Carlos Castillo-Chavez Joaquin Bustoz Jr. Professor Arizona State University

04/22/23 Arizona State University

Carlos Castillo-ChavezJoaquin Bustoz Jr. ProfessorArizona State University

Tutorials 3: Epidemiological Mathematical Modeling, The Case of Tuberculosis.

Mathematical Modeling of Infectious Diseases: Dynamics and Control (15 Aug - 9 Oct 2005)Jointly organized by Institute for Mathematical Sciences, National University of Singapore and Regional Emerging Diseases Intervention (REDI) Centre, Singapore

http://www.ims.nus.edu.sg/Programs/infectiousdiseases/index.htm

Singapore, 08-23-2005














Primary Collaborators:Juan Aparicio (Universidad Metropolitana, Puerto Rico)Angel Capurro (Universidad de Belgrano, Argentina, deceased)Zhilan Feng (Purdue University)Wenzhang Huang (University of Alabama)Baojung Song (Montclair State University)


Our work on TB Aparicio, J., A. Capurro and C. Castillo-Chavez, “On the long-term dynamics and re-

emergence of tuberculosis.” In: Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, IMA Volume 125, 351-360, Springer-Veralg, Berlin-Heidelberg-New York. Edited by Carlos Castillo-Chavez with Pauline van den Driessche, Denise Kirschner and Abdul-Aziz Yakubu, 2002

Aparicio J., A. Capurro and C. Castillo-Chavez, “Transmission and Dynamics of Tuberculosis on Generalized Households” Journal of Theoretical Biology 206, 327-341, 2000

Aparicio, J., A. Capurro and C. Castillo-Chavez, Markers of disease evolution: the case of tuberculosis, Journal of Theoretical Biology, 215: 227-237, March 2002.

Aparicio, J., A. Capurro and C. Castillo-Chavez, “Frequency Dependent Risk of Infection and the Spread of Infectious Diseases.” In: Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, IMA Volume 125, 341-350, Springer-Veralg, Berlin-Heidelberg-New York. Edited by Carlos Castillo-Chavez with Pauline van den Driessche, Denise Kirschner and Abdul-Aziz Yakubu, 2002

Berezovsky, F., G. Karev, B. Song, and C. Castillo-Chavez, Simple Models with Surprised Dynamics, Journal of Mathematical Biosciences and Engineering, 2(1): 133-152, 2004.

Castillo-Chavez, C. and Feng, Z. (1997), To treat or not to treat: the case of tuberculosis, J. Math. Biol.


Our work on TB

Castillo-Chavez, C., A. Capurro, M. Zellner and J. X. Velasco-Hernandez, “El transporte publico y la dinamica de la tuberculosis a nivel poblacional,” Aportaciones Matematicas, Serie Comunicaciones, 22: 209-225, 1998

Castillo-Chavez, C. and Z. Feng, “Mathematical Models for the Disease Dynamics of Tuberculosis,” Advances In Mathematical Population Dynamics - Molecules, Cells, and Man (O. , D. Axelrod, M. Kimmel, (eds), World Scientific Press, 629-656, 1998.

Castillo-Chavez,C and B. Song: Dynamical Models of Tuberculosis and applications, Journal of Mathematical Biosciences and Engineering, 1(2): 361-404, 2004.

Feng, Z. and C. Castillo-Chavez, “Global stability of an age-structure model for TB and its applications to optimal vaccination strategies,” Mathematical Biosciences, 151,135-154, 1998

Feng, Z., Castillo-Chavez, C. and Capurro, A.(2000), A model for TB with exogenous reinfection, Theoretical Population Biology

Feng, Z., Huang, W. and Castillo-Chavez, C.(2001), On the role of variable latent periods in mathematical models for tuberculosis, Journal of Dynamics and Differential Equations .


Our work on TB

Song, B., C. Castillo-Chavez and J. A. Aparicio, Tuberculosis Models with Fast and Slow Dynamics: The Role of Close and Casual Contacts, Mathematical Biosciences 180: 187-205, December 2002

Song, B., C. Castillo-Chavez and J. Aparicio, “Global dynamics of tuberculosis models with density dependent demography.” In: Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods and Theory, IMA Volume 126, 275-294, Springer-Veralg, Berlin-Heidelberg-New York. Edited by Carlos Castillo-Chavez with Pauline van den Driessche, Denise Kirschner and Abdul-Aziz Yakubu, 2002


OutlineBrief Introduction to TBLong-term TB evolutionDynamical models for TB transmissionThe impact of social networks – cluster modelsA control strategy of TB for the U.S.: TB and HIV


Long History of Prevalence

• TB has a long history.

• TB transferred from animal-populations.

• Huge prevalence.

• It was a one of the most fatal diseases.


• Pathogen? Tuberculosis Bacilli (Koch, 1882).• Where? Lung.• How? Host-air-host• Immunity? Immune system responds quickly

Transmission Process


• Bacteria invades lung tissue• White cells surround the invaders

and try to destroy them.• Body builds a wall of cells and

fibers around the bacteria to confine them, forming a small hard lump.

Immune System Response


• Bacteria cannot cause more damage as long as the confining walls remain unbroken.

• Most infected individuals never progress to active TB.

• Most remain latently-infected for life.

• Infection progresses and develops into active TB in less than 10% of the cases.

Immune System Response


Current Situations• Two million people around the world die of TB

each year.• Every second someone is infected with TB today.• One third of the world population is infected with

TB (the prevalence in the US around 10-15% ).• Twenty three countries in South East Asia and Sub

Saharan Africa account for 80% total cases around the world.

• 70% untreated actively infected individuals die.


Reasons for TB Persistence

• Co-infection with HIV/AIDS (10% who are HIV positive are also TB infected)

• Multi-drug resistance is mostly due to incomplete treatment

• Immigration accounts for 40% or more of all new recent cases.


Basic Model Framework

• N=S+E+I+T, Total population• F(N): Birth and immigration rate• B(N,S,I): Transmission rate (incidence)• B`(N,S,I): Transmission rate (incidence)

Sμ

I Ir1 TE kE

TμId )( +μEμ

S)(NF ),,( ISNB

Er 2

),,(' ITNB


Model Equations

€

dSdt =F(N)−β CS I

N −μI,dEdt =β CS I

N −(μ +k+r2)E+β'CT IN ,

dIdt =kE−(μ +d+r1)E,dTdt =r2E+r1I−β' CT I

N −μT,N =S+E+I +T


R0

Probability of surviving to infectious stage:

Average successful contact rate

Average infectious period

krk++ 2μ

Cβ

dr ++ 1

1μ

€

R0 = βCμ +r1+d ⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

kμ +r2+k ⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟


Phase Portraits


Bifurcation Diagram

0R

*I

n Bifurcatio calTranscriti Global

1


Fast and Slow TB (S. Blower, et al., 1995)

Sμ

IE kE

Id)( +μEμ

SSIp β)1( −

SIpβ

Λ


.

,)1(

,

IdIkE SIpdtdI

EkE SIpdtdE

S SI dtdS

μβ

μβ

μβ

−−+=

−−−=

−−=Λ

Fast and Slow TB


What is the role of long and variable latent periods?

(Feng, Huang and Castillo-Chavez. JDDE, 2001)


A one-strain TB model with a distributed period of latency

Assumption

Let p(s) represents the fraction of individuals who are still in the latent class

at infection age s, and

Then, the number of latent individuals at time t is:

and the number of infectious individuals at time t is:


The model


The reproductive number

Result: The qualitative behavior is similar to that of the ODE model.

Q: What happens if we incorporate resistant strains?


What is the role of long and variable latent periods?

(Feng, Hunag and Castillo-Chavez, JDDE, 2001)

A one-strain TB model

1/k is the latency period


Bifurcation Diagram

0R

*I


1


A TB model with exogenous reinfection(Feng, Castillo-Chavez and Capurro. TPB, 2000)


Exogenous Reinfection

Sμ

IE kE

Id )( +μEμ

S NIcSβ

c NISpβ

Λ T

Tμ

rI

NIcTσβ

E


The model


Basic reproductive number is

Note: R0 does not depend on p.A backward bifurcation occurs at some pc (i.e., E* exists for R0 <

1)

Backward bifurcation Number of infectives I vs. time


Backward Bifurcation


Dynamics depends on initial values


A two-strain TB model(Castillo-Chavez and Feng, JMB, 1997)

Drug sensitive strain TB - Treatment for active TB: 12 months - Treatment for latent TB: 9 months - DOTS (directly observed therapy strategy) - In the US bout 22% of patients currently fail to complete their

treatment within a 12-month period and in some areas the failure rate reaches 55% (CDC, 1991)

Multi-drug resistant strain TB - Infection by direct contact - Infection due to incomplete treatment of sensitive TB - Patients may die shortly after being diagnosed - Expensive treatment


A diagram for two-strain TB transmission

S L1 I1 T

L2

I2

μ

Λ

μ μ+d1 μ

μ

μ+d2

k1 pr2 (1-(p+q))r2

qr2

K2

r1β1

β2 β*

β*

β’’

r2 is the treatment rate for individuals with active TB q is the fraction of treatment failure



The two-strain TB model

r2 is the treatment rate for individuals with active TB q is the fraction of treatment failure


Reproductive numbers

For the drug-sensitive strain:

For the drug-resistant strain:


Equilibria and stability

There are four possible equilibrium points:

E1 : disease-free equilibrium (always exists)

E2 : boundary equilibrium with L2 = I2 = 0 (R1 > 1; q = 0)

E3 : interior equilibrium with I1 > 0 and I2 > 0 (conditional)

E4 : boundary equilibrium with L1 = I1 = 0 (R2 > 1)

Stability dependent on R1 and R2

q=0

q>0

Sensitive TB only

Coexistence

Resistant TB only

Resistant TB only

Coexistence

Bifurcation diagram


q >0

Fraction of infections vs time

q = 0

Resistant TB only

Sensitive TB only

Resistant TB only

Coexistence TB-free


Contour plot of the fraction of resistant TB, J/N, vs treatment rate r2 and fraction of treatment failure q


Optimal control strategies of TB through treatment of sensitive TB

Jung, E., Lenhart, S. and Feng, Z. (2002), Optimal control of treatments in a two-strain tuberculosis model, Discrete and

Continuous Dynamical Systems

“Case holding", which refers to activities and techniques used to ensure regularity of drug intake for a duration adequate to achieve a cure

“Case finding", which refers to the identification (through screening, for example) of individuals latently infected with sensitive TB who are at high risk of developing the disease and who may benefit from preventive intervention

These preventive treatments will reduce the incidence (new cases per unit of time) of drug sensitive TB and hence indirectly reduce the incidence of drug resistant TB


A diagram for two-strains TB transmission with controls

S L1 I1 T

L2

I2

μ

Λ

μ μ+d1 μ

μ

μ+d2

k1(1-u2)pr2

(1-(1-u2)(p+q))r2(1-u2) qr2

K2

r1u1β1

β2 β*

β*

β’’


u1(t): Effort to identify and treat typical TB individuals1-u2(t): Effort to prevent failure of treatment of active TB0 < u1(t), u2(t) <1 are Lebesgue integrable functions

The two-strain system with time-dependent controls

(Jung, Lenhart and Feng. DCDSB, 2002)


Objective functional

B1 and B2 are balancing cost factors. We need to find an optimal control pair, u1 and u2, such that

where

ai, bi are fixed positive constants, and tf is the final time.



Numerical Method: An iteration method Jung, E., Lenhart, S. and Feng, Z. (2002), Optimal control of treatments in a two-strain tuberculosis model, Discrete and


1. Guess the value of the control over the simulated time.

2. Solve the state system forward in time using the Runge-Kutta scheme.

3. Solve the adjoint system backward in time using the Runge-Kutta scheme using the solution of the state equations from 2.

4. Update the control by using a convex combination of the previous control and the value from the characterization.

5. Repeat the these process of until the difference of values of unknowns at the present iteration and the previous iteration becomes negligibly small.


Optimal control strategiesJung, E., Lenhart, S. and Feng, Z. (2002), Optimal control of treatments in a two-strain tuberculosis model, Discrete and


u1(t)u2(t)

without control

With controlTB cases(L2+I2)/N

Control


Controls for various population sizes Jung, E., Lenhart, S. and Feng, Z. (2002), Optimal control of treatments in a two-strain tuberculosis model, Discrete and


u1(t)

u2(t)


Demography

€

dNdt =F(N)−μN −dI,dEdt =β C(N −E−I) I

N −(μ +k+r2)E,dIdt =kE−(μ +d+r1)I.

€

F(N)=rN, Exponential GrowthF(N)=rN 1−N

K ⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟, Logistic Growth

F(N)=Λ, a constant

Results: More than one Threshold Possible


Bifurcation Diagram--Not Complete or Correct

Picture

0R

*I


1


Demography and Epidemiology

€

R0 = βCμ +r1+d ⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

kμ +r2 +k ⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟


Demography

Where:

*2 durR μ−=

))((2))()(())((42)((*

rnrmCkdCdkrmCrnrmdcrnrmCkdCdkrmcrnrmdu

−−+−+−−−++−+

= ββββββ


1R

Bifurcation Diagram(exponential growth )

μ=r

1

0→N

0→I

0→NI

0R

∞→I

0→NI

*uNI → ⎩⎨

⎧>∞→<→

)1( )1( 0

2

2

RIRN

1


Logistic Growth

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

++++= krìk

drìâCR

210

0

0

1

2 1R

Rrkdì

kdìrR*

−++++

=


Logistic Growth (cont’d)

If R2* >1

• When R0 1, the disease dies out at an exponential rate. The decay rate is of the order of R0 – 1.

• Model is equivalent to a monotone system. A general version of Poincaré-Bendixson Theorem is used to show that the endemic state (positive equilibrium) is globally stable whenever R0 >1.

• When R0 1, there is no qualitative difference between logistic and exponential growth.


Bifurcation Diagram

0R

*I


1


Particular Dynamics(R0 >1 and R2

* <1)

All trajectories approach theorigin. Global attraction isverified numerically by randomly choosing5000 sets of initial conditions.


Particular Dynamics(R0 >1 and R2

* <1)

All trajectories approach theorigin. Global attraction isverified numerically by randomly choosing5000 sets of initial conditions.


Conclusions on Density-dependent Demography

• Most relevant population growth patterns handled with the examples.

• Qualitatively all demographic patterns have the same impact on TB dynamics.

• In the case R0<1, both exponential growth and logistic grow lead to the exponential decay of TB cases at the rate of R0-1.

• When parameters are in a particular region, theoretically model predicts that TB could regulate the entire population.

•However, today, real parameters are unlikely to fall in that region.


A fatal disease

• Leading cause of death in the past, accounted for one third of all deaths in the 19th century.

• One billion people died of TB during the 19th and early 20th centuries.


Per Capita Death Rate of TB


Non Autonomous Model

Here, N(t) is a known function of t or it comes from data (time series). The death rates are known functions of time, too.


Births and immigration adjusted to fit data


Life Expectancy in Years


Incidence = k E


Incidence of TB since 1850


Conclusions• Contact rates increased--people move

massively to cities• Life span increased in part because of

reduce impact of TB-induced mortality• Prevalence of TB high• Progression must have slow down

dramatically

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Carlos Castillo-Chavez Joaquin Bustoz Jr. Professor Arizona State University

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