchers investigated the dynamics of interacting populations
with self and cross-diffusion [21, 176, 221]. However untill now, little work has been
done to study the importance of cross-diffusion phenomena for the spatial spread
and transmission of infectious diseases.
The present study is a numerical investigation of the effects of cross-diffusion on
the transmission dynamics of Severe Acute Respiratory Syndrome (SARS). Here
the self diffusion is defined as the effect of the population pressure on the diffusion
of its own compartmental population and cross-diffusion as the density pressure
on compartmental population because of movement in another compartment. So
here, each individual is affected by pressures based on its own population and the
population of the other compartments, due to mixing between different compart-
ments. The spatial epidemic model with both self-and cross-diffusion along with
the parameters of the model, are defined in Sec. 2. Stability of point equilibrium
and bifurcation analysis of the model are derived in Sec. 3. Numerical scheme
and solutions are given in Sec. 4 and 5. In Sec. 6 the results are summarised and
concluded. In order to differentiate between two types of diffusion simple diffusion
has been referred as self diffusion in this chapter.
4.2 The SEIJR epidemic model
4.2.1 Equations
This model is based on the SEIJR model discussed in Chapter 3 with the inclusion
of self and cross-diffusion in the equations governing the system. Cross-diffusion
is introduced in susceptible (S) and exposed to disease (E) populations only. The
59
Table 4.1: Interpretation of parameters (per day)
Parameter Description Values
Π Rate of inflow of susceptible individuals into region 3.3× 10−5b
β Transmission rate 0.75a
µ Rate of natural mortality 3.4× 10−5b
l Relative measure of reduced risk among diagnosed 0.38a
κ Rate of progression from exposed to infected 0.33a
q Relative measure of infectiousness for exposed individuals 0.1a
α Rate of progression from infective to diagnosed 0.33a
γ1 Recovery rate of infected individuals 0.125a
γ2 Recovery rate of diagnosed individuals 0.2a
δ SARS induced mortality rate .006a
a(Chowell G. et al. [45]), b (Gummel A. B. et al. [95] )
total population proportion is to be N where N = S + E + I + J +R.
∂S
∂t= −β
(I + qE + lJ)
NS − µS +Π+ d1
∂2S
∂x2+ dse
∂2E
∂x2(4.1)
∂E
∂t= β
(I + qE + lJ)
NS − (µ+ κ)E + d2
∂2E
∂x2+ des
∂2S
∂x2(4.2)
∂I
∂t= κE − (µ+ α + γ1 + δ)I + d3
∂2I
∂x2(4.3)
∂J
∂t= αI − (µ+ γ2 + δ)J + d4
∂2J
∂x2(4.4)
∂R
∂t= γ1I + γ2J − µR + d5
∂2R
∂x2(4.5)
where the variables S, E, I, J and R denote the proportions of susceptible, exposed,
infected, diagnosed and recovered individuals respectively. d1, d2, d3, d4 and d5 are
self diffusion constants while dse and des are cross-diffusion constants. Table 4.1
provides description and the values of the parameters.
60
4.2.2 Initial and boundary conditions
The domain of all the calculations is taken to be [−2, 2]. Boundary and initial
conditions are chosen as follows:
∂S(−2, t)
∂x=
∂E(−2, t)
∂x=
∂I(−2, t)
∂x=
∂J(−2, t)
∂x=
∂R(−2, t)
∂x= 0 (4.6)
∂S(2, t)
∂x=
∂E(2, t)
∂x=
∂I(2, t)
∂x=
∂J(2, t)
∂x=
∂R(2, t)
∂x= 0 (4.7)
(i)
S0 = 0.98Sech(5x− 1), − 2 ≤ x ≤ 2.
E0 = 0, − 2 ≤ x ≤ 2.
I0 = 0.02Sech(5x− 1), − 2 ≤ x ≤ 2.
J0 = 0, − 2 ≤ x ≤ 2.
R0 = 0, − 2 ≤ x ≤ 2.
(ii)
S0 ={
0.96Sech(15x), − 2 ≤ x ≤ 2.
E0 = 0, − 2 ≤ x ≤ 2.
I0 =
0, − 2 ≤ x < −.6,
0.04, − .6 ≤ x ≤ .6,
0, .6 < x ≤ 2.
J0 = 0, − 2 ≤ x ≤ 2.
R0 = 0, − 2 ≤ x ≤ 2.
The initial conditions are graphed in Fig. 4.1. Under initial condition (i) large
proportions of the susceptible and infected populations are concentrated towards
the right half of the main domain. For initial condition (ii) most susceptible S
are near the middle of the domain [−2, 2] and infectious individuals widely spread
around the middle of the domain.
Four cases with different pairs of values for the cross-diffusion coefficient des and
61
t = 0
t = 0
(i)-2 -1 0 1 2
x
0.2
0.4
0.6
0.8
1.0
1.2S,I
t = 0
t = 0
(ii)
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0S,I
Figure 4.1: Initial Conditions (i) and (ii).
dse have been chosen, as given in Table 4.2. Numerical solutions are given for all
these cases.
Table 4.2: Cases for cross-diffusion
Case dse des
(a) 0.00000 0.00000
(b) 0.01250 0.00000
(c) 0.01250 0.00350
(d) −0.0125 0.00350
4.3 Stability analysis
4.3.1 Reproduction number and disease-free equilibrium
(DFE)
The variation matrix for the system of equations (4.1) - (4.5) and the conditions
for disease-free equilibrium are chosen to be the same as in the previous chapter.
Thus the expression for the reproduction number, without diffusion is as follows:
RI=q(µ+α+γ1+δ)(µ+γ2+δ)+κ(µ+γ2+δ)+lακ
(µ+α+γ1+δ)(µ+γ2+δ)(κ+µ).
62
4.3.2 Stability of endemic equilibrium with cross-diffusion
To calculate the small perturbations S1(x, t), E1(x, t), I1(x, t),J1(x, t) and R1(x, t),
the equations are linearized about the point of equilibrium P ∗(S∗, E∗, I∗, J∗, R∗)
as described in [35, 201].
∂S1
∂t= a11S1 + a12E1 + a13I1 + a14J1 + a15R1 + d1
∂2S1
∂x2+ dse
∂2E1
∂x2(4.8)
∂E1
∂t= a21S1 + a22E1 + a23I1 + a24J1 + a25R1 + d2
∂2E1
∂x2+ des
∂2S1
∂x2(4.9)
∂I1∂t
= a31S1 + a32E1 + a33I1 + a34J1 + a35R1 + d3∂2I1∂x2
(4.10)
∂J1∂t
= a41S1 + a42E1 + a43I1 + a44J1 + a45R1 + d4∂2J1∂x2
(4.11)
∂R1
∂t= a51S1 + a52E1 + a53I1 + a54J1 + a55R1 + d5
∂2R1
∂x2(4.12)
where a11, a12, a13 etc are the elements of the variational matrix V ∗ calculated using
the same method as described in [198] and are same as given in Chapter 3. Assume
a Fourier series solution exists for equations (4.8) - (4.12) of the form:
S1(x, t) =∑k
Skeλt cos(kx) (4.13)
E1(x, t) =∑k
Ekeλt cos(kx) (4.14)
I1(x, t) =∑k
Ikeλt cos(kx) (4.15)
J1(x, t) =∑k
Jkeλt cos(kx) (4.16)
R1(x, t) =∑k
Rkeλt cos(kx) (4.17)
where k = nπ2, (n = 1, 2, 3, · · · · · · ) is the wave number for the node n. Substituting
the values of S1, E1, I1, J1, R1 in the equations (4.8) - (4.12), the equations are
transformed into∑k
(a11 − d1k2 − λ)Sk +
∑k
(a12 − dsek2)Ek +
∑k
a13Ik +∑k
a14Jk +∑k
a15Rk = 0
(4.18)∑k
(a21 − desk2)Sk +
∑k
(a22 − d2k2 − λ)Ek +
∑k
a23Ik +∑k
a24Jk +∑k
a25Rk = 0
(4.19)
63
∑k
a32Ek +∑k
(a33 − d3k2 − λ)Ik = 0 (4.20)
∑k
a43Ik +∑k
(a44 − d4k2 − λ)Jk = 0 (4.21)
∑k
a53Ik +∑k
a54Jk +∑k
(a55 − d5k2 − λ)Rk = 0 (4.22)
The Variational matrix V d for the equations (4.18) - (4.22) is
V d =
a11 − d1k2 a12 − dsek
2 a13 a14 a15
a21 − desk2 a22 − d2k
2 a23 a24 a25
0 a32 a33 − d3k2 0 0
0 0 a43 a44 − d4k2 0
0 0 a53 a54 a55 − d5k2
Where a11, a12, a13,... are same as given in Chapter 3. The characteristic equation
for the variational matrix V d is given as
λ5 + q1λ4 + q2λ
3 + q3λ2 + q4λ+ q5 = 0 (4.23)
where q1, q2, q3, q4 and q5 are calculated with the same technique as used in [198].
Routh-Hurwitz Conditions for stability are given as:
C1 : q1 > 0,
C2 : q5 > 0,
C3 : q1q2 − q3 > 0,
C4 : q1q2q3 + q1q5 − (q23 + q21q4) > 0,
C5 : (q1q4 − q5)(q1q2q3 − q23 − q21q4) + q21q4q5 − (q5(q1q2 − q3)2q1q
25) > 0.
The numerical results of L.H.S of Routh-Hurwitz Conditions on the point of
equilibrium, P1 = (0.331581, 0.000064, 0.000046, 0.000074, 0.607887) are given in
Table 4.3 for all cases taken in Table 4.2.
64
Table 4.3: Routh-Hurwitz criteria with and without cross-diffusion
Case dse des C1 C2 C3 C4 C5 Stability
(a) 0.0000 0.0000 1.066 4.9 × 10−8 0.295 0.0048 8.1 × 10−7 Stable
(b) 0.0125 0.0000 1.013 1.8 × 10−8 0.242 0.0016 4.1 × 10−8 Stable
(c) 0.0125 0.0035 1.066 4.4 × 10−8 0.296 0.0048 −7.2 × 10−7 Unstable
(d) −0.0125 0.0035 1.013 1.7 × 10−8 0.242 0.0016 −1.2 × 10−7 Unstable
4.3.3 Reproduction number with diffusion
Variational matrix method is used to calculate reproduction number with diffusionRd
I . The variational matrix with diffusion for P0 = (1, 0, 0, 0, 0) is given as followes:
Vd
=
−(d1k2 + µ) −(dsek
2 + qβ) −β −lβ 0
−desk2 qβ − (d2k
2 + κ + µ) β lβ 0
0 κ −(d3k2 + α + γ1 + δ + µ) 0 0
0 0 α −(d4k2 + γ2 + δ + µ) 0
0 0 γ1 γ2 −(d5k2 + µ)
It is observed that last eigenvalue −(d5k2 + µ) of variational matrix V d is negative
and all entries above it are zero. This allows to eliminate the last row and column.
So, the reduced matrix is given as follows:
V d =
−(d1k
2 + µ) −(dsek2 + qβ) −β −lβ
−desk2 qβ − (d2k
2 + κ+ µ) β lβ
0 κ −(d3k2 + α + γ1 + δ + µ) 0
0 0 α −(d4k2 + γ2 + δ + µ)
The characteristic equation for the above matrix is given as :
λ4 + p1λ3 + p2λ
2 + p3λ1 + p4 = 0 (4.24)
Where p1, p2, p3 and p4 are calculated as in [198] and are given in appendix A.4.
The Routh-Hurwitz criterion for the stability is given as in [200]. The Routh Hur-
witz condition p4 > 0 gives the following expression for reproduction number, RdI
with diffusion:
RdI =
k2(A+B + C) +D
(d1k2 + µ)(d3k2 + α + γ1 + δ + µ)(d4k2 + γ2 + δ + µ)(d2k2 + κ+ µ)(4.25)
where
A = (desdsek2 + d1qβ + desqβ)(α + γ1 + δ + µ)(γ2 + δ + µ)
65
B = (d3(d4k2 + γ2 + δ + µ)(des(dsek
4 + k2qβ) + qβ(d1k2 + µ)) − d4(β(d1k
2 +
µ)(κ+ q(α+ γ1+ δ+µ))+ desk2(dsek
2(α+ γ1+ δ+µ)+β(κ+ q(α+ γ1+ δ+µ)))))
C = (d1 + des)βκ(lα + γ2 + δ + µ)
D = βµ(q(α + γ1 + δ + µ)(γ2 + δ + µ) + κ(lα + γ2 + δ + µ))
The values ofRdI for the cases (a) - (d) as defined in Table 4.2 are given in Table4.4.
Table 4.4: Reproduction number
Cases Value of RdI
(a) 2.6
(b) 3.95
(c) 3.96
(d) 1.31
4.3.4 Excited mode and bifurcation value
The first excited mode of the oscillation n is calculated by the same technique as
used in [35]. In Case (a), n = 1 represents the first mode of excitation as being
closest to the β-axis of the curve given by equation (4.26). Similarly, n = 1 is the
first mode of excitation for Cases (b)− (d).
f(β) = (q1q4 − q5)(q1q2q3 − q23 − q21q4) + q21q4q5 − (q5(q1q2 − q3)2q1q
25). (4.26)
Bifurcation values of the transmission coefficient β, recovery rate of infected indi-
viduals γ1 and recovery rate of diagnosed individuals γ2 are given in Table 4.5. It
is observed that the bifurcation value of the transmission coefficient, β increases
in Cases (a) and (b) and decreases in Cases (c) and (d). So the system remains
stable for higher values in Cases (a) and (b) while in Cases (c) and (d), the system
becomes stable at smaller bifurcation values. For the infected and diagnosed recov-
ery coefficients, γ1 and γ2, the system becomes stable at smaller bifurcation values
66
n=1
0.5 1.0 1.5 2.0Β
2.´ 10-6
4.´ 10-6
6.´ 10-6
8.´ 10-6
0.00001
fHΒL
n=2
0.5 1.0 1.5 2.0Β
0.0002
0.0004
0.0006
0.0008
0.0010
0.0012
fHΒL
n=3
0.5 1.0 1.5 2.0Β
-0.005
0.005
0.010
0.015
0.020
0.025
fHΒL
Figure 4.2: Determination of first excited mode with β as an unknown parameter.
Table 4.5: Bifurcation value of β, γ1 and γ2 with cross-diffusion
Cases Value Considered Bifurcation Value
β γ1 γ2 β γ1 γ2
(a) 0.75 0.125 0.20 0.776 0.109 0.182
(b) 0.75 0.125 0.20 0.869 0.108 0.181
(c) 0.75 0.125 0.20 0.727 0.180 0.279
(d) 0.75 0.125 0.20 0.734 0.174 0.269
in Case (a) and (b). In Cases (c) and (d), the system becomes stable at higher
bifurcation values. The corresponding bifurcation diagrams are given in appendix
A.4.
4.4 Numerical scheme
The operator splitting technique [244] has been used to solve the SEIJR model
equations. The equations are divided into two groups of sub equations. The first
group comprises the nonlinear reaction equations to be used for the first half-time
67
step:1
2
∂S
∂t= −β
(I + qE + lJ)
NS − µS +Π (4.27)
1
2
∂E
∂t= β
(I + qE + lJ)
NS − (µ+ κ)E (4.28)
1
2
∂I
∂t= κE − (µ+ α + γ1 + δ)I (4.29)
1
2
∂J
∂t= αI − (µ+ γ2 + δ)J (4.30)
1
2
∂R
∂t= γ1I + γ2J − µR (4.31)
The second group consists of the linear diffusion equations, to be used for the
second half-time step as follows:
1
2
∂S
∂t= d1
∂2S
∂x2+ dse
∂2E
∂x2(4.32)
1
2
∂E
∂t= d2
∂2E
∂x2+ des
∂2S
∂x2(4.33)
1
2
∂I
∂t= d3
∂2I
∂x2(4.34)
1
2
∂J
∂t= d4
∂2J
∂x2(4.35)
1
2
∂R
∂t= d5
∂2R
∂x2(4.36)
The solution of the first half-time step is given in Chapter 3. For the second half-
time step,
Sj+1i = S
j+ 12
i +∆t
(∆x)2(d1(S
j+ 12
i−1 −2Sj+ 1
2i +S
j+ 12
i+1 )+dse(Ej+ 1
2i−1 −2E
j+ 12
i +Ej+ 1
2i+1 )) (4.37)
Ej+1i = E
j+ 12
i +∆t
(∆x)2(d2(E
j+ 12
i−1 −2Ej+ 1
2i +E
j+ 12
i+1 )+des(Sj+ 1
2i−1 −2S
j+ 12
i +Sj+ 1
2i+1 )) (4.38)
Ij+1i = I
j+ 12
i + d3∆t
(∆x)2(I
j+ 12
i−1 − 2Ij+ 1
2i + I
j+ 12
i+1 ) (4.39)
J j+1i = J
j+ 12
i + d4∆t
(∆x)2(J
j+ 12
i−1 − 2Jj+ 1
2i + J
j+ 12
i+1 ) (4.40)
Rj+1i = R
j+ 12
i + d5∆t
(∆x)2(R
j+ 12
i−1 − 2Rj+ 1
2i +R
j+ 12
i+1 ) (4.41)
The stability condition satisfied by the numerical method described above is given
as:dn∆t
(∆x)2≤ 0.5, n = 1, 2, 3, 4, 5. (4.42)
In each case, ∆x = 0.1, ∆t = 0.03, d1 = 0.025, d2 = 0.01, d3 = 0.001, d4 = 0.0 and
d5 = 0.0 are used.
68
4.5 Numerical solutions
Four cases, with different pairs of values of the cross-diffusion coefficients des and
dse, have been chosen, as given in Table 4.2. Numerical solutions are given for all
these cases.
4.5.1 Numerical solution for initial condition (i)
Fig. 4.3, shows the numerical solution with initial condition (i) and self diffusion
for Case (a). In the first five days a significant proportion of the susceptible pop-
ulation moves to the exposed class and the effective domain of susceptible alters
from [−1, 1.5] to [−2, 2], producing a pulse with peak value 0.222857. At t = 10
days almost the whole proportion of susceptible have been exposed to the disease.
After that the proportion of susceptible is not apparent in Fig. 4.3, but it has
pulses with peak values 0.005952, 0.000013, 0.000023 at t = 10, t = 15 and t = 20
days respectively. Initially the exposed are mainly confined to domain [−1, 1.5]
producing a pulse with peak value 0.118055 at t = 5 days. At t = 10 days, the
exposed population is spread accross domain [−2, 2] producing a pulse with peak
value 0.068133. This is followed by a rapid decrease in the exposed proportion and
thus at t = 15 days, a very small proportion of exposed is observed, producing a
pulse with peak value 0.008858. Infection grows with the passage of time. In the
first five days of disease infected are concentrated in domain [−1, 1.5] producing a
pulse with peak value 0.046294. The proportion of the infected spreads across the
domain [−2, 2] with the passage of time and at t = 10, produces a pulse with peak
value 0.077894. After this there is a fall in the infected population proportion and
at t = 20, there is a very low level of infected, producing a pulse with peak value
0.002717 confined to domain [−2, 2]. The diagnosed proportion is fairly small in
the first five days of the disease producing a pulse with peak value 0.026952 in the
domain [−1, 1.5]. After t = 5 days, there is a sudden increase in the diagnosed
proportion of population across the entire domain [−2, 2]. The proportion of the
diagnosed population produces a pulse with peak value 0.105364 at t = 10. After
that, there is a gradual decrease in the diagnosed population proportion producing
69
a pulse with peak value 0.072124 at t = 15 days. In the last five days of study of
disease a rapid decrease is observed in diagnosed population proportion, producing
a pulse with peak of only 0.028397 at t = 20 days. There is a small recovery in the
first five days of the disease but in the next fifteen days recovery rapidly increases
in proportion. At t = 10 days, the peak value of the pulse of proportion of recov-
ered population reaches 0.173848. At t = 15 days, peak value of pulse produced by
recovered population reaches 0.320015. At t = 20 days, the pulse attains a peak
value 0.383669 with the domain extending to [−2, 2].
Fig. 4.4, shows the output for initial condition (i) for Case (b). Here cross-diffusion,
dse, is introduced among the susceptible population. In this case, due to the den-
sity pressure of the exposed population, susceptible move in the direction of low
concentration of the exposed population because of cross-diffusion in the system.
This situation causes a change in the transmission of the disease such that at
t = 5 days, the peak value of the pulse of the susceptible population proportion is
0.217763 and this peak value remains steady in the domain [−0.4, 0.7]. The peak
value of pulses produced for the exposed proportion are 0.116768, 0.060374 and
0.007727 at t = 5, t = 10 and t = 15 days, respectively. At t = 5 days, the peak
value of the infected population pulse is 0.046061 in the domain [−1, 1.5]. The
pulse produced by infected proportion keeps on increasing and reaches peak value
0.071505 at t = 10 days in domain [−2, 2]. At t = 15 days the proportion of popu-
lation infected decreases with the peak value 0.015325. At t = 20, there is a further
decrease in infected proportion producing pulse with peak value 0.002382. Here,
the intensities of the proportions of diagnosed and infected are slightly lower for
Case (a). This indicates a slow transmission pattern of the disease. The recovered
population follows the same pattern as in Case (a) except that the pulses produced
here have lower peak values.
Fig. 4.5, shows the output with initial condition (i) for Case (c). In this case the
susceptible and exposed populations cause motility to the lower concentration of
each other due to cross-diffusion in the system. This generates a different pattern
of disease transmission by giving rise to oscillations. There is a sudden fall in the
population proportion of susceptible, producing a pulse with peak value 0.329880 at
70
t = 5 days in the domain [−1.5, 2]. The susceptible population move to the exposed
compartment quite quickly and after t = 10 days, there is a negligible proportion
of susceptible left. The proportion of population exposed accordingly increases in
the first five days of the disease with the production of oscillations. The first pulse
appears in the domain [−2, 0.2] with value 0.068716. The lowest amplitude of this
pulse is observed to be 0.018522. The next pulse appears in the domain [0.2, 2]
with same peak value 0.068716. At t = 10 days, exposed population spread across
the domain [−2, 2] producing a single pulse with peak value 0.074724. In next five
days the proportion of the exposed population suddenly reduces producing a pulse
with peak value 0.008375 at t = 15 days. Infection slowly grows in the first five
days, producing oscillations with the first pulse having peak value 0.026041 in the
domain [−2, 0.2]. This pulse reaches the lowest amplitude of 0.005 and the second
pulse appear in the domain [0.2, 2] with peak value 0.026041. There is a sudden
increase in the infected population at t = 10 days, with pulses having peak value
0.067428 in the domain [−1, 1]. There is a sudden increase in the proportion of
diagnosed population between t = 5 and t = 10 days. Oscillations are generated
here too, producing pulses with peak value 0.074977 in the domain [−1, 1] at t = 10
days. Proportion of recovered population also shows oscillations, producing pulses
having highest amplitude of 0.276379 at t = 20 days.
Fig. 4.6, shows the output for initial condition (i) for Case (d). In this case the
negative value of the diffusivity coefficient, dse, shows that density pressure of the
exposed causes the mobility of the susceptible towards higher concentration of the
exposed population. This also has significant impact on the transmission of disease.
This case also gives rise to oscillations as in Case (c). There are however slight
differences in the behavior of the oscillations as compared to Case (c). For example
the lowest amplitude of the first pulse is higher in this case than with Case (c) in
almost all compartments. Here at t = 10 exposed population produce pulses in the
interval [−1, 1] with little difference in their peak values. Also the domain of the
exposed population narrows down slightly at t = 10 days, as compared to Case (c).
At t = 15 days the exposed and infected remain steady in the domain [−0.4, 0.8]
unlike for Case (c). Exposed and recovered populations behave in the same way as
71
t = 0
t = 5
t = 10 t = 15 t = 20
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0
S
t = 5
t = 10
t = 20t=15
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
0.10
0.12E
t = 0t = 20
t=15
t=5
t=10
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
I
t = 0
t = 20
t=5
t=15
t=10
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
0.10
0.12J
t = 0
t = 5
t = 10
t = 15t = 20
t=0-2 -1 0 1 2
x
0.1
0.2
0.3
0.4R
Figure 4.3: Solutions with initial condition (i) for Case (a).
for Case (c), apart from differences peak values of the pulses along the domain of
activity. Moreover, the effective domain of all compartments reduce as compared
to Case (c).
4.5.2 Numerical solution for initial condition (ii)
Fig. 4.7, shows the results with initial condition (ii) for Case (a). A sudden fall
in the proportion of susceptible is observed at t = 5 producing pulses with peak
value 0.005466. Susceptible proportions produce pulse with peak values 0.000213,
0.000025 and 0.000048 at t = 10, t = 15 and t = 20 days respectively. At t = 5
days, the exposed population spreads across domain [−1.2, 1.2] producing a pulse
with peak value 0.043322. The exposed population proportion shows rapid de-
crease and produces a pulse with peak value 0.005844 at t = 10 days with the
72
t = 0
t = 5
t = 10
t = 20-2 -1 1 2
x
0.2
0.4
0.6
0.8
1.0
S
t = 5
t = 10
t=15
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
0.10
0.12E
t = 0
t = 20
t=10
t=5
t=15
-2 -1 0 1 2x
0.02
0.04
0.06
0.08I
t = 15
t = 20
t=10
t=5
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
0.10J
t = 0
t = 5
t = 15
t = 20
t=10
-2 -1 0 1 2x
0.1
0.2
0.3
0.4R
Figure 4.4: Solutions with initial condition (i) for Case (b).
73
t = 0
t = 5
t = 20-2 -1 1 2
x
0.2
0.4
0.6
0.8
1.0
S
t = 5t = 10
t = 20t=15
-2 -1 0 1 2x
0.02
0.04
0.06
0.08E
t = 0
t = 20
t=5
t=10
-2 -1 0 1 2x
0.02
0.04
0.06
0.08I
t = 20
t=10
t=15
t=5
-2 -1 0 1 2x
0.02
0.04
0.06
0.08J
t = 5
t = 15
t = 20
t=10
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25
0.30R
Figure 4.5: Solutions with initial condition (i) for Case (c).
74
t = 0
t = 5
t = 20-2 -1 1 2
x
0.2
0.4
0.6
0.8
1.0
S
t = 5
t = 10
t = 20
t=15
-2 -1 0 1 2x
0.02
0.04
0.06
0.08E
t = 0
t = 20
t=10
t=5t=15
-2 -1 0 1 2x
0.02
0.04
0.06
0.08I
t = 20
t=10
t=15
t=20
t=5
-2 -1 0 1 2x
0.02
0.04
0.06
0.08J
t = 0
t = 5
t = 15
t = 20
t=10
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25
0.30R
Figure 4.6: Solutions with initial condition (i) for Case (d).
75
main concentration of the population in the domain [−1.8, 1.8]. Infection reduces
in first five days of the disease, producing a pulse with peak value 0.032972 in the
domain [−1, 1]. At t = 10, infected population spreads in the domain [−1.5, 1.5]
producing a pulse with peak value 0.010391. At t = 5 days, proportion of the
population diagnosed with SARS in the domain [−1, 1] is producing a pulse with
peak value 0.033966. This diagnosed proportion of the population spreads to do-
main [−1.4, 1.4] at t = 10 days, producing a pulse with peak value 0.032229. After
that there is a quick decrease in the proportion of diagnosed, which spread across
domain [−1.5, 1.5] producing a pulse with peak value 0.013849 at t = 15 days.
This proportion reduce to 0.004733 at t = 20 days. Recovery initially occurs in
the domain [−1, 1] at t = 5, producing a pulse with peak value 0.045636. There is
a further increase in the recovered at t = 10 producing a pulse with a peak value
0.106593 in the domain [−1.2, 1.2]. At t = 20 there is maximum recovery producing
a pulse with peak value 0.147773. Here, the population of recovered concentrates
in domain [−1.5, 1.5].
Fig. 4.8, shows the output for the initial condition (ii) and Case (b). Values of
the susceptible proportion over time are very low and thus difficult to observe in
the graph. The proportion of susceptible is, however, higher in magnitude than for
Case (a) in the first ten days and lower in the next ten days of the disease. The
peak value of the pulse produced by the exposed proportion of the population at
t = 5 is 0.039203 in the domain [−1.2, 1.2]. It undergoes a decline over the next five
days, producing a pulse with peak value 0.004513 in the domain [−2, 2] . There is
a reduction in infected population between t = 5 and t = 10 days, producing pulses
with peak values 0.031539 and 0.008701 respectively with an increase in length of
effective domain compared to Case (a). At t = 15 and t = 20 days the propor-
tion of population infected is small, producing pulses with peak value 0.001352
and 0.002333 respectively. At t = 5 the proportion of the population diagnosed
has a pulse with peak value 0.033472. At t = 10 days, the peak value of pulse
for the diagnosed is 0.029586 which decreases further at t = 15 days, to a pulse
with peak value 0.012348. At this stage the diagnosed population spread over the
whole domain [−2, 2]. At t = 5 days, the proportion of the population recovered
76
has a pulse with peak value 0.053113 in the domain [−1, 1]. At t = 10 days, the
recovered population spread over the domain [−1.4, 1.4], and the pulse has a higher
peak value 0.102424. At t = 15 and t = 20 days, the proportion of the population
recovered spreads to the domain [−1.8, 1.8] and has pulses with increased peak
values 0.129854 and 0.139435. The improvement in recovery is greatest between
t = 5 and t = 10 days, after which the improvement slows.
Fig. 4.9, shows the output for initial condition (ii) and Case (c). The introduction
of cross-diffusion in the susceptible and exposed populations causes a change in
the transmission of the disease. The susceptible are found to be greater at t = 5
days, than for Case (a) and in the last fifteen days of the disease, the proportion
is less than for Case (a). At t = 5 days, the effective domain of the exposed pop-
ulation proportion is [−1.5, 1.5] and the pulse has peak value 0.037877. At t = 10
days, there is an increase in the domain from [−1.5, 1.5] to [−2, 2] with a decline
in the exposed population giving a pulse with peak value 0.004453. At t = 5,
the infected population has two pulses with peak value 0.027584 in the interval
[−1.2, 1.2]. At t = 10 there is a single pulse with peak value 0.008502, which is
quite low compared to Case (a). In the first five days, the proportion of population
diagnosed has two pulses in the domain [−1.2, 0] and [0, 1.2] with common peak
value 0.030418. At t = 10 and t = 15 days the peak values of the pulses reduce to
0.027121 and 0.011435 with domain extended to [−1.6, 1.6]. These peak values of
pulses are lower than for Case (a), with significant difference at t = 15 days. The
recovered population also oscillates with increase in the peak values of its pulses
with the passage of time. The effective domain of the recovered population is wider
compared to Case (a).
Fig. 4.10, shows the output for the initial condition (ii) and Case (d). Here again
the negative value of the diffusivity coefficient, dse, from susceptible to exposed,
population density pressure of the exposed causes the motility of the susceptible
towards higher concentration of the exposed population. This case also gives rise
to oscillations (as in the previous Case (c)), with significant differences in the peaks
of pulses observed at various time steps. In this case the diagnosed population has
pulses with higher peak values at t = 10 days, compared to Case (c). At t = 15
77
t = 0
t = 5 t = 10 t = 15 t = 20
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0
S
t = 20t=15
t=10
t=5
-2 -1 0 1 2x
0.01
0.02
0.03
0.04
0.05E
t = 20
t=0
t=5
t=10
t=15
-2 -1 0 1 2x
0.01
0.02
0.03
0.04
I
t = 0
t = 20
t=5
t=10
t=15
-2 -1 0 1 2x
0.005
0.010
0.015
0.020
0.025
0.030
0.035J
t = 0t = 10
t=5
t=10
t=20
t=15
-2 -1 0 1 2x
0.05
0.10
0.15
R
Figure 4.7: Solutions with initial condition (ii) for Case (a).
there is a smooth curve for diagnosed in the interval [−1.2, 1.2]. The effective do-
main of population in all compartments is narrower than in Case (c). Recovery
in this case is comparatively slower than in Case (a). There are oscillations pro-
duced in this case too. At t=5 days, two pulses are produced with shared peak
value 0.042747 in effective domain [−0.8, 0.8]. At t = 10 days, two pulses are also
produced with peak value 0.099205 in the domain [−1.2, 1.2]. At t = 20 days, the
proportion of population recovered concentrates inside the domain [−1.4, 1.4] with
smaller peak values of pulses than for Case (a).
78
t = 0
-2 -1 1 2x
0.2
0.4
0.6
0.8
1.0
S
t = 10
t=5
t=15
-2 -1 0 1 2x
0.01
0.02
0.03
0.04
0.05E
t = 0
t=5
t=10t=15
-2 -1 0 1 2x
0.01
0.02
0.03
0.04I
t = 20
t=5
t=10
t=15
-2 -1 0 1 2x
0.005
0.010
0.015
0.020
0.025
0.030
0.035J
t = 10
t=20
t=15
t=10
t=5
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
0.10
0.12
0.14
R
Figure 4.8: Solutions with initial condition (ii) for Case (b).
79
t = 0
t = 20-2 -1 1 2
x
0.2
0.4
0.6
0.8
1.0
S
t = 10
t=5
t=15
-2 -1 0 1 2x
0.01
0.02
0.03
0.04
0.05E
t = 0
t=5
t=10
t=15
-2 -1 0 1 2x
0.01
0.02
0.03
0.04
I
t = 20
t=5
t=10
t=15
-2 -1 0 1 2x
0.005
0.010
0.015
0.020
0.025
0.030
0.035J
t = 10
t=20
t=15
t=10
t=5
-2 -1 0 1 2x
0.05
0.10
0.15
R
Figure 4.9: Solutions with initial condition (ii) for Case (c).
80
t = 0
-2 -1 1 2x
0.2
0.4
0.6
0.8
1.0
S
t = 10
t=5
t=15
-2 -1 0 1 2x
0.01
0.02
0.03
0.04
0.05E
t = 0
t=5
t=10
t=15
-2 -1 0 1 2x
0.01
0.02
0.03
0.04
I
t = 20
t=5
t=10
t=15
-2 -1 0 1 2x
0.005
0.010
0.015
0.020
0.025
0.030
0.035J
t=20
t=15
t=10
t=5
-2 -1 0 1 2x
0.05
0.10
0.15
R
Figure 4.10: Solutions with initial condition (ii) for Case (d).
81
4.6 Discussion
In this paper SEIJR model for SARS described in Chapter 1 is considered, with
self and cross-diffusion included in the system. Two different initial conditions
are taken for the population distribution. Differential equations governing the sys-
tem are solved numerically using the operator splitting technique with forward
and central difference schemes. The stability of solutions with and without cross-
diffusion is established using Routh-Hurwitz conditions. Four different cases with
cross-diffusion coefficients in the susceptible and exposed compartments are chosen,
with a view to see the effect on the spread of disease.
Bifurcation values of the transmission coefficient β and recovery coefficients γ1 and
γ2 are obtained. It is observed that in Case (b), the system remains stable for a
higher value of β as than in the other cases. Case (a) has bifurcation value of β
higher than Cases (c) and (d). Case (c) has the lowest bifurcation value. The bi-
furcation values for cases (c) and (d) is less than the considered value of β = 0.75.
This shows that when the susceptible and exposed population cross-diffuse, the
system is destabilised for a smaller value of β. Case (c) gives the highest values
of bifurcation for the infected and diagnosed recovery coefficients γ1 and γ2. This
shows that with positive cross-diffusion in the susceptible and exposed compart-
ments, the system stabilises for higher values of the recovery coefficients γ1 and γ2.
With negative cross-diffusion in the susceptible compartment, the system gets sta-
bilised slightly earlier than in Case (c) for the recovery coefficients γ1 and γ2. With
no cross-diffusion in the exposed compartment in Case (b), the system stabilises
much earlier than in the other cases. This implies that with positive cross-diffusion
in the susceptible, a proportion of susceptible move towards the lower concentra-
tions of exposed, thus stabilises the system earlier compared to systems without
cross-diffusion, for recovery coefficients γ1 and γ2.
Graphs of numerical solutions for the initial condition (i) are shown in Figs. (4.3)-
(4.6). For initial condition (i) with cross-diffusion in the susceptible proportion of
population, Case (b), the transmission of disease to the exposed population pro-
duces pulse with lower peak value than for Case (a), without cross-diffusion, at
t = 10 days, of disease as shown in Fig. 4.4. Thus cross-diffusion slows down
82
the transmission of disease to reach its peak value. With cross-diffusion in both
susceptible and exposed populations, oscillations appear in the exposed population
at t = 5 days. Here, two loops are generated in the given domain [−2, 2]. At t = 10
days, the exposed population reduces to one pulse with peak value higher than at
t = 5 days. This resulted in two pulses in the infected at t = 5 days with the
prominent peak values. However at t = 10 days there are two pulses in the infected
proportion of population with no prominent peak values. As a result of oscillations
produced in the exposed population, oscillations are generated in infected, diag-
nosed and recovered proportions of the population. In the case of cross-diffusion
in susceptible and exposed populations, the intensity of disease reduces right from
the stage of exposed to diagnosed. Thus, in this case the recovery process slows.
With cross-diffusion in the exposed proportion of the population, oscillations with
small pulses are produced in all compartments of the population. There is a dip
in proportion of the population exposed near the middle of the domain at t = 5
days because of cross-diffusion in the system. Due to cross-diffusion in the system,
the exposed proportion of the population starts moving away from the middle of
domain to areas of low density, thus creating a dip in this proportion around the
middle of the domain at t = 5 days. With negative cross-diffusion in the susceptible
compartment, the peak values of the pulses of the exposed proportion are lower
than for Case (c) in the first five days, but higher after day ten of the disease. As
an effect of negative diffusion in the susceptible population, the proportion of pop-
ulation infected is higher after day five. Recovery is a little higher with negative
cross-diffusion in the system. The introduction of cross-diffusion in the susceptible
proportion of the population has not helped to contain the disease remarkably.
With cross-diffusion in both the susceptible and exposed proportions of the popu-
lation, small pockets of population have been generated in each compartment. In
the first five days the proportion of the population in the exposed compartment is
lower in Case (d) than in Case (c). After day five of the disease, transmission from
the infected to the diagnosed compartment is slightly higher in Case (d) than in
Case (c). This shows that in Case (d), there is comparatively fast transmission of
disease.
83
For Initial condition (ii), numerical solutions are shown in Figs. (4.7)- (4.10), for
Cases (a) − (d). In the absence of cross-diffusion, the largest spread of infection
goes to the domain [−1.8, 1.8]. When positive cross-diffusion is introduced in the
susceptible compartment the transmission of disease slows and the proportion of
susceptible in the first ten days of disease increases. The intensity of the exposed
and infected populations also goes down, especially in the first ten days, whereas the
domain of proportion expands. When positive cross-diffusion is included in both
the exposed and the susceptible compartments, oscillations are produced where
exposed at t = 5 days doesn’t show any oscillation as in condition (i). The pro-
portion of susceptible is higher in the first five days. Infection spreads slower with
lower peaks of the proportion than for Case (a), and the domain of the proportion
does not change as much as in condition (i). When negative cross-diffusion occurs
in the susceptible while the exposed have positive cross-diffusion, domains of the
population reduces in after t = 5 days. Here the infected population attains higher
peaks than in Case (a). The proportion of population exposed is higher in Case (d)
for all days of the disease under study i.e t = 20 days. The domain of the exposed
population is significantly smaller in Case (d) at t = 5 and t = 10 days than for
Case (c). The proportion of infected shows higher peaks in the first fifteen days
in Case (d), compared with Case (c). The main domain of the infected proportion
is shortened to [−1, 1] in Case (d). The proportions of diagnosed follow the same
transmission pattern as the infected and have the same differences in Cases (d)
and (c). The proportion of the population recovered is higher in Case (d) than in
Case (c) after day five of the disease. The domain of concentration of recovered
proportion is smaller in Case (d) in the last fifteen days of study of the disease than
for Case (c).
The numerical values of reproduction number with and without diffusion are given
in Table 4.4. The numerical value for the reproduction number is 2.83 without
diffusion as given in Chapter 3. Table 4.4 shows that with the inclusion of self-
diffusion, the value of the reproduction number decreases as in Case (a). The
inclusion of positive cross-diffusion in the susceptible compartment, as in Case (b),
and in the susceptible and exposed compartments as in Case (c), cause significant
84
increase in the reproduction number. This leads to an acceleration of the spread of
disease. In Case (d), where cross-diffusion in the exposed compartment is negative,
diffusion of the susceptible towards high concentrations of the exposed has reduced
the reproduction number. Thus the spread of the disease is slowed.
It is now proposed to include treatment compartment in the SEIJR model. In
the view of that all the parameters governing the system of equations have been
re-estimated using SARS data for the 2003 outbreak in Hong Kong, in next chap-
ter.
85
Chapter 5
Parameter Estimation with
Uncertainty and Sensitivity
Analysis for the SARS Outbreak
in Hong Kong
5.1 Introduction
Mathematical modeling of real-life processes often requires the estimation of un-
known parameters. The problem of parameter estimation belongs to the class of
inverse problems in which the knowledge of the dynamical system is derived from
the input as well as output data values of the system. This process is empirical
in nature and often ill-posed because, in many instances, it is possible that some
different model can be fitted to the same response. This process of determining
the unknown parameters of a mathematical model from noisy data based on input
and output values is termed as parameter estimation. Parameter estimation is an
important step in the development of systems biology, as it helps to obtain predic-
tions from computational models of biological systems.
Parameter estimation in dynamic systems is a wide area involving many different
aspects of mathematical as well as statistical analysis. Parameters are estimated
from a data fit depending on their type such as epidemiological or demographic,
86
and, with the resulting model, predictions are made that can be tested with further
experiments. There are a number of difficulties and complexities involved in the
estimation of parameters of non-linear systems. On the basis of prior theoretical
knowledge the structure of a model is suggested defining the state variables and pa-
rameters. The parameter estimation criteria (hardly ever a single criterion) reflect
the desired properties of the estimates. A great deal of effort in recent decades has
gone into developing parameter estimation with the help of algorithms having good
theoretical properties (Norton, 1986; Soderstrom and Stoica, 1989; Ljung, 1999). A
Few of them are Maximum Likelihood Estimator (MLE), Bayes Estimators (BE),
Principal Differential Analysis (PDA), Methods of Moments Estimators (MME),
Minimum Variance Unbiased Estimator (MV UE), Particle Filter, Maximum a
Posteriori (MAP ), Minimum Mean Squared Error (MMSE), Best Linear Unbi-
ased Estimator (BLUE), Markov Chain Monte Carlo (MCMC) method, Kalman
Filter and Ensemble Kalman Filter (EnKF ) [144, 174]. The most used algorithms
to solve the non-linear inverse problems are based on deterministic and stochastic
methods.
Once the parameter estimates have been computed by the means of optimization,
it is very important to know how reliable they are and what is the quality of the
parameter estimates especially if the parameter values are used to draw biological
conclusions from the model. According to Jakeman et al. [127], “Uncertainty in
models stems from incomplete system understanding (which processes to include,
which processes to interact); from imprecise, finite and often sparse data and mea-
surements; and from uncertainty in the baseline inputs and conditions for model
runs, including predicted inputs.” Whereas Marino et al. states that [161], “Un-
certainty analysis is performed to investigate the uncertainty in the model output
generated from uncertainty in parameter inputs. Sensitivity analysis naturally fol-
lows uncertainty analysis as it assesses how variations in model outputs can be
apportioned, qualitatively or quantitatively, to different input sources.” This type
of uncertainty is termed epistemic or subjective or reducible type B uncertainty.
Epistemic uncertainty derives from a lack of knowledge about the adequate value
for a parameter/input/quantity that is assumed to be constant throughout model
87
analysis [160]. Numerous approaches to uncertainty and sensitivity analysis are
available, such as [105]
• Differential analysis, which involves approximating a model with a
Taylor series and then using variance propagation formulas to obtain
uncertainty and sensitivity analysis results.
• Response surface methodology, which is based on using classical
experimental designs to select points for use in developing a response
surface replacement for a model and then using this replacement model
in subsequent uncertainty and sensitivity analyses based on Monte
Carlo simulation and variance propagation.
• The Fourier amplitude sensitivity test (FAST ) and other variance
decomposition procedures, which involve the determination of
uncertainty and sensitivity analysis results on the basis of the variance
of model predictions and the contributions of individual variables to
this variance.
• Fast probability integration, which is primarily an uncertainty analysis
procedure used to estimate the tails of uncertainty distributions for
model predictions.
• Sampling-based (i.e. Monte Carlo) methods, that are based on the
formation and exploration of a probabilistically based mapping from
analysis inputs to analysis results.
Sampling based methods using random and Latin Hypercube Sampling (LHS)
are focused in this study for sensitivity and uncertainty analysis. The calculation
of uncertainty values with large number of parameters is based on formulation of
the multi-dimensional parameter space. It is done with an aim to find an
appropriate and computationally effectual way such as Latin Hypercube
Sampling. It was McKay et al. [113] who extended Latin Square sampling and
developed Latin Hypercube Sampling (LHS). Whereas Iman et al. [113],
improved it. Latin Hypercube Sampling (LHS) belongs to the group of Monte
Carlo sampling.
88
Donnelly et al. [57], was one of the leading groups who assessed the epidemiology
of SARS in Hong Kong. They estimated the key epidemiological parametric
distributions using the integrated data bases constructed from several sources
containing information about epidemiological, demographic and clinical variables
which provided the base line for the parameters of SARS models by Chowell et
al. [45, 46], Gummel et al. [95], Riley et al. [191], Yan et al. [242] and Lipsitich et
al. [152]. Chowell et al. [45, 46], fitted an SEIJR model to the data from
Toronto, Hong Kong and Singapore outbreaks, calculated the reproductive
number and used uncertainty and sensitivity analysis of reproductive number to
assess the role that model parameters play in outbreak control. Riley et al. [191]
and Lipsitch et al. [152], developed comparatively complex dynamical models for
the SARS transmission in Hong Kong and Singapore. But these models provided
the researchers with enough information and understanding to estimate many
influential parameters that can access the danger of disease spread in future.
Wallinga and Teunis [230] developed a likelihood-based estimation procedure that
infers the temporal pattern of effective reproduction numbers from an observed
epidemic curve. Zhou et al. [254], formulated a discrete mathematical model to
investigate the transmission of SARS and estimated the parameters of the model
on the basis of statistical data. Numerical simulations have been carried out to
describe the transmission process for SARS in China. Wanga and Ruanb [231]
proposed a mathematical model to simulate the SARS outbreak in Beijing by
estimating the reproduction number and estimated certain important
epidemiological parameters using the available data after simplifying the model in
suspect-probable and single-compartment model.
The model considered here is an extension of the SEIJR open population SARS
model presented in Chapter 3. It consists of six sub-populations compartments,
namely susceptible(S), exposed(E), infected(I), diagnosed(J) and recovered(R)
along with treatment class(T ) named as SEIJTR model. The aim of this chapter
is to estimate the finite set of parameters for the new SEIJTR. To calculate the
parameters for the new model SEIJTR data from the SARS outbreak in Hong
Kong [152] is used. Uncertainty and sensitivity analysis of the estimated
89
parameters are also investigated on the basis of local derivative and sampling
techniques in the context of a deterministic dynamical system.
5.2 Model formulation
The SEIJTR model is considered to study the SARS epidemic which occurred
in Hong Kong in 2003. This model of the SARS epidemic consists of a system of
non-linear ordinary differential equations:
dS
dt= πΛ− β
(I + qE + lJ)
NS − µS, (5.1)
dE
dt= (1− π)Λ + β
(I + qE + lJ)
NS − (µ+ κ)E, (5.2)
dI
dt= κE − (µ+ α + δ)I, (5.3)
dJ
dt= αI − (µ+ γ1 + δ + ζ)J, (5.4)
dT
dt= ζJ − (γ2 + µ+ δ(1− θ))T, (5.5)
dR
dt= γ1J + γ2T − µR. (5.6)
With initial conditions
S(0) = S0, E(0) = E0, I(0) = I0, J(0) = J0, T (0) = T0, R(0) = R0 and
N = S + E + I + J + T +R. The parameter estimation problem given by the
system (5.1)- (5.6) is basically set up using the objective function and is given as
as follows:
Φ = g(t,Φ,Ψ), Φ(t0) = Φ0, (5.7)
where t denotes time (independent variable), Φ denotes the vector of dependent
variable (S,E,I,J,T,R) and Ψ represents vector of unknown parameters (Λ, β, l, κ,
α, γ1, γ2, ζ and δ). Then according to [217] if the solution of Eq. (5.7) for ith
component at the time ti is denoted by Φi(ti,Ψ), then the ith residual can be
given as:
ri = Φi − g(ti,Ψ) (5.8)
90
Table 5.1: Biological definition of parameters and state variables
Parameter Description
Λ Rate at which the new recruits enter the population
π Proportion of new recruits into the population that are susceptible
1− π Proportion of new recruits into the population that are are exposed
β Transmission Rate
µ Rate of natural mortality
l Relative measure of reduced risk among diagnosed
κ Rate of progression from exposed to the infectives
q Relative measure of infectiousness for exposed individuals
α Rate of progression from infective to diagnosed
γ1 Natural recovery rate
γ2 Recovery rate in treatment class
ζ Treatment Rate
δ SARS induced mortality rate
θ Effectiveness of the drugs as a reduction factor in disease-induced death of
infectious individuals(0 ≤ θ ≤ 1)
S0 Initial Susceptible Population
E0 Initial Exposed Population
I0 Initial Infected Population
J0 Initial Diagnosed Population
T0 Initial Treated Population
R0 Initial Recovered Population
91
5.2.1 Reproduction number RIT for SARS
In epidemiology, the basic reproduction number RIT is the number of individuals
infected by a single infected individual during his or her entire infectious period,
in a population which is entirely susceptible. If RIT < 1, then disease dies out,
but if RIT > 1 infection spreads and causes epidemic. The reproduction number
of SEIJTR model that represents the effect of isolation after diagnosis and
treatment into standard SEIR model is calculated. The detailed calculation for
reproduction number RIT is given in appendix A.5 and the formula is given as
follows:
RIT =Λβ(q(µ+ δ + α)(µ+ δ + γ1 + ζ) + κ(µ+ δ + γ1 + ζ + lα))
µ(µ+ κ)(µ+ δ + α)(µ+ δ + γ1 + ζ)(5.9)
5.2.2 Epidemiological data
Parameters estimation leads to a comparison with experimental data. Often a
model contains the parameters that need to be adjusted to obtain a best fit to the
data. The accuracy of the parameter estimate depends on the nature of the data,
the noise in the data and the structure of the model. In some circumstances a
small error in the data will cause a vastly magnified error in the parameters [220].
In 2003, a SARS (Severe Acute Respiratory Syndrome) epidemic spread globally.
The data considered here [121] is taken from one of the pandemic waves of SARS
in Hong Kong. It consists of daily reported infected and recovered. The SARS
epidemic in Hong Kong went through three phases appearing in a teaching
hospital then in a community followed by eight hospitals and 170 housing states
[152]. The data on infection is of dates from 17 March to 12 July 2003. Fig. 5.1
shows the infection data per day.
5.2.3 Parameter estimation
Estimation of parameters is a non-linear problem as the system consists of
non-linear ODES with respect to parameters and system state variables. The
solution of non-linear problems require a numerical solver and an optimizer to
92
0 10 20 30 40 50 60 70 80 900
20
40
60
80
100
120
140
160
Time (Days)
Infe
cted
Indi
vidu
las
Figure 5.1: SARS infected incidence data, Hong Kong 2003.
estimate the parameters of system. The series of tests for the selection of better
solver and optimizer in [144] suggests that Runge-Kutta methods provides quite
accurate solutions with better convergence for the solution of initial value
problem as compared to all avialable numerical methods and
Levenberg-Marquardt works as best optimizer [144]. Hence Dormand-Prince
Pairs (Runge-Kutta methods) method has been selected as solver for the system
of non-linear ordinary differential and the least square optimization problem is
solved using Levenberg-Marquardt method. MATLAB software has been used for
calculation throughout. Various steps of the calculations are as follows:
• System of ordinary differential equations is solved using the random
choosen initial values of parameters and system state variables with
MATLAB ode45 routine.
• The model output is then compared with field data. Levenberg-Marquardt
optimization algorithm is then used to estimate a new set of parameter
values that better fits the field data.
• The system of ordinary differential equations is again solved numerically
then using the new parameters values estimated with the help of
93
Levenberg-Marquardt optimization algorithm. The model output is again
compared with the field data.
• This comparison of the updated parameter values and field data is continued
till convergence criterion is obtained. About 1010 and 1015 simulations for
models have been run with more than a thousand values chosen randomly.
There are 13 unknown parameters and 6 state variables in the SEIJTR model.
Among these, the value of natural death rate µ, is obtained from the available
literature. The proportion of new recruits π and effectiveness of drug θ are
assumed. The remaining parameters are estimated from field data. The estimated
parameters are given in Table 5.2.
5.2.4 Validation statistics
With a view to examine possible errors in the estimated values of parameters as
well as to check their reliability, model validation has been performed. Many
statistical tools for model validation are available and among them the primary
and most applicable tool is graphical residual analysis that compares the model
and the field data. Fig. 5.2, shows the predicted model for infected population
and Fig. 5.3, shows the initial (the basic solution) and the final fit obtained after
optimization to the available SARS data. The residual plot for the prediction
model and field data as well as correlogram of residuals for the output infected
population of the corresponding estimates are shown in Fig. 5.4. In the initial
days of epidemic, a partial pattern of the residuals can be seen as a result of
difference of field data values from the model values. In the last phase of disease,
values in field data are quite small. This leads to negative values of residual
indicating reliability of estimated values of the parameters.
The concept of randomness is critically important for all measurement
procedures. Many of the statistical results cannot be assured without checking
the randomness of the model. Autocorrelation plot or correlogram is the best way
to check the randomness of the residuals. From Fig. 5.4, it is clear that
autocorrelation of residuals of corresponding estimates are inside the 99%
94
Table 5.2: Estimated parameters value for model
Parameter Value(per days) Source
Λ 0.00002 Estimated
π 0.85 Assumed
β 0.24 Estimated
µ .000035 [114]
l 0.65 Estimated
κ 0.195 Estimated
q 0.1 [57]
α 0.238 Estimated
γ1 0.046 Estimated
γ2 0.05 Estimated
ζ 0.2 Estimated
δ 0.024 Estimated
θ 0.25 Assumed
S0 18440 Estimated
E0 19 Estimated
I0 18 Estimated
J0 0 Estimated
T0 0 Estimated
R0 0 Estimated
95
0 10 20 30 40 50 60 70 80
5
10
15
20
25
30
35
40Predicted output SEIJTR Model
SA
RS
Inci
denc
e
Time (day)
Figure 5.2: Predicted model of SARS.
confidence interval. This gives evidence of randomness. Since adjacent
observations do not co-relate and so there is no significant autocorrelations. The
model fits the data well.
0 10 20 30 40 50 60 70 800
10
20
30
40
50
60
Time (day)
Infe
cted
Pop
ulat
ion
DATAModel Fit
(a)
0 10 20 30 40 50 60 70 800
10
20
30
40
50
60
Infected Population. (1−step pred)
Time (day)
Infe
cted
Pop
ulat
ion
DATAModel Fit
(b)
Figure 5.3: The Model fitted to the data for the infected individuals : Initial (a)
and final fit (b).
96
0 10 20 30 40 50 60 70 80
−15
−10
−5
0
5
10
15
20
25
Prediction error
SA
RS
Inci
denc
e
Time (day)
(a)
0 10 20 30 40 50 60 70 80 90−0.4
−0.2
0
0.2
0.4
0.6
0.8
1Correlation function of residuals. Output Infected Population
lag
(b)
Figure 5.4: Residual (a) and residual correlation (b).
5.3 Uncertainty and sensitivity analysis
Uncertainty and sensitivity analysis is a fundamental component of
epidemiological modeling. It is based on the uncertainty in output derived from
the uncertainty in the input and the relationships between the uncertainty of
output and the uncertainty in the individual input. An effort to understand the
nature of the real field data and the uncertainty associated with it, is the key to
evaluating the parameters with the greater accuracy. The uncertainty of the
model parameters and the inconsistency in the model output can be determined
by the covariance matrix. All covariance matrices are symmetrical. The absolute
values in the leading diagonal of the covariance matrix provides the information
about accuracy. The inter-connections of the parameters can be studied having a
glance at the other values in the matrix. This matrix provides the variability in
the parameter estimation that interpret relations in uncertainties of
measurements. The matrix∑
shows the covariance matrix for the SEIJTR
97
model.
∑=
Λ β l κ α γ1 γ2 ζ δ
Λ 1.227 0.6277 6.830 −0.5926 13.316 −28.31 9.834 21.93 −1.237
β 0.6277 0.4485 4.181 −0.3782 8.577 −21.12 6.266 20.15 −0.7925
l 6.830 4.181 23.57 −4.025 25.08 −27.53 24.36 26.78 −8.386
κ −0.5926 −0.3782 −4.025 0.3468 −7.869 20.94 −5.646 −19.78 0.7241
α 13.32 8.577 25.08 −7.869 25.88 −28.85 26.18 28.53 −16.43
γ1 −28.31 −21.12 −27.53 27.28 −28.85 30.57 −28.03 −29.55 24.01
γ2 9.833 6.266 24.36 −5.646 26.18 −28.03 21.56 27.98 −11.80
ζ 21.93 20.15 26.78 −19.79 28.53 −29.55 27.98 29.02 −28.03
δ −1.237 −0.7925 −8.386 0.7241 −16.43 24.01 −11.80 −28.03 1.512
(5.10)
The leading diagonal elements of the matrix∑
represent the variance of each
parameter. The rate of recovery in the diagnosed class γ1, has the largest variance
(30.57) followed by the treatment rate, ζ (29.01), recovery rate in treatment class,
γ2 (21.56), rate of progression from infective to diagnosed, α (25.88), relative
measure of reduced risk among diagnosed, l (23.57), SARS induced mortality
rate, δ (1.512), Net inflow of individuals, Λ (1.227), transmission rate, β (0.4485)
and rate of progression from exposed to infectivs, κ (0.3468). If the covariance
between two parameters is positive then it implies that the two parameters move
in the same direction but if the covariance between two parameters is negative it
shows they move in the opposite direction. Calculated relations of covariance for
all parameters of the model are given in Table 5.3. An uncertainty analysis is
performed to determine how the uncertainty in the selection of input factor
(parameter) causes variability in the model output(s).The uncertainty in the
output results of prediction models due to the variability in the model inputs is
investigated through sensitivity analysis. Sensitivity analysis identifies the
importance of parameters based on the variability in their uncertainty
contributing to the variability in the outcome. In fact sensitivity analysis
determines the robustness of the model predictions to parameter values because
of errors in data collection and assumed values of parameters.
SARS is considered as one of the short term and fast spreading infectious
diseases. Its control basically depends on the study of those factors or parameters
which directly effect the reproduction number RIT . So with a precise knowledge
of the sensitivity of the factors effecting reproduction number, transmission of a
disease can be controlled. In order to identify key parameters in the transmission
of SARS, we have conducted the sensitivity analysis based on two methods. The
98
Table 5.3: Covariance relations among parameters of SEIJTR model
Covariance Direction Covariance Direction Covariance Direction
CΛ,β +ve Cβ,γ2 +ve Cκ,ζ −ve
CΛ,l +ve Cβ,ζ +ve Cκ,δ +ve
CΛ,κ −ve Cβ,δ −ve Cα,γ1 −ve
CΛ,α +ve Cl,κ −ve Cα,γ2 +ve
CΛ,γ1 −ve Cl,α +ve Cα,ζ +ve
CΛ,γ2 +ve Cl,γ1 −ve Cα,δ −ve
CΛ,ζ +ve Cl,γ2 +ve Cγ1,γ2 −ve
CΛ,δ −ve Cl,ζ +ve Cγ1,ζ −ve
Cβ,l +ve Cl,δ −ve Cγ1,δ +ve
Cβ,κ −ve Cκ,α −ve Cγ2,ζ +ve
Cβ,α +ve Cκ,γ1 +ve Cγ2,δ −ve
Cβ,γ1 −ve Cκ,γ2 −ve Cζ,δ −ve
first method is based on fixed point estimates of model parameters and the
second method is based on uncertainty in the model parameter estimation.
5.3.1 Sensitivity indices of RIT
Sensitivity analysis using the fixed point estimation has been applied to
determine the relative importance of different parameters responsible for the
SARS transmission related to the reproductive number RIT [42]. The normalized
forward sensitivity index of a variable p, that depends differentially on a
parameter x, is defined as [186]:
Υpx =
∂p
∂x× x
p(5.11)
The sensitivity indices have been calculated for the reproduction numbers RIT for
all the parameters using the definition given in Eq. (5.11). Expressions for
normalized sensitivity indices for all parameters are given in appendix A.5. The
calculated numerical values for each parameters are as given in Table 5.4. The
sensitivity index of RIT with respect to parameter Λ and β is 1, being not
99
Table 5.4: Parameters’ sensitivity analysis
Parameters Sensitivity Index for RIT Percentage Change
Λ +1.00000 −1.00000
π −−−−− −−−−−
β +1.00000 −1.00000
µ −1.00032 +0.99968
l +0.27682 −3.61242
κ −0.085780 +11.6577
q +0.08595 −11.6341
α −0.55023 +1.81742
γ1 −0.04273 +23.4049
γ2 −−−−− −−−−−
ζ −0.210844 +4.74284
δ −0.11009 +9.08299
θ −−−−− −−−−−
100
dependent on any parameter value. Table 5.4 shows that the sensitivity indices of
Λ, β, l, q are positive and µ, κ, α, γ1, δ and ζ are negative. Table 5.4 shows
corresponding percentage changes in the parameters with change of 1% in the
value of RIT . The value of RIT decreases 1% with the decrease in the value of Λ,
β, l, q, by 1%, 1%, 3.61242% and 11.6341%, respectively. While in order to
decrease RIT by 1% we need to increase µ, κ, α, γ1 and δ and ζ by 0.99968%,
11.6577%, 1.81742%, 23.4049%, 9.0829% and 4.74284% respectively.
5.3.2 Partial rank correlation coefficient (PRCC)
In this section, sensitivity analysis is performed to find out the most important
parameters or factors in contributing the variability in the output of the
reproduction number based on the uncertainty in their estimation. PRCC
technique has been used here for the parameter ranking. It is a powerful
sensitivity measure for non-linear but monotonic relationships between input
parameters and output of model, as long as small or no correlation exists between
the inputs. PRCCs are appropriate for determining the most effective
parameters but not for evaluating how much change appears in the outcome by
changing the value of the input parameter. However, the signs (negative or
positive) of PRCCs can suggest the direction of change in the outcome variable
due to the change in the input parameter [166].
There are eight parameters Λ, β, α, δ, l, ζ, κ, γ1, involved in Eq. (5.9), whose
estimated values are given in Table 5.2. In order to examine the sensitivity of the
estimated parameters, all parameters are assumed to be random variables with a
corresponding density probability functions. The assumption of the probability
distribution functions is a critical decision which is based on the biological
understanding, and information of the natural history of the concerned disease.
The assumed probability distributions for SARS using available information
[57, 85] are described in Table 5.5 and their plots are shown in Fig. 5.5. Random
Sampling (RS) and Latin Hypercube Sampling (LHS) have been used to
generate the samples for the parameters. Random Sampling is generally
preferable to univariate approaches for uncertainty and sensitivity analysis. In
101
Table 5.5: Probability distribution functions (PDF ) for parameters
Parameters Type of Distribution PDF Parameter Values Source
Λ Exponential Mean= .000021 Assumed
β Exponential Mean=.24 Assumed
l Beta a= 1,b= 2 Assumed
1/κ Gamma a= 2.4, b= 2.8 [46]
1/α Gamma a= 8.3, b= 2.7 [46]
1/γ1 Gamma a= 1.8, b= 2.6 [46]
ζ Beta a= 2, b= 1 Assumed
1/δ Gamma a= 2.25, b=15.5 [46]
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
I
Pro
babi
lity
Dis
trib
utio
n F
unct
ion
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Zeta
Pro
babi
lity
Dis
trib
utio
n F
unct
ion
Figure 5.5: Plots of probability distributions for all parameters generated with
10, 000 sample size.
102
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
I
RIT
Figure 5.6: Scatter plots for the basic reproduction number and eight sampled
input parameters values with 10, 000 random samples.
Random Sampling each sample element is generated independently of all other
sample elements. The more efficient technique is Latin Hypercube Sampling
(LHS) which was introduced to the field of disease modeling by Blower [24].
Both sampling techniques are used to select 10, 000 and 20, 000 samples with each
probability distribution. The scatter plots comparing the basic reproduction
number RIT and eight parameters with 10,000 and 20,000 samples are shown in
the Fig. 5.6 and Fig. 5.7, for Random and Latin Hypercube Sampling. Some
parameters show clear and some partial linear monotonic relationship between
the reproduction number RIT and the parameters. The partial rank correlation
coefficients (PRCCs) are calculated using technique given in [113] between the
values of each of the eight parameters and the values of RIT in order to rank the
103
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
I
RIT
Figure 5.7: Scatter plots for the basic reproduction number and eight sampled
input parameters values with 10, 000 LHS samples.
104
Table 5.6: Estimates of partial rank correlation coefficients
Parameters Λ β l κ α γ1 ζ δ
RSa1 +0.9168 +0.9153 +0.2579 +0.1088 +0.5569 +0.0427 −0.1438 +0.3104
LHSb1 +0.9149 +0.9144 +0.25383 +0.1420 +0.5594 +0.0464 −0.1536 +0.2944
RSa2 +0.9158 +0.9156 +0.2530 +0.1121 +0.5523 +0.0515 −0.1429 +0.3095
LHSb2 +0.9156 +0.9152 +0.2564 +0.1257 +0.56322 +0.0329 −.1455 +0.2927
Importance Λ β α δ l ζ κ γ1
RSa1 = 10, 000 Random Samples, LHSb1 = 10, 000 Latin Hypercube Samples
RSa2 = 20, 000 Random Samples, LHSb2 = 20, 000 Latin Hypercube Samples
parameters according to the magnitude of their effect on RIT . The higher the
value of the PRCCs, the greater effect of input parameter is on the magnitude of
RIT . The order of these PRCCs directly aggregate to the level of statistical
influence, the corresponding input parameter has on the variability of the
reproduction number RIT . It is based on its own estimation uncertainty. Table
5.6, represents the estimated partial rank correlation coefficients for all considered
eight parameters. All results satisfy the condition of significance i.e (p < 0.05).
(a) (b)
Figure 5.8: PRCCs for the full range of parameters from Table 5.6 for LHSb1 =
10, 000 (a) and RSa1 = 10, 000 (b).
Random Sampling shows a strong correlation between RIT and Λ and β with
+0.9168 and +0.9153 using 10, 000 samples and +0.9158, +0.9156 using 20,000
samples respectively as shown in Table 5.6. Moderate correlation is observed
between α and δ with RIT with the corresponding values as +0.5569 and +0.3104
with 10, 000 and +0.5523, +0.3095 with 20, 000 samples. Weak correlation is
105
(a) (b)
Figure 5.9: PRCCs for the full range of parameters from Table 5.6 LHSb2 = 20, 000
(a) and RSa2 = 20, 000 (b).
observed between l, ζ, κ and γ1 with RIT with corresponding values +0.2579,
−0.1438, 0.1088 and +0.0427 using 10, 000 samples and +0.2530, −0.1429,
+0.1121 and +0.0515 with 20, 000 samples.
With Latin Hypercube Sampling, Λ and β correlate with RIT strongly with the
highest correlation coefficient with values +0.9149 and +0.9144 using 10, 000
samples and +0.9156, +0.9152 using 20, 000 samples respectively. Moderate
correlation of α and δ with RIT and the corresponding values are +0.5594 and
+0.2944 with 10, 000 and +0.5632, +0.2927 with 20, 000 samples, similarly again
weak correlation is observed between l, ζ, κ and γ1 with RIT with corresponding
values +0.2538, −0.1536, +0.1420 and 0.0464 using 10, 000 samples and +0.2564,
−0.1455, +0.1257 and +0.0329 with 20, 000 samples as shown in Table 5.6.
Tornado plots of partial rank correlation coefficients, indicating the importance of
each parameter’s uncertainty in contributing to the variability in the time to
eradicate infection are shown in Fig. 5.8 and 5.9. Values of PRCCs and the
corresponding parameters under consideration are shown on x-axis and y-axis
respectively. Here treatment rate ζ moves in the negative direction while Λ, β, α,
δ, l, κ and γ1 move in the positive direction showing that with the increase in
treatment rate ζ, RIT will decrease in magnitude and with the increase in Λ, β,
α, δ, l, κ and γ1, RIT will increase. It is shown in Fig. 5.8 and 5.9, that the net
106
inflow rate of individuals Λ and the transmission rate β plays the most important
role in the transmission of disease. Other parameters in order of decreasing
importance follow as α, δ, l, ζ, κ, γ1.
5.3.3 Factor prioritization by reduction of variance
Factor prioritization is a vast concept representing a group of statistical
approaches for classifying the importance of variables in contributing to
particular outcomes. After being popular in various disciplines variance based
measures for factor prioritization are becoming famous in computational
modeling. The main target of this method is to determine factor which can lead
to the maximum reduction in the variance of outcome. After identifying first
factor the second most important factor in reducing the variance is determined
this procedure continues till all independent input factors are ranked. The
concept of importance is thus particularly associated to a reduction of the
variance of the outcome. The sensitivity index attain values between 0 and 1.
The higher the value of the sensitivity index the more important is the random
variable. Variance based measures, such as the sensitivity index just defined, are
compact, and easy to understand and convey. This is an appropriate measure of
sensitivity to use to rank the input factors in order of importance even if the
input factors are correlated [113]. Furthermore, this method is completely
’model-free’. The sensitivity index is interpreted as being the proportion of the
total variance attributable to the random variable [113].
Factor prioritization by reduction of variance method has been used to calculate
the sensitivity [113] and the results obtained are shown in Table 5.7. As shown in
Table 5.7, when Random Sampling is selected Λ and β shows the maximum
variability in the output RIT with value 46.6268% and 45.7693% with 10, 000
samples and 46.12%, 45.9968% with 20, 000 samples. The next parameters in
variability are α, δ, l, ζ, k, γ1 with values 5.4738%, 0.9135%, 0.6753%, 0.3466%,
0.1714%, 0.0233% with 10, 000 samples and 5.6176%, 0.9968%, 0.7357%, 0.3479%,
0.1545%, 0.0308% with 20, 000 samples.
When Latin hyper cube sampling is considered, the parameter Λ shows again
107
Table 5.7: Percentage values of sensitivity index based on reduction of variance
Outcome for RT Λ β l κ α γ1 ζ δ
RSa1 46.6268% 45.7693% 0.6753% 0.1714% 5.4738% 0.0233% 0.3466% 0.9135%
LHSb1 46.351% 46.0087% 0.6799% 0.1803% 5.2876% 0.0311% 0.3031% 1.1583%
RSa2 46.12% 45.9968% 0.7357% 0.1545% 5.6176% 0.0308% 0.3479% 0.9968%
LHSb2 46.9666% 45.2418% 0.7461% 0.1591% 5.7453% 0.0331% 0.2929% 0.8152%
RSa1 = 10, 000 Random Samples, LHSb1 = 10, 000 Latin Hypercube Samples
RSa2 = 20, 000 Random Samples, LHSb2 = 20, 000 Latin Hypercube Samples
Figure 5.10: Pie chart of factor prioritization sensitivity indices LHSb1 and RSa1.
high variability with value 46.351% followed by β causing variability with
46.0087%, α with 5.2876%, δ with 1.1583%, l with 0.6799%, ζ with 0.3031% κ
with 0.1803% and γ1 for 0.0311% with 10, 000 samples. Similarly for the 20, 000
samples calculation shows Λ causing the highest variability 46.9666% and γ1
accounts for the least variability 0.0331% in the reproduction number RIT as
shown in the Table 5.7. Pie charts of factor prioritization sensitivity indices are
shown in Fig. 5.10 and Fig 5.11, which clearly shows the dominance of the net
inflow rate of the individuals Λ and transmission rate of disease β for the model.
In Fig. 5.10 and Fig. 5.11, δ, l, ζ, κ, γ1 are put together under title “Other” as
the sensitivity indices of these parameters are small compared to α, β and Λ.
108
Figure 5.11: Pie chart of factor prioritization sensitivity indices LHSb2 and RSa2.
5.4 Discussion
In order to investigate the effects of transmission of dynamics of SARS along
with treatment rate and the net flow of the individuals to the region of the
disease, a deterministic model SEIJTR has been constructed. First of all,
parameters involved in the model have been estimated based on the best fit to the
field data [121], published as daily reports on SARS epidemic in Hong Kong in
2003 by world health organization. A numerical method called Dormand-Prince
pairs method has been used as system solver for the non-linear differential model
SEIJTR and Levenberg-Marquardt technique has been used as the least square
optimizer in order to determine the best fit to the field data. MATLAB has been
used for all the calculations. A large number of simulations have been run to
estimate the parameters. The estimation of some parameters is obtained from the
demographic information on the city of Hong Kong. Different graphical and
numerical methods have been used to verify the estimation. Autocorrelation of
the residuals are within the confidence interval as shown in Fig. 5.4. This shows
that there is no correlation among residuals hence the estimates are reliable and
the model fits the data well. Uncertainty in the estimation of parameters is
investigated, followed by sensitivity analysis. The covariance matrix∑
provides
information on uncertainties in parameter estimation based on the covariance
between them. The information about interrelationship of the parameter given by
109
covariance matrix helps to determine the uncertainties in their measurements. It
is shown in Table 5.3 that the net inflow rate of individuals Λ moves in the same
direction as β, l, α, γ2 and ζ and moves in the opposite direction of κ, γ1 and δ.
Transmission rate of disease β have positive relation with l, α, γ2 and ζ but
negative relation with κ, γ1, and δ. The third most influential parameter α moves
in the same direction as γ2, l and ζ while in the opposite direction of κ, γ1, and δ.
For sensitivity analysis all the newly estimated parameters appearing in the
equation for reproductive number RIT are considered. Methods of Perturbation
of fixed point, Partial rank correlation and Factor prioritization by reduction of
variance have been applied for the calculation of sensitivity analysis.
• Method of finding the sensitivity indices based on the perturbation of fixed
point shows that the most important parameter for RIT are the rate at
which the new recruits enter the population (net inflow rate of individuals)
Λ and transmission rate, β as shown in Table 5.4. In order to decrease RIT
by 1% it is necessary to decrease Λ and β by 1%. Rate of progression from
infective to diagnosed class α is the third influential parameter. In order to
obtain 1% decrease in the value of RIT it is necessary to increase α by
1.81742% other important parameters are µ, l, ζ, δ q, κ and γ1 respectively.
• Partial Rank Correlation Coefficients of the parameters are calculated using
Random and Latin hyper cube sampling technique for two independently
generated samples of size 10, 000, 20, 000 each and the results are compared
as shown in Table 5.6. For both approaches, the rate at which the new
recruits enter the population, Λ and transmission rate, β have the highest
correlation with RIT , showing that these are the most important
parameters for RIT . The order of the importance of the remaining
parameters is α, δ, l, ζ, κ, γ1 respectively. The positive sign of PRCC
indicates that if we increase a parameter with positive sign, the
reproduction number RIT also increases. The negative sign with the value
of parameter suggests that if we increase it, reproduction number RIT will
decrease. Only the treatment rate, ζ shows the negative correlation with
RIT . All other parameters show positive correlation. Hence with the
110
increase in ζ, the reproduction number will decrease and thus slow down
the disease transmission. Also β and Λ have the greatest potential to make
the epidemic worse on increasing. However ζ is the parameter with the
maximum potential to decrease the intensity of epidemic when maximized.
• The results obtained in Table 5.7, using the factor prioritization variance
based technique, with the inclusion of both sampling techniques, determine
that the parameter Λ, the rate of new recruits entering the population, is
the cause of maximum variability in the reproduction number RIT .
Parameter β, transmission rate, is the next for accounting the variability in
RIT . Where as parameter γ1, natural recovery rate, causes the least
variability in the reproduction number RIT .
On the basis of sensitivity analysis using all the techniques, rate of new recruits
entering the population, Λ, transmission rate, β and rate of progression from
infective to diagnosed, α have been shown in Tables 5.4, 5.6 and 5.7, to be the
most influential parameters for the reproduction number. In order to control the
initial growth of the disease it is very important to control the net inflow of the
individuals to the concerned region and the transmission rate to control the
transmission of the disease. After the transmission rate, β diagnosis rate, α is the
most important factor to develop the control strategies. Faster movement of the
infective to diagnosed will reduce the transmission of disease. Although there was
not any specific treatment available at the time of SARS epidemic, but
sensitivity analysis comes up with the result that treatment rate, ζ is the
parameter with the potential to reduce the epidemic when it is maximized as
shown in Figs. 5.8 and 5.9.
In the next chapter SARS model with treatment (SEIJTR) has been
investigated numerically to study the impact of treatment on the transmission
dynamics of disease.
111
Chapter 6
Numerical Study of SARS Model
with Treatment (SEIJTR) and
Diffusion in the System
6.1 Introduction
Recent years have seen an increasing trend of utilizing mathematical models for
the prediction of and insight in infectious diseases. These models are considered
as conceptual tools to explain the behavior of disease at different scales, and allow
us to understand the spread of infection in the real world and the impact of
various factors on disease dynamics. The key concepts associated with
mathematical modeling, such as basic and effective reproduction number,
generation time, epidemic growth rates, mortality rates, transmission rates,
incubation periods, heterogeneities, disease transmission routes, risk factors for
diseases spread and pre-clinical infectiousness play significant roles in the
epidemiological analysis and control of diseases. The process of modeling in
epidemiology has, at its heart, the same underlying philosophy and goals as
ecological modelling. Both endeavors share the ultimate aim of attempting to
understand the prevalence and distribution of a species, together with the factors
that determine incidence, spread, and persistence [4, 64].
The spatio-temporal spread of infectious diseases is the most significant area of
112
epidemic modelling. Accurate and precise mathematical models enable scientists
to understand the risk factors of disease transmission and to develop workable
control strategies for possible future outbreaks. There are obvious public health
and/or economic benefits in understanding the infectious dynamics of diseases in
humans, animals, and plants. Furthermore, it is well understood how important
the spatial aspect of these dynamics is to understand disease spread [147]. In the
case of emerging and re-emerging outbreaks of an infectious disease, it is crucial
to quantify the characteristics of a disease in order to estimate the potential
threat. Accurate estimation of these characteristics relies on modified
epidemiological information.
This chapter is based on the numerical study of a mathematical model to
investigate the transmission of SARS. This model includes exposed, infected,
diagnosed, treatment and recovered classes (compartment). Diffusion has been
included in the system to examine its role in transmission of the disease. The
compartmental model for SARS transmission is given in Section 2. The numerical
scheme to solve the model is described in Section 3. The stability of numerical
solutions with and without diffusion is analyzed in Section 4. Section 5 shows
numerical simulations. Further discussion and conclusion are given in Section 6.
6.2 SEIJTR epidemic model
6.2.1 Equations
This model of SARS consists of the following system of non-linear partial
differential equations.
∂S
∂t= πΛ− β
(I + qE + lJ)
NS − µS + d1
∂2S
∂x2(6.1)
∂E
∂t= (1− π)Λ + β
(I + qE + lJ)
NS − (µ+ κ)E + d2
∂2E
∂x2(6.2)
∂I
∂t= κE − (µ+ α + δ)I + d3
∂2I
∂x2(6.3)
∂J
∂t= αI − (µ+ γ1 + δ + ζ)J + d4
∂2J
∂x2(6.4)
113
∂T
∂t= ζJ − (γ2 + µ+ δ(1− θ))T + d5
∂2T
∂x2(6.5)
∂R
∂t= γ1J + γ2T − µR + d6
∂2R
∂x2(6.6)
with initial conditions
S(0) = S0, E(0) = E0, I(0) = I0, J(0) = J0, T (0) = T0 and R(0) = R0 where
S,E, I, J, T and R represent susceptible, exposed, infected, diagnosed, treated
and recovered classes respectively and N denotes the total population,
N = S +E + I + J + T +R. d1, d2, d3, d4, d5 and d6 are the diffusivity constants.
Table 6.1 and Table 6.2, provide, respectively the description and the values of
the parameters.
To scale the population size in each compartment by the total population sizes by
substituting s = S/N , e = E/N , i = I/N , j = J/N , t1 = T/N , r = R/N
Π = Λ/N in the system of equations (6.1) - (6.6). After simplification replacing s
by S, e by E, i by I, j by J , t1 by T and r by R, the following dimensionless
system of equations is obtained:
∂S
∂t= −β(I+qE+lJ)S+(π−S)Π+γ1IS+δ(I+J+(1−θ)T )S+d1
∂2S
∂x2(6.7)
∂E
∂t= (1−π)Π+β(I+qE+ lJ)S−(Π+κ)E+δ(I+J+(1−θ)T )E+d2
∂2E
∂x2(6.8)
∂I
∂t= κE − (α + δ +Π)I + δ(I + J + (1− θ)T )I + d3
∂2I
∂x2(6.9)
∂J
∂t= αI − (Π + γ1 + ζ)J + δ(I + J + (1− θ)T )J + d4
∂2J
∂x2(6.10)
∂T
∂t= ζJ − (Π+γ2)T + δ(I+J +(θ− 1)+ (1− θ)T )T +d5
∂2T
∂x2(6.11)
∂R
∂t= γ1J + γ2T −ΠR+ δ(I + J + (1− θ)T )R+ d6
∂2R
∂x2(6.12)
where S + E + I + J + T +R = 1. The detail of calculation is given in appendix
A.6
6.2.2 Initial and boundary conditions
The domain of all the calculations is considered as [−2, 2]. Boundary and initial
conditions are chosen as follows:
∂S(−2, t)
∂x=
∂E(−2, t)
∂x=
∂I(−2, t)
∂x=
∂J(−2, t)
∂x=
∂T (−2, t)
∂x=
∂R(−2, t)
∂x= 0
(6.13)
114
Table 6.1: Biological definition of parameters
Parameter Description
Λ Rate at which new recruits enter the population
π Proportion of new recruits into the population that are susceptible
(the complementary proportion are infective)
β Transmission coefficient
µ Rate of natural mortality
l Relative measure of reduced risk among diagnosed
κ Rate of progression from exposed to the infectives
q Relative measure of infectiousness for exposed individuals
α Rate of progression from infective to diagnosed
γ1 Natural recovery rate
γ2 Recovery due to treatment
ζ Treatment rate
δ SARS-induced mortality rate
θ Effectiveness of drugs as a reduction factor in disease-induced death of
infectious individuals(0 ≤ θ ≤ 1)
115
Table 6.2: Parameters’ value for model
Parameter Value (per day) Source
Λ 0.00002 [175]
π 0.85 [175]
β 0.24 [175]
µ .000035 [95]
l 0.65 [175]
κ 0.195 [175]
q 0.1 [57]
α 0.238 [175]
γ1 0.046 [175]
γ2 0.05 [175]
ζ 0.2 [175]
δ 0.024 [175]
θ 0.25 [175]
116
∂S(2, t)
∂x=
∂E(2, t)
∂x=
∂I(2, t)
∂x=
∂J(2, t)
∂x=
∂T (2, t)
∂x=
∂R(2, t)
∂x= 0 (6.14)
(i)
S0 = 0.98Sech(5x− 1), − 2 ≤ x ≤ 2.
E0 = 0, − 2 ≤ x ≤ 2.
I0 = 0.02Sech(5x− 1), − 2 ≤ x ≤ 2.
J0 = 0, − 2 ≤ x ≤ 2.
T0 = 0, − 2 ≤ x ≤ 2.
R0 = 0, − 2 ≤ x ≤ 2.
(ii)
S0 = 0.97 exp(−5(x− 1)2), − 2 ≤ x ≤ 2.
E0 = 0, − 2 ≤ x ≤ 2.
I0 = 0.03 exp(−5(x+ 1)2), − 2 ≤ x ≤ 2.
J0 = 0, − 2 ≤ x ≤ 2.
T0 = 0, − 2 ≤ x ≤ 2.
R0 = 0, − 2 ≤ x ≤ 2.
(iii)
S0 ={
0.96Sech(15x), − 2 ≤ x ≤ 2,
E0 = 0, − 2 ≤ x ≤ 2.
I0 =
0, − 2 ≤ x < −0.6,
0.04, − 0.6 ≤ x ≤ 0.6,
0, 0.6 < x ≤ 2.
J0 = 0, − 2 ≤ x ≤ 2.
T0 = 0, − 2 ≤ x ≤ 2.
R0 = 0, − 2 ≤ x ≤ 2.
117
t = 0
t = 0(i)
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0S,I
t = 0
t = 0
(ii)
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0S,I
Figure 6.1: Initial conditions (i) and (ii).
t = 0(iii)
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0S,I
Figure 6.2: Initial condition (iii).
118
Figs. 6.1 and 6.2 show the initial “population distributions” for S and I. A larger
susceptible and a smaller infected proportion is concentrated towards the right
half of the main domain in initial condition (i). In initial condition (ii), I has
high concentration in the left half of the domain [−2, 2] and population S has
concentration on the right half of the domain [−2, 2]. In the initial condition (iii)
susceptible S exists in high concentration around the middle of domain [−2, 2]
with infected also around the middle but beyond the domain of S.
6.3 Numerical scheme
Operator splitting method has been used to solve the SEIJTR model. According
to this technique the system of equations is divided into non-linear reaction
equations and linear diffusion equations [244]. The non-linear reaction equations
to be used for the first half-time step are given as:
1
2
∂S
∂t= πΠ−ΠS − β(I + qE + lJ)S + δ(I + J + (1− θ1)T )S (6.15)
1
2
∂E
∂t= (1− π)Π + β(I + qE + lJ)S − (Π + κ)E + δ(I + J + (1− θ)T )E (6.16)
1
2
∂I
∂t= κE − (α + δ +Π)I + δ(I + J + (1− θ)T )I (6.17)
1
2
∂J
∂t= αI − (Π + γ1 + ζ)J + δ(I + J + (1− θ)T )J (6.18)
1
2
∂T
∂t= ζJ − (Π + γ2)T + δ(I + J + (θ − 1) + (1− θ)T )T (6.19)
1
2
∂R
∂t= γ1J + γ2T − ΠR + δ(I + J + (1− θ)T )R (6.20)
The second group consists of the linear diffusion equations, to be used for the
second half-time step as follows:
1
2
∂S
∂t= d1
∂2S
∂x2(6.21)
1
2
∂E
∂t= d2
∂2E
∂x2(6.22)
1
2
∂I
∂t= d3
∂2I
∂x2(6.23)
1
2
∂J
∂t= d4
∂2J
∂x2(6.24)
119
1
2
∂T
∂t= d5
∂2T
∂x2(6.25)
1
2
∂R
∂t= d6
∂2R
∂x2(6.26)
Applying the forward Euler scheme the non-linear equations transform to
Sj+ 1
2i = Sj
i +∆t(πΠ−ΠSji −β(Iji +qEj
i + lJ ji )S
ji +δ(Iji +J j
i +(1−θ)T ji )S
ji ) (6.27)
Ej+ 1
2i = Ej
i+∆t((1−π)Π+β(Iji+qEji+lJ j
i )Sji−(Π+κ)Ej
i+δ(Iji+J ji+(1−θ)T j
i )Eji )
(6.28)
Ij+ 1
2i = Iji +∆t(κEj
i − (α + δ +Π)Iji + δ(Iji + J ji + (1− θ)T j
i )Iji ) (6.29)
Jj+ 1
2i = J j
i +∆t(αIji − (Π+γ1+ ζ)J ji + δ(Iji +J j
i +(1− θ)T ji )J
ji ) (6.30)
Tj+ 1
2i = T j
i+∆t(ζJ ji−(Π+γ2)T
ji+δ(Iji+J j
i+(θ−1)+(1−θ)T ji )T
ji ) (6.31)
Rj+ 1
2i = Rj
i +∆t(γ1Jji +γ2T
ji −ΠRj
i +δ(Iji +J ji +(1−θ)T j
i )Rji ) (6.32)
where Sji , E
ji , I
ji , J
ji , T
ji and Rj
i are the approximated values of S, E, I, J , T and
R at position −2 + i∆x, for i = 0, 1, . . . and time j∆t, j = 0, 1, . . . and Sj+ 1
2i ,
Ej+ 1
2i , I
j+ 12
i , Jj+ 1
2i , T
j+ 12
i , and Rj+ 1
2i denote their values at the first half-time step.
Similarly, for the second half-time step, the linear equations transform as
Sj+1i = S
j+ 12
i + d1∆t
(∆x)2(S
j+ 12
i−1 − 2Sj+ 1
2i + S
j+ 12
i+1 ) (6.33)
Ej+1i = E
j+ 12
i + d2∆t
(∆x)2(E
j+ 12
i−1 − 2Ej+ 1
2i + E
j+ 12
i+1 ) (6.34)
Ij+1i = I
j+ 12
i + d3∆t
(∆x)2(I
j+ 12
i−1 − 2Ij+ 1
2i + I
j+ 12
i+1 ) (6.35)
J j+1i = J
j+ 12
i + d4∆t
(∆x)2(J
j+ 12
i−1 − 2Jj+ 1
2i + J
j+ 12
i+1 ) (6.36)
T j+1i = T
j+ 12
i + d5∆t
(∆x)2(T
j+ 12
i−1 − 2Tj+ 1
2i + T
j+ 12
i+1 ) (6.37)
Rj+1i = R
j+ 12
i + d6∆t
(∆x)2(R
j+ 12
i−1 − 2Rj+ 1
2i +R
j+ 12
i+1 ) (6.38)
The stability condition satisfied by the numerical method described above is
given as:dn∆t
(∆x)2≤ 0.5, n = 1, 2, 3, 4, 5, 6. (6.39)
In each case, ∆x = 0.1, d1 = 0.025, d2 = 0.01, d3 = 0.001, d4 = 0.0, d5 = 0.0,
d6 = 0.0 and ∆t = 0.03 are used.
120
6.4 Stability analysis
6.4.1 Disease-free equilibrium (DFE)
The basic reproduction number RIT is considered to be the threshold parameter
for any DFE and is defined as “the number of secondary cases which one case
would produce in a completely susceptible population”. The probability of
infecting a susceptible individual during one contact, duration of the infectious
period, and the number of new susceptible individuals contacted per unit of time
are the main factors in calculation of the reproduction number. Therefore RIT
may vary remarkably for different infectious diseases and also for the same disease
in different populations. The variational matrix of the system of equations (6.1) -
(6.6) at the disease-free equilibrium P0 = (1, 0, 0, 0, 0, 0), giving:
V 0 =
−Π −qβ −β + δ −lβ + δ (1− θ)δ 0
0 (qβ − κ− Π) β lβ 0 0
0 κ −(α + δ +Π) 0 0 0
0 0 α −(γ1 + ζ +Π) 0 0
0 0 0 ζ −(γ2 − (θ − 1)δ +Π) 0
0 0 0 γ1 γ2 −Π
The variational matrix V 0 can be written as
V 0 =
A11 A12
A21 A22
where
A11 =
−Π −qβ
0 (qβ − κ− Π)
, A12 =
−β + δ −lβ + δ (1− θ)δ 0
β lβ 0 0
,
A21 =
0 κ
0 0
0 0
0 0
and
121
A22 =
−(α + δ +Π) 0 0 0
α −(γ1 + ζ +Π) 0 0
0 ζ −(γ2 − (θ − 1)δ +Π) 0
0 γ1 γ2 −Π
.
The stability of the point of equilibrium, P0(1, 0, 0, 0, 0, 0) depends on the
characteristic of the eigenvalues of the matrices A11 and A22. The eigenvalues of
matrix A22, −(α+ δ +Π), −(γ1 + ζ +Π), −(γ2 − δ(θ− 1) +Π), −Π are all clearly
negative. The eigenvalues of the matrix A11 are qβ − κ−Π, and −Π. The second
eigenvalue is clearly negative. Also shown in 3.4.2 RIT < 1 implies that
qβ − κ− Π < 0.
where RIT = β (q(α + δ + Π) (γ1 + ζ + Π) +κ (lα + γ1 + ζ + Π))(κ + Π) (α + δ + Π) (γ1 + ζ + Π)
This shows that P0 is stable for RIT < 1. In the same way we can illustrate the
stability of the endemic point for RIT > 1. The expression for RIT is derived in
the same way as given in appendix A.5.
6.4.2 Endemic equilibrium without diffusion
The variational matrix of the system of equations (6.1) - (6.6) at
P ∗(S∗, E∗, I∗, J∗, T ∗, R∗), is given by
V ∗ =
a11 a12 a13 a14 a15 a16
a21 a22 a23 a24 a25 a26
a31 a32 a33 a34 a35 a36
a41 a42 a43 a44 a45 a46
a51 a52 a53 a54 a55 a56
a61 a62 a63 a64 a65 a66
where
a11 = −β(I + qE + lJ) + δ(I + J + (1− θ)T )− Π, a12 = −qβS, a13 = −βS + δS,
a14 = −lβS + δS, a15 = δ(1− θ)S, a21 = β(I + qE + lJ),
a22 = qβS + δ(I + J + (θ − 1)T )− κ− Π,
a23 = βS + δE, a24 = lβS + δE, a25 = δ(1− θ)E, a32 = κ,
122
Table 6.3: Values of LHS of Routh-Hurwitz criteria of equilibrium without diffusion
Case Equil. Pt. C1 C2 C3 C4 C5 C6 Stability
1 P1 0.760 4.7× 10−13 0.167 0.011 0.2× 10−4 2.7× 10−8 Stable
2 P2 0.761 3.4× 10−13 0.127 0.012 2.4× 10−4 2.2× 10−8 Stable
3 P3 0.702 4.1× 10−13 0.009 0.009 1.4× 10−4 2.3× 10−8 Stable
4 P4 0.759 6.2× 10−13 0.126 0.011 1.7× 10−4 3.3× 10−8 Stable
5 P5 0.818 5.1× 10−13 0.156 0.014 2.6× 10−4 3.2× 10−8 Stable
a33 = −(Π + α + δ) + δI + δ(I + J + (1− θ)T ), a34 = δI, a35 = δ(1− θ)I,
a43 = α + δJ , a44 = −(Π + γ1 + ζ) + δJ + δ(I + J + (1− θ)T ), a45 = δ(θ − 1)J ,
a53 = δT , a54 = δT + ζ,
a55 = −(Π + γ2) + δ(1− θ)T + δ(I + J + (1− θ)T + (θ − 1)),
a63 = δR, a64 = γ1 + δR, a65 = γ2 + δ(1− θ)R, a66 = δ(I + J + (1− θ)T )− Π,
a16 = a26 = a31 = a36 = a41 = a42 = a46 = a51 = a52 = a56 = a61 = a62 = 0.
The characteristic equation for P ∗(S∗, E∗, I∗, J∗, T ∗, R∗) can be written as
λ6 + p1λ5 + p2λ
4 + p3λ3 + p4λ
2 + p5λ+ p6 = 0 (6.40)
where p1, p2, p3, p4, p5, p6 and the Routh-Hurwitz conditions are calculated on
the basis of [167]. This calculation is given in appendix A.6 whereas their
expressions are given as:
C1 : p1 > 0
C2 : p6 > 0
C3 :p1p2−p3
p1> 0
C4 :p1p2p3−p23−p21p4−p1p5
p1p2−p3> 0
C5 :=p23p4−p2p3p5+p25+p21(p
24−p2p6)+p1(p22p5−p2p3p4−2p4p5+p3p6)
p23+p21p4−p1(p2p3+p5)> 0
C6 :=
p23p4p5−p2p3p25+p35−p33p6+p31p26+p21(p
24p5−p3p4p6−2p2p5p6)+p1(p22p
25+p2p3(p3p6−p4p5)+p5(3p3p6−2p4p5))
p23p4−p2p3p5+p25+p21(p24−p2p6)+p1(p22p5−p2p3p4−2p4p5+p3p6)
>
0
Here, P1, P2, P3, P4, and P5 are the points of equilibrium given as:
P1 = (0.474068, 0.000068, 000050, 0.000049, 000143, 0.600255),
P2 = (0.586992, 0.000056, 0.000042, 0.000040, 0.000119, 0.471351),
123
P3 = (0.420033, 0.000073, 0.000071, 0.000051, 0.000151, 0.661921),
P4 = (0.388376, 0.000076, 0.000056, 0.000055, 0.000160, 0.698072),
P5 = (0.514339, 0.000063, 0.000038, 0.000046, 0.000136, 0.554296).
6.4.3 Endemic equilibrium with diffusion
To calculate the small perturbations S1(x, t), E1(x, t), I1(x, t), J1(x, t), T1(x, t)
and R1(x, t) the equations (6.1) - (6.6) are linearised about the point of
equilibrium P ∗(S∗, E∗, I∗, J∗, T ∗, R∗) as described in [35, 201], giving
∂S1
∂t= a11S1 + a12E1 + a13I1 + a14J1 + a15T1 + a16R1 + d1
∂2S1
∂x2(6.41)
∂E1
∂t= a21S1 + a22E1 + a23I1 + a24J1 + a25T1 + a26R1 + d2
∂2E1
∂x2(6.42)
∂I1∂t
= a31S1 + a32E1 + a33I1 + a34J1 + a35T1 + a36R1 + d3∂2I1∂x2
(6.43)
∂J1∂t
= a41S1 + a42E1 + a43I1 + a44J1 + a45T1 + a46R1 + d4∂2J1∂x2
(6.44)
∂T1
∂t= a51S1 + a52E1 + a53I1 + a54J1 + a55T1 + a56R1 + d5
∂2T1
∂x2(6.45)
∂R1
∂t= a61S1 + a62E1 + a63I1 + a64J1 + a65T1 + a66R1 + d6
∂2R1
∂x2(6.46)
where a11, a12, a13 ... etc are the elements of the variational matrix V ∗ calculated
using the method described in [198]. We assume the existence of a Fourier series
solution of equations (6.41) - (6.46), of form:
S1(x, t) =∑k
Skeλt cos(kx) (6.47)
E1(x, t) =∑k
Ekeλt cos(kx) (6.48)
I1(x, t) =∑k
Ikeλt cos(kx) (6.49)
J1(x, t) =∑k
Jkeλt cos(kx) (6.50)
T1(x, t) =∑k
Tkeλt cos(kx) (6.51)
R1(x, t) =∑k
Rkeλt cos(kx) (6.52)
124
where k = nπ2, (n = 1, 2, 3, · · · · · · ) is the wave number for node n. Substituting
the values of S1, E1, I1, J1, T1 and R1 into the equations (6.41) -(6.46), the
equations are transformed into∑k
(a11 − d1k2 − λ)Sk +
∑k
a12Ek +∑k
a13Ik +∑k
a14Jk +∑k
a15Tk = 0 (6.53)
∑k
a21Sk +∑k
(a22 − d2k2 − λ)Ek +
∑k
a23Ik +∑k
a24Jk +∑k
a25Tk = 0 (6.54)
∑k
a32Ek+∑k
(a33−d3k2−λ)Ik+
∑k
a34Jk+∑k
a35Tk = 0 (6.55)
∑k
a43Ik+∑k
(a44−d4k2−λ)Jk+
∑k
a45Tk = 0 (6.56)
∑k
a53Ik+∑k
a54Jk+∑k
(a55−d5k2−λ)Tk = 0 (6.57)
∑k
a63Ik+∑k
a64Jk+∑k
a65Tk+∑k
(a66−d6k2−λ)Rk = 0 (6.58)
The Variational matrix V for the equations (6.53) - (6.58)
V =
a11 − d1k2 a12 a13 a14 a15 0
a21 a22 − d2k2 a23 a24 a25 0
0 a32 a33 − d3k2 a34 a35 0
0 0 a43 a44 − d4k2 a45 0
0 0 a53 a54 a55 − d5k2 0
0 0 a63 a64 a65 a66 − d6k2
The characteristic equation for the variational matrix V is given as
λ6 + q1λ5 + q2λ
4 + q3λ3 + q4λ+ q2 + q5λ+ q1 + q6 = 0 (6.59)
where q1, q2, q3, q4, q5 and q6 are calculated by the technique used in [198].
The Routh-Hurwitz Conditions are given as:
C1 : q1 > 0
C2 : q6 > 0
C3 :q1q2−q3
q1> 0
C4 :q1q2q3−q23−q21q4−q1q5
q1q2−q3> 0
C5 :=q23q4−q2q3q5+q25+q21(q
24−q2q6)+q1(q22q5−q2q3q4−2q4q5+q3q6)
q23+q21q4−q1(q2q3+q5)> 0
125
Table 6.4: Values of LHS of Routh-hurwitz criteria of equilibrium with diffusion
Case Equil. Pt. C1 C2 C3 C4 C5 C6 Stability
1 P1 0.849 4.5× 10−10 0.216 0.023 0.0011 2.2× 10−5 Stable
2 P2 0.850 5.1× 10−10 0.217 0.023 0.0012 2.5× 10−5 Stable
3 P3 0.790 3.3× 10−10 0.187 0.019 0.0008 1.6× 10−5 Stable
4 P4 0.849 4.2× 10−10 0.215 0.022 0.0010 1.9× 10−5 Stable
5 P5 0.908 5.9× 10−10 0.245 0.027 0.0013 2.8× 10−5 Stable
C6 :=
q23q4q5−q2q3q25+q35−q33q6+q31q26+q21(q
24q5−q3q4q6−2q2q5q6)+q1(q22q
25+q2q3(q3q6−q4q5)+q5(3q3q6−2q4q5))
q23q4−q2q3q5+q25+q21(q24−q2q6)+q1(q22q5−q2q3q4−2q4q5+q3q6)
> 0
6.4.4 Excited mode and bifurcation value
The technique used in chapter 3 is used to calculate the the first excited mode of
the oscillation n. According to the description of mode of excitation for the curve
given in Fig. 6.3, n = 1 represents the first mode of excitation as being closest to
the β-axis.
f(β) =A+B + (CD + E)
F. (6.60)
where A = p23p4p5 − p2p3p25 + p35 − p33p6 + p31p
26, B = p21(p
24p5 − p3p4p6 − 2p2p5p6),
C = p1(p22p
25 + p2p3), D = −p4p5 + p3p6, E = p1p5(−2p4p5 + 3p3p6).
F = (p23p4 − p2p3p5 + p25 + p21(p24 − p2p6) + p1(−p2p3p4 + p22p5 − 2p4p5 + p3p6))
It is observed that the bifurcation value of transmission coefficient, β and rate of
progression from infective to diagnosed, α increases with diffusion as compared to
the system without diffusion. The corresponding bifurcation diagrams for β and
α are shown in appendix A.6.
6.5 Numerical solutions
Five cases with different values of β, the transmission coefficient and α, rate of
progression from infective to diagnosed, are summarised in Table 6.5. For all the
cases given in Table 6.5, numerical solutions are calculated for the SEIJTR
126
n=1
0.1 0.2 0.3 0.4 0.5 0.6Β
-0.00002
0.00002
0.00004
0.00006
fHΒL
n=2
0.2 0.4 0.6 0.8Β
-0.0001
0.0001
0.0002
0.0003
fHΒL
n=30.2 0.4 0.6 0.8Β
0.0002
0.0004
0.0006
0.0008
0.0010
fHΒL
Figure 6.3: Determination of first excited mode with β as an unknown parameter.
Table 6.5: Bifurcation values of β and α.
Case Values Considered Bifurcation Values
Without Diffusion With Diffusion
β α β α β α
1 0.242 0.238 0.313 0.068 0.356 0.111
2 0.182 0.238 0.253 0.080 0.288 0.099
3 0.242 0.179 0.306 0.045 0.349 0.089
4 0.303 0.238 0.383 0.053 0.435 0.118
5 0.242 0.298 0.321 0.088 0.387 0.129
127
model both in the absence and presence of diffusion in the system.
6.5.1 Solutions of SEIJTR model in the absence of
diffusion (Case 1)
Fig. 6.4, shows the numerical solution for initial condition (i) in the absence of
diffusion. It can be observed that the susceptible population decreases abruptly in
the first five days of disease, with the concentration of the population fluctuating
near the edges of domain [−1, 1]. At t = 10, 15, 20 days, the number of susceptible
decays with the passage of time. The decrease in susceptible population continues
between days t = 5 and t = 20, but move slowly than in the first five days. In the
first five days of the onset of the disease, maximum population is exposed to
SARS within the domain [−.6, 1]. After t = 5 days, exposed population
proportion shows a rapid decrease till t = 10 days. With the passage of time, the
exposed population proportion keeps on decreasing slowly, with concentration
confined to the domain [−1, 1] at t = 20 days. In the first five days of onset of
SARS, most of the exposed get infected. There is a very large increase in the
proportion of infective in the first 5 days, concentrated in the domain [−.6, 1].
After t = 5 days, infective proportion decreases quickly till t = 15 days. From
t = 15 days to t = 20 days, the infected population proportion decreases slowly
but remains higher than the initial infected population proportion. Infectives, are
largely confined to the main concentration region [−1, 1] throughout the period of
prevailing disease. As soon as the infection spreads and some of the population
becomes infective, the proportion of diagnosed cases increases quite quickly from
t = 0 to t = 5 days. This trend continues but at a slower pace, for the next five
days within the domain of concentration [−1, 1]. The diagnosed population
proportion decreases quickly after t = 10 days of the disease, but there is still a
substantial proportion of diagnosed at t = 20. Once the infected are diagnosed,
they start entering the treatment compartment. A small increase in the
population being treated can be observed in the first five days of disease. From
t = 5 days to t = 10 days, there is a sharp increase in the proportion of
population in the treatment class. This proportion keeps on increasing but at a
128
low pace and reaches its maximum at t = 15 days. After that, the treated
population proportion decreases slowly with most of this population remaining
confined to the domain [−1, 1]. At t = 20 days, a large proportion of the
population is still in the treatment compartment. Recovery is slow in the first five
days of spread of SARS but increases rapidly in the next five days. A sharp
increase in the proportion of recovered individuals is noticeable between
t = 10, 15 and 20 days. The highest recovery is observed at t = 20 days.
Fig. 6.5, shows the output for initial condition (ii) in the absence of diffusion.
The susceptible population, which is mainly concentrated in the domain [0, 2],
shows sharp decline in the first five days of the spread of SARS. After five days,
concentration of susceptible is confined to the edges of the domain [0, 2]. A small
decrease is observed from t = 5 days onward (not visible in Fig. 6.5). The
exposed population grows in proportion remarkably fast in the first five days,
showing that the rate of infection is very high during this period. Almost half of
the susceptible population gets exposed to the disease in the first five days. Then
the proportion of the population that are exposed rapidly decreases in the next
five days. After t = 10 days, a slower decrease in exposed population proportion
occurs. Most of the population becomes infective in the first five days and the
domain of concentration of infective moves from [−2, 0] to [0, 2]. After t = 5 days,
the infective proportion starts decreasing slowly till t = 10 days in the domain
[0, 2]. A sharp decline in infective proportion is observed in the whole domain
[−2, 2] between t = 10 days and t = 15 days, which continues at slow pace
thereafter. The diagnosed proportion increases over the first five days. Between
t = 5 and t = 10 days, an increase in diagnosed population proportion is noticed
in the domain [0, 2]. In the same period, a much smaller proportion of diagnosed
is also observed in the domain [−2, 0]. A small proportion of the population is in
the treatment class after five days and this proportion increases quickly in the
next five days. The proportion of the population in treatment class reaches its
maximum in t = 15 days and after that it decreases. There is negligible recovery
in the first five days followed by a quicker recovery in the next five days in
domain [−1.5, 2], particularly the right half of the domain [−2, 2]. The recovery
129
t = 0
t = 5
t=10
t=10t=5
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0S
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.1
0.2
0.3
0.4E
t = 0
t = 5
t = 10
t = 20
t = 15
t=0
-2 -1 0 1 2x
0.1
0.2
0.3
0.4I
t = 0
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.1
0.2
0.3
0.4J
t = 0t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.1
0.2
0.3
0.4T
t = 0
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.1
0.2
0.3
0.4
0.5R
Figure 6.4: Solutions for initial condition (i) without diffusion.
130
proportion is greatest at t = 20 days.
Fig. 6.6, shows the results with initial condition (iii) in the absence of diffusion.
In the first five days of the spread of SARS, the susceptible proportion reduces
quickly to very low concentration around the edges at x = −0.1 and x = 0.1. In
the first five days, most of the susceptible population gets exposed to the disease.
After that, there is a quick decrease in the exposed population proportion until
t = 10 days. This decreasing trend continues until t = 20 days in the domain
[−.1, .1]. The infected population proportion increases quickly at a slower pace in
the domain [−.1, .1] from its initial concentration in domain [−.5, .5] and reaches
its peak value at t = 5 days. After that, infection decreases until t = 20 days.
The diagnosed population proportion increases considerably in the first five days
and is concentrated in the domain [−.5, .5]. The maximum diagnosed proportion
occurs ten days after the onset of SARS. After t = 10 days, the diagnosed
proportion of the population decreases. In the beginning, diagnosed individuals
move slowly to the treatment class but a considerable increase in the proportion
of the treatment population is observed between t = 5 and t = 10 days. Recovery
is extremely slow in the first five days but during the next five days a substantial
increase in the recovered population is observed, which continues till t = 20 days.
6.5.2 Solutions of SEIJTR model in the presence of
diffusion (Case 1)
Fig. 6.7, shows the output for initial condition (i) in the presence of diffusion.
The impact of the diffusion on the solution is well observed, as the proportion of
the susceptible reduces rapidly in the first five days, with a negligible proportion
of susceptible at t = 5 days in the domain [−2,−.5] and [.5, 2]. Most of these
susceptible in domain [−1.5, 1.5] are exposed to SARS during the first five days.
A sharp decrease in the exposed proportion of population between t = 5 days and
t = 10 days is observed, with the spread of the exposed population in the domain
[−2, 2]. The infected proportion of the population is maximised within t = 5 days
of spread of SARS in the domain [−1.5, 1.5]. In the next five days, infection not
131
t = 0
t = 5t = 10
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0S
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.1
0.2
0.3
0.4
0.5E
t = 0
t = 5
t = 10
t = 20
t = 15
t=15
-2 -1 0 1 2x
0.1
0.2
0.3
0.4I
t = 5
t = 10
t = 20
t = 15
t=5
t=10
-2 -1 0 1 2x
0.1
0.2
0.3
0.4J
t = 5
t = 10
t = 20
t = 15
t=10 t=15
-2 -1 0 1 2x
0.1
0.2
0.3
0.4T
t = 5
t = 10
t = 20
t = 15
t=15t=10
t=5
-2 -1 0 1 2x
0.1
0.2
0.3
0.4
0.5R
Figure 6.5: Solutions for initial condition (ii) without diffusion.
132
t = 0
t = 5
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0S
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.1
0.2
0.3
0.4E
t = 0
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.1
0.2
0.3
0.4I
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.1
0.2
0.3
0.4J
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.1
0.2
0.3
0.4T
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.1
0.2
0.3
0.4R
Figure 6.6: Solutions for initial condition (iii) without diffusion.
133
only decreases but also spreads rapidly in the domain [−2, 2]. The diagnosed
population proportion peaks in the first ten days of the disease and is spread
across whole domain [−2, 2]. The treatment class attains its peak value fifteen
days after the spread of SARS. This is similar to the case without diffusion but
with, different peak values. Due to the presence of diffusion, recovery spreads in
the domain [−2, 2]. The recovered population proportion increases rapidly after
t = 5 days and spreads to the domain [−1.5, 1.5]. Maximum population is
recovered in t = 20 days.
Fig. 6.8, shows the solution for initial condition (ii) and with diffusion.
Susceptible move from the domain [0, 2] to the domain [−1.5, 0] at t = 5 days and
then to the domain [−2,−.5] at t = 10 days (not clearly visible in Fig. 6.8 ). The
exposed population spread in the domain [−.5, 2], attaining its maximum
proportion at t = 5 days and then declining sharply in the next five days,
spreading in the domain [−1, 2]. After that there is slow decrease during the next
ten days. Concentration of infected population remains mostly confined to the
domain [−.5, 2] with a peak value at t = 5 days. After that, the infection
proportion reduces gradually. At t = 20 days a very small proportion of the
population is infected. The diagnosed population proportion also spreads in the
domain [−.5, 2] with its maximum occurring at t = 10 days. A sharp increase in
the treated population is observed from t = 5 days to t = 10 days in the domain
[−2, 2], mostly in the domain [−.5, 2]. The treated proportion of the population
attains its peak value fifteen days after the spread of the disease and then reduces
slowly. After fifteen days recovery seems to propagate in the larger domain [−2, 2]
but mostly in the domain [−.5, 2]. As compared to without diffusion case for
recovered population, peak values are smaller but domain of concentration
spreads.
Fig. 6.9, shows the solution for initial condition (iii) with diffusion. There is a
quick decline in the susceptible population proportion during the initial five days
of the onset of disease. Maximum population exposure to SARS occurs in the
domain [−1, 1] in the first five days. Between t = 5 and t = 10 days, the peak
value of the exposed population proportion decreases while the exposed spread in
134
t = 0
t = 5t=5
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0
S
t = 5
t = 10
t = 15t = 20
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25E
t = 5
t = 10
t = 15t = 20
t=0
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25I
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25J
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25T
t = 5
t = 10
t = 15
t = 20
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25R
Figure 6.7: Solutions for initial condition (i) with diffusion.
135
t = 0
t = 5
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0S
t = 5
t = 10
t = 20 t = 15
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25
0.30E
t = 0
t = 5
t = 10
t = 20
t = 15
t=5
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25
0.30I
t = 5
t = 10
t = 20
t = 15
t=10
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25
0.30J
t = 5
t = 10
t = 20
t = 15
t=10 t=15t=5
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25
0.30T
t = 5
t = 10
t = 20
t = 15
t=20
t=15
t=10
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25
0.30R
Figure 6.8: Solutions for initial condition (ii) with diffusion.
136
Table 6.6: Peak values of susceptible(S), exposed(E), infective(I) and recovered(R)
(without diffusion)
Case t S(i) S(ii) S(iii) E(i) E(ii) E(iii) I(i) I(ii) I(iii) R(i) R(ii) R(iii)
1 00 0.980 0.970 0.960 0.000 0.000 0.000 0.020 0.030 0.040 0.000 0.000 0.000
05 0.034 0.039 0.009 0.406 0.455 0.393 0.324 0.313 0.322 0.024 0.014 0.026
10 0.009 0.007 0.004 0.133 0.149 0.128 0.199 0.213 0.194 0.145 0.120 0.150
15 0.004 0.003 0.002 0.041 0.047 0.040 0.082 0.090 0.079 0.305 0.275 0.309
20 0.002 0.002 0.001 0.013 0.014 0.012 0.029 0.032 0.028 0.437 0.408 0.441
2 00 0.980 0.970 0.960 0.000 0.000 0.000 0.020 0.030 0.040 0.000 0.000 0.000
05 0.053 0.070 0.009 0.423 0.479 0.409 0.324 0.309 0.322 0.021 0.012 0.024
10 0.012 0.014 0.007 0.138 0.157 0.133 0.204 0.219 0.199 0.139 0.113 0.145
15 0.005 0.006 0.002 0.043 0.049 0.042 0.085 0.094 0.082 0.299 0.267 0.305
20 0.003 0.002 0.001 0.013 0.015 0.013 0.029 0.034 0.029 0.433 0.403 0.437
3 00 0.980 0.970 0.960 0.000 0.000 0.000 0.020 0.030 0.040 0.000 0.000 0.000
05 0.034 0.039 0.009 0.407 0.373 0.394 0.456 0.352 0.372 0.019 0.011 0.022
10 0.009 0.007 0.004 0.134 0.271 0.129 0.150 0.284 0.266 0.124 0.102 0.129
15 0.004 0.003 0.004 0.042 0.131 0.041 0.048 0.141 0.127 0.276 0.247 0.281
20 0.002 0.002 0.001 0.013 0.053 0.013 0.015 0.058 0.052 0.411 0.382 0.415
4 00 0.980 0.970 0.960 0.000 0.000 0.000 0.020 0.030 0.040 0.000 0.000 0.000
05 0.008 0.037 0.008 0.217 0.439 0.383 0.191 0.315 0.322 0.025 0.016 0.028
10 0.007 0.006 0.002 0.129 0.144 0.125 0.195 0.208 0.191 0.149 0.125 0.154
15 0.003 0.002 0.001 0.040 0.045 0.039 0.079 0.087 0.077 0.308 0.279 0.313
20 0.002 0.001 0.001 0.012 0.014 0.012 0.028 0.031 0.027 0.439 0.412 0.443
5 00 0.980 0.970 0.960 0.000 0.000 0.000 0.020 0.030 0.040 0.000 0.000 0.000
05 0.035 0.039 0.009 0.406 0.456 0.393 0.284 0.279 0.281 0.028 0.016 0.031
10 0.009 0.007 0.004 0.132 0.148 0.127 0.152 0.147 0.165 0.161 0.134 0.166
15 0.004 0.003 0.002 0.041 0.046 0.039 0.056 0.054 0.061 0.323 0.293 0.328
20 0.002 0.002 0.001 0.012 0.014 0.012 0.018 0.017 0.020 0.452 0.424 0.456
the domain [−1.5, 1.5]. Infection peaks at t = 5 days and spreads in the domain
[−1.5, 1.5] at t = 10 days. Diffusion also causes diagnosed, treated and recovered
population proportions to spread in the domain [−1.5, 1.5] with peak values at
t = 10, 15, and 20 days respectively.
6.5.3 Other cases
The graphical output for Cases 2, 3, 4 and 5 is not shown because of its similarity
to Case 1. The numerical results of all cases are summarized in Tables 6.6 and
6.7. Here Sj, Ej, Ij and Rj for j = (i), (ii) and (iii) represent peak values of the
proportion of susceptible, exposed, infected and recovered population in the
domain [−2, 2] in the absence (Table 6.6) and presence (Table 6.7) of diffusion for
the initial population distributions (i), (ii) and (iii). The following description is
based on the results given in Tables 6.6 and 6.7.
Moving from Case 1 to Case 2, there is a decrease in the transmission coefficient
137
t = 0
t = 5
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0S
t = 5
t = 10t = 20
t = 15
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
0.10E
t = 0
t = 5
t = 10
t = 20t = 15
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
0.10I
t = 5t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
0.10J
t = 5
t = 10
t = 15
t=20
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
0.10T
t = 5
t = 10
t = 20
t=15
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
0.10R
Figure 6.9: Solutions for initial condition (iii) with diffusion.
138
Table 6.7: Peak values of susceptible(S), exposed(E), infective(I) and recovered(R)
(with diffusion)
Case t S(i) S(ii) S(iii) E(i) E(ii) E(iii) I(i) I(ii) I(iii) R(i) R(ii) R(iii)
1 00 0.980 0.970 0.960 0.000 0.000 0.000 0.020 0.030 0.040 0.000 0.000 0.000
05 .00986 .01191 .00706 .22313 .29394 .06802 .18704 .20155 .05063 .01379 .00794 .00613
10 .00108 .00173 .00129 .05818 .07859 .01705 .09902 .12357 .02796 .08119 .07235 .02591
15 .00013 .00067 .00035 .01584 .02141 .00467 .03612 .04704 .01034 .16517 .16339 .04998
20 .00008 .00032 .00017 .00445 .00593 .00138 .01168 .01541 .00343 .23444 .24169 .07034
2 00 0.980 0.970 0.960 0.000 0.000 0.000 0.020 0.030 0.040 0.000 0.000 0.000
05 .01416 .01892 .01175 .23344 .31042 .07408 .18182 .19248 .04489 .01161 .00590 .00552
10 .00171 .00252 .00201 .06135 .08350 .01909 .10112 .12661 .02878 .07472 .06503 .02318
15 .00021 .00096 .00054 .01679 .02284 .00530 .03754 .04916 .01113 .15645 .15331 .04589
20 .00009 .00044 .00023 .00473 .00635 .00157 .01226 .01626 .00378 .22495 .23083 .06566
3 00 0.980 0.970 0.960 0.000 0.000 0.000 0.020 0.030 0.040 0.000 0.000 0.000
05 .00996 .01204 .00712 .22348 .29444 .06814 .21454 .22479 .05868 .01100 .00621 .00499
10 .00109 .00174 .00130 .05851 .07904 .01711 .13599 .16516 .03819 .06908 .06095 .02217
15 .00013 .00068 .00035 .01600 .02165 .00469 .05891 .07484 .01665 .14766 .14526 .04453
20 .00008 .00032 .00017 .00451 .00602 .00138 .02229 .02882 .00641 .21674 .22296 .06451
4 00 0.980 0.970 0.960 0.000 0.000 0.000 0.020 0.030 0.040 0.000 0.000 0.000
05 .00759 .00839 .00490 .21664 .28347 .06427 .19018 .20684 .05379 .01543 .00955 .00669
10 .00075 .00132 .00092 .05618 .07547 .01587 .09765 .12152 .02743 .08576 .07762 .02799
15 .00009 .00052 .00025 .01523 .02049 .00431 .03522 .04567 .00987 .17129 .17053 .05301
20 .00007 .00025 .00014 .00427 .00567 .00127 .01132 .01487 .00323 .24110 .24937 .07383
5 00 0.980 0.970 0.960 0.000 0.000 0.000 0.020 0.030 0.040 0.000 0.000 0.000
05 .00997 .01206 .00719 .22333 .29429 .06821 .16394 .18103 .043998 .01612 .00938 .00704
10 .00109 .00175 .00131 .05807 .07846 .01709 .07468 .09523 .02127 .09009 .08086 .02859
15 .00013 .00068 .00035 .01576 .02129 .00468 .02411 .03193 .00701 .17671 .17542 .05352
20 .00008 .00032 .00017 .00442 .00589 .00138 .00712 .00949 .00213 .24523 .25310 .07389
139
from β = 0.242 to β = 0.182, as shown in Table 6.5. As a result, the susceptible
population proportion show an increase in peak values for initial population
distributions (i)− (iii) with and without diffusion. There is quite a significant
increase in the first ten days of SARS as compared to Case 1. The proportions of
population exposed also show a significant increase from t = 10 to t = 15 days as
compared to Case 1. Infection grows in the last ten days of the disease without
diffusion but with diffusion infective population show an increase as compared to
Case 1 from t = 10 to t = 20 days. There is a moderate decrease in the peak
value of the recovered population proportion with all initial conditions.
In Case 3, the rate of progression from infective to diagnosed, α, is decreased
from α = 0.238 to α = 0.179 as compared to Case 1, while keeping values of β
same. There is no change in the susceptible population proportion in the absence
of diffusion but, with diffusion, a small increase appears at t = 5 and t = 10 days
of disease as compared to Case 1. Initial conditions (i) and (iii) show a small
increase in the proportion of exposed both with and without diffusion while
initial condition (ii) shows a decrease at t = 5 days and great increase in next
t = 10 days without diffusion and small increase with diffusion. In Case 3 infected
proportion values are higher than for Case 1 both with and without diffusion.
Peak values of the recovered population proportion are lower for Case 3 than
Case 1.
In Case 4, there is an increase in the transmission coefficient, β, from β = 0.242
to β = 0.303 as compared to Case 1. As a result, peak susceptible proportion
values in Case 4 decrease in comparison to Case 1 for initial population
distributions (i)− (iii), with and without diffusion. There is also a decrease in
exposed population proportion as a result of the reduction in the proportion of
susceptible under all initial conditions. In particular in the absence of diffusion it
occurs in the first five days of SARS, where population distribution with initial
condition (i) shows significant decrease in susceptible population as compared to
initial conditions (ii) and (iii) while in the presence of diffusion this decrease can
be observed on all days. A decrease in proportion of infected population has been
observed in the emergence of SARS as compared to Case 1, in the absence of
140
diffusion. This decrease is quite significant in the case of initial condition (i).
With diffusion in the system, the peak values of infected proportion are higher for
Case 4 than Case 1 under all initial conditions. Only very small increase in
recovered proportions from Case 1 to Case 4, has been observed under all three
conditions (i), (ii) and (iii), without diffusion because there is not much
movement in population due to absence of diffusion in the system. With diffusion
in the system, initial conditions (i) and (ii) reflect a significant increase in
recovery from Case 1 to Case 4, while initial condition (iii) shows a slight
increase. It is because of movement of population from higher to lower density
due to diffusion in the system. In comparison to Case 2, it has been observed that
there are lower proportions of susceptible and exposed in Case 4. Also in
comparison to Case 2, proportion of infective is slightly less with diffusion as
compared to without diffusion in the system during the first five days of disease.
As compared to Case 2, there is an increase in the recovered population both
with and without diffusion in the system.
In Case 5, as compared to Case 1 the rate of progression from infective to
diagnosed, α is increased from α = 0.238 to α = 0.298. In the absence of diffusion
this has not affected susceptible and exposed proportion values. With the
inclusion of diffusion in the system, a small increase is observed in the susceptible
and exposed population proportion in the early stage of disease. A significant
reduction in the proportion of infected individuals has been noticed both with
and without diffusion, showing that if infection is diagnosed earlier, the
population move to diagnosed compartment quickly for treatment. A large
increase in the recovered population proportion is observed both with and
without diffusion as compared to Case 1. In comparison to Case 3, a lower
proportion of exposed and infective are observed in Case 5. The proportion of the
population recovered, both with and without diffusion, has been observed to be
higher in Case 5 as compared to Case 3 .
141
Table 6.8: Basic reproduction number RIT
Case β α Value of RIT
1 0.242 0.238 1.6148
2 0.182 0.238 1.2212
3 0.242 0.179 1.8668
4 0.303 0.238 2.0354
5 0.242 0.298 1.4561
6.6 Discussion
The SEIJTR model for numerical study of the SARS epidemics is used with
diffusion and treatment included in the system to explore the effects of their
availability on the spread of disease. Three different initial conditions have been
used to examine the effects on transmission of the disease under different
population distributions. Operator splitting technique is used to calculate the
numerical solutions of the differential equations. The Routh-Hurwitz criterion is
used to check stability of points of equilibrium. The models under investigation
have two possible equilibria, namely the disease-free equilibrium and endemic
equilibria. Reproduction numbers RIT for the various cases considered (RIT > 1
) are given in Table 6.8. A study of bifurcation values of the transmission
coefficient, β and rate of progression from infective to diagnosed, α, as shown in
Table 6.5, indicates that the system remain stable with higher values of β and α
with diffusion in comparison to the system without diffusion.
It is observed from the values given in Table 6.8 that transmission coefficient, β
and rate of progression from infective to diagnosed, α, are quite sensitive
parameters having significant impact on the reproductive number RIT . If the
transmission coefficient, β is decreased as in Case 2, the reproduction number
decreases significantly even though it is still greater than 1. This causes slow
transmission of infection and thus higher peak values in the infected compartment
as shown in Tables 6.6 and 6.7 from t = 10 to t = 20 days. A decrease in infective
142
to diagnosed coefficient, α as in Case 3, causes a significant increase in the value
of the basic reproduction number RIT . Here a significant increase in the peak
value of the infected proportion can be seen at t= 20 days. An increased value of
transmission coefficient β, as in Case 4, gives the maximum value of the
reproduction number RIT . Here, transmission of the infection becomes fastest in
all initial conditions, as shown in the rate of decrease of values at various time
steps shown in Tables 6.6 and 6.7. There are also slightly lower peak values of
infected population proportion at t= 20 days, as compared to the original
situation depicted in Case 1. An increase in the rate of progression from infective
to diagnosed, α, as given in Case 5 causes a slight decrease in the value of the
basic reproduction number.
In initial condition (i), without diffusion, infectives are concentrated in the
domain [−1, 1]. With and without diffusion in the system, infection spreads
quickly in the first five days of onset of disease. With diffusion in the system,
infected population spreads to the edges of the domain [−2, 2] in the first ten
days, where initially there were no infectives. But a decrease in the peak values of
the infected proportion occurs, showing that diffusion causes a decrease in the
intensity of disease. Maximum population of diagnosed then enters treatment
class at the day fifteen of the disease. In initial condition (ii), susceptible and
infected proportions are in different domains initially. Here, infected not only
increase with the passage of time but also move from domain [−2, 0] to domain
[0, 2] without diffusion. With diffusion in the system, the infected spread to
almost in the whole domain [−2, 2]. Under initial condition (iii), there is a
significant shift as the infected population move to the domain [−0.1, 0.1]
significantly from the initial domain [−0.6, 0.6]. Diffusion again causes the
infection to spread, in the domain [−1.5, 1.5] but with reduced peak values.
The SEIJTR model considered here has been extended further in the next
chapter with the inclusion of quarantine of exposed population in the system of
equations. For that a model with the inclusion of another compartment named as
quarantined is considered. The new model is named as SEQIJTR model where
the suspected cases of SARS are kept under quarantined.
143
Chapter 7
Simulating the Effect of
Quarantine on Isolation
Treatment Model for SARS
Epidemic
7.1 Introduction
Mathematical models are the most efficient epidemiological tools for
understanding and determining the factors that cause and encourage epidemic.
These models allow us to estimate the effects of the measures and strategies
before their physical application, saving time and cost and helping to decide
which factors should be focused on more in order to control an epidemic. At the
time of emergence of a new infectious disease, when there is no effective
treatment and vaccine, isolation of known cases and quarantine of their contacts
are the two most effective strategies. The chances of spread of an infection
increase with the delay in diagnosis of infected individuals. In the history of
epidemiology, quarantine was first practised in 1377 for a plague epidemic, when
an official order of isolation for 30 days for ships and 40 days for land travelers
was passed by the Rector of the seaport of Ragusa [83]. According to Sattenspiel
[204] “In recent years the word quarantine has been used to refer to two related
144
but distinct activities, especially when applied to human activities and behaviors.
In most of the historical literature, quarantine usually refers to attempts to limit
flow of goods and people between different places. However, the same word has
also been used to refer to attempts to keep infectious individuals isolated from
everyone else. Both situations focus on limiting potential interactions between
infectious material or individuals and susceptible material or individuals, but the
intended targets of any rules or regulations are very different.” She also states
that “The term quarantine is derived from the Italian words quarantins and
quaranta giorni, which refer to the 40-days period during which ships, their
goods, crew, and passengers were isolated in the Port of Venice during the 14th
and 15th centuries [162]. The Italian authorities believed that an isolation period
of 40 days would be sufficient to dissipate the causes of infections.” The early
typical quarantine not only focused on sea travelers but also involved
arrangements to reduce population movement in and out of affected communities.
Over the last hundred years, quarantine has been used to control the transmission
of many infectious epidemics like cholera, leprosy, plague, yellow fever,
tuberculosis, diphtheria, smallpox, ebola, typhus, lassa fever, mumps and measles
[58]. During the outbreak of SARS epidemic in 2003, a new European institute
for research on controlling and developing quarantine and isolation measures for
SARS epidemic was designed [83]. Mathematical models including quarantine
effects answers numerous questions about using quarantine as a control strategy.
For example, does quarantine make any difference to the transmission of disease?
Is quarantine of the exposed population is more important than quarantine of the
infected population? Is there a difference in effectiveness between model with
quarantine and the model without quarantine and with treatment once the
infection is diagnosed? In deterministic compartmental models to study
quarantine effects, a new compartment usually referred to as Q is included in the
model. This compartment represents the isolated individuals that are removed
from the susceptible, exposed or infected (depending on model) compartment.
For some moderate infectious diseases, people who stay home from work or school
are considered to be in quarantine. But in the case of some deadly diseases,
145
quarantined individuals are those who are forced to live in isolation. Also, it is
assumed that there is no mixing of quarantined people with the rest of population
[112].
In 1995, Feng and Thieme [70] formulated a susceptible, infected, quarantined
and recovered (SIQR) model considering “quarantine-adjusted incidence”. They
concluded that addition of a quarantine compartment to the basic SIR
compartmental model can give rise to periodic solutions, explaining the oscillating
behavior of the disease considered in their studies. Five years later, Feng and
Thieme [71, 72] further developed a susceptible, exposed, infected, quarantined
and recovered (SEIQR) compartmental model with a general incidence term for
quarantine. They calculated the length of periods for quarantine in which
endemic equilibrium destablizes and can extinct and persist. A generalization of
the work of Feng and Thieme’s [70, 71, 72] was presented in the model of Hethcote
et al. [112]. They added quarantine to the basic SIR and SIS compartmental
models and compared the new SIQR and SIQS models on the basis of three
types of incidence. They investigated the equilibria, thresholds and stabilities of
the models. They found that only the cases with “quarantine adjusted incidence”
have unstable spiral endemic equilibria for few parameters and that periodic
solutions with “Hopf bifurcation surface” can arise in these cases.
At the time of the worldwide spread of severe acute respiratory syndrome
(SARS) in 2003 the major task for health authorities was to apply control
measures that could control the spread of the disease. Mass quarantine was
considered as the main control measure to reduce transmission of the infection.
The main concern in this respect was whether or not quarantine and isolation
were sufficient to decrease disease transmission? [50]. Many mathematical models
were formulated to answer this fundamental question afterwards. Fraser et al.
[78] investigated the time during infecting and for appearance of infection for an
individual, with the help of mathematical modelling. They also concluded that,
in the case of SARS simple public control measures are enough to control the
spread of disease. Many mathematical model studies have illustrated that
quarantined and isolation of a small number of individuals from the infected
146
population can lead to reduced transmission of SARS [184, 20, 189] but this is
economically expensive [216, 104, 96]. Day et al. [50] investigated useful
quarantine for SARS epidemic in all possible circumstances. They stated that “
Our results indicate that there are three main requirements for quarantine to
substantially reduce the number of infections that occur during a disease
outbreak. These are the following: 1) a large disease reproduction number in the
presence of isolation alone; 2) a large proportion of infections generated by an
individual that can be prevented through quarantine, q; and 3) a large probability
that an asymptomatic infected individual will get placed into quarantine before
he/she develops symptoms and is isolated, q.”
In this chapter a numerical study of the previously developed susceptible,
exposed, infected, diagnosed, treated and recovered (SEIJTR) model in Chapter
5 is extended to an SEQIJTR model including a quarantine compartment with a
view to study the effect of quarantine on disease transmission, for SARS
outbreak. Investigation is done with the inclusion of diffusion in the system. In
Sec. 2 the numerical scheme is derived. In Sec. 3 stability analyses for
disease-free and endemic equilibrium are performed. Bifurcation values, modes of
excitation and reproduction number in the absence and presence of diffusion is
also estimated. In Sec. 4 numerical solutions are calculated. The results are
summarised in Sec. 5.
7.2 The SEQIJTR epidemic model
7.2.1 Equations
This model of SARS consists of the following system of nonlinear partial
differential equations.
∂S
∂t= πΛ− β
(I + qE + pQ+ lJ)
NS − µS + d1
∂2S
∂x2(7.1)
∂E
∂t= (1− π)Λ + β
(I + qE + pQ+ lJ)
NS − (µ+ κ1 + κ2)E + d2
∂2E
∂x2(7.2)
∂Q
∂t= κ1E − (µ+ σ)Q+ d3
∂2Q
∂x2(7.3)
147
∂I
∂t= κ2E − (µ+ α + δ1)I + d4
∂2I
∂x2(7.4)
∂J
∂t= αI − (µ+ γ1 + δ2 + ζ)J + σQ+ d5
∂2J
∂x2(7.5)
∂T
∂t= ζJ − (µ+ γ2 + δ2(1− θ))T + d6
∂2T
∂x2(7.6)
∂R
∂t= γ1J + γ2T − µR + d7
∂2R
∂x2(7.7)
with initial conditions
S(0) = S0, E(0) = E0, Q(0) = Q0, I(0) = I0, J(0) = J0, T (0) = T0 and R(0) = R0
where S,E,Q, I, J, T and R represent susceptible, exposed, quarantined, infected,
diagnosed, treated and recovered classes, respectively, and N denotes the total
population, N = S + E +Q+ I + J + T +R. d1, d2, d3, d4, d5 ,d6 and d7 are the
diffusivity constants. Table 7.1 and Table 7.2, provide, respectively, the
description and values of the parameters. To scale the population size in each
compartment by the total population sizes by substituting s1 = S/N , e1 = E/N ,
q1 = Q/N , i1 = I/N , j1 = J/N , t1 = T/N , r1 = R/N Π = Λ/N giving the system
of equations (7.8) - (7.14). After simplification replacing s1 by S, e1 by E, q1 by
Q, i1 by I, j1 by J, t1 by T and r1 by R, the following dimensionless system of
equations is obtained:
∂S
∂t= −β(I+qE+pQ+ lJ)S−Π(S−π)+δ1IS+δ2(J+(1−θ)T )S+d1
∂2S
∂x2(7.8)
∂E
∂t= (1−π)Π+β(I+qE+pQ+lJ)S−(Π+κ1+κ2)E+δ1EI+δ2(J+(1−θ)T )E+d2
∂2E
∂x2
(7.9)∂Q
∂t= κ1E− (Π+ σ)Q+ δ1IQ+ δ2(J + (1− θ)T )Q+ d3
∂2J
∂x2(7.10)
∂I
∂t= κ2E − (Π + α + δ1)I + δ1I
2 + δ2(J + (1− θ)T )I + d4∂2I
∂x2(7.11)
∂J
∂t= σQ+αI−(Π+δ2+γ1+ζ)J+δ1IJ+δ2(J+(1−θ)T )J+d5
∂2J
∂x2(7.12)
∂T
∂t= ζJ−(Π+γ2+δ2)T+δ1IT +δ2(J+θ+(1−θ)T )T+d6
∂2T
∂x2(7.13)
∂R
∂t= γ1J + γ2T + δ1IR+ δ2(J + (1− θ)T )R−ΠR+ d7
∂2R
∂x2(7.14)
where S + E +Q+ I + J + T +R = 1
148
Table 7.1: Biological definition of parameters
Parameter Description
Λ Rate at which new recruits enter the population
π Proportion of new recruits into the population that are susceptible
β Transmission coefficient
µ Rate of natural mortality
l Relative measure of reduced risk among diagnosed
q Relative measure of infectiousness for exposed individuals
p Relative measure of reduced risk among quarantined individuals
κ1 Rate of progression from exposed to quarantined
κ2 Rate of progression from exposed to infective
σ Rate of progression from quarantine to diagnosed
α Rate of progression from infective to diagnosed
γ1 Natural recovery rate
γ2 Recovery due to treatment
ζ Treatment rate
δ1 SARS-induced mortality rate for infected
δ2 SARS-induced mortality rate for diagnosed and treated
θ Effectiveness of drugs as a reduction factor in disease-induced death of infected
149
Table 7.2: Parameters values for model
Parameter Value Source
Λ 0.00002 per day [175]
π 0.85000 [175]
β 0.24000 [175]
µ .000035 [114]
l 0.65000 [175]
q 0.10000 [57]
p 0.25000 [197]
κ1 0.10000 [95]
κ2 0.19500 [175]
α 0.23800 [175]
σ 0.15700 [57]
γ1 0.04600 [175]
γ2 0.05000 [175]
ζ 0.200000 [175]
δ1 0.04200 [150]
δ2 0.02400 [175]
θ 0.25000 [175]
150
7.2.2 Initial and boundary conditions
The domain of all the calculations is taken as [−2, 2]. Boundary and initial
conditions are chosen as follows:
∂S(−2, t)
∂x=
∂E(−2, t)
∂x=
∂Q(−2, t)
∂x=
∂I(−2, t)
∂x=
∂J(−2, t)
∂x=
∂T (−2, t)
∂x=
∂R(−2, t)
∂x= 0
(7.15)∂S(2, t)
∂x=
∂E(2, t)
∂x=
∂Q(2, t)
∂x=
∂I(2, t)
∂x=
∂J(2, t)
∂x=
∂T (2, t)
∂x=
∂R(2, t)
∂x= 0
(7.16)
(i)
S0 = 0.98Sech(5x− 1), − 2 ≤ x ≤ 2.
E0 = 0, − 2 ≤ x ≤ 2.
Q0 = 0, − 2 ≤ x ≤ 2.
I0 = 0.02Sech(5x− 1), − 2 ≤ x ≤ 2.
J0 = 0, − 2 ≤ x ≤ 2.
T0 = 0, − 2 ≤ x ≤ 2.
R0 = 0, − 2 ≤ x ≤ 2.
(ii)
S0 = 0.96Sech(5x− 1), − 2 ≤ x ≤ 2.
E0 = 0, − 2 ≤ x ≤ 2.
Q0 = 0, − 2 ≤ x ≤ 2.
I0 =
0, − 2 ≤ x < −0.6,
0.04, − 0.6 ≤ x ≤ 0.6,
0, 0.6 < x ≤ 2.
J0 = 0, − 2 ≤ x ≤ 2.
T0 = 0, − 2 ≤ x ≤ 2.
R0 = 0, − 2 ≤ x ≤ 2.
The graphs of initial conditions are shown in Fig. 7.1
151
t = 0
(i)-2 -1 0 1 2
x
0.2
0.4
0.6
0.8
1.0S,I
t = 0
(ii)-2 -1 0 1 2
x
0.2
0.4
0.6
0.8
1.0S,I
Figure 7.1: Initial conditions (i) and (ii).
7.2.3 Numerical scheme
The SEQIJTR model is solved with operator splitting technique by dividing the
system into non-linear reaction equations and linear diffusion equations [244].
The non-linear reaction equations to be used for the first half-time are:
1
2
∂S
∂t= −β(I+qE+pQ+lJ)S−Π(S−π)+δ1IS+δ2(J+(1−θ)T )S (7.17)
1
2
∂E
∂t= (1−π)Π+β(I+qE+pQ+lJ)S−(Π+κ1+κ2)E+δ1EI+δ2(J+(1−θ)T )E
(7.18)1
2
∂Q
∂t= κ1E−(Π+σ)Q+δ1IQ+δ2(J+(1−θ)T )Q (7.19)
1
2
∂I
∂t= κ2E−(Π+α+δ1)I+δ1I
2+δ2(J+(1−θ)T )I (7.20)
1
2
∂J
∂t= σQ+αI−(Π+δ2+γ1+ζ)J+δ1IJ+δ2(J+(1−θ)T )J (7.21)
1
2
∂T
∂t= ζJ−(Π+γ2+δ2)T+δ1IT+δ2(J+θ+(1−θ)T )T (7.22)
1
2
∂R
∂t= γ1J+γ2T+δ1IR+δ2(J+(1−θ)T )R−ΠR (7.23)
The linear diffusion equations to be used for the second half-time step are:
1
2
∂S
∂t= d1
∂2S
∂x2(7.24)
1
2
∂E
∂t= d2
∂2E
∂x2(7.25)
1
2
∂Q
∂t= d3
∂2Q
∂x2(7.26)
1
2
∂I
∂t= d4
∂2I
∂x2(7.27)
152
1
2
∂J
∂t= d5
∂2J
∂x2(7.28)
1
2
∂T
∂t= d6
∂2T
∂x2(7.29)
1
2
∂R
∂t= d7
∂2R
∂x2(7.30)
Applying the forward Euler scheme, the non-linear equations transform to
Sj+ 1
2i = −β(Iji+qEj
i+pQji+lJ j
i )Sji−Π(Sj
i−π)+δ1IjiS
ji+δ2(J
ji+(1−θ)T j
i )Sji (7.31)
Ej+ 1
2i = (1−π)Π+β(Iji+qEj
i+pQji+lJ j
i )Sji−(Π+κ1+κ2)E
ji+δ1E
jiI
ji+δ2(J
ji+(1−θ)T )Ej
i
(7.32)
Qj+ 1
2i = κ1E
ji−(Π+σ)Qj
i+δ1IjiQ
ji+δ2(J
ji+(1−θ)T j
i )Qji (7.33)
Ij+ 1
2i = κ2E
ji − (Π+α+ δ1)I
ji + δ1I
2ji + δ2(J
ji +(1− θ)T j
i )Iji (7.34)
Jj+ 1
2i = σQj
i+αIji−(Π+δ2+γ1+ζ)J ji+δ1I
jiJ
ji+δ2(J
ji+(1−θ)T j
i )Jji (7.35)
Tj+ 1
2i = ζJ j
i − (Π+γ2+δ2)Tji +δ1I
jiT
ji +δ2(J
ji +θ+(1−θ)T j
i )Tji (7.36)
Rj+ 1
2i = γ1J
ji + γ2T
ji + δ1I
jiR
ji + δ2(J
ji + (1− θ)T j
i )Rji −ΠRj
i (7.37)
where Sji , E
ji , Q
ji , I
ji , J
ji , T
ji and Rj
i are the approximated values of S, E, Q, I, J ,
T and R at position −2+ iδx, for i = 0, 1, . . . and time jδt, j = 0, 1, . . . and Sj+ 1
2i ,
Ej+ 1
2i , Q
j+ 12
i , Ij+ 1
2i , J
j+ 12
i , Tj+ 1
2i and R
j+ 12
i denote their values at the first half-time
step. Similarly, for the second half-time step, the linear equations transform as
Sj+1i = S
j+ 12
i + d1δt
(δx)2(S
j+ 12
i−1 − 2Sj+ 1
2i + S
j+ 12
i+1 ) (7.38)
Ej+1i = E
j+ 12
i + d2δt
(δx)2(E
j+ 12
i−1 − 2Ej+ 1
2i + E
j+ 12
i+1 ) (7.39)
Qj+1i = Q
j+ 12
i + d3δt
(δx)2(Q
j+ 12
i−1 − 2Qj+ 1
2i +Q
j+ 12
i+1 ) (7.40)
Ij+1i = I
j+ 12
i + d4δt
(δx)2(I
j+ 12
i−1 − 2Ij+ 1
2i + I
j+ 12
i+1 ) (7.41)
J j+1i = J
j+ 12
i + d5δt
(δx)2(J
j+ 12
i−1 − 2Jj+ 1
2i + J
j+ 12
i+1 ) (7.42)
T j+1i = T
j+ 12
i + d6δt
(δx)2(T
j+ 12
i−1 − 2Tj+ 1
2i + T
j+ 12
i+1 ) (7.43)
Rj+1i = R
j+ 12
i + d7δt
(δx)2(R
j+ 12
i−1 − 2Rj+ 1
2i +R
j+ 12
i+1 ) (7.44)
153
The stability condition satisfied by the numerical method described above is
given as:dnδt
(δx)2≤ 0.5, n = 1, 2, 3, 4, 5, 6, 7. (7.45)
In each case, δx = 0.1, d1 = 0.025, d2 = 0.01, d3 = 0.0, d4 = 0.001, d5 = 0.0,
d6 = 0.0, d7 = 0.0 and δt = 0.03 are used.
7.3 Stability Analysis
7.3.1 Reproduction number without diffusion
For all the susceptible under consideration, RQIT represents the reproduction
number, i.e. the average number of secondary cases the infection is transmitted to
during a typical individuals infectious period in a situation where all persons are
susceptible. If RQIT < 1, the disease dies out and if RQIT > 1, then introduction
of one infected individual in susceptible, will spread the disease. The
reproduction number may vary remarkably for different infectious diseases and
also for the same disease in different populations. For a disease-free equilibrium
the reproduction number should always be less than one. The reproduction
number for the SEQIJTR model is calculated using the generation matrix
method [228]. The derivation is given in appendix A.7 whereas the expression for
of the reproduction number RQIT is given by:
RQIT = β(a+b+c+d)(κ1+κ2+Π)(α+δ1+Π)(γ1+δ2+ζ+Π)(σ+Π)
a = q(α + δ1 +Π)(γ1 + δ2 + ζ +Π)(σ +Π).
b = κ2(γ1 + δ2 + ζ +Π)(σ +Π).
c = pκ1(α + δ1 +Π)(γ1 + δ2 + ζ +Π).
d = l(κ1(δ1 +Π)σ + α(κ1σ + κ2(σ +Π))).
In the situation where there are no quarantine measures, but isolation and
treatment are still available i.e where p = 1, κ1 = 0 and σ = 0, the reproduction
number expression reduces to the following:
RIT = β(q(α+δ1+Π)(γ1+δ2+ζ+Π)+κ2(γ1+δ2+ζ+Π)+lακ2)(κ2+Π)(α+δ1+Π)(γ1+δ2+ζ+Π)
154
and in the absence of any treatment for disease, ζ = 0. Hence the expression for
the reproduction number is:
RI =β(q(α+δ1+Π)(γ1+δ2+Π)+κ2(γ1+δ2+Π)+lακ2)
(κ2+Π)(α+δ1+Π)(γ1+δ2+Π)
while in the absence of quarantine, treatment and isolation of infected individuals
after diagnosis i.e α = 0 and δ2 = 0, the RI changes to the basic reproduction
number R0, and is given as :
R0 =β(q(δ1+Π)+κ2)(κ2+Π)(δ1+Π)
The values of the quantities are given in Table 7.4
7.3.2 Disease-free equilibrium (DFE)
The variational matrix of the system of equations (7.8) - (7.14) at the disease-freeequilibrium P0 = (1, 0, 0, 0, 0, 0, 0), is:
V0
=
−Π −qβ −pβ (−β + δ1) (−lβ + δ2) (δ2(1 − θ)) 0
0 (qβ − κ1 − κ2 − Π) pβ β lβ 0 0
0 κ1 −(σ + Π) 0 0 0 0
0 κ2 0 −(α + δ1 + Π) 0 0 0
0 0 σ α −(γ1 + δ2 + ζ + Π) 0 0
0 0 0 0 ζ −(γ2 + δ2(θ − 1) + Π) 0
0 0 0 0 γ1 γ2 −Π
It is observed that the first eigenvalue −Π is negative and all entries below it arezero. This allows the elimination of the first row and column. The last eigenvalue−Π is also negative and all entries above it are zero, so the last row and columnmay be eliminated, giving:
V01 =
(qβ − κ1 − κ2 − Π) pβ β lβ 0
κ1 −(σ + Π) 0 0 0
κ2 0 −(α + δ1 + Π) 0 0
0 σ α −(γ1 + δ2 + ζ + Π) 0
0 0 0 ζ −(γ2 + δ2(θ − 1) + Π)
Now, the third eigenvalue of the variational matrix in the last column and last
row of the above matrix, −(γ2 + δ2(θ − 1) + Π) is negative and the entries above
are zero so, after eliminating the last row and column a fourth order variational
matrix is obtained:
V02 =
(qβ − κ1 − κ2 − Π) pβ β lβ
κ1 −(σ + Π) 0 0
κ2 0 −(α + δ1 + Π) 0
0 σ α −(γ1 + δ2 + ζ + Π)
The characteristic equation det(V 0 − λI) = 0 gives
λ4 + p1λ3 + p2λ
2 + p3λ1 + p4 = 0 (7.46)
155
The Routh-Hurwitz conditions used to check the stability here are (i)p1 > 0,
(ii)p4 > 0 and (iii)p2 − p3p1
> 0 and (iv)p1p2p3 − p23 − p21p4 > 0 where expressions
for p1, p2, p3 and p4 are given in appendix A.7.
(i)p1 = α− qβ + γ1 + δ1 + δ2 + ζ + κ1 + κ2 + 4Π + σ
(ii)p4 = (α + δ1 +Π)(γ1 + δ2 + ζ +Π)(κ1 + κ2 +Π)(Π + σ)− lβ(ακ2Π+ ακ1σ +
δ1κ1σ + ακ2σ + κ1Πσ)− qβ(α+ δ1 +Π)(γ1 + δ2 + ζ +Π)(σ +Π)− pβκ1(α+ δ1 +
Π)(γ1 + δ2 + ζ +Π)− β(κ2Π+ κ2σ)(γ1 + δ2 + ζ +Π)
(iii)p2 − p3p1
= qαβγ1 + qβγ1δ1+ qαβδ2+ qβδ1δ2 + qαβζ + qβδ1ζ + pαβκ1−αγ1κ1+
pβγ1κ1 + pβδ1κ1 − γ1δ1κ1 − αδ2κ1 + pβδ2κ1 − δ1δ2κ1 − αζκ1 + pβζκ1 − δ1ζκ1 +
lαβκ2−αγ1κ2+βγ1κ2− γ1δ1κ2− αδ2κ2+βδ2κ2− δ1δ2κ2−αζκ2+βζκ2− δ1ζκ2+
2qαβΠ−2αγ1Π+2qβγ1Π+2qβδ1Π−2γ1δ1Π−2αδ2Π+2qβδ2Π−2δ1δ2Π−2αζΠ+
2qβζΠ−2δ1ζΠ−2ακ1Π+2pβκ1Π−2γ1κ1Π−2δ1κ1Π−2δ2κ1Π−2ζκ1Π−2ακ2Π+
2βκ2Π− 2γ1κ2Π− 2δ1κ2Π− 2δ2κ2Π− 2ζκ2Π− 3αΠ2 + 3qβΠ2 − 3γ1Π2 − 3δ1Π
2 −
3δ2Π2 − 3ζΠ2 − 3κ1Π
2 − 3κ2Π2 − 4Π3 + qαβσ − αγ1σ + qβγ1σ + qβδ1σ − γ1δ1σ −
αδ2σ+qβδ2σ−δ1δ2σ−αζσ+qβζσ−δ1ζσ−ακ1σ+ lβκ1σ−γ1κ1σ−δ1κ1σ−δ2κ1σ−
ζκ1σ− ακ2σ+ βκ2σ− γ1κ2σ− δ1κ2σ− δ2κ2σ− ζκ2σ− 2αΠσ+ 2qβΠσ− 2γ1Πσ−
2δ1Πσ−2δ2Πσ−2ζΠσ−2κ1Πσ−2κ2Πσ−3Π2σ+(α− qβ+γ1+ δ1+ δ2+ ζ+κ1+
κ2+4Π+σ)(γ1δ1+δ1δ2+δ1ζ−pβκ1+γ1κ1+δ1κ1+δ2κ1+ζκ1−βκ2+γ1κ2+δ1κ2+
δ2κ2+ζκ2+3γ1Π+3δ1Π+3δ2Π+3ζΠ+3κ1Π+3κ2Π+6Π2+γ1σ+δ1σ+δ2σ+ζσ+
κ1σ+κ2σ+3Πσ−qβ(α+γ1+δ1+δ2+ζ+3Π+σ)+α(γ1+δ2+ζ+κ1+κ2+3Π+σ))
(iv)p1p2p3−p23−p21p4 = −(α− qβ+γ1+ δ1+ δ2+ ζ+ κ1+κ2+4Π+σ)2(αγ1κ1Π+
γ1δ1κ1Π+ αδ2κ1Π+ δ1δ2κ1 Π+ αζκ1Π+ δ1ζκ1Π− lαβκ2Π+ αγ1κ2Π− βγ1κ2Π+
γ1δ1κ2Π+αδ2κ2Π−βδ2κ2Π+δ1δ2κ2Π+αζκ2Π−βζκ2Π+δ1ζκ2Π+αγ1Π2+γ1δ1Π
2+
αδ2Π2 + δ1δ2Π
2 + αζΠ2 + δ1ζΠ2 + ακ1Π
2 + γ1κ1Π2 + δ1κ1Π
2 + δ2κ1Π2 + ζκ1Π
2 +
ακ2Π2−βκ2Π
2+ γ1κ2Π2+ δ1κ2Π
2+ δ2κ2Π2+ ζκ2Π
2+αΠ3+ γ1Π3+ δ1Π
3+ δ2Π3+
ζΠ3 + κ1Π3 + κ2Π
3 +Π4 − pβκ1(α+ δ1 +Π)(γ1 + δ2 + ζ +Π)− lαβκ1σ+αγ1κ1σ−
lβδ1κ1σ+γ1δ1κ1σ+αδ2κ1σ+δ1δ2κ1σ+αζκ1σ+δ1ζκ1σ−lαβκ2σ+αγ1κ2σ−βγ1κ2σ+
γ1 δ1κ2σ+αδ2κ2σ−βδ2κ2σ+δ1δ2κ2σ+αζκ2σ−βζκ2σ+δ1ζκ2σ+αγ1Πσ+γ1δ1Πσ+
αδ2Πσ+δ1δ2Πσ+αζΠσ+δ1ζΠσ+ακ1Πσ− lβκ1Πσ+γ1κ1Πσ+δ1κ1Πσ+δ2κ1Πσ+
ζκ1Πσ+ακ2Πσ−βκ2Πσ+γ1κ2Πσ+ δ1κ2Πσ+ δ2κ2Πσ+ ζκ2Πσ+αΠ2σ+γ1Π2σ+
δ1Π2σ+δ2Π
2σ+ζΠ2σ+κ1Π2σ+κ2Π
2σ+Π3σ−qβ(α+δ1+Π)(γ1+δ2+ζ+Π)(Π+
156
σ))+(α−qβ+γ1+δ1+δ2+ζ+κ1+κ2+4Π+σ)(γ1δ1+δ1δ2+δ1ζ−pβκ1+γ1κ1+δ1κ1+
δ2κ1+ζκ1−βκ2+γ1κ2+δ1κ2+δ2κ2+ζκ2+3γ1Π+3δ1Π+3δ2Π+3ζΠ+3κ1Π+3κ2Π+
6Π2+γ1σ+δ1σ+δ2σ+ζσ+κ1σ+κ2σ+3Πσ−qβ(α+γ1+δ1+δ2+ζ+3Π+σ)+α(γ1+
δ2+ζ+κ1+κ2+3Π+σ))(αγ1κ1+γ1δ1κ1+αδ2κ1+δ1δ2κ1+αζκ1+δ1ζκ1− lαβκ2+
αγ1κ2− βγ1κ2+γ1δ1κ2+αδ2κ2−βδ2κ2+ δ1δ2κ2+αζκ2−βζκ2+ δ1ζκ2+2αγ1Π+
2γ1δ1Π+2αδ2Π+2δ1δ2Π+2αζΠ+2δ1ζΠ+2ακ1Π+2γ1κ1Π+2δ1κ1Π+2δ2κ1Π+
2ζκ1Π+ 2ακ2Π− 2βκ2Π+ 2γ1κ2Π+ 2δ1κ2Π+ 2δ2κ2Π+ 2ζκ2Π+ 3αΠ2 + 3γ1Π2 +
3δ1Π2+3δ2Π
2+3ζΠ2+3κ1Π2+3κ2Π
2+4Π3−pβκ1(α+γ1+δ1+δ2+ζ+2Π)+αγ1σ+
γ1δ1σ+αδ2σ+δ1δ2σ+αζσ+δ1ζσ+ακ1σ− lβκ1σ+γ1κ1σ+δ1κ1σ+δ2κ1σ+ζκ1σ+
ακ2σ− βκ2σ+ γ1κ2σ+ δ1κ2σ+ δ2κ2σ+ ζκ2σ+2αΠσ+2γ1Πσ+2δ1Πσ+2δ2Πσ+
2ζΠσ+2κ1Πσ+2κ2Πσ+3Π2σ− qβ(δ1δ2+ δ1ζ +2δ1Π+2δ2Π+2ζΠ+3Π2+ δ1σ+
δ2σ+ζσ+2Πσ+γ1(δ1+2Π+σ)+α(γ1+δ2+ζ+2Π+σ)))−(αγ1κ1+γ1δ1κ1+αδ2 κ1+
δ1δ2κ1+αζκ1+ δ1ζκ1− lαβκ2+αγ1κ2−βγ1κ2+γ1δ1κ2+αδ2κ2−βδ2κ2+ δ1δ2κ2+
αζκ2−βζκ2+δ1ζκ2+2αγ1Π+2γ1δ1Π+2αδ2Π+2δ1δ2Π+2αζΠ+2δ1ζΠ+2ακ1Π+
2γ1κ1Π+2δ1κ1Π+2δ2κ1Π+2ζκ1Π+2ακ2Π−2βκ2Π+2γ1κ2Π+2δ1κ2Π+2δ2κ2Π+
2ζκ2Π+3αΠ2+3γ1Π2+3δ1Π
2+3δ2Π2+3ζΠ2+3κ1Π
2+3κ2Π2+4Π3− pβκ1(α+
γ1+ δ1+ δ2+ ζ+2Π)+αγ1σ+γ1δ1σ+αδ2σ+ δ1δ2σ+αζσ+ δ1ζσ+ακ1σ− lβκ1σ+
γ1κ1σ+δ1κ1σ+δ2κ1σ+ζκ1σ+ακ2σ−βκ2σ+γ1κ2σ+δ1κ2σ+δ2κ2σ+ζκ2σ+2αΠσ+
2γ1Πσ+2δ1Πσ+2δ2Πσ+2ζΠσ+2κ1Πσ+2κ2Πσ+3Π2σ− qβ(δ1δ2+ δ1ζ+2δ1Π+
2δ2Π+2ζΠ+3Π2+δ1σ+δ2σ+ζσ+2Πσ+γ1(δ1+2Π+σ)+α(γ1+δ2 +ζ+2Π+σ)))2
7.3.3 Endemic equilibrium without diffusion
The variational matrix of the system of equations (7.8)-(7.14) at
P ∗(S∗, E∗, Q∗, I∗, J∗, T ∗, R∗), is given by
V ∗ =
a11 a12 a13 a14 a15 a16 a17
a21 a22 a23 a24 a25 a26 a27
a31 a32 a33 a34 a35 a36 a37
a41 a42 a43 a44 a45 a46 a47
a51 a52 a53 a54 a55 a56 a57
a61 a62 a63 a64 a65 a66 a67
a71 a72 a73 a74 a75 a76 a77
157
where
a11 = (−I∗ − J∗l − E∗q − pQ∗)β + I∗δ1 + δ2(J∗ + T ∗(1− θ))− Π, a12 = −qS∗β,
a13 = −pS∗β, a14 = −S∗β + S∗δ1, a15 = −lS∗β + S∗δ2, a16 = S∗δ2(1− θ),
a21 = (I∗ + J∗l + E∗q + pQ∗)β,
a22 = qS∗β + I∗δ1 + δ2(J∗ + T ∗(1− θ))− κ1 − κ2 − Π,
a23 = pS∗β, a24 = S∗β + E∗δ1, a25 = lS∗β + E∗δ2, a26 = E∗δ2(1− θ),
a32 = κ1, a33 = I∗δ1 + δ2(J∗ + T ∗(1− θ))− Π− σ, a34 = δ1Q
∗,
a35 = δ2Q∗, a36 = δ2(1− θ)Q∗, a42 = κ2,
a44 = −α− δ1 + 2I∗δ1 + δ2(J∗ + T ∗(1− θ))− Π), a45 = I∗δ2, a46 = δ2(1− θ)I∗,
a53 = σ, a54 = α+ J∗δ1, a55 = −γ1 + I∗δ1 − δ2 + J∗δ2 − ζ + δ2(J∗ + T ∗(1− θ))−Π,
a56 = J∗δ2(1− θ), a64 = T ∗δ1, a65 = T ∗δ2 + ζ,
a66 = −γ2 + I∗δ1 − δ2 + T ∗δ2(1− θ) + δ2(J∗ + T ∗(1− θ) + θ)− Π,
a74 = R∗δ1, a75 = γ1 +R∗δ2, a76 = γ2 +R∗δ2(1− θ),
a77 = I∗δ1 + δ2(J∗ + T ∗(1− θ))− Π,
a17 = a27 = a31 = a37 = a41 = a43 = a47 = a51 = a52 = 0,
a57 = a61 = a62 = a63 = a67 = a71 = a72 = a73 = 0.
The characteristic equation for P ∗(S∗, E∗, Q∗, I∗, J∗, T ∗, R∗) can be written as
λ7 + p1λ6 + p2λ
5 + p3λ4 + p4λ
3 + p5λ2 + p6λ
1 + p7 = 0 (7.47)
where the expression for p1, p2, p3, p4, p5, p6 ,p7 and the Routh-Hurwitz conditions
are calculated using method described in [84]. The Routh-Hurwitz condition are:
C1 : p1 > 0,
C2 : p7 > 0,
C3 : p2 − p3p1
> 0,
C4 :p23+p21p4−p1(p2p3+p5)
p3−p1p2> 0,
C5 :p23p4+p25+p21(p
24−p2p6)−p3(p2p5+p7)+p1(p22p5−2p4p5+p3p6+p2(p7−p3p4))
p23+p21p4−p1(p2p3+p5)> 0,
C6 :A1+A2
(p23p4+p25+p21(p24−p2p6)−p3(p2p5+p7)+p1(p22p5−2p4p5+p3p6+p2(p7−p3p4+)))
> 0,
C7 :A1+A2+A3
B1+B2> 0.
where
A1 = p35−p33p6+p31p26−p3p5(p2p5+2p7)+p23(p4p5+p2p7)+p21(p
24p5−2p6(p2p5+p7),
A2 =
p4(p2p7−p3p6))+p1(3p3p5p6−2p4p25+p27+p22(p
25−p3p7)+p2(p
23p6−p3p4p5+p5p7)),
158
Table 7.3: Values for Routh-Hurwitz criteria of equilibrium
Diffusion C1 C2 C3 C4 C5 C6 C7 Stability
No 1.057 6.5× 10−14 0.340 0.051 3.1× 10−3 6.5× 10−5 4.9× 10−9 Stable
Yes 1.145 8.8× 10−11 0.405 0.070 6.2× 10−3 2.7× 10−4 5.3× 10−6 Stable
A3 = p35p6 − p33p26 + p31p
36 − p4p
25p7 + p2p5p
27 − p37 + p23(p4p5p6 − p24p7 + 2p2p6p7)−
p21(p34p7 − p24p5p6 + p26(2p2p5 + 3p7) + p4p6(p3p6 − 3p2p7)),
A4 = −p3(p22p
27 + p7(3p5p6 − 2p4p7) + p2p5(p5p6 − p4p7)) + p1(2p
24p5p7 + p32p
27 −
p4p6(2p25 + p3p7) + p22(p
25p6 − p4p5p7 − 2p3p6p7),
A5 = 3p6(p3p5p6 + p27) + p2(p23p
26 + p7(p5p6 − 3p4p7) + p3p4(p4p7 − p5p6)),
B1 = p35 − p33p6 + p31p26 − p3p5(p2p5 + 2p7) + p23(p4p5 + p2p7) + p21(p
24p5 − 2p6(p2p5 +
p7) + p4(p2p7 − p3p6)) ,
B2 = p1(3p3p5p6 − 2p4p25 + p27 + p22(p
25 − p3p7) + p2(p
23p6 − p3p4p5 + p5p7)).
The derivation of Routh-Hurwitz conditions is given in appendix A.7. The
numerical values for the Routh-Hurwitz conditions are given in Table 7.3 at the
endemic point of equilibrium
P1 = (0.55678, 3.86×10−5, 2.46×10−5, 2.69×10−5, 3.80×10−5, 1.12×10−4, 0.44298).
7.3.4 Endemic equilibrium with diffusion
To calculate the small perturbations S1(x, t), E1(x, t), Q1(x, t), I1(x, t), J1(x, t),
T1(x, t) and R1(x, t) the equations (7.8) - (7.14) are linearised about the point of
equilibrium P ∗(S∗, E∗, Q∗, I∗, J∗, T ∗, R∗) as described in [35, 201] giving
∂S1
∂t= a11S1 + a12E1 + a13Q1 + a14I1 + a15J1 + a16T1 + a17R1d1
∂2S1
∂x2(7.48)
∂E1
∂t= a21S1 + a22E1 + a23Q1 + a24I1 + a25J1 + a26T1 + a27R1 + d2
∂2E1
∂x2(7.49)
∂Q1
∂t= a31S1 + a32E1 + a33Q1 + a34I1 + a35J1 + a36T1 + a37R1 + d3
∂2Q1
∂x2(7.50)
∂I1∂t
= a41S1 + a42E1 + a43Q1 + a44I1 + a45J1 + a46T1 + a47R1 + d4∂2I1∂x2
(7.51)
∂J1∂t
= a51S1 + a52E1 + a53Q1 + a54I1 + a55J1 + a56T1 + a57R1 + d5∂2J1∂x2
(7.52)
159
∂T1
∂t= a61S1 + a62E1 + a63Q1 + a64I1 + a65J1 + a66T1 + a67R1 + d6
∂2T1
∂x2(7.53)
∂R1
∂t= a71S1 + a72E1 + a73Q1 + a74I1 + a75J1 + a76T1 + a77T1 + d7
∂2R1
∂x2(7.54)
where a11, a12, a13 ... are the elements of the variational matrix V ∗ as calculated in
the previous section 7.3.3. The existence of a Fourier series solution of the
following form for Eqs. (7.48) - (7.54) is assumed:
S1(x, t) =∑k
Skeλt cos(kx) (7.55)
E1(x, t) =∑k
Ekeλt cos(kx) (7.56)
Q1(x, t) =∑k
Qkeλt cos(kx) (7.57)
I1(x, t) =∑k
Ikeλt cos(kx) (7.58)
J1(x, t) =∑k
Jkeλt cos(kx) (7.59)
T1(x, t) =∑k
Tkeλt cos(kx) (7.60)
R1(x, t) =∑k
Rkeλt cos(kx) (7.61)
where k = nπ2, (n = 1, 2, 3, · · · · · · ) is the wave number for node n. Substituting
the values of S1, E1, Q1 I1, J1, T1 and R1 into equations (7.48) -(7.54), the
equations are transformed into:∑k
(a11−d1k2−λ)Sk+
∑k
a12Ek+∑k
a13Qk+∑k
a14Ik+∑k
a15Jk+∑k
a16Tk = 0
(7.62)∑k
a21Sk+∑k
(a22−d2k2−λ)Ek+
∑k
a23Qk+∑k
a24Ik+∑k
a25Jk+∑k
a26Tk = 0
(7.63)∑k
a32Ek +∑k
(a33 − d3k2 − λ)Qk +
∑k
a34Ik +∑k
a35Jk∑k
a36Tk = 0 (7.64)
∑k
a42Ek +∑k
(a44 − d4k2 − λ)Ik +
∑k
a45Jk +∑k
a46Tk = 0 (7.65)
∑k
a53Qk +∑k
a54Ik +∑k
(a55 − d5k2 − λ)Jk +
∑k
a56Tk = 0 (7.66)
∑k
a64Ik +∑k
a65Jk +∑k
(a66 − d6k2 − λ)Tk = 0 (7.67)
160
∑k
a74Ik +∑k
a75Jk +∑k
a76Tk +∑k
(a77 − d7k2 − λ)Rk = 0 (7.68)
The Variational matrix Vd for the equations (7.62) - (7.68) is
Vd =
a11 − d1k2 a12 a13 a14 a15 a16 0
a21 a22 − d2k2 a23 a24 a25 a26 0
0 a32 a33 − d3k2 a34 a35 a36 0
0 a42 0 a44 − d4k2 a45 a46 0
0 0 a53 a54 a55 − d5k2 a56 0
0 0 0 a64 a65 a66 − d6k2 0
0 0 0 a74 a75 a76 a77 − d7k2
The characteristic equation for the variational matrix Vd is the same as given in
equation (7.47), where as p1, p2, p3..., p7 are calculated again, using the above
variational matrix Vd using method described in [84]. The numerical results for
the Routh-Hurwitz conditions for endemic equilibrium are given in Table 7.3. It
is observed that at the point of equilibrium, P1, Routh-Hurwitz conditions for
stability are satisfied in the presence of diffusion in the system.
7.3.5 Reproduction number with diffusion
Variational matrix method is used to calculate reproduction number with
diffusion RdQIT . The variational matrix with diffusion for P0 = (1, 0, 0, 0, 0, 0, 0) is
as follows:
V 0d =
[V 0d1
V 0d2
]
V 0d1
=
−(Π + d1k2) −qβ −pβ −(β − δ1)
0 qβ − κ1 − κ2 − Π− d2k2 pβ β
0 κ1 −(Π + σ)− d3k2 0
0 κ2 0 −(α + δ1 +Π)− d4k2
0 0 σ α
0 0 0 ζ
0 0 0 0
161
V 0d2
=
−(lβ − δ2) δ2(1− θ) 0
lβ 0 0
0 0 0
0 0 0
−(γ1 + δ2 + ζ +Π+ d5k2) 0 0
ζ −(γ2 + δ2(1− θ) + Π + d6k2) 0
γ1 γ2 −Π− d7k2
It is observed that the first eigenvalue −(d1k
2 +Π) of the variational matrix V 0d is
negative and all entries below it are zero. This allows us to eliminate the first row
and column. So the reduced matrix is:
V 0d =
[V 0d1
V 0d2
]
V 0d1
=
qβ − (κ1 + κ2 +Π+ d2k2) pβ β
κ1 −(Π + σ + d3k2) 0
κ2 0 −(α + δ1 +Π+ d4k2)
0 σ α
0 0 ζ
0 0 0
V 0d2
=
lβ 0 0
0 0 0
0 0 0
−(γ1 + δ2 + ζ +Π+ d5k2) 0 0
ζ −(γ2 + δ2(1− θ) + Π + d6k2) 0
γ1 γ2 −(Π + d7k2)
Now, the last eigenvalue −(Π + d7k
2) is again negative and all entries above it are
zero, so the last row and column can be eliminated. The reduced matrix is:
V0d =
(qβ − κ1 − κ2 − Π − d2k2) pβ β lβ 0
κ1 −(Π + σ + d3k2) 0 0 0
κ2 0 −(α + δ1 + Π + d4k2) 0 0
0 σ α −(γ1 + δ2 + ζ + Π + d5k2) 0
0 0 0 ζ −(γ2 + δ2(1 − θ) + Π + d6k2)
Again, the third eigenvalue of the variational matrix given in the last column
and last row −(d6k2 + γ2 + δ2(1− θ) + Π) is negative and the entries above it are
162
zero, so after eliminating the last row and column, a 4th order variational matrix
V 0d is obtained.
V0d =
qβ − κ1 − κ2 − Π − d2k
2 pβ β lβ
κ1 −(Π + σ) − d3k2 0 0
κ2 0 −(α + δ1 + Π) − d4k2 0
0 σ α −(γ1 + δ2 + ζ + Π) − d5k2
Characteristic equation det(V 0d − λI) = 0 is given as
λ4 + q1λ3 + q2λ
2 + q3λ1 + q4 = 0 (7.69)
Where q’s are given in appendix A.7. The Routh-Hurwitz condition p4 > 0 is
satisfied only for RdQIT < 1. So, the expression for the reproduction number with
diffusion in the system is given by:
RdQIT = β(A+B+C+D)
(d4k2+α+δ1+Π)(d5k2+γ1+δ2+ζ+Π)(d2k2+κ1+κ2+Π)(d3k2+Π+σ),
where A = l(d3k2ακ2 + κ1(d4k
2 + δ1 +Π)σ + α(κ1σ + κ2(Π + σ))),
B =
κ2(d3k2(d5k
2+γ1+δ2+ζ)+Π(1+d3k2+d5k
2+γ1+δ2+ζ+Π)+(d5k2+γ1+δ2+ζ)σ),
C = pκ1((d4k2 + α+ δ1 +Π)(d5k
2 + γ1 + δ2 + ζ +Π)),
D = q(((d5k2 + γ1 + δ2 + ζ +Π)(d3k
2(d4k2 + α + δ1 +Π) + (d4k
2 + α +Π)(Π +
σ) + δ1(Π + σ))) + δ1).
The numerical values of all reproduction numbers without diffusion are given in
Table 7.4. It is observed that, in the absence of any control measures and
treatment, the reproduction number of SARS, R0 is quite high. If isolation of
the infected population is introduced into the system, the reproduction number
RI halves. With the additional presence of any treatment of SARS, the
reproduction number RIT halves again. The value of the reproduction number
RQIT reduces further if quarantine measures are introduced.
In the presence of diffusion in the compartment, where the mixing of the
individuals is maximized, the reproduction number, RdQIT is 1.47506. This value
is higher than RQIT , the corresponding value without diffusion. The value of
RdQIT is smaller to that of RIT , where there is no quarantine. Hence, introducing
the diffusion at the beginning of the infection opposes the effects of quarantine
measures. Although it is assumed that there is no diffusion in the quarantine
163
compartment, the diffusion in the other compartments increases the reproduction
number and eventually reduces the effectiveness of the quarantine measures.
Table 7.4: Value of reproduction number (without diffusion)
Reproduction Number Value
R0 5.88256
RI 2.89739
RIT 1.4833
RQIT 1.30855
7.3.6 Excited mode and bifurcation value
The technique from Chapter 3 is used to calculate the first excited mode of
oscillation n. According to the description of mode of excitation the curve.
f(β) =2.72611 ∗ 10−16 − 2.10836 ∗ 10−15β + 6.03344 ∗ 10−15β2 − 8.0138 ∗ 10−15β3 + 4.99653 ∗ 10−15β4 − 1.29674 ∗ 10−15β5 + 1.08455 ∗ 10−16β6
3.45506 ∗ 10−8 − 2.14726 ∗ 10−7β + 4.38499 ∗ 10−7β2 − 3.49595 ∗ 10−7β3 + 1.02242 ∗ 10−7β4 − 9.04955 ∗ 10−9β5
(7.70)
where n = 1 represents the first mode of excitation as being closest to the β-axis
as shown in Fig. 7.2. It is observed that bifurcation value of β, α, ζ and σ
increase in the presence of diffusion. So, the system remains stable for larger
values if diffusion included. On the other hand, bifurcation values of β is higher
than the bifurcation value of β for SEIJTR as given in Chapter 6 model but
lower in the presence of diffusion. The bifurcation values of α is lower in the
present SEQIJTR model than for SEIJTR (Chapter 6) both, with and without
diffusion. The corresponding bifurcation diagrams for β, α ,ζ and σ are shown in
appendix A.7.
7.4 Numerical solutions
Two initial population distributions are considered and solved numerically, with
and without diffusion. Graphs and descriptions of these solutions are given below:
164
n=1
0.2 0.4 0.6 0.8 1.0Β
-1.´ 10-8
-5.´ 10-9
5.´ 10-9
1.´ 10-8
1.5´ 10-8
fHΒL
n=2
0.2 0.4 0.6 0.8 1.0Β
-1.´10-8
-5.´10-9
5.´10-9
1.´10-8
1.5´10-8
f HΒL
n=3
0.2 0.4 0.6 0.8 1.0Β
-1.´ 10-8
-5.´ 10-9
5.´ 10-9
1.´ 10-8
1.5´ 10-8
fHΒL
Figure 7.2: Determination of first excited mode with β as an unknown parameter.
Table 7.5: Bifurcation value of influential parameters
Parameters Value Considered Bifurcation Value
Without Diffusion With Diffusion
β 0.242 0.33224 0.362239
α 0.238 0.04879 0.07952
ζ 0.200 0.01619 0.05062
σ 0.157 0.01615 0.02636
165
7.4.1 Numerical solution without diffusion
Fig. 7.3, shows the output for initial condition (i) without diffusion for the
SEQIJTR system. At t = 0 a small proportion of infected are introduced to the
system. As soon as the disease spreads the proportion of susceptible reduces
dramatically, the peak proportion of susceptible dropping from 0.98 to 0.03488
within five days in the domain [−1, 1.5], where a large number of the population
get exposed to SARS. In the next fifteen days the peak proportions of susceptible
0.01102, 0.00459 and 0.00278 on days t = 10, t = 15 and t = 20 respectively. After
the first five days of disease the exposed population concentrated in the domain
[−0.6, 1.0] with peak value 0.25885. In the next five days a rapid decrease in the
exposed population is observed with peak proportion value 0.04860 in the domain
[−1, 1.5]. At t = 15 and t = 20 there are very small proportions of exposed left,
with peak values 0.00899 and 0.00167. Over time, from the exposed compartment
a proportion of the population that is not knowing to be infected yet is removed
and kept under observation in the quarantine compartment. Fig. 7.3, shows that
in the beginning of the disease, the proportion observed in the quarantine class is
maximized. At t = 5 days the peak proportion of the quarantined population in
the domain [−0.45, 0.85] is 0.17154. After five days the proportion of the
population quarantined decreases and the main concentration of the population
in the domain [−1.0, 1.5]. The peak values of proportion of population
quarantined at t = 10, t = 15 and t = 20 days are 0.11979, 0.05915 and 0.02601.
The proportion of the exposed population that becomes infected with SARS
moves to the infected compartment. The first five days of study of the SARS
disease are observed to be very important for exposed, quarantined and infected
compartments. In the first five days the proportion of infected increases rapidly
and attains its peak value, 0.25039 at t = 5 days. Thereafter, the proportion of
population infected decreases and the peak at t = 10 days is almost half of the
peak value in the first five days, being 0.11139 in the domain [−1, 1.5]. In the
next ten days, the proportion of infected keeps on decreasing with very low peaks
of 0.03350 and 0.00862 at t = 15 and t = 20 days, respectively. Once a
quarantined individual is diagnosed with SARS it is moved to the diagnosed
166
compartment as, for the simplification of the model, it is assumed that the
population in the quarantine compartment develop the disease at the end. The
individuals in the infected compartment diagnosed with SARS are isolated and
moved to the diagnosed compartment. The diagnosed population proportion
rapidly increase in the first five days of the disease and at t = 5 days, attains the
peak value 0.20568 across the domain [−0.4, 0.8]. This proportion keeps
increasing in the next five days and at t = 10 attains the peak value 0.23601 in
the domain [−1, 1.25]. After ten days the proportion diagnosed decreases rapidly,
with the peak values of the proportion having 0.13043 and 0.05602 at t = 15 and
t = 20 days, respectively. There is a only a small proportion of the population
admitted for treatment in the first five days of disease, so the peak value of the of
treated proportion of the population is 0.08539 at t = 5, with principal domain
[−0.4, 0.8]. Between t = 5 and t = 10 days, treated proportion increases further
and the peak proportion at t = 10 is 0.31772. This increase continues in the next
five days of study with a peak value of 0.41515 at t = 15. After that, the
proportion treated starts decreasig slowly, and at t = 20 days the peak of the
treated proportion is 0.38180. Negligible recovery is observed at t = 5 days across
the domain [−0.2, 0.6]. An increase in the recovered population is observed in the
next fifteen days with the recovered being concentrated in the domain [−0.8, 1.2].
The peak values of the recovered proportion of the population at t = 10, t = 15
and t = 20 days are 0.16649, 0.35276 and 0.52586.
Fig. 7.4, shows the output of initial condition (ii) in the absence of diffusion. The
peak values for the graphs in Fig. 7.4, are given in Table 7.6.
Table 7.6: Peak values for initial condition (ii) without diffusion
t S E Q I J T R
t = 00 0.96000 0.00000 0.00000 0.04000 0.00000 0.00000 0.00000
t = 05 0.00896 0.24929 0.16856 0.24887 0.21033 0.09204 0.03091
t = 10 0.00553 0.04681 0.11681 0.10886 0.23331 0.32254 0.17166
t = 15 0.00076 0.00867 0.05755 0.03257 0.12772 0.41538 0.35811
t = 20 0.00084 0.00161 0.02527 0.00835 0.05462 0.37974 0.53039
167
t = 0
t = 5t = 10
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0S
t = 5
t = 10
t = 15
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25
0.30E
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.05
0.10
0.15
0.20Q
t = 0
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25
0.30I
t = 0
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25J
t = 0
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.1
0.2
0.3
0.4
T
t = 0
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.1
0.2
0.3
0.4
0.5
R
Figure 7.3: Solutions for initial condition (i) without diffusion.
168
t = 0
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0S
t = 5
t = 10
t = 15
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25
E
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.05
0.10
0.15
0.20Q
t = 0
t = 5
t = 10
t = 20 t = 15
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25I
t = 0
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25J
t = 5
t = 10
t = 20
t=15
-2 -1 0 1 2x
0.1
0.2
0.3
0.4
T
t = 0
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.1
0.2
0.3
0.4
0.5
R
Figure 7.4: Solutions for initial condition (ii) without diffusion.
169
7.4.2 Numerical solution with diffusion
Fig. 7.5, shows the output for initial condition (i), when the diffusion is
introduced in the SEQIJTR model. In the first five days of the spread of the
disease, susceptible move to the other compartment rapidly. At t = 5, t = 10,
t = 15 and t = 20 days the peak proportions in the susceptible compartment are
0.01069, 0.00154, 0.00034 and 0.00020, respectively. The exposed population
rapidly increases in the first five days across the domain [−1.4, 1.6] with peak
proportion value 0.14742 at t = 5 days. The next five days of the study show a
significant decrease in exposed with a peak proportion of 0.02186 across the
domain [−2, 2]. In the next ten days negligible proportions of exposed population
are observed, with peak values 0.00343 and 0.00586 at t = 15 and t = 20. The
proportion in the quarantined compartment rapidly increases in the first five
days. The peak value of the proportion is 0.10207 at t = 5 days, mainly
concentrated in the domain [−1.2, 1.5]. This is followed by rapid decrease in the
next fifteen days across the domain [−1.6, 2], with peak values 0.06492, 0.02984
and 0.01252 at t = 10, t = 15, and t = 20 days. The infected population
proportion quickly increases over the first five days, attaining peak value 0.14795
at t = 5, across the domain [−1.2, 1.6]. A major decrease is observed in the next
five days and at t = 10 the peak value of the infected proportion across the larger
domain [−1.6, 2] has dropped to 0.05708. At t = 15 and t = 20 lower peaks are
observed across the same domain i.e 0.01517 and 0.00354 respectively. The
proportion diagnosed has peak value 0.12121 at t = 5 days with principal domain
[−1, 1.2]. At t = 10 days the peak diagnosed proportion across the effective
domain [−1.5, 1.8] is 0.12882. That is the maximum diagnosed population
proportion of the study. After that, the values decrease rapidly and at t = 15 and
t = 20 the peak proportions of diagnosed are 0.06563 and 0.02655, respectively.
The treated proportions peak at t = 15 days, with these treated concentrated in
the domain [−1.6, 2]. The maximum of treated population proportions are
0.04955, 0.17665, 0.21799 and 0.19196 at t = 5, t = 10, t = 15 and t = 20 days,
respectively. Recovery increases significantly after t = 5 days and spreads across
the domain [−1.4, 1.6] within the next fifteen days. Maximum recovery is
170
observed at t = 20 days, where the peak value of the proportion is 0.268532.
Fig. 7.6, shows the output of the initial condition (ii) with diffusion. The peak
values for the graphs in Fig. 7.6, are given in Table 7.7.
Table 7.7: Peak values for initial condition (ii) without diffusion
t S E Q I J T R
t = 00 0.96000 0.00000 0.00000 0.04000 0.00000 0.00000 0.00000
t = 05 0.00789 0.05115 0.02311 0.04255 0.03207 0.01669 0.00618
t = 10 0.00169 0.00742 0.01677 0.01742 0.03512 0.04873 0.02686
t = 15 0.00064 0.00122 0.00777 0.00468 0.01794 0.05842 0.05133
t = 20 0.00039 0.00025 0.00325 0.00113 0.00722 0.05061 0.07173
7.5 Discussion
SEQIJTR model, with and without diffusion is defined and used to simulate the
transmission of SARS in 2003. A quarantined compartment is added to the
SEIJTR model given in Chapter 5 and 6, in order to construct the new model.
The numerical study of this model shows the effects of quarantine on disease
dynamics. At the time of the SARS epidemic, the two most influential
non-medical interventions were isolation of the infected and quarantine of the
exposed. So, it is important to study the effect of quarantine especially for
situations where medical interventions like treatment and vaccination are not
very effective. Two different initial conditions have been used to study
numerically their effects on transmission of the disease under different population
distributions. The operator splitting technique is used to calculate numerical
solutions of the differential equations. Expressions for the seventh-order
Routh-Hurwitz criteria are calculated and then used to check the stability of two
possible equilibria for the model, namely the disease-free and endemic equilibria.
The reproduction number RQIT is calculated with and without diffusion. For the
case without diffusion generation matrix method is used. While for with diffusion
case a new method involving the Routh-Hurwitz stability conditions and the
171
t = 0
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0S
t = 5
t = 10
t=15
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
0.10
0.12
0.14
E
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
0.10
0.12Q
t = 0
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.05
0.10
0.15
I
t = 0
t = 5
t = 10
t = 15
t=20
-2 -1 0 1 2x
0.02
0.04
0.06
0.08
0.10
0.12
0.14
J
t = 0
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25T
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.05
0.10
0.15
0.20
0.25
R
Figure 7.5: Solutions for initial condition (i) with diffusion.
172
t = 0
t = 5t=5
-2 -1 0 1 2x
0.2
0.4
0.6
0.8
1.0S
t = 5
t = 10
t =15
-2 -1 0 1 2x
0.01
0.02
0.03
0.04
0.05
0.06E
t = 5
t = 20
t = 15
t=10
-2 -1 0 1 2x
0.005
0.010
0.015
0.020
0.025Q
t = 0
t = 5
t = 10
t = 20
t = 15
-2 -1 0 1 2x
0.01
0.02
0.03
0.04
0.05I
t = 0
t = 5
t = 10
t = 15
t=20
-2 -1 0 1 2x
0.01
0.02
0.03
0.04J
t = 0
t = 5
t = 10
t = 30
-2 -1 0 1 2x
0.01
0.02
0.03
0.04
0.05
0.06T
t = 5
t = 10
t = 15
t=20
-2 -1 0 1 2x
0.02
0.04
0.06
0.08R
Figure 7.6: Solutions for initial condition (ii) with diffusion.
173
variational matrix is developed to calculate the expression for reproductive
number RdQIT . In disease-free equilibrium (DFE) the stability conditions are
satisfied for RQIT < 1. For endemic equilibrium RQIT and RdQIT are calculated.
The numerical value of the reproduction number without diffusion RQIT as given
in Table 7.4 is less than the numerical value of the reproduction number RdQIT
(with diffusion), showing the negative effect of diffusion on the system even
though it is assumed that there is no diffusion in the quarantined compartment.
The reproduction number without effective quarantine in the system is higher
than with quarantine measures as given in Table 7.4. This shows that the absence
of quarantine measures or the presence of diffusion both increase the infection, as
compared to the system with quarantine measures and no diffusion. The
bifurcation values of influential parameters are calculated, with and without
diffusion in the system, and are given in the Table 7.5. In the absence of
diffusion, the bifurcation value of β increases negligibly with the introduction of
quarantine compartment to the model, showing that the system SEQIJTR
system will be stable for larger transmission coefficient values than the SEIJTR
model studied earlier in Chapter 6. On the other hand, in the presence of
diffusion, SEIJTR system (Chapter 6) remains stable for higher values of β than
the SEQIJTR system. This shows the negative effect of diffusion on quarantine
measures. The bifurcation values of rate of progression from infected to
diagnosed, α, in the SEQIJTR model, both in the presence and absence of
diffusion, are lower as compared to the SEIJTR model.
The numerical solutions of the new SEQIJTR model are compared with those
for the previously-developed model SEIJTR (Chapter 6), in order to understand
the effects of quarantine on disease dynamics. In the absence of diffusion and
with initial condition (i), the proportion of population exposed is observed to be
significantly lower in the SEQIJTR model than in the model without a
quarantined class, SEIJTR. At t = 5 days, the peak value is nearly half of the
peak value under treatment model studied in Chapter 6 whereas, at t = 10 days,
the peak value of the proportion for SEQIJTR is even less than half of the
SEIJTR value. This trend continues in the following ten days of the disease
174
study. The present SEQIJTR model shows that a significant proportion of the
exposed population moves to quarantine compartment where infective but not
infectious population is isolated and quarantined. This immensely effect the
transmission of SARS by reducing the infected population by restricting the
entry to the infected class for those individuals who are not infectious yet.
Finally, it effects the recovery by increasing the recovery immensely. The
difference between the infected proportion of the population in the models grows
until the maximum difference occurs at t = 10 days and then, in the next ten
days, difference starts on decreasing, but the infected proportion in SEQIJTR
model is less than that of the SEIJTR model on all days under study. This
shows that using quarantine measures in the beginning of the disease affects the
prorogation of the disease. The proportion of the population under treatment is
higher than with the treatment model in Chapter 6. This difference keeps on
increasing with the passage of time until the maximum difference is observed at
t = 15 days. Recovery is higher in the present model than the model without
quarantine (Chapter 6) on all days under study, the largest difference in recovery
between the models occurring at day 20. Under initial condition (ii), very similar
results are observed whereas the difference in recovery under the two models is
even more pronounced, with SEQIJTR recovery even higher under condition (ii)
than under (i).
When diffusion is introduced to the system for initial conditions (i) and (ii), the
proportion of population exposed is lower than for the treatment isolation model
in Chapter 6. Although the proportion of exposed is lower in the model under
study, the difference between the proportions for the two models is less in the
presence of diffusion. It is very important to note that, although the diffusion in
the quarantined class is assumed to be zero, the diffusion in other compartments
affects the transmission of disease so much that it reduces the intensity of the
population in the quarantined compartment and spreads the population across a
larger domain than in the without diffusion case. The infected population
proportion is comparatively lower than that in the treatment model. As
compared to the without diffusion case for the same SEQIJTR model, the
175
intensity of infection is almost half. The differences are more significant in the
first ten days of the disease than the last ten days of disease under study. The
numerical results of the SEQIJTR model shown in Fig. 7.5, indicate that at
t = 5 the peak value of the diagnosed population is significantly higher than for
the SEIJTR model in Chapter 6, but at t = 10 the diagnosed population is
significantly lower than for the treatment model whereas at t = 15 and t = 20
there is negligible difference between diagnosed proportions. The proportion of
population recovered with diffusion, in both conditions, is greater under
SEQIJTR model than in the SEIJTR model (in Chapter 6). In the last ten
days the difference is higher than in the first ten days for condition (i) whereas
condition (ii) shows no difference in recovery in the first ten days and a negligible
difference in the last ten days for the two models in the presence of diffusion.
176
Chapter 8
Conclusions
The aim of this work is to study the transmission dynamics of the SARS disease.
First, a model consisting of susceptible, exposed, infected, diagnosed and
recovered (SEIJR) compartments is chosen. In Chapter 3, a SEIJR model is
considered with the inclusion of diffusion. Four different initial conditions are
taken for the population distribution. The equations governing the system are
solved numerically using the operator splitting method. The reproduction
number RI is calculated for the disease. It is shown that in the disease-free
equilibrium, the disease dies out for RI < 1, but prevails for endemic equilibrium,
where RI > 1, as shown in Table 3.11. Stability of solutions with and without
diffusion is established using the Routh-Hurwitz conditions. The parameters
transmission coefficient, β, recovery coefficient in infectious class, γ1, and recovery
coefficient in diagnosed, γ2, are varied to observe the effects on the spread of
disease. Effect on the spread of disease is examined for four cases involving
different values of transmission coefficient β, recovery rates γ1 and γ2. It is
observed that the system can be stable for large values of transmission coefficient
β and small values of recovery rates γ1 and γ2 in the presence of diffusion. Thus
the spread of disease will not increase in the presence of diffusion even with larger
value of transmission coefficient β and smaller value of recovery rates γ1 and γ2.
In order to examine the effects of diffusion further, cross-diffusion along with
self-diffusion are introduced to the SEIJR model in Chapter 4. Two different
177
initial conditions are taken for the population distribution. Stability of solutions
with and without cross-diffusion is also established here using the Routh-Hurwitz
conditions. Four different cases of cross-diffusion coefficients in the susceptible
and exposed compartments are chosen in order to see their effects on the spread
of disease. Bifurcation values of the transmission coefficient, β and recovery
coefficients γ1 and γ2 are obtained. It is observed that when both populations i.e
susceptible and exposed, cross diffuse, the system is destabilised for a smaller
value of β. It is also observed that with a positive value of cross-diffusion in the
susceptible and exposed compartments, the system stabilises for higher values of
the recovery coefficients γ1 and γ2.
In order to investigate the effect on the transmission dynamics of SARS with the
inclusion of a treatment compartment in the system a SEIJTR model has been
constructed in Chapter 5. First of all, parameters involved in the model are
estimated based on the best fit to the field data [121] published as daily reports
by the World Health Organization when the SARS epidemic outbreak occurred
in Hong Kong in 2003. A numerical method called the Dormand-Prince Pairs
method has been used as system solver for the non-linear differential model
SEIJTR, and the Levenberg-Marquardt technique has been used as the least
squares optimiser for determining the best fit to the field data. MATLAB has
been used for all the calculations. A large number of simulations and
demographic information based on the city of Hong Kong have been used to
estimate the parameters. Different graphical and numerical methods have been
used to verify the estimation. Autocorrelations of the residuals are within the
confidence interval, as shown in Fig. 5.4. The model considered here thus
captures the dynamical behavior of the SARS epidemic field data [121].
After estimating the required parameters in Chapter 5, a SEIJTR model is used
to obtain numerical solutions in Chapter 6, with the inclusion of diffusion and
treatment in the system. Three different initial population distributions have
been chosen to examine their effects on transmission of the disease. The models
178
under investigation have two possible equilibria, disease-free and endemic. The
reproduction numbers RIT for the various cases considered (RIT > 1 ) are given
in Table 6.8. A study of bifurcation values of the transmission coefficient, β and
rate of progression from infective to diagnosed, α, as shown in Table 6.5, indicates
that the system remains stable for higher values of β and α with diffusion than
without diffusion. It is observed from the values given in Table 6.8 that β and α,
have significant impact on the basic reproductive number RIT . If the value of β is
decreased, as in Case 2, the reproduction number RIT decreases significantly even
though it is still greater than 1. This causes slow transmission of infection and
thus higher peak values in the infected compartment, from t = 10 to t = 20 days
as shown in Tables 6.6 and 6.7. A decrease in the value of α, as in Case 3, causes
a significant increase in the value of the basic reproduction number RIT . Thus
there is an increase in transmission of the infection under all initial population
distributions considered, as shown in Tables 6.6 and 6.7. There are also slightly
lower peak values of infected population proportion at t= 20 days, as compared to
the original situation depicted in Case 1. An increase in the value of α, as given
in Case 5, causes a slight decrease in the value of the basic reproduction number.
Finally, quarantine of the exposed population is introduced in Chapter 7. A
“quarantined” compartment is added to SEIJTR model from Chapter 6, giving
a SEQIJTR model which is used to simulate transmission of the SARS
epidemic with and without diffusion in Hong Kong in 2003. The numerical study
of this model shows the effects of quarantine on the disease dynamics. At the
time of the SARS epidemic, the two most influential non-medical interventions
were isolation of the infected and quarantine of the exposed. Hence it is
important to study the effect of quarantine, especially when medical interventions
for disease like treatment and vaccination, are not very effective. Two different
initial population distributions have been used to study numerically the effects on
transmission of the disease. Expressions for the seventh order Routh-Hurwitz
criteria are calculated and then used to check the stability of two possible
equilibria for the model, namely disease-free and endemic. The reproduction
179
number RQIT is calculated with and without diffusion. For the case without
diffusion the generation matrix method is used. For the case with diffusion, the
Routh-Hurwitz stability conditions and the variational matrix are used to
calculate an expression for the reproductive number RdQIT . In the disease-free
equilibrium (DFE) the stability conditions are satisfied for RQIT < 1. For the
endemic equilibrium, RQIT and RdQIT are calculated. The numerical value of the
reproduction number without diffusion, RQIT is less than that of the
reproduction number RdQIT with diffusion, as given in Table 7.4. This shows the
negative effect of diffusion on the system, although it is assumed that there is no
diffusion in the quarantined compartment. The value of the reproduction number
RIT is similar to the value of RdQIT . This shows that the absence of quarantine
measures or presence of diffusion both increase infections as compared to the
system with quarantine measures and without diffusion. The bifurcation values of
influential parameters are calculated with and without diffusion in the system,
and are given in Table 7.5. In the absence of diffusion, the bifurcation value of β
increases negligibly with the introduction of a quarantine compartment to the
model, showing that system SEQIJTR is stable for large value of transmission
coefficient, β, than the SEIJTR model studied in Chapter 6. On the other hand,
in the presence of diffusion the SEIJTR system remains stable for larger values
of β than the SEQIJTR system, thus showing the negative effect of diffusion on
quarantine measures.
In summary it can be concluded that
• Initial population distribution plays a crucial role in the spread of disease.
• Diffusion is significant in reducing the intensity of disease.
• Increased recovery of the infectives through intervention is effective in
reducing the spread of disease during the initial days of the onset of disease.
• Increased recovery of the diagnosed is more effective in reducing spread of
disease during the last days of the disease.
180
• Introduction of positive and negative cross-diffusion to the system leads to,
respectively increase and decrease in the domains of the susceptible and
exposed proportions. This in turn affects the domains in other
compartments of the system. Thus, positive cross-diffusion in the system
expands and negative cross-diffusion restricts the spread of disease.
• Introduction of negative cross-diffusion to the system lowers the
reproduction number significantly. This indicates lower intensity of the
infection, thus making the transmission of disease slower.
• Implementation of quarantine measures, along with isolation and treatment,
affects disease dynamics by slowing disease transmission and thus reducing
the proportion infected.
• System without diffusion shows greater recovery with quarantine measures
than system with diffusion.
• Quarantine is most effective during the initial days of the onset of disease.
• Even though treatment and isolation are necessary for recovery, with the
addition of quarantine earlier recovery is possible in diseases like SARS
which occur for short periods.
Recommendations for possible future work related to the work done this thesis:
• The self-diffusion model can be further studied on the basis of age and
gender, as medical findings show SARS is sensitive to age and gender where
the diffusivity terms are functions of age and gender.
• Develop a model for SARS vaccination, so that the public health sector can
measure vaccines efficacy before the implementation of vaccines, should
they become available in the future.
• Study cross-diffusion among susceptible, exposed and infected in the model
to see the effects on transmission of disease. Also the effects of self and
cross-diffusion can be studied for a vaccination model.
181
• The models can be extended by introducing patches or edges in the spatial
domain, in order to study the effects on the spread of disease of restricted
movement of infected from one region to another within a boundary.
182
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Appendix
Appendix A.3
Bifurcation value and diagram for transmission
coefficient β for Case(1)
The Routh-Hurwitz criterion for stability gives:
p1 = 1.00317− 0.0352016β = f1(β),
p5 = 3.68074 ∗ 10−11 + 4.29617 ∗ 10−11β = f2(β),
p1p2 − p3 = 0.287979− 0.110303β + 0.00496585β2 = f3(β),
p1p2p3 + p1p5 − (p23 + p21p4) =
0.00917341− 0.0157348β + 0.00483904β2 − 0.00021073β3 = f4(β),
(p1p4 − p5)(p1p2p3 − p23 − p21p4) + p21p4p5 − (p5(p1p2 − p3)2 + p1p
25) =
1.99247 ∗ 10−8 − 3.65302 ∗ 10−8β +1.45989 ∗ 10−8β2 − 1.79598 ∗ 10−9β3 +8.44811 ∗
10−11β4 − 1.32948 ∗ 10−12β5 = f5(β).
f5(β) = 0, gives the bifurcation value of β for which the point of equilibrium
moves from stable to unstable equilibrium. For f5(β) = 0, β = 0.750435. Other
values of β either being large numbers or negative, are neglected. Thus for any
value greater than β = 0.750435 as shown in Fig. 1 for case 1, the point of
equilibrium will be unstable. Similarly the bifurcation value of β is calculated for
other cases with and without diffusion. Figs. 1 and 2, shows the bifurcation
diagrams for β with and without diffusion.
Bifurcation value and diagram for recovery rate
γ1 Case(1)
The Routh-Hurwitz criterion for stability gives: p1 = 0.851769 + γ1 = f1(γ1),
i
p5 = 4.10822 ∗ 10−11 + 2.23572 ∗ 10−10γ1 = f2(γ1),
p1p2 − p3 = 0.134729 + 0.522435γ1 + 0.512735γ21 = f3(γ1),
p1p2p3 + p1p5 − (p23 + p21p4) =
−0.00106088 + 0.00440029γ1 + 0.0289846γ21 + 0.0324131γ3
1 = f4(γ1),
(p1p4 − p5)(p1p2p3 − p23 − p21p4) + p21p4p5 − (p5(p1p2 − p3)2 + p1p
25) =
−8.50633 ∗ 10−10 − 5.36645 ∗ 10−9γ1 + 5.07547 ∗ 10−8γ21 + 3.06905 ∗ 10−7γ3
1 +
5.24123 ∗ 10−7γ41 + 2.82745 ∗ 10−7γ5
1 = f5(γ1).
f5(γ1) = 0, gives the bifurcation value of γ1 for which the point of equilibrium
moves from stable to unstable equilibrium. For f5(γ1) = 0, γ1 = 0.124708. Other
values of γ1 either being small numbers or negative, are neglected. Thus for any
value less than γ1 = 0.124708 as shown in Fig. 3, for case 1, the point of
equilibrium will be unstable. Similarly the bifurcation value of γ1 is calculated for
other cases with and without diffusion. Figs. 3 and 4, shows the bifurcation
diagrams for γ1 with and without diffusion.
Bifurcation value and diagram for recovery
coefficients γ2
The bifurcation values of γ2 are also calculated in same way for all cases as for γ1.
The bifurcation diagrams for γ2 with and without diffusion are given in Figs. 5
and 6 respectively.
The coefficient of characteristic equation 3.29
p1 = −a11 − a22 − a33 − a44 − a55;
p2 =
−a12a21+a11a22−a13a31−a23a32+a11a33+a22a33−a14a41−a24a42−a34a43+a11a44+
a22a44+a33a44−a15a51−a25a52−a35a53−a45a54+a11a55+a22a55+a33a55+a44a55;
p3 = a13a22a31 − a12a23a31 − a13a21a32 + a11a23a32 + a12a21a33 − a11a22a33 +
a14a22a41 − a12a24a41 + a14a33a41 − a13a34a41 − a14a21a42 + a11a24a42 + a24a33a42 −
a23a34a42 − a14a31a43 − a24a32a43 + a11a34a43 + a22a34a43 + a12a21a44 − a11a22a44 +
ii
a13a31a44 + a23a32a44 − a11a33a44 − a22a33a44 + a15a22a51 − a12a25a51 + a15a33a51 −
a13a35a51 + a15a44a51− a14a45a51 − a15a21a52 + a11a25a52 + a25a33a52 − a23a35a52 +
a25a44a52 − a24a45a52 − a15a31a53 − a25a32a53 + a11a35a53 + a22a35a53 + a35a44a53 −
a34a45a53 − a15a41a54 − a25a42a54 − a35a43a54 + a11a45a54 + a22a45a54 + a33a45a54 +
a12a21a55 − a11a22a55 + a13a31a55 + a23a32a55 − a11a33a55 − a22a33a55 + a14a41a55 +
a24a42a55 + a34a43a55 − a11a44a55 − a22a44a55 − a33a44a55;
p4 = a14a23a32a41 − a13a24a32a41 − a14a22a33a41 + a12a24a33a41 + a13a22a34a41 −
a12a23a34a41 − a14a23a31a42 + a13a24a31a42 + a14a21a33a42 − a11a24a33a42 −
a13a21a34a42+a11a23a34a42+a14a22a31a43−a12a24a31a43−a14a21a32a43+a11a24a32a43+
a12a21a34a43 − a11a22a34a43 − a13a22a31a44 + a12a23a31a44 + a13a21a32a44 −
a11a23a32a44 − a12a21a33a44 + a11a22a33a44 + a15a23a32a51 − a13a25a32a51 −
a15a22a33a51+a12a25a33a51+a13a22a35a51−a12a23a35a51+a15a24a42a51−a14a25a42a51+
a15a34a43a51−a14a35a43a51−a15a22a44a51+a12a25a44a51−a15a33a44a51+a13a35a44a51+
a14a22a45a51 − a12a24a45a51 + a14a33a45a51 − a13a34a45a51 − a15a23a31a52 +
a13a25a31a52+a15a21a33a52−a11a25a33a52−a13a21a35a52+a11a23a35a52−a15a24a41a52+
a14a25a41a52+a25a34a43a52−a24a35a43a52+a15a21a44a52−a11a25a44a52−a25a33a44a52+
a23a35a44a52−a14a21a45a52+a11a24a45a52+a24a33a45a52−a23a34a45a52+a15a22a31a53−
a12a25a31a53−a15a21a32a53+a11a25a32a53+a12a21a35a53−a11a22a35a53−a15a34a41a53+
a14a35a41a53−a25a34a42a53+a24a35a42a53+a15a31a44a53+a25a32a44a53−a11a35a44a53−
a22a35a44a53−a14a31a45a53−a24a32a45a53+a11a34a45a53+a22a34a45a53+a15a22a41a54−
a12a25a41a54+a15a33a41a54−a13a35a41a54−a15a21a42a54+a11a25a42a54+a25a33a42a54−
a23a35a42a54−a15a31a43a54−a25a32a43a54+a11a35a43a54+a22a35a43a54+a12a21a45a54−
a11a22a45a54 + a13a31a45a54 + a23a32a45a54 − a11a33a45a54 − a22a33a45a54 −
a13a22a31a55+a12a23a31a55+a13a21a32a55−a11a23a32a55−a12a21a33a55+a11a22a33a55−
a14a22a41a55+a12a24a41a55−a14a33a41a55+a13a34a41a55+a14a21a42a55−a11a24a42a55−
a24a33a42a55+a23a34a42a55+a14a31a43a55+a24a32a43a55−a11a34a43a55−a22a34a43a55−
a12a21a44a55+a11a22a44a55−a13a31a44a55−a23a32a44a55+a11a33a44a55+a22a33a44a55;
p5 = −a15a24a33a42a51 + a14a25a33a42a51 + a15a23a34a42a51 − a13a25a34a42a51 −
a14a23a35a42a51 + a13a24a35a42a51 + a15a24a32a43a51 − a14a25a32a43a51 −
a15a22a34a43a51 + a12a25a34a43a51 + a14a22a35a43a51 − a12a24a35a43a51 −
a15a23a32a44a51 + a13a25a32a44a51 + a15a22a33a44a51 − a12a25a33a44a51 −
iii
a13a22a35a44a51 + a12a23a35a44a51 + a14a23a32a45a51 − a13a24a32a45a51 −
a14a22a33a45a51 + a12a24a33a45a51 + a13a22a34a45a51 − a12a23a34a45a51 +
a15a24a33a41a52 − a14a25a33a41a52 − a15a23a34a41a52 + a13a25a34a41a52 +
a14a23a35a41a52 − a13a24a35a41a52 − a15a24a31a43a52 + a14a25a31a43a52 +
a15a21a34a43a52 − a11a25a34a43a52 − a14a21a35a43a52 + a11a24a35a43a52 +
a15a23a31a44a52 − a13a25a31a44a52 − a15a21a33a44a52 + a11a25a33a44a52 +
a13a21a35a44a52 − a11a23a35a44a52 − a14a23a31a45a52 + a13a24a31a45a52 +
a14a21a33a45a52 − a11a24a33a45a52 − a13a21a34a45a52 + a11a23a34a45a52 −
a15a24a32a41a53 + a14a25a32a41a53 + a15a22a34a41a53 − a12a25a34a41a53 −
a14a22a35a41a53 + a12a24a35a41a53 + a15a24a31a42a53 − a14a25a31a42a53 −
a15a21a34a42a53 + a11a25a34a42a53 + a14a21a35a42a53 − a11a24a35a42a53 −
a15a22a31a44a53 + a12a25a31a44a53 + a15a21a32a44a53 − a11a25a32a44a53 −
a12a21a35a44a53 + a11a22a35a44a53 + a14a22a31a45a53 − a12a24a31a45a53 −
a14a21a32a45a53 + a11a24a32a45a53 + a12a21a34a45a53 − a11a22a34a45a53 +
a15a23a32a41a54 − a13a25a32a41a54 − a15a22a33a41a54 + a12a25a33a41a54 +
a13a22a35a41a54 − a12a23a35a41a54 − a15a23a31a42a54 + a13a25a31a42a54 +
a15a21a33a42a54 − a11a25a33a42a54 − a13a21a35a42a54 + a11a23a35a42a54 +
a15a22a31a43a54 − a12a25a31a43a54 − a15a21a32a43a54 + a11a25a32a43a54 +
a12a21a35a43a54 − a11a22a35a43a54 − a13a22a31a45a54 + a12a23a31a45a54 +
a13a21a32a45a54 − a11a23a32a45a54 − a12a21a33a45a54 + a11a22a33a45a54 −
a14a23a32a41a55 + a13a24a32a41a55 + a14a22a33a41a55 − a12a24a33a41a55 −
a13a22a34a41a55 + a12a23a34a41a55 + a14a23a31a42a55 − a13a24a31a42a55 −
a14a21a33a42a55 + a11a24a33a42a55 + a13a21a34a42a55 − a11a23a34a42a55 −
a14a22a31a43a55 + a12a24a31a43a55 + a14a21a32a43a55 − a11a24a32a43a55 −
a12a21a34a43a55 + a11a22a34a43a55 + a13a22a31a44a55 − a12a23a31a44a55 −
a13a21a32a44a55 + a11a23a32a44a55 + a12a21a33a44a55 − a11a22a33a44a55.
The coefficient of characteristic equation 3.45
q1 = −a11 − a22 − a33 − a44 − a55 + d1k2 + d2k
2 + d3k2 + d4k
2 + d5k2;
q2 = −a12a21 + a11a22 − a13a31 − a23a32 + a11a33 + a22a33 − a14a41 − a24a42 −
iv
a34a43 + a11a44 + a22a44 + a33a44 − a15a51 − a25a52 − a35a53 − a45a54 + a11a55 +
a22a55 + a33a55 + a44a55 − a22d1k2 − a33d1k
2 − a44d1k2 − a55d1k
2 − a11d2k2 −
a33d2k2 − a44d2k
2 − a55d2k2 − a11d3k
2 − a22d3k2 − a44d3k
2 − a55d3k2 − a11d4k
2 −
a22d4k2 − a33d4k
2 − a55d4k2 − a11d5k
2 − a22d5k2 − a33d5k
2 − a44d5k2 + d1d2k
4 +
d1d3k4 + d2d3k
4 + d1d4k4 + d2d4k
4 + d3d4k4 + d1d5k
4 + d2d5k4 + d3d5k
4 + d4d5k4;
q3 = a13a22a31−a12a23a31−a13a21a32+a11a23a32+a12a21a33−a11a22a33+a14a22a41−
a12a24a41 + a14a33a41 − a13a34a41 − a14a21a42 + a11a24a42 + a24a33a42 − a23a34a42 −
a14a31a43 − a24a32a43 + a11a34a43 + a22a34a43 + a12a21a44 − a11a22a44 + a13a31a44 +
a23a32a44 − a11a33a44 − a22a33a44 + a15a22a51 − a12a25a51 + a15a33a51 − a13a35a51 +
a15a44a51 − a14a45a51 − a15a21a52 + a11a25a52 + a25a33a52 − a23a35a52 + a25a44a52 −
a24a45a52 − a15a31a53 − a25a32a53 + a11a35a53 + a22a35a53 + a35a44a53 − a34a45a53 −
a15a41a54 − a25a42a54 − a35a43a54 + a11a45a54 + a22a45a54 + a33a45a54 + a12a21a55 −
a11a22a55 + a13a31a55 + a23a32a55 − a11a33a55 − a22a33a55 + a14a41a55 + a24a42a55 +
a34a43a55−a11a44a55−a22a44a55−a33a44a55−a23a32d1k2+a22a33d1k
2−a24a42d1k2−
a34a43d1k2 + a22a44d1k
2 + a33a44d1k2 − a25a52d1k
2 − a35a53d1k2 − a45a54d1k
2 +
a22a55d1k2 + a33a55d1k
2 + a44a55d1k2 − a13a31d2k
2 + a11a33d2k2 − a14a41d2k
2 −
a34a43d2k2 + a11a44d2k
2 + a33a44d2k2 − a15a51d2k
2 − a35a53d2k2 − a45a54d2k
2 +
a11a55d2k2 + a33a55d2k
2 + a44a55d2k2 − a12a21d3k
2 + a11a22d3k2 − a14a41d3k
2 −
a24a42d3k2 + a11a44d3k
2 + a22a44d3k2 − a15a51d3k
2 − a25a52d3k2 − a45a54d3k
2 +
a11a55d3k2 + a22a55d3k
2 + a44a55d3k2 − a12a21d4k
2 + a11a22d4k2 − a13a31d4k
2 −
a23a32d4k2 + a11a33d4k
2 + a22a33d4k2 − a15a51d4k
2 − a25a52d4k2 − a35a53d4k
2 +
a11a55d4k2 + a22a55d4k
2 + a33a55d4k2 − a12a21d5k
2 + a11a22d5k2 − a13a31d5k
2 −
a23a32d5k2 + a11a33d5k
2 + a22a33d5k2 − a14a41d5k
2 − a24a42d5k2 − a34a43d5k
2 +
a11a44d5k2+a22a44d5k
2+a33a44d5k2−a33d1d2k
4−a44d1d2k4−a55d1d2k
4−a22d1d3k4−
a44d1d3k4−a55d1d3k
4−a11d2d3k4−a44d2d3k
4−a55d2d3k4−a22d1d4k
4−a33d1d4k4−
a55d1d4k4−a11d2d4k
4−a33d2d4k4−a55d2d4k
4−a11d3d4k4−a22d3d4k
4−a55d3d4k4−
a22d1d5k4−a33d1d5k
4−a44d1d5k4−a11d2d5k
4−a33d2d5k4−a44d2d5k
4−a11d3d5k4−
a22d3d5k4 − a44d3d5k
4 − a11d4d5k4 − a22d4d5k
4 − a33d4d5k4 + d1d2d3k
6 + d1d2d4k6 +
d1d3d4k6+d2d3d4k
6+d1d2d5k6+d1d3d5k
6+d2d3d5k6+d1d4d5k
6+d2d4d5k6+d3d4d5k
6;
q4 = a14a23a32a41 − a13a24a32a41 − a14a22a33a41 + a12a24a33a41 + a13a22a34a41 −
a12a23a34a41−a14a23a31a42+a13a24a31a42+a14a21a33a42−a11a24a33a42−a13a21a34a42+
v
a11a23a34a42+a14a22a31a43−a12a24a31a43−a14a21a32a43+a11a24a32a43+a12a21a34a43−
a11a22a34a43−a13a22a31a44+a12a23a31a44+a13a21a32a44−a11a23a32a44−a12a21a33a44+
a11a22a33a44+a15a23a32a51−a13a25a32a51−a15a22a33a51+a12a25a33a51+a13a22a35a51−
a12a23a35a51+a15a24a42a51−a14a25a42a51+a15a34a43a51−a14a35a43a51−a15a22a44a51+
a12a25a44a51−a15a33a44a51+a13a35a44a51+a14a22a45a51−a12a24a45a51+a14a33a45a51−
a13a34a45a51−a15a23a31a52+a13a25a31a52+a15a21a33a52−a11a25a33a52−a13a21a35a52+
a11a23a35a52−a15a24a41a52+a14a25a41a52+a25a34a43a52−a24a35a43a52+a15a21a44a52−
a11a25a44a52−a25a33a44a52+a23a35a44a52−a14a21a45a52+a11a24a45a52+a24a33a45a52−
a23a34a45a52+a15a22a31a53−a12a25a31a53−a15a21a32a53+a11a25a32a53+a12a21a35a53−
a11a22a35a53−a15a34a41a53+a14a35a41a53−a25a34a42a53+a24a35a42a53+a15a31a44a53+
a25a32a44a53 − a11a35a44a53 − a22a35a44a53 − a14a31a45a53 − a24a32a45a53 +
a11a34a45a53+a22a34a45a53+a15a22a41a54−a12a25a41a54+a15a33a41a54−a13a35a41a54−
a15a21a42a54+a11a25a42a54+a25a33a42a54−a23a35a42a54−a15a31a43a54−a25a32a43a54+
a11a35a43a54+a22a35a43a54+a12a21a45a54−a11a22a45a54+a13a31a45a54+a23a32a45a54−
a11a33a45a54 − a22a33a45a54 − a13a22a31a55 + a12a23a31a55 + a13a21a32a55 −
a11a23a32a55−a12a21a33a55+a11a22a33a55−a14a22a41a55+a12a24a41a55−a14a33a41a55+
a13a34a41a55+a14a21a42a55−a11a24a42a55−a24a33a42a55+a23a34a42a55+a14a31a43a55+
a24a32a43a55 − a11a34a43a55 − a22a34a43a55 − a12a21a44a55 + a11a22a44a55 −
a13a31a44a55 − a23a32a44a55 + a11a33a44a55 + a22a33a44a55 + a24a33a42d1k2 −
a23a34a42d1k2 − a24a32a43d1k
2 + a22a34a43d1k2 + a23a32a44d1k
2 − a22a33a44d1k2 +
a25a33a52d1k2 − a23a35a52d1k
2 + a25a44a52d1k2 − a24a45a52d1k
2 − a25a32a53d1k2 +
a22a35a53d1k2 + a35a44a53d1k
2 − a34a45a53d1k2 − a25a42a54d1k
2 − a35a43a54d1k2 +
a22a45a54d1k2 + a33a45a54d1k
2 + a23a32a55d1k2 − a22a33a55d1k
2 + a24a42a55d1k2 +
a34a43a55d1k2 − a22a44a55d1k
2 − a33a44a55d1k2 + a14a33a41d2k
2 − a13a34a41d2k2 −
a14a31a43d2k2 + a11a34a43d2k
2 + a13a31a44d2k2 − a11a33a44d2k
2 + a15a33a51d2k2 −
a13a35a51d2k2 + a15a44a51d2k
2 − a14a45a51d2k2 − a15a31a53d2k
2 + a11a35a53d2k2 +
a35a44a53d2k2 − a34a45a53d2k
2 − a15a41a54d2k2 − a35a43a54d2k
2 + a11a45a54d2k2 +
a33a45a54d2k2 + a13a31a55d2k
2 − a11a33a55d2k2 + a14a41a55d2k
2 + a34a43a55d2k2 −
a11a44a55d2k2 − a33a44a55d2k
2 + a14a22a41d3k2 − a12a24a41d3k
2 − a14a21a42d3k2 +
a11a24a42d3k2 + a12a21a44d3k
2 − a11a22a44d3k2 + a15a22a51d3k
2 − a12a25a51d3k2 +
a15a44a51d3k2 − a14a45a51d3k
2 − a15a21a52d3k2 + a11a25a52d3k
2 + a25a44a52d3k2 −
vi
a24a45a52d3k2 − a15a41a54d3k
2 − a25a42a54d3k2 + a11a45a54d3k
2 + a22a45a54d3k2 +
a12a21a55d3k2 − a11a22a55d3k
2 + a14a41a55d3k2 + a24a42a55d3k
2 − a11a44a55d3k2 −
a22a44a55d3k2 + a13a22a31d4k
2 − a12a23a31d4k2 − a13a21a32d4k
2 + a11a23a32d4k2 +
a12a21a33d4k2 − a11a22a33d4k
2 + a15a22a51d4k2 − a12a25a51d4k
2 + a15a33a51d4k2 −
a13a35a51d4k2 − a15a21a52d4k
2 + a11a25a52d4k2 + a25a33a52d4k
2 − a23a35a52d4k2 −
a15a31a53d4k2 − a25a32a53d4k
2 + a11a35a53d4k2 + a22a35a53d4k
2 + a12a21a55d4k2 −
a11a22a55d4k2 + a13a31a55d4k
2 + a23a32a55d4k2 − a11a33a55d4k
2 − a22a33a55d4k2 +
a13a22a31d5k2 − a12a23a31d5k
2 − a13a21a32d5k2 + a11a23a32d5k
2 + a12a21a33d5k2 −
a11a22a33d5k2 + a14a22a41d5k
2 − a12a24a41d5k2 + a14a33a41d5k
2 − a13a34a41d5k2 −
a14a21a42d5k2 + a11a24a42d5k
2 + a24a33a42d5k2 − a23a34a42d5k
2 − a14a31a43d5k2 −
a24a32a43d5k2 + a11a34a43d5k
2 + a22a34a43d5k2 + a12a21a44d5k
2 − a11a22a44d5k2 +
a13a31a44d5k2 + a23a32a44d5k
2 − a11a33a44d5k2 − a22a33a44d5k
2 − a34a43d1d2k4 +
a33a44d1d2k4 − a35a53d1d2k
4 − a45a54d1d2k4 + a33a55d1d2k
4 + a44a55d1d2k4 −
a24a42d1d3k4 + a22a44d1d3k
4 − a25a52d1d3k4 − a45a54d1d3k
4 + a22a55d1d3k4 +
a44a55d1d3k4 − a14a41d2d3k
4 + a11a44d2d3k4 − a15a51d2d3k
4 − a45a54d2d3k4 +
a11a55d2d3k4 + a44a55d2d3k
4 − a23a32d1d4k4 + a22a33d1d4k
4 − a25a52d1d4k4 −
a35a53d1d4k4 + a22a55d1d4k
4 + a33a55d1d4k4 − a13a31d2d4k
4 + a11a33d2d4k4 −
a15a51d2d4k4 − a35a53d2d4k
4 + a11a55d2d4k4 + a33a55d2d4k
4 − a12a21d3d4k4 +
a11a22d3d4k4 − a15a51d3d4k
4 − a25a52d3d4k4 + a11a55d3d4k
4 + a22a55d3d4k4 −
a23a32d1d5k4 + a22a33d1d5k
4 − a24a42d1d5k4 − a34a43d1d5k
4 + a22a44d1d5k4 +
a33a44d1d5k4 − a13a31d2d5k
4 + a11a33d2d5k4 − a14a41d2d5k
4 − a34a43d2d5k4 +
a11a44d2d5k4 + a33a44d2d5k
4 − a12a21d3d5k4 + a11a22d3d5k
4 − a14a41d3d5k4 −
a24a42d3d5k4 + a11a44d3d5k
4 + a22a44d3d5k4 − a12a21d4d5k
4 + a11a22d4d5k4 −
a13a31d4d5k4 − a23a32d4d5k
4 + a11a33d4d5k4 + a22a33d4d5k
4 − a44d1d2d3k6 −
a55d1d2d3k6−a33d1d2d4k
6−a55d1d2d4k6−a22d1d3d4k
6−a55d1d3d4k6−a11d2d3d4k
6−
a55d2d3d4k6−a33d1d2d5k
6−a44d1d2d5k6−a22d1d3d5k
6−a44d1d3d5k6−a11d2d3d5k
6−
a44d2d3d5k6−a22d1d4d5k
6−a33d1d4d5k6−a11d2d4d5k
6−a33d2d4d5k6−a11d3d4d5k
6−
a22d3d4d5k6 + d1d2d3d4k
8 + d1d2d3d5k8 + d1d2d4d5k
8 + d1d3d4d5k8 + d2d3d4d5k
8;
q5 = −a15a24a33a42a51 + a14a25a33a42a51 + a15a23a34a42a51 − a13a25a34a42a51 −
a14a23a35a42a51 + a13a24a35a42a51 + a15a24a32a43a51 − a14a25a32a43a51 −
a15a22a34a43a51 + a12a25a34a43a51 + a14a22a35a43a51 − a12a24a35a43a51 −
vii
a15a23a32a44a51 + a13a25a32a44a51 + a15a22a33a44a51 − a12a25a33a44a51 −
a13a22a35a44a51 + a12a23a35a44a51 + a14a23a32a45a51 − a13a24a32a45a51 −
a14a22a33a45a51 + a12a24a33a45a51 + a13a22a34a45a51 − a12a23a34a45a51 +
a15a24a33a41a52 − a14a25a33a41a52 − a15a23a34a41a52 + a13a25a34a41a52 +
a14a23a35a41a52 − a13a24a35a41a52 − a15a24a31a43a52 + a14a25a31a43a52 +
a15a21a34a43a52 − a11a25a34a43a52 − a14a21a35a43a52 + a11a24a35a43a52 +
a15a23a31a44a52 − a13a25a31a44a52 − a15a21a33a44a52 + a11a25a33a44a52 +
a13a21a35a44a52 − a11a23a35a44a52 − a14a23a31a45a52 + a13a24a31a45a52 +
a14a21a33a45a52 − a11a24a33a45a52 − a13a21a34a45a52 + a11a23a34a45a52 −
a15a24a32a41a53 + a14a25a32a41a53 + a15a22a34a41a53 − a12a25a34a41a53 −
a14a22a35a41a53 + a12a24a35a41a53 + a15a24a31a42a53 − a14a25a31a42a53 −
a15a21a34a42a53 + a11a25a34a42a53 + a14a21a35a42a53 − a11a24a35a42a53 −
a15a22a31a44a53 + a12a25a31a44a53 + a15a21a32a44a53 − a11a25a32a44a53 −
a12a21a35a44a53 + a11a22a35a44a53 + a14a22a31a45a53 − a12a24a31a45a53 −
a14a21a32a45a53 + a11a24a32a45a53 + a12a21a34a45a53 − a11a22a34a45a53 +
a15a23a32a41a54 − a13a25a32a41a54 − a15a22a33a41a54 + a12a25a33a41a54 +
a13a22a35a41a54 − a12a23a35a41a54 − a15a23a31a42a54 + a13a25a31a42a54 +
a15a21a33a42a54 − a11a25a33a42a54 − a13a21a35a42a54 + a11a23a35a42a54 +
a15a22a31a43a54 − a12a25a31a43a54 − a15a21a32a43a54 + a11a25a32a43a54 +
a12a21a35a43a54 − a11a22a35a43a54 − a13a22a31a45a54 + a12a23a31a45a54 +
a13a21a32a45a54 − a11a23a32a45a54 − a12a21a33a45a54 + a11a22a33a45a54 −
a14a23a32a41a55 + a13a24a32a41a55 + a14a22a33a41a55 − a12a24a33a41a55 −
a13a22a34a41a55 + a12a23a34a41a55 + a14a23a31a42a55 − a13a24a31a42a55 −
a14a21a33a42a55 + a11a24a33a42a55 + a13a21a34a42a55 − a11a23a34a42a55 −
a14a22a31a43a55 + a12a24a31a43a55 + a14a21a32a43a55 − a11a24a32a43a55 −
a12a21a34a43a55 + a11a22a34a43a55 + a13a22a31a44a55 − a12a23a31a44a55 −
a13a21a32a44a55 + a11a23a32a44a55 + a12a21a33a44a55 − a11a22a33a44a55 +
a25a34a43a52d1k2 − a24a35a43a52d1k
2 − a25a33a44a52d1k2 + a23a35a44a52d1k
2 +
a24a33a45a52d1k2 − a23a34a45a52d1k
2 − a25a34a42a53d1k2 + a24a35a42a53d1k
2 +
a25a32a44a53d1k2 − a22a35a44a53d1k
2 − a24a32a45a53d1k2 + a22a34a45a53d1k
2 +
a25a33a42a54d1k2 − a23a35a42a54d1k
2 − a25a32a43a54d1k2 + a22a35a43a54d1k
2 +
viii
a23a32a45a54d1k2 − a22a33a45a54d1k
2 − a24a33a42a55d1k2 + a23a34a42a55d1k
2 +
a24a32a43a55d1k2 − a22a34a43a55d1k
2 − a23a32a44a55d1k2 + a22a33a44a55d1k
2 +
a15a34a43a51d2k2 − a14a35a43a51d2k
2 − a15a33a44a51d2k2 + a13a35a44a51d2k
2 +
a14a33a45a51d2k2 − a13a34a45a51d2k
2 − a15a34a41a53d2k2 + a14a35a41a53d2k
2 +
a15a31a44a53d2k2 − a11a35a44a53d2k
2 − a14a31a45a53d2k2 + a11a34a45a53d2k
2 +
a15a33a41a54d2k2 − a13a35a41a54d2k
2 − a15a31a43a54d2k2 + a11a35a43a54d2k
2 +
a13a31a45a54d2k2 − a11a33a45a54d2k
2 − a14a33a41a55d2k2 + a13a34a41a55d2k
2 +
a14a31a43a55d2k2 − a11a34a43a55d2k
2 − a13a31a44a55d2k2 + a11a33a44a55d2k
2 +
a15a24a42a51d3k2 − a14a25a42a51d3k
2 − a15a22a44a51d3k2 + a12a25a44a51d3k
2 +
a14a22a45a51d3k2 − a12a24a45a51d3k
2 − a15a24a41a52d3k2 + a14a25a41a52d3k
2 +
a15a21a44a52d3k2 − a11a25a44a52d3k
2 − a14a21a45a52d3k2 + a11a24a45a52d3k
2 +
a15a22a41a54d3k2 − a12a25a41a54d3k
2 − a15a21a42a54d3k2 + a11a25a42a54d3k
2 +
a12a21a45a54d3k2 − a11a22a45a54d3k
2 − a14a22a41a55d3k2 + a12a24a41a55d3k
2 +
a14a21a42a55d3k2 − a11a24a42a55d3k
2 − a12a21a44a55d3k2 + a11a22a44a55d3k
2 +
a15a23a32a51d4k2 − a13a25a32a51d4k
2 − a15a22a33a51d4k2 + a12a25a33a51d4k
2 +
a13a22a35a51d4k2 − a12a23a35a51d4k
2 − a15a23a31a52d4k2 + a13a25a31a52d4k
2 +
a15a21a33a52d4k2 − a11a25a33a52d4k
2 − a13a21a35a52d4k2 + a11a23a35a52d4k
2 +
a15a22a31a53d4k2 − a12a25a31a53d4k
2 − a15a21a32a53d4k2 + a11a25a32a53d4k
2 +
a12a21a35a53d4k2 − a11a22a35a53d4k
2 − a13a22a31a55d4k2 + a12a23a31a55d4k
2 +
a13a21a32a55d4k2 − a11a23a32a55d4k
2 − a12a21a33a55d4k2 + a11a22a33a55d4k
2 +
a14a23a32a41d5k2 − a13a24a32a41d5k
2 − a14a22a33a41d5k2 + a12a24a33a41d5k
2 +
a13a22a34a41d5k2 − a12a23a34a41d5k
2 − a14a23a31a42d5k2 + a13a24a31a42d5k
2 +
a14a21a33a42d5k2 − a11a24a33a42d5k
2 − a13a21a34a42d5k2 + a11a23a34a42d5k
2 +
a14a22a31a43d5k2 − a12a24a31a43d5k
2 − a14a21a32a43d5k2 + a11a24a32a43d5k
2 +
a12a21a34a43d5k2 − a11a22a34a43d5k
2 − a13a22a31a44d5k2 + a12a23a31a44d5k
2 +
a13a21a32a44d5k2 − a11a23a32a44d5k
2 − a12a21a33a44d5k2 + a11a22a33a44d5k
2 +
a35a44a53d1d2k4 − a34a45a53d1d2k
4 − a35a43a54d1d2k4 + a33a45a54d1d2k
4 +
a34a43a55d1d2k4 − a33a44a55d1d2k
4 + a25a44a52d1d3k4 − a24a45a52d1d3k
4 −
a25a42a54d1d3k4 + a22a45a54d1d3k
4 + a24a42a55d1d3k4 − a22a44a55d1d3k
4 +
a15a44a51d2d3k4 − a14a45a51d2d3k
4 − a15a41a54d2d3k4 + a11a45a54d2d3k
4 +
a14a41a55d2d3k4 − a11a44a55d2d3k
4 + a25a33a52d1d4k4 − a23a35a52d1d4k
4 −
ix
0.2 0.4 0.6 0.8 1.0Β
5.´ 10-9
1.´ 10-8
1.5´ 10-8
2.´ 10-8
fHΒL
0.2 0.4 0.6 0.8 1.0Β
1.´ 10-8
2.´ 10-8
3.´ 10-8
fHΒL
0.2 0.4 0.6 0.8 1.0Β
-5.´ 10-9
5.´ 10-9
1.´ 10-8
1.5´ 10-8
2.´ 10-8
2.5´ 10-8
3.´ 10-8
fHΒL
0.2 0.4 0.6 0.8 1.0Β
5.´ 10-9
1.´ 10-8
1.5´ 10-8
2.´ 10-8
fHΒL
Figure 1: Bifurcation diagram for β without diffusion for Case(1)-(4)
a25a32a53d1d4k4 + a22a35a53d1d4k
4 + a23a32a55d1d4k4 − a22a33a55d1d4k
4 +
a15a33a51d2d4k4 − a13a35a51d2d4k
4 − a15a31a53d2d4k4 + a11a35a53d2d4k
4 +
a13a31a55d2d4k4 − a11a33a55d2d4k
4 + a15a22a51d3d4k4 − a12a25a51d3d4k
4 −
a15a21a52d3d4k4 + a11a25a52d3d4k
4 + a12a21a55d3d4k4 − a11a22a55d3d4k
4 +
a24a33a42d1d5k4 − a23a34a42d1d5k
4 − a24a32a43d1d5k4 + a22a34a43d1d5k
4 +
a23a32a44d1d5k4 − a22a33a44d1d5k
4 + a14a33a41d2d5k4 − a13a34a41d2d5k
4 −
a14a31a43d2d5k4 + a11a34a43d2d5k
4 + a13a31a44d2d5k4 − a11a33a44d2d5k
4 +
a14a22a41d3d5k4 − a12a24a41d3d5k
4 − a14a21a42d3d5k4 + a11a24a42d3d5k
4 +
a12a21a44d3d5k4 − a11a22a44d3d5k
4 + a13a22a31d4d5k4 − a12a23a31d4d5k
4 −
a13a21a32d4d5k4 + a11a23a32d4d5k
4 + a12a21a33d4d5k4 − a11a22a33d4d5k
4 −
a45a54d1d2d3k6 + a44a55d1d2d3k
6 − a35a53d1d2d4k6 + a33a55d1d2d4k
6 −
a25a52d1d3d4k6 + a22a55d1d3d4k
6 − a15a51d2d3d4k6 + a11a55d2d3d4k
6 −
a34a43d1d2d5k6 + a33a44d1d2d5k
6 − a24a42d1d3d5k6 + a22a44d1d3d5k
6 −
a14a41d2d3d5k6 + a11a44d2d3d5k
6 − a23a32d1d4d5k6 + a22a33d1d4d5k
6 −
a13a31d2d4d5k6+a11a33d2d4d5k
6−a12a21d3d4d5k6+a11a22d3d4d5k
6−a55d1d2d3d4k8−
a44d1d2d3d5k8 − a33d1d2d4d5k
8 − a22d1d3d4d5k8 − a11d2d3d4d5k
8 + d1d2d3d4d5k10.
x
0.2 0.4 0.6 0.8 1.0Β
0.00001
0.00002
0.00003
0.00004
fHΒL
0.2 0.4 0.6 0.8 1.0Β
0.00002
0.00004
0.00006
0.00008
fHΒL
0.2 0.4 0.6 0.8 1.0Β
0.00001
0.00002
0.00003
0.00004
0.00005
0.00006
fHΒL
0.2 0.4 0.6 0.8 1.0Β
0.00001
0.00002
0.00003
0.00004
fHΒL
Figure 2: Bifurcation diagram for β with diffusion Case(1)-(4)
0.05 0.10 0.15 0.20Γ1
-1.´ 10-9
1.´ 10-9
2.´ 10-9
3.´ 10-9
fHΓ1L
0.05 0.10 0.15 0.20Γ1
-1.´ 10-9
1.´ 10-9
2.´ 10-9
3.´ 10-9
4.´ 10-9
5.´ 10-9
6.´ 10-9
fHΓ1L
0.1 0.2 0.3 0.4Γ1
5.´ 10-9
1.´ 10-8
1.5´ 10-8
2.´ 10-8
2.5´ 10-8
fHΓ1L
0.1 0.2 0.3 0.4Γ1
1.´ 10-8
2.´ 10-8
3.´ 10-8
fHΓ1L
Figure 3: Bifurcation diagram for γ1 without diffusion Case(1)-(4)
xi
0.05 0.10 0.15 0.20Γ2
-2.´ 10-7
2.´ 10-7
4.´ 10-7
6.´ 10-7
8.´ 10-7
fHΓ2L
0.05 0.10 0.15 0.20Γ2
-4.´ 10-7
-3.´ 10-7
-2.´ 10-7
-1.´ 10-7
1.´ 10-7
fHΓ2L
0.05 0.10 0.15 0.20Γ2
-4.´ 10-7
-2.´ 10-7
2.´ 10-7
4.´ 10-7
6.´ 10-7
8.´ 10-7
1.´ 10-6
fHΓ2L
0.05 0.10 0.15 0.20Γ2
-2.´ 10-7
2.´ 10-7
4.´ 10-7
6.´ 10-7
8.´ 10-7
fHΓ2L
Figure 4: Bifurcation diagram for γ1 with diffusion Case(1)-(4)
0.05 0.10 0.15 0.20Γ2
-4.´ 10-10
-2.´ 10-10
2.´ 10-10
fHΓ2L
0.05 0.10 0.15 0.20Γ2
-8.´ 10-10
-6.´ 10-10
-4.´ 10-10
-2.´ 10-10
2.´ 10-10
fHΓ2L
0.05 0.10 0.15 0.20Γ2
-6.´ 10-10
-4.´ 10-10
-2.´ 10-10
2.´ 10-10
4.´ 10-10
fHΓ2L
0.05 0.10 0.15 0.20Γ2
-4.´ 10-10
-2.´ 10-10
2.´ 10-10
fHΓ2L
Figure 5: Bifurcation diagram for γ2 without diffusion Case(1)-(4)
xii
0.05 0.10 0.15 0.20Γ2
-2.´ 10-7
2.´ 10-7
4.´ 10-7
6.´ 10-7
8.´ 10-7
fHΓ2L
0.05 0.10 0.15 0.20Γ2
-4.´ 10-7
-3.´ 10-7
-2.´ 10-7
-1.´ 10-7
1.´ 10-7
fHΓ2L
0.05 0.10 0.15 0.20Γ2
-4.´ 10-7
-2.´ 10-7
2.´ 10-7
4.´ 10-7
6.´ 10-7
8.´ 10-7
1.´ 10-6
fHΓ2L
0.05 0.10 0.15 0.20Γ2
-2.´ 10-7
2.´ 10-7
4.´ 10-7
6.´ 10-7
8.´ 10-7
fHΓ2L
Figure 6: Bifurcation diagram for γ2 with diffusion Case(1)-(4)
Appendix A.4
The coefficient of characteristic equation (4.24)
q1 = −d1k2 − d2k
2 − d3k2 − d4k
2 − α + qβ − γ1 − γ2 − 2δ − κ− 4µ;
q2 = −d1d2k4−d1d3k
4−d2d3k4−d1d4k
4−d2d4k4−d3d4k
4+desdsek4−d1k
2α−d2k2α−
d4k2α+d1k
2qβ+d3k2qβ+d4k
2qβ+desk2qβ+qαβ−d1k
2γ1−d2k2γ1−d4k
2γ1+qβγ1−
d1k2γ2 − d2k
2γ2 − d3k2γ2 −αγ2 + qβγ2 − γ1γ2 − 2d1k
2δ− 2d2k2δ− d3k
2δ− d4k2δ−
αδ+2qβδ− γ1δ− γ2δ− δ2 − d1k2κ− d3k
2κ− d4k2κ−ακ+ βκ− γ1κ− γ2κ− 2δκ−
3d1k2µ−3d2k
2µ−3d3k2µ−3d4k
2µ−3αµ+3qβµ−3γ1µ−3γ2µ−6δµ−3κµ−6µ2;
q3 = −d1d2d3k6 − d1d2d4k
6 − d1d3d4k6 − d2d3d4k
6 + d3desdsek6 + d4desdsek
6 −
d1d2k4α− d1d4k
4α− d2d4k4α + desdsek
4α+ d1d3k4qβ + d1d4k
4qβ + d3d4k4qβ +
d3desk4qβ + d4desk
4qβ + d1k2qαβ + d4k
2qαβ + desk2qαβ − d1d2k
4γ1 − d1d4k4γ1 −
d2d4k4γ1 + desdsek
4γ1 + d1k2qβγ1 + d4k
2qβγ1 + desk2qβγ1 − d1d2k
4γ2 − d1d3k4γ2 −
d2d3k4γ2 + desdsek
4γ2 − d1k2αγ2 − d2k
2αγ2 + d1k2qβγ2 + d3k
2qβγ2 + desk2qβγ2 +
qαβγ2 − d1k2γ1γ2 − d2k
2γ1γ2 + qβγ1γ2 − 2d1d2k4δ− d1d3k
4δ− d2d3k4δ− d1d4k
4δ−
d2d4k4δ+2desdsek
4δ−d1k2αδ−d2k
2αδ+2d1k2qβδ+d3k
2qβδ+d4k2qβδ+2desk
2qβδ+
qαβδ− d1k2γ1δ− d2k
2γ1δ+ qβγ1δ− d1k2γ2δ− d2k
2γ2δ+ qβγ2δ− d1k2δ2− d2k
2δ2+
qβδ2−d1d3k4κ−d1d4k
4κ−d3d4k4κ−d1k
2ακ−d4k2ακ+d1k
2βκ+d4k2βκ+desk
2βκ+
xiii
lαβκ− d1k2γ1κ− d4k
2γ1κ− d1k2γ2κ− d3k
2γ2κ−αγ2κ+ βγ2κ− γ1γ2κ− 2d1k2δκ−
d3k2δκ−d4k
2δκ−αδκ+βδκ−γ1δκ−γ2δκ−δ2κ−2d1d2k4µ−2d1d3k
4µ−2d2d3k4µ−
2d1d4k4µ− 2d2d4k
4µ− 2d3d4k4µ+ 2desdsek
4µ− 2d1k2αµ− 2d2k
2αµ− 2d4k2αµ+
2d1k2qβµ+ 2d3k
2qβµ+ 2d4k2qβµ+ 2desk
2qβµ+ 2qαβµ− 2d1k2γ1µ− 2d2k
2γ1µ−
2d4k2γ1µ+2qβγ1µ−2d1k
2γ2µ−2d2k2γ2µ−2d3k
2γ2µ−2αγ2µ+2qβγ2µ−2γ1γ2µ−
4d1k2δµ− 4d2k
2δµ− 2d3k2δµ− 2d4k
2δµ− 2αδµ+4qβδµ− 2γ1δµ− 2γ2δµ− 2δ2µ−
2d1k2κµ− 2d3k
2κµ− 2d4k2κµ− 2ακµ+2βκµ− 2γ1κµ− 2γ2κµ− 4δκµ− 3d1k
2µ2−
3d2k2µ2 − 3d3k
2µ2 − 3d4k2µ2 − 3αµ2 +3qβµ2 − 3γ1µ
2 − 3γ2µ2 − 6δµ2 − 3κµ2 − 4µ3;
q4 = d4desdsek6α + d4desk
4qαβ + d4desdsek6γ1 + d4desk
4qβγ1 + desdsek4αγ2 +
desk2qαβγ2+ desdsek
4γ1γ2+ desk2qβγ1γ2+ d4desdsek
6δ+ desdsek4αδ+ d4desk
4qβδ+
desk2qαβδ + desdsek
4γ1δ + desk2qβγ1δ + desdsek
4γ2δ + desk2qβγ2δ + desdsek
4δ2 +
desk2qβδ2 + d4desk
4βκ+ desk2lαβκ+ desk
2βγ2κ+ desk2βδκ+ d4desdsek
6µ−
d2d4k4αµ+ desdsek
4αµ+ d4desk4qβµ+ d4k
2qαβµ+ desk2qαβµ− d2d4k
4γ1µ+
desdsek4γ1µ+ d4k
2qβγ1µ+ desk2qβγ1µ+ desdsek
4γ2µ− d2k2αγ2µ+ desk
2qβγ2µ+
qαβγ2µ− d2k2γ1γ2µ+ qβγ1γ2µ− d2d4k
4δµ+ 2desdsek4δµ− d2k
2αδµ+ d4k2qβδµ+
2desk2qβδµ+qαβδµ−d2k
2γ1δµ+qβγ1δµ−d2k2γ2δµ+qβγ2δµ−d2k
2δ2µ+qβδ2µ−
d4k2ακµ+ d4k
2βκµ+ desk2βκµ+ lαβκµ− d4k
2γ1κµ−αγ2κµ +βγ2κµ− γ1γ2κ µ−
d4k2δκµ−αδκ µ+βδκµ−γ1δκ µ−γ2δκµ−δ2κµ −d2d4k
4µ2+desdsek4µ2−d2k
2αµ2−
d4k2αµ2+d4k
2qβµ2+desk2qβµ2+qαβµ2−d2k
2γ1µ2−d4k
2γ1µ2+qβγ1µ
2−d2k2γ2µ
2−
αγ2µ2+ qβγ2µ
2− γ1γ2µ2− 2d2k
2δµ2− d4k2δµ2−αδµ2+2qβδµ2− γ1δµ
2− γ2δµ2−
δ2µ2−d4k2κµ2−ακµ2+β κµ2−γ1κµ
2−γ2κ µ2−2δκµ2−d2k2µ3−d4k
2µ3−αµ3+
qβµ3−γ1µ3−γ2µ
3−2δµ3−κµ3−µ4−d1k2(−d4k
2qαβ−d4k2qβγ1−qαβγ2−qβγ1γ2−
d4k2qβδ−qαβδ−qβγ1δ−qβγ2δ−qβδ2+d4k
2ακ−d4k2βκ−lαβκ+d4k
2γ1κ+αγ2κ−
β γ2κ+γ1γ2κ+d4k2δκ+αδκ−β δκ+γ1δκ+γ2δ κ+δ2κ+d4k
2αµ−d4k2qβµ−qαβµ+
d4k2γ1µ− qβγ1µ+αγ2µ− qβγ2µ+ γ1γ2µ+ d4k
2δµ+αδµ− 2qβδµ+ γ1δµ+ γ2 δµ+
δ2µ+ d4k2κµ+ακµ−β κµ+ γ1κµ+ γ2κµ +2δκµ+ d4k
2µ2+αµ2− qβµ2+ γ1µ2+
γ2µ2+2δµ2+κµ2+µ3−d3k
2(qβ−κ−µ)(d4k2+γ2+ δ+µ)+d2k
2(d3k2+α+γ1+δ+
µ)(d4k2 +γ2+δ+µ))+d3k
2(d4k2+γ2+δ+µ)(des(dsek
4+k2qβ)−µ(d2k2−qβ+κ+µ)).
Bifurcation diagrams for β, γ1 and γ2 with
cross-diffusion
xiv
(a)
0.5 1.0 1.5 2.0Β
0.00002
0.00004
0.00006
fHΒL
(b)
0.5 1.0 1.5 2.0Β
0.00002
0.00004
0.00006
fHΒL
(c)
0.5 1.0 1.5 2.0Β
5.´ 10-6
0.00001
0.000015
0.00002
fHΒL
(d)
0.5 1.0 1.5 2.0Β
5.´ 10-6
0.00001
0.000015
0.00002
fHΒL
Figure 7: Bifurcation diagram for β for Case(a)-(d)
The bifurcation diagrams for transmission rate β and the recovery rates γ1 and γ2
with cross-diffusion are given in Figs. 7, 8 and 9.
xv
(a)
0.02 0.04 0.06 0.08 0.10 0.12 0.14Γ1
5.´ 10-8
1.´ 10-7
1.5´ 10-7
fHΓ1L
(b)
0.02 0.04 0.06 0.08 0.10 0.12 0.14Γ1
5.´ 10-8
1.´ 10-7
1.5´ 10-7
fHΓ1L
(c)
0.05 0.10 0.15 0.20Γ1
-1.´ 10-7
-5.´ 10-8
5.´ 10-8
1.´ 10-7
fHΓ1L
(d)
0.05 0.10 0.15 0.20Γ1
-1.´ 10-7
-5.´ 10-8
5.´ 10-8
1.´ 10-7
1.5´ 10-7
fHΓ1L
Figure 8: Bifurcation diagram for γ1 for Case(a)-(d)
(a)
0.05 0.10 0.15 0.20Γ2
-2.´ 10-8
2.´ 10-8
4.´ 10-8
6.´ 10-8
8.´ 10-8
1.´ 10-7
fHΓ2L
(b)
0.05 0.10 0.15 0.20Γ2
-2.´ 10-8
2.´ 10-8
4.´ 10-8
6.´ 10-8
8.´ 10-8
1.´ 10-7
fHΓ2L
(c)
0.05 0.10 0.15 0.20 0.25 0.30Γ2
-1.´ 10-7
-5.´ 10-8
5.´ 10-8
1.´ 10-7
fHΓ2L
(d)
0.05 0.10 0.15 0.20 0.25 0.30Γ2
-1.´ 10-7
-5.´ 10-8
5.´ 10-8
1.´ 10-7
1.5´ 10-7
2.´ 10-7
fHΓ2L
Figure 9: Bifurcation diagram for γ2 for Case(a)-(d)
xvi
Appendix A.5
The expression of basic reproduction number RIT
for SEIJTR system
The derivation of the basic reproduction number is based on the linearization of
the ODE model about disease free equilibrium. Following Diekmann and
Heesterbeek [52] the reproduction number is calculated using the next generation
matrix of the system at the disease free equilibrium. The disease free equilibrium
for this model is (S = Λ/µ, E = 0, I = 0, J = 0, T = 0, R = 0). The infected
classes are E, J , I and T . Then the matrices F and V chosen for the system 5.1 -
5.6 are as follows:
F =
β (I+qE+lJ)
NS
0
0
0
and V =
−(1− π)Λ + (µ+ κ)E
−κE + (µ+ α + δ)I
−αI + (µ+ γ1 + δ + ζ)J
−ζJ + (µ+ γ2 + δ(1− θ))T
F =
qβ β lβ 0
0 0 0 0
0 0 0 0
0 0 0 0
and
V =
µ+ κ 0 0 0
−κ µ+ α + δ 0 0
0 −α µ+ γ1 + δ + ζ 0
0 0 −ζ µ+ γ2 + δ(1− θ)
Thus, the basic reproduction number is given by the dominant eigenvalue of
F.V −1
xvii
ρ(F.V −1) = RIT = Λβ(q(µ+δ+α)(µ+δ+γ1+ζ)+κ(µ+δ+γ1+ζ+lα))µ(µ+κ)(µ+δ+α)(µ+δ+γ1+ζ)
.
Expressions of Sensitivity indices of RIT based on
perturbation of fixed points estimations
As the reproduction number RIT is a function of ten parameters Λ, β, µ, l, κ, q,
α, γ1, ζ and δ, where
RIT = Λβ(q(µ+δ+α)(µ+δ+γ1+ζ)+κ(µ+δ+γ1+ζ+lα))µ(µ+κ)(µ+δ+α)(µ+δ+γ1+ζ)
.
So using Eq. (5.11)
ΥRITx = ∂RIT
∂x× x
RIT
ΥRITΛ = ∂RIT
∂Λ× Λ
RIT=
β(q(α+δ+µ)(γ1+δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ))µ(α+δ+µ)(γ1+δ+ζ+µ)(κ+µ)
× µ(α+δ+µ)(γ1+δ+ζ+µ)(κ+µ)β(q(α+δ+µ)(γ1+δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ))
;
ΥRITβ = ∂RIT
∂β× β
RIT=
Λq(α+δ+µ)(γ1+δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ))µ(α+δ+µ)(γ1+δ+ζ+µ)(κ+µ)
× µ(α+δ+µ)(γ1+δ+ζ+µ)(κ+µ)Λ(q(α+δ+µ)(γ1+δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ))
;
ΥRITµ = ∂RIT
∂µ× µ
RIT=
−1 + βΛ(κ+q(α+δ+µ)+q(γ1+δ+ζ+µ))µ(α+δ+µ)(γ1+δ+ζ+µ)(κ+µ)
− βΛ(q(α+δ+µ)(γ1+δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ))µ(α+δ+µ)(γ1+δ+ζ+µ)(κ+µ)2
−βΛ(q(α+δ+µ)(γ1+δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ))
µ(α+δ+µ)(γ1+δ+ζ+µ)2(κ+µ)− βΛ(q(α+δ+µ)(γ1+δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ))
µ(α+δ+µ)2(γ1+δ+ζ+µ)(κ+µ);
ΥRITl = ∂RIT
∂l× l
RIT= lακ
q(α+δ+µ)(γ1+δ+ζ+ µ)+κ(lα+γ1+δ+ζ+µ);
ΥRITκ = ∂RIT
∂κ× κ
RIT= − κ(q(α+δ+µ)(γ1+δ+ζ+µ)−µ(lα+γ1+δ+ζ +µ]))
(κ+µ)(q(α+δ+µ)(γ1+ δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ));
ΥRITq = ∂RIT
∂q× q
RIT= q(α+δ+µ)(γ1+δ+ζ+ µ)
q(α+δ+µ)(γ1+δ+ζ+ µ)+κ(lα+γ1+δ+ζ+ µ);
ΥRITα = ∂RIT
∂α× α
RIT= − ακ(γ1+δ−lδ+ζ+µ−lµ)
(α+δ+µ)(q(α+δ+µ)(γ1+δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ));
ΥRITγ1
= ∂RIT
∂γ1× γ1
RIT= − lαγ1κ
(γ1+δ+ζ+µ)(q(α+δ+µ)(γ1+δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ));
xviii
ΥRITζ = ∂RIT
∂ζ× ζ
RIT= − lαζκ
(γ1+δ+ζ+µ)(q(α+δ+µ)(γ1+δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ));
ΥRITδ = ∂RIT
∂δ× δ
RIT= − δκ((γ1+δ+ζ+µ)2+lα(α+γ1+2δ+ζ+2µ))
(α+δ+µ)(γ1+δ+ζ+ µ)(q(α+δ+µ)(γ1+δ+ ζ+µ)+κ(lα+γ1+δ+ ζ+µ));
xix
Appendix A.6
Non-dimensionalization of SEIJTR system
(6.1)-(6.6)
In order to analyze in terms of proportions of susceptible, exposed, infected,
diagnosed, treated and recovered individuals it is assumed that
s =S
N(1)
e =E
N(2)
i =I
N(3)
j =J
N(4)
t1 =T
N(5)
r =R
N(6)
Differentiate Equations (1)-(6) with respect to time, t gives
Nds
dt=
dS
dt− s
dN
dt(7)
Nde
dt=
dE
dt− e
dN
dt(8)
Ndi
dt=
dI
dt− i
dN
dt(9)
Ndj
dt=
dJ
dt− j
dN
dt(10)
Ndt1dt
=dT
dt− t1
dN
dt(11)
Ndr
dt=
dR
dt− r
dN
dt(12)
where
N = S+E+I+J+T+R,dN
dt= Λ−µN−δ(I+J+(1−θ)T ) (13)
Now from Equation (7)
Nds
dt=
dS
dt− s
dN
dt
xx
= −(I + lJ + qE)Sβ
N+ πΛ− Sµ− s(Λ− µN − δ(I + J + (1− θ)T ))
= −(iN + ljN + qeN)sNβ
Nµ− s(Λ− µN − δ(iN + jN + (1− θ)t1N)) + πΛ
= N [−β(i+ lj + qe)s+ δ(i+ j + (1− θ)t1)s−πΛ
N− sΛ
N]
Therefore
ds
dt= −β(i+ lj + qe)s+ δ(i+ j + (1− θ)t1)s+
πΛ
N− sΛ
N(14)
Now from Equation (8)
Nde
dt=
dE
dt− e
dN
dt
= β(I + lJ + qE)
NS + (1− π)Λ− E(κ+ µ)− e(Λ− µN − δ(I + J + (1− θ)T )
= β(iN + ljN + qeN)
NsN+(1−π)Λ−eN(κ+µ)−e(Λ−µN−δ(iN+jN+(1−θ)t1N)
= N [β(i+ lj + eq)s+ δe(i+ j + (1− θ)t1)− eκ− eΛ
N+ (1− π)
Λ
N]
Therefore
de
dt= β(i+ lj + eq)s+ δe(i+ j + (1− θ)t1)− eκ− eΛ
N+ (1− π)
Λ
N(15)
Now from Equation (9)
Ndi
dt=
dI
dt− i
dN
dt
= κE − I(α + δ + µ)− i(Λ− µN − δ(I + J + (1− θ)T )
xxi
= κeN − iN(α + δ + µ)− i(Λ− µN − δ(iN + jN + (1− θ)t1N)
= N [−iα−iδ+i2δ+ijδ+it1δ−it1δθ+eκ−(iΛ
N−iµ+eiµ+i2µ+ijµ+irµ+isµ+it1µ]
Thereforedi
dt= κe− (α+ δ +
Λ
N)i+ δ(i+ j + (1− θ)t1)i (16)
Now from Equation (10)
Ndj
dt=
dJ
dt− j
dN
dt
= αI − (γ1 + δ + ζ + µ)J − j(Λ− µN − δ(I + J + (1− θ)T )
= αiN − (γ1 + δ + ζ + µ)jN − j(Λ− µN − δ(iN + jN + (1− θ)t1N)
= N [αi− (Λ
N+ γ1 + ζ)j + δ(i+ j + (1− θ)t1)j]
Thereforedj
dt= αi− (
Λ
N+ γ1 + ζ)j + δ(i+ j + (1− θ)t1)j (17)
Now from Equation (11)
Ndt1dt
=dT
dt− t1
dN
dt
= ζJ − (γ2 + δ(1− θ) + µ)T − t1(Λ− µN − δ(I + J + (1− θ)T )
= ζjN − (γ2 + δ(1− θ) + µ)t1N − t1(Λ− µN − δ(iN + jN + (1− θ)t1N)
= N [ζj − (Λ
N+ γ2)t1 + δ(i+ j + (θ − 1) + (1− θ)t1)t1]
xxii
Therefore
dt
dt= ζj − (
Λ
N+ γ2)t1 + δ(i+ j + (θ − 1) + (1− θ)t1)t1 (18)
Now from Equation (12)
Ndr
dt=
dR
dt− r
dN
dt
= γ1J + γ2T − µR− r(Λ− µN − δ(I + J + (1− θ)T )
= γ1jN + γ2t1N − µrN − r(Λ− µN − δ(iN + jN + (1− θ)tN)
= N [γ1j + γ2t1 −Λ
Nr + δ(i+ j + (1− θ)t1)r]
Thereforedr
dt= γ1j + γ2t1 −
Λ
Nr + δ(i+ j + (1− θ)t1)r (19)
After replacing s by S, e by E, i by I, j by J , t1 by T , r by R, and ΛN
by Π
Equations (14)-(19) can be written as
dS
dt= −β(I + qE + lJ)S + (π − S)Π + γ1IS + δ(I + J + (1− θ)T )S (20)
dE
dt= (1− π)Π + β(I + qE + lJ)S − (Π + κ)E + δ(I + J + (1− θ)T )E (21)
dI
dt= κE − (α + δ +Π)I + δ(I + J + (1− θ)T )I (22)
dJ
dt= αI − (Π + γ1 + ζ)J + δ(I + J + (1− θ)T )J (23)
dT
dt= ζJ − (Π + γ2)T + δ(I + J + (θ − 1) + (1− θ)T )T (24)
dR
dt= γ1J + γ2T − ΠR + δ(I + J + (1− θ)T )R (25)
and
S + E + I + J + T +R = 1 (26)
xxiii
The Routh-Hurwitz stability condition
Consider an nth order characteristic equation of the form:
λn + p1λn−1 + p2λ
n−2 + p3λn−3 + · · · · · ·+ pn = 0 (27)
Then the roots of the characteristic Equation (27) lie in the left half of the
complex plane iff the first column of the following table are nonzero and have the
same sign [167]. The Routh Table is:
Routh Table
r1,1 r1,2 r1,3 r1,4 r1,5 · · ·
r2,1 r2,2 r2,3 r2,4 r2,5 · · ·
r3,1 r3,2 r3,3 r3,4 · · ·
r4,1 r4,2 r4,3 · · ·...
......
rn+1,1
where
r1,1 = 1; r1,2 = p2; r1,3 = p4; · · · · · · · · ·
r2,1 = p1; r2,2 = p3; r2,3 = p5; · · · · · · · · ·
and other rows are
ri,1 = ri−2,2 − ri−2,1
ri−1,1ri−1,2; ri,2 = ri−2,3 − ri−2,1
ri−1,1ri−1,3; ri,3 = ri−2,4 − ri−2,1
ri−1,1ri−1,4;
· · · · · · · · · (i > 2)
Stability conditions for sixth order polynomial
equation
The sixth order characteristic equation is of the form:
λ6 + p1λ5 + p2λ
4 + p3λ3 + p4λ
2 + p5λ+ p6 = 0 (28)
Then the Routh Table is: where
xxiv
Routh Table
r1,1 r1,2 r1,3 r1,4 r1,5 r1,6
r2,1 r2,2 r2,3 r2,4 r2,5 r2,6
r3,1 r3,2 r3,3 r3,4 r3,5
r4,1 r4,2 r4,3 r4,4
r5,1 r5,2 r5,3
r6,1 r6,2
r7,1
r1,1 = 1; r1,2 = p2; r1,3 = p4; r1,4 = p6;r1,5 = 0; r1,6 = 0; r2,1 = p1; r2,2 = p3;
r2,3 = p5; r2,4 = 0; r2,5 = 0;r2,6 = 0;
r3,1 = r1,2 − r1,1r2,1
r2,2 =p1p2−p3
p1;
r3,2 = r1,3 − r1,1r2,1
r2,3 =p1p4−p5
p1;
r3,3 = r1,4 − r1,1r2,1
r2,4 = p6;
r3,4 = r1,5 − r1,1r2,1
r2,5 = 0;
r4,1 = r2,2 − r2,1r3,1
r3,2 =p23+p21p4−p1(p2p3+p5)
p3−p1p2;
r4,2 = r2,3 − r2,1r3,1
r3,3 =P1p2p5−p3p5−p21p6
p1p2−p3;
r4,3 = r2,4 − r2,1r3,1
r3,4 = 0;
r4,4 = r2,5 − r2,1r3,1
r3,5 = 0;
r5,1 = r3,2 −r3,1r4,1
r4,2
=p23p4 − p2p3p5 + p25 + p21(p
24 − p2p6) + p1(−p2p3p4 + p22p5 − 2p4p5 + p3p6)
p23 + p21p4 − p1(p2p3 + p5);
r5,2 = r3,3 −r3,1r4,1
r4,3 = p6;
r5,3 = r3,4 −r3,1r4,1
r4,4 = 0;
r6,1 = r4,2 −r4,1r5,1
r5,2 =
p23p4p5 − p2p3p25 + p35 − p33p6 + p31p
26 + p21(p
24p5 − p3p4p6 − 2p2p5p6) + p1(p
22p
25 + p2p3(−p4p5 + p3p6) + p5(−2p4p5 + 3p3p6))
(p23p4 − p2p3p5 + p25 + p21(p24 − p2p6) + p1(−p2p3p4 + p22p5 − 2p4p5 + p3p6))
;
xxv
0.2 0.4 0.6 0.8 1.0 1.2 1.4Β
-2.´ 10-8
2.´ 10-8
4.´ 10-8
fHΒL
0.1 0.2 0.3 0.4 0.5Α
1.´ 10-8
2.´ 10-8
3.´ 10-8
4.´ 10-8
5.´ 10-8
fHΑL
Figure 10: Bifurcation diagrams for without diffusion
r6,2 = r4,3 −r4,1r5,1
r5,3 = 0;
r7,1 = r5,2 −r5,1r6,1
r6,2 = p6;
Therefore the Routh-Hurwitz criterion [167] for stability gives
C1 : p1 > 0
C2 : p6 > 0
C3 :p1p2−p3
p1> 0
C4 :p1p2p3−p23−p21p4−p1p5
p1p2−p3> 0
C5 :=p23p4−p2p3p5+p25+p21(p
24−p2p6)+p1(p22p5−p2p3p4−2p4p5+p3p6)
p23+p21p4−p1(p2p3+p5)> 0
C6 :=
p23p4p5−p2p3p25+p35−p33p6+p31p26+p21(p
24p5−p3p4p6−2p2p5p6)+p1(p22p
25+p2p3(p3p6−p4p5)+p5(3p3p6−2p4p5))
p23p4−p2p3p5+p25+p21(p24−p2p6)+p1(p22p5−p2p3p4−2p4p5+p3p6)
>
0
Bifurcation diagrams for β and α with and
without diffusion
The bifurcation diagrams for transmission rate, β and rate of progression from
infective to diagnosed, α with and without diffusion are given in in Figs. (10)-
(11) for Case (1). The remaining cases shows similar results.
xxvi
0.2 0.4 0.6 0.8 1.0Β
-0.0001
-0.00005
0.00005
fHΒL
0.2 0.4 0.6 0.8 1.0Α
0.00005
0.0001
fHΑL
Figure 11: Bifurcation diagrams with diffusion
Appendix A.7
The expression of basic reproduction number
RQIT for SEIQJTR system
The disease free equilibrium for this model is (S = 1, E = 0, Q = 0 I = 0, J = 0,
T = 0, R = 0). The infected classes are E, Q J , I and T . Then the matrices F
and V chosen for the system 7.8 - 7.14 are as follows:
F =
β((I + qE + pQ+ lJ)S)
0
0
0
0
and V =
−(1− π)Π + (Π + κ1 + κ2)E − δ1EI − δ2(J + (1− θ)T )E
−κ1E + (Π + σ)Q− δ1IQ− δ2(J + (1− θ)T )Q
−κ2E + (Π + α + δ1)I − δ1I2 − δ2(J + (1− θ)T )I
−σQ− αI + (δ2 +Π+ γ1 + ζ)J − δ1IJ − δ2(J + (1− θ)T )J
−ζJ + (Π + γ2 + δ2)T − δ1IT − δ2(J + (1− θ)T + θ)T
xxvii
F =
qβ pβ β lβ 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
and
V =
κ1 + κ2 +Π 0 0 0 0
−κ1 Π+ σ 0 0 0
−κ2 0 α + δ1 +Π 0 0
0 −σ −α γ1 + δ2 + ζ +Π 0
0 0 0 −ζ γ2 + δ2 − δ2θ +Π
Thus, the basic reproduction number is given by the dominant eigenvalue of
F.V −1
ρ(F.V −1) = RQIT = β(a+b+c+d)(κ1+κ2+Π)(α+δ1+Π)(γ1+δ2+ζ+Π)(σ+Π)
a = q(α + δ1 +Π)(γ1 + δ2 + ζ +Π)(σ +Π).
b = κ2(γ1 + δ2 + ζ +Π)(σ +Π).
c = pκ1(α + δ1 +Π)(γ1 + δ2 + ζ +Π).
d = l(κ1(δ1 +Π)σ + α(κ1σ + κ2(σ +Π))).
The coefficient of characteristic equation (7.46)
p1 = α− qβ + γ1 + δ1 + δ2 + ζ + κ1 + κ2 + 4Π + σ;
p2 = γ1δ1+δ1δ2+δ1ζ−pβκ1+γ1κ1+δ1κ1+ δ2κ1+ζκ1−βκ2+ γ1κ2+δ1κ2+δ2κ2+
ζκ2+3γ1Π+3δ1Π+3δ2Π+3ζΠ+3κ1Π+3κ2Π+6Π2+γ1σ+δ1σ+ δ2σ+ζσ+κ1σ+
κ2σ+3Πσ− qβ(α+ γ1 + δ1 + δ2+ ζ +3Π+σ)+α(γ1 + δ2+ ζ +κ1 +κ2 +3Π+σ);
p3 = αγ1κ1 + γ1δ1κ1 + αδ2κ1 + δ1δ2κ1 + αζκ1 + δ1ζκ1 − lαβκ2 +αγ1κ2 − βγ1κ2 +
γ1δ1κ2 + αδ2κ2 − βδ2κ2 + δ1δ2κ2 + αζκ2 − β ζκ2 + δ1ζκ2 + 2αγ1Π+ 2γ1δ1Π+
2αδ2Π+2δ1δ2Π+2αζΠ+2δ1ζΠ+2ακ1Π+2γ1κ1Π+2δ1κ1Π+2δ2κ1Π+2ζκ1Π+
xxviii
2ακ2Π− 2βκ2Π+ 2γ1κ2Π+ 2δ1κ2Π+ 2δ2κ2Π+ 2ζκ2Π+ 3αΠ2 + 3γ1Π2 + 3δ1Π
2 +
3δ2Π2 + 3ζΠ2 + 3κ1Π
2 + 3κ2Π2 + 4Π3 − pβκ1(α+ γ1 + δ1 + δ2 + ζ + 2Π)+ αγ1σ+
γ1 δ1σ + αδ2σ + δ1 δ2σ + αζσ + δ1ζ σ + ακ1σ − lβκ1σ + γ1κ1σ + δ1κ1σ + δ2κ1σ +
ζ κ1σ + ακ2σ − βκ2 σ + γ1κ2σ + δ1κ2 σ + δ2κ2σ + ζκ2σ + 2αΠσ + 2γ1Πσ +
2δ1Πσ+2δ2Πσ+2ζΠσ+2κ1Πσ+2κ2Πσ+3Π2σ− qβ(δ1δ2 + δ1ζ +2δ1Π+ 2δ2Π+
2ζΠ+ 3Π2 + δ1σ + δ2σ + ζσ + 2Πσ + γ1(δ1 + 2Π+ σ) + α(γ1 + δ2 + ζ + 2Π+ σ))
p4 = αγ1κ1Π+γ1δ1κ1Π+αδ2κ1Π+δ1δ2κ1Π+αζκ1Π+δ1ζκ1Π−lαβκ2Π+αγ1κ2Π−
βγ1κ2Π+γ1δ1κ2Π+αδ2κ2Π−βδ2κ2Π+δ1δ2κ2Π+αζκ2Π−βζκ2Π+δ1ζκ2Π+αγ1Π2+
γ1δ1Π2 + αδ2Π
2 + δ1δ2Π2 + αζΠ2 + δ1ζΠ
2 + ακ1Π2 + γ1κ1Π
2 + δ1κ1Π2 + δ2κ1Π
2 +
ζκ1Π2+ακ2Π
2−βκ2Π2+γ1κ2Π
2+δ1κ2Π2+δ2κ2Π
2+ζ κ2Π2+αΠ3+γ1Π
3+δ1Π3+
δ2Π3 +ζΠ3+κ1Π
3+κ2Π3+Π4−pβκ1(α+δ1+Π)(γ1+δ2+ζ+Π)−lαβκ1σ+αγ1κ1σ−
lβδ1κ1σ+γ1δ1κ1σ+αδ2κ1σ+ δ1δ2κ1σ+αζκ1σ+δ1ζκ1σ−lαβκ2σ+αγ1κ2σ−βγ1κ2σ+
γ1δ1κ2σ+αδ2κ2σ−β δ2κ2σ+δ1δ2κ2σ+αζκ2σ−βζκ2σ+δ1ζκ2σ+αγ1Πσ+γ1δ1Πσ+
αδ2Πσ+δ1δ2Πσ+αζΠσ+δ1ζΠσ+ακ1Πσ− lβκ1Πσ+γ1κ1Πσ+δ1κ1Πσ+δ2κ1Πσ+
ζκ1Πσ+ακ2Πσ−βκ2Πσ+γ1κ2Πσ+ δ1κ2Πσ+ δ2κ2Πσ+ ζκ2Πσ+αΠ2σ+γ1Π2σ+
δ1Π2σ+δ2Π
2σ+ζΠ2σ+κ1Π2σ+κ2Π
2σ+Π3σ−qβ(α+δ1+Π)(γ1+δ2+ζ+Π)(Π+σ)
Stability conditions for seventh order polynomial
equation
The sixth order characteristic equation is of the form:
λ7 + p1λ6 + p2λ
5 + p3λ4 + p4λ
3 + p5λ2 + p6λ+ p7 = 0 (29)
Then the Routh Table is: where
r1,1 = 1; r1,2 = p2; r1,3 = p4; r1,4 = p6;r1,5 = 0; r1,6 = 0;r1,7 = 0;r2,1 = p1; r2,2 = p3;
r2,3 = p5; r2,4 = p7; r2,5 = 0;r2,6 = 0;r2,7 = 0;
r3,1 = r1,2 − r1,1r2,1
r2,2 =p1p2−p3
p1;
r3,2 = r1,3 − r1,1r2,1
r2,3 =p1p4−p5
p1;
r3,3 = r1,4 − r1,1r2,1
r2,4 =p1p6−p7
p1;
r3,4 = r1,5 − r1,1r2,1
r2,5 = 0;
r3,5 = r1,6 − r1,1r2,1
r2,6 = 0;
r3,6 = r1,7 − r1,1r2,1
r2,7 = 0;
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Routh Table
r1,1 r1,2 r1,3 r1,4 r1,5 r1,6 r1,7
r2,1 r2,2 r2,3 r2,4 r2,5 r2,6 r2,7
r3,1 r3,2 r3,3 r3,4 r3,5 r3,6
r4,1 r4,2 r4,3 r4,4 r4,5
r5,1 r5,2 r5,3 r5,4
r6,1 r6,2 r6,3
r7,1 r7,2
r8,1
r4,1 = r2,2 − r2,1r3,1
r3,2 =p23+p21p4−p1(p2p3+p5)
p3−p1p2;
r4,2 = r2,3 − r2,1r3,1
r3,3 =(p1p2−p3)p5+p1(p7−p1p6)
p1p2−p3;
r4,3 = r2,4 − r2,1r3,1
r3,4 = p7;
r4,4 = r2,5 − r2,1r3,1
r3,5 = 0;
r4,5 = r2,6 − r2,1r3,1
r3,6 = 0;
r5,1 = r3,2 −r3,1r4,1
r4,2
=p23p4 + p25 + p21(p
24 − p2p6)− p3(p2p5 + p7) + p1(p
22p5 − 2p4p5 + p3p6 + p2(−p3p4 + p7))
p23 + p21p4 − p1(p2p3 + p5);
r5,2 = r3,3 −r3,1r4,1
r4,3 = p6 −p7p1
+(−p1p2 + p3)
2p7p1(p23 + p21p4 − p1(p2p3 + p5))
;
r5,3 = r3,4 −r3,1r4,1
r4,4 = 0;
r5,4 = r3,5 −r3,1r4,1
r4,5 = 0;
r6,1 = r4,2 −r4,1r5,1
r5,2
=A1 + A2
(p23p4 + p25 + p21(p24 − p2p6)− p3(p2p5 + p7) + p1(p22p5 − 2p4p5 + p3p6 + p2(p7 − p3p4+)))
;
xxx
r6,2 = r4,3 −r4,1r5,1
r5,3 = p7;
r6,3 = r4,4 −r4,1r5,1
r5,4 = 0;
r7,1 = r5,2 −r5,1r6,1
r6,2
=A1 + A2 + A3
B1 +B2
;
r7,2 = r5,3 −r5,1r6,1
r6,3 = 0;
r8,1 = r6,2 −r6,1r7,1
r7,2 = p7.
Therefore the Routh-Hurwitz criterion [167] for stability gives
C1 : p1 > 0,
C2 : p7 > 0,
C3 : p2 − p3p1
> 0,
C4 :p23+p21p4−p1(p2p3+p5)
p3−p1p2> 0,
C5 :p23p4+p25+p21(p
24−p2p6)−p3(p2p5+p7)+p1(p22p5−2p4p5+p3p6+p2(p7−p3p4))
p23+p21p4−p1(p2p3+p5)> 0,
C6 :A1+A2
(p23p4+p25+p21(p24−p2p6)−p3(p2p5+p7)+p1(p22p5−2p4p5+p3p6+p2(p7−p3p4+)))
> 0,
C7 :A1+A2+A3
B1+B2> 0.
where
A1 = p35−p33p6+p31p26−p3p5(p2p5+2p7)+p23(p4p5+p2p7)+p21(p
24p5−2p6(p2p5+p7),
A2 =
p4(p2p7−p3p6))+p1(3p3p5p6−2p4p25+p27+p22(p
25−p3p7)+p2(p
23p6−p3p4p5+p5p7)),
A3 = p35p6 − p33p26 + p31p
36 − p4p
25p7 + p2p5p
27 − p37 + p23(p4p5p6 − p24p7 + 2p2p6p7)−
p21(p34p7 − p24p5p6 + p26(2p2p5 + 3p7) + p4p6(p3p6 − 3p2p7)),
A4 = −p3(p22p
27 + p7(3p5p6 − 2p4p7) + p2p5(p5p6 − p4p7)) + p1(2p
24p5p7 + p32p
27 −
p4p6(2p25 + p3p7) + p22(p
25p6 − p4p5p7 − 2p3p6p7),
A5 = 3p6(p3p5p6 + p27) + p2(p23p
26 + p7(p5p6 − 3p4p7) + p3p4(p4p7 − p5p6)),
B1 = p35 − p33p6 + p31p26 − p3p5(p2p5 + 2p7) + p23(p4p5 + p2p7) + p21(p
24p5 − 2p6(p2p5 +
p7) + p4(p2p7 − p3p6)) ,
xxxi
0.2 0.4 0.6 0.8 1.0Β
-1.´ 10-8
-5.´ 10-9
5.´ 10-9
1.´ 10-8
1.5´ 10-8
fHΒL
0.02 0.04 0.06 0.08 0.10Α
-1.´ 10-9
-5.´ 10-10
5.´ 10-10
1.´ 10-9
fHΑL
0.01 0.02 0.03 0.04 0.05Ζ
-4.´ 10-10
-2.´ 10-10
2.´ 10-10
4.´ 10-10
6.´ 10-10
8.´ 10-10
fHΖL
0.01 0.02 0.03 0.04 0.05Σ
-5.´ 10-10
5.´ 10-10
1.´ 10-9
fHΣL
Figure 12: Bifurcation diagrams without diffusion
B2 = p1(3p3p5p6 − 2p4p25 + p27 + p22(p
25 − p3p7) + p2(p
23p6 − p3p4p5 + p5p7)).
Bifurcation diagrams for β, α ζ and σ with and
without diffusion for SEQIJTR system
The bifurcation diagrams for transmission rate, β, rate of progression from
infective to diagnosed, α, treatment rate, ζ and rate of progression from
quarantine to diagnosed, σ with and without diffusion are given in in Figs. 12
and 13.
xxxii
0.2 0.4 0.6 0.8 1.0Β
-0.00002
-0.00001
0.00001
fHΒL
0.02 0.04 0.06 0.08 0.10 0.12 0.14Α
-2.´ 10-6
-1.´ 10-6
1.´ 10-6
2.´ 10-6
fHΑL
0.02 0.04 0.06 0.08 0.10Ζ
-1.5´ 10-6
-1.´ 10-6
-5.´ 10-7
5.´ 10-7
1.´ 10-6
1.5´ 10-6
fHΖL
0.01 0.02 0.03 0.04 0.05Σ
-1.´ 10-6
-5.´ 10-7
5.´ 10-7
1.´ 10-6fHΣL
Figure 13: Bifurcation diagrams with diffusion
xxxiii
List of Publication and Conference Presentations
⋆ Naheed A., Singh M., Lucy D., Numerical study of SARS epidemic model
with the inclusion of diffusion in the system, Applied Mathematics and
Computation 229 (2014), 480-498.
⋆ Naheed A., Singh M. and Lucy D., Parameter estimation with uncertainty
and sensitivity analysis for the SARS outbreak in Hong Kong, 4th IMA
Conference on numerical linear algebra and optimization, (2014),
Birmingham University, United Kingdom.
⋆ Naheed A., Singh M., Richards D. and Lucy D., Numerical study of SARS
model with treatment (SEIJTR) and diffusion in the System, ANZIAM
Conference, (2014), The Millennium Hotel, Rotorua, New Zealand.
xxxiv