TB Cluster Models, Time Scales and Relations to HIV

Jun, 2002 MTBI Cornell University

TB Cluster Models, Time Scales and Relations to HIV

Carlos Castillo-Chavez

Department of Biological Statistics and Computational Biology

Department of Theoretical and Applied Mechanics

Cornell University, Ithaca, New York, 14853


Outline

• A non-autonomous model that incorporates the impact of HIV on TB dynamics.

• Model to test CDC’s TB control goals.• Casual versus close contacts and their impact on TB.• Time scales and singular perturbation approaches in

the study of the dynamics of TB.


TB in the US(1953-1999)


Reemergence of TB

• New York City and San Francisco had recent outbreaks.• Cost of control the outbreak in NYC alone was estimated to be about 1 billion.• Observed national TB case rate increase.• TB reemergence became an international issue.• CDC sets control goal in 1989.


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)

S

I Ir1 TE kE

TId )( E

S)(NF ),,( ISNB

Er 2

),,(' ITNB


Model Equations

,'

,)(

,')(

, )(

12

1

2

TIESN

TNI CTIrEr

dtdT

ErdkEdtdI

NICTErk

NI CS

dtdE

I NI CSNF

dtdS


CDC Short-Term Goal:

3.5 cases per 100,000 by 2000.

Has CDC met this goal?

CDC Long-term Goal:

One case per million by 2010.

Is it feasible?

TB control in the U.S.


Model Construction

Since d has been approximately equal to zero over the past 50 years in the US, we only consider

dINFdtdN )(

Hence, N can be computed independently of TB.

).(NFdtdN


Non-autonomous model (permanent latent class of TB introduced)

.))()(())(())((

))()((

,))()(()(

))((

321

,2212

11211

IrtdtLtAvLtAkdtdI

LtAvrtpLdt

dL

LrkptAttN

IILLtN dtdL

known. are )( ),( ,)( ,)( tdtAttN


Effect of HIV

Otherwise. ,0

1983; tif ,)1983()1983()(3

21

tExpttA


Upper Bound and Lower Bound For Epidemic Threshold

31 rdrpkkR

31 rdrpkkR

If R<1, L1(t), L2(t) and I(t) approach zero;If R>1, L1(t), L2(t) and I(t) all have lower positive boundary;If (t) and d(t) are time-independent, R and R areEqual to R0 .


Parameter estimation and simulation setup

Parameter Estimation

0.22

c 10

k 0.001

r1 0.05

r2 0.05

r3 0.65

p 0.1

Initial Values

I(0) 87423

L1(0) 106

L2(0) 106


N(t) is from census data and population projection

Parameter estimation and simulation setup


RESULTS


CONCLUSIONS


CONCLUSIONS


CDC’s Goal Delayed

• Impact of HIV.

• Lower curve does not include HIV impact;

• Upper curve represents the case rate when HIV is included;

• Both are the same before 1983. Dots represent real data.


Regression approach

20006.00597.03970.11Log

:Equation Regression

XXY

A Markov chain model supports the same result


Cluster Models

• Incorporates contact type (close vs. casual) and focus on the impact of close and prolonged contacts.

• Generalized households become the basic epidemiological unit rather than individuals.

• Use natural epidemiological time-scales in model development and analysis.


Close and Casual contacts

Close and prolonged contacts are likely to be responsible for the generation of most new cases of secondary TB infections. “A high school teacher who also worked at library infected the students in her classroom but not those who came to the library.”

Casual contacts also lead to new cases of active TB. WHO gave a warning in 1999 regarding air travel. It announced that flights of more than 8 hours pose a risk for TB transmission.


Transmission Diagram

In1E1S

I

2S

1S

2E

2S 1E2E2kE

2

22 N

EknE

I

2knE

2

2

2 NSknE

1S


• Basic epidemiological unit: cluster (generalized household)

• Movement of kE2 to I class brings nkE2 to N1 population, where by assumptions nkE2(S2 /N2) go to S1 and nkE2(E2/N2) go to E1

• Conversely, recovery of I infectious bring nI back to N2 population, where nI (S1 /N1)= S1 go to S2 and nI (E1 /N1)= E1 go to E2

Key Features


Basic Cluster Model


Basic Reproductive Number

where

is the expected number of infections produced by one infectious individual within his/her cluster.

denotes the fraction who survives the latency period and become active cases.

nQ0

kkf

fQk

knRc00


Diagram of Extended Cluster Model

In1E1S

I

2S

1Sp

2E

1S2S 1E

2E 2kE

2

22 N

EknE

I

2knE

2

22 N

SknE

nNISp 2

2)1(

nNISp 2

1)1(


(n)

Both close casual contacts are included in the extended model. The risk of infection per susceptible, , is assumed to be a nonlinear function of the average cluster size n. The constant p measures the average proportion of the time that an “individual spends in a cluster.


R0 Depends on n in a non-linear fashion


Role of Cluster Size

E(n) denotes the ratio of within cluster to between cluster transmission. E(n) increases and reaches its maximum value at

The cluster size n* is defined as optimal as it maximizes the relative impact of within to between cluster transmission.

L

L

p

pK

Kn

11

*


Hoppensteadt’s Theorem(1973)

Full system Reduced system

where x Rm, y Rn and is a positive real parameter near zero (small parameter). Five conditions must be satisfied (not listed here) to apply the theorem. In addition, it is shown that if the reduced system has a globally asymptotically stable equilibrium then the full system has a g.a.s. equilibrium whenever 0< <<1.


Two time Scales

• Latent period is long and roughly has the same order of magnitude as that associated with the life span of the host.

• Average infectious period is about six months (wherever there is treatment, is even shorter).


Rescaling

Time is measured in average infectious periods (fast time scale), that is, = k t. The state variables are rescaled as follow:

Where / is the asymptotic carrying capacity.


Rescaled Model


Rescaled Model


Dynamics on Slow Manifold

Solving for the quasi-steady states y1, y2 and y3 in

terms of x1 and x2 gives

Substituting these expressions into the equations for

x1 and x2 lead to the equations of motion on the slow manifold.


Slow Manifold Dynamics

Where is the number of secondary

infections produced by one infectious individual in a

population where everyone is susceptible


Theorem

If Rc0 1,the disease-free equilibrium (1,0) is globally

asymptotically stable. While if Rc0 > 1, (1,0) is unstable and

the endemic equilibrium

is globally asymptotically stable.

This theorem characterizes the dynamics on the slow manifold


Dynamics for Full System

Theorem: For the full system, disease-free equilibrium is globally asymptotically stable whenever R0

c <1; while R0c >1

there exists a unique endemic equilibrium which is globally asymptotically stable.

Proof approach: Construct Lyapunov function for the case R0

c <1; for the case R0c >1, we use Hoppensteadt’s

Theorem.A similar result can be found in Z. Feng’s 1994, Ph.D. dissertation.


Bifurcation Diagram

Global bifurcation diagram when 0<<<1 where denotes the ratio between rate of progression to active TB and the average life-span of the host (approximately).

0R

*I

n Bifurcatio

calTranscriti Global

1


Numerical Simulations


Conclusions from cluster models

• TB has slow dynamics but the change of epidemiological units makes it possible to identify non-traditional fast and slow dynamics.• Quasi steady assumptions (adiabatic elimination of parameter) are valid (Hoppensteadt’s theorem).• The impact of close and casual contacts can be study using this approach as long as progression rates from the latently to the actively-infected stages are significantly different.


Conclusions from cluster models

• Singular perturbation theory can be used to study the global asymptotic dynamics.

• Optimal cluster size highlights the relative impact of close versus casual contacts and suggests alternative mechanisms of control.

• The analysis of the system for the case where the small parameter is not small has not been carried out. Simulations suggest a wider range.

Documents

TB Cluster Models, Time Scales and Relations to HIV