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CHAPTER 4

SIMPLE SIS MODEL WITH VACCINATION

4.1 Introduction

A vaccine is a biological preparation that improves immunity to a particular disease. Note

that vaccine do not guarantee complete protection from a disease because it have to depend s on

many factors, for example the type of disease, age of the particular individual, as some

individuals are non-responders to certain vaccines and etc. Some vaccine may also wear off

after a period of time and wont last for whole life. This SIS model with vaccine is rather simple,

but nevertheless it gives insight into some of the plausible consequences of public health policies

(Kribs-Zaleta and Velasco-Hernandez, 1999).

4.2 SIS Model with Vaccination

Since the individuals in the SIS model that recovered may be infected again, the vaccine

is an important factor to be considered for preventing an occurrence of endemic. Although the

vaccine does not guarantee complete protection, in this model we consider that the vaccine is

totally effective and the susceptible individuals will gain immunity to the disease after the

vaccination is done, but the vaccine will wear off after a period of time, where its effect will start

to wane. The assumption above may not be truly realistic, but it can simplify the models

equation for easier case study about the effect of vaccine to the classic SIS model. A new

compartment which is )t(V , the vaccinated susceptible individuals at time t is considered in this

model.

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Figure 4.1: Compartmental chart for SIS model with vaccine

From Figure 4.1 we can see that there is no flow of vaccinated individuals back to the

group of infective individuals due to the earlier assumption regarding complete immunity after

vaccination. Thus the progression of the disease from the standpoint of an individual is illustrated

as follows:

Susceptible Infective Susceptible

Vaccinated Susceptible

4.2.1 Formulating epidemiology models

Since the population size N is fixed, the total population is )t(V)t(S)t(IN ++= where N

is a constant. There are two new parameters to be taken into account for the new compartment

S I

V

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V(t), which is , the vaccination rate and the , the rate of the vaccine wearing off. Based on the

earlier assumption, the vaccines effect will wane off and there is a proportion of vaccinated

individuals who will become susceptible again at time t, with .10

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4.2.2 Basic Reproduction Number, R(0)

Same concept with the basic reproduction number for SIS model, for an endemic to be

occur, at the beginning of the endemic the rate of change of the infective individuals should more

than zero, otherwise the number of infective individual will keep on decreasing and the endemic

will dies out.

0IN

IS

dt

dI>=

0IN

S

>

At the beginning of the endemic, the number of susceptible individuals is approximately

equal to the total population because there only have a small group of infective individuals and

there is no individuals are vaccinated, thus N1N)0(S .

0)0(IN

)0(S

>

0>

1>

Hence we can say that the endemic will not occur unless 1>

, the basic reproduction number

can be define as

=)0(R .

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4.3 Numerical Solution

We seek a numerical solution rather than analytical solution because this models

equation is unsolvable. From the previous chapter we know that the Runge-Kutta method is a

workable numerical method to be used. The Second-Order Runge-Kutta method is chosen for

plotting the solution throughout this whole chapter because we will use the time step equal to 0.1

for all the simulations afterwards (which means the error of the numerical solution will be very

small, thus the Second-Order Runge-Kutta method is good enough). According to Second-Order

Runge-Kutta method (Vetterling and Press, 1992):

VISN

S

dt

dS)S,t(F

++==

IN

IS

dt

dI)I,t(G

==

VSdt

dV)V,t(H ==

Then the numerical solution for all time, t is:

j)t(S)tt(S 2+=+

k)t(I)tt(I 2+=+

m)t(V)tt(V 2+=+

where

t))t(S,t(Fj1=

t)j2

1)t(S,tt(Fj 12 ++= .

t))t(I,t(Gk1 =

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t)k2

1)t(I,tt(Gk 12 ++=

t))t(V,t(Hm1 =

t)m21)t(V,tt(Hm 22 ++=

4.4 Steady state or the endemic equilibrium points

Since we cannot find the analytical solution for this model, perhaps we find the endemic

equilibrium points to forecast the final state of the endemic. The equilibria are points where the

variables do not change with time (Medlock, 2010), which means 0dt

dS= 0

dt

dI, = 0

dt

dV, = at time

t. Thus from equation (4.1), (4.2), (4.3) we get

0VISN

S=++

(4.4)

0IN

IS

=

(4.5)

0VS =

(4.6)

4.4.1 Disease-free equilibrium point

We know that if the disease-free equilibrium occurs, that means the endemic will dies out

at the end. Thus the equilibrium point is zero at that time. From equation (4.5)

0IN

S **

=

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We know it is either 0I* = or 0

N

S*

=

. Since we want to find the disease-free equilibrium

point first, suppose 0I

*

= . From equation (4.6),

S

V

**

= . Based on the early assumption, the

population size, VISN ++= for all time t, thus at the moment where endemic equilibrium

occur, VISN*** ++= where N is a constant.

)VIN(V

*** =

+

=N

V*

since 0I* =

+=

NNS

*

+=

Nby the condition 0I

* = , VNS ** = .

Thus if the disease-free equilibrium occur, the equilibrium point is )V,S,I(*** = )

N,

N,0(

++.

4.4.2 Vaccine Reproduction Number, )(R

The stability of the disease-free equilibrium points above is not yet known, which means

that we dont know whether the solution will converge to the equilibrium point or diverge away

from it. A model with multiple variables is stable if and only if the real part of the eigon values of

the models Jacobian Matrix are less than zero (Jones, 2008). Thus we form a Jacobian Matrix

about the equilibrium points of I(t) and V(t) first.

Let I)VIN(N

I)V,I(F

=

V)VIN()V,I(G =

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=

N

I)VIN(

NI

F

N

I

V

F =

=I

G

)(V

G+=

Jacobian derivative of F and G (Medlock, 2010) is

=)V,I(

V

G)V,I(

I

G

)V,I(V

F)V,I(I

F

)V,I(J

Disease-free equilibrium

+

+

=+

)(

0)

N,0(J

Now let 0|I)V,I(J|22

=

where I22 is identity matrix and is the eigon values.

0

)(

0|I)V,I(J|

22

=

+

+

=

)(1 += and clearly it is less than zero since 10

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1)(