Jun,2002MTBI Cornell University Carlos Castillo-Chavez Department of Biological Statistics and Computational Biology Department of Theoretical and Applied

Jun,2002 MTBI Cornell University

Carlos Castillo-Chavez

Department of Biological Statistics and Computational Biology

Department of Theoretical and Applied Mechanics

Cornell University, Ithaca, New York, 14853

Dynamics Models for Tuberculosis Transmission and Control


• TB has a history as long as the human race.

• TB appears in the history of nearly every culture.

• TB was probably transferred from animals to humans.

• TB thrives in dense populations.

• It was the most important cause of death up to the middle of the 19th century.

Ancient disease


• Causative agent:Tuberculosis Bacilli (Koch, 1882).

• Preferred habitat Lung.

• Main Mode of transmissionHost-air-host.

• Immune ResponseImmune system tends to respond

quickly to initial invasion.

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 Caricature


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

• Most infected individuals never develop active TB (that is, become infectious).

• Most remain latently-infected for life.

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

Immune System Response Caricature


TB was the main cause of mortality

• 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.• TB’s nicknames: White Death,

Captain of Death, Time bomb


Per Capita Death Rate of TB


Current Situation

• Two to three million people around the world die of TB each year.

• Someone is infected with TB every second.• One third of the world population is infected

with TB ( the prevalence in the US is 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.


TB in the US


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.

• Lack of public knowledge about modes of TB transmission and prevention.


Earliest Models

• H.T. Waaler, 1962

• C.S. ReVelle, 1967

• S. Brogger, 1967

• S.H. Ferebee, 1967


Epidemiological Classes


Parameters


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


Epidemiology(Basic Reproductive Number, R0)

The expected number of secondary infections

produced by a “typical” infectious individual during his/her entire infectious periodwhen introduced in a population of mostly susceptibles at a demographic steady state.

• Sir Ronald Ross (1911)• Kermack and McKendrick (1927)


Epidemiology(Basic Reproductive Number, R0)

Frost (1937) wrote “…it is not necessary that transmission be immediately and completely prevented. It is necessary only that the rate of transmission be held permanently below the level at which a given number of infection spreading (i.e. open) cases succeed in establishing an equivalent number to carry on the succession”


R0

• Probability of surviving the latent stage:

• Average effective contact rate

• Average effective infectious period

krk

2

C

dr 1

1

krk

drC

210R


Demography

.)(

,)()(

,)(

1

2

IrdkEdtdI

ErkNIIEN C

dtdE

dINNFdtdN

Growth Logistic ,1)(

Growth lExponentia ,)(

KNrNNF

rNNF

F(N)=, Linear Growth


Exponential Growth(Three Thresholds)

The Basic Reproductive Number is

krk

drC

210R


Demography and Epidemiology

krrk

drrC

211R


Demography

Where

*2 durR

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

rnrmCkdCdkrmCrnrmdcrnrmCkdCdkrmcrnrmd

u


1R

Bifurcation Diagram(exponential growth )

r

1

0N

0I

0NI

0R

I

0NI

*uNI

)1(

)1( 0

2

2

RI

RN

1


Logistic Growth

krμ

kdrμ

βCR21

0

0

0

1

2 1R

Rrkdμ

kdμ

rR*


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 the 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

n Bifurcatio

calTranscriti Global

1


Particular Dynamics(R0 >1 and R2

* <1)

All trajectories approach the origin. Global attraction is verified numerically by randomly choosing 5000 sets of initial conditions.


Fast and Slow TB

(S. Blower, et al., 1995)

S

IE kE

Id)( E

SSIp )1(

SIp


.

,)1(

,

IdIkE SIpdtdI

EkE SIpdtdE

S SI dtdS

Fast and Slow TB


Variable Latency Period (Z. Feng, et al,2001)

p(s): proportion of infected (noninfectious) individuals who became infective s unit of time ago and who are still infected (non infectious).

Number of exposed from 0 to t who are alive and still in the E class

Number of those who progress to infectious from 0 to t and who are still alive in I class at time t


Variable Latency Period(differentio-integral model)

• E0(t): # of individuals in E class at t=0 and still in E class at time t

• I0: # of individuals in I class at t=0 and still in I class at time t


Exogenous Reinfection

S

IE kE

Id )( E

S NIcS

c NISp

T

T

rI

NIcT


TIESN

TNIcTrI-

dtdT

IrdkENIc Ep

dtdI

NIc TEk

NIc Ep

NIc S

dtdE

SNIc S

dtdS

.

,

,)(

,)(

,

Exogenous Reinfection


Backward Bifurcation


atjaat jtBacaatriatjdadt

atiaratklatidadt

)μ(a))l(t,a(kδv(t,a)σj(t,a)at stBacaatldadt

atvtBaatvaatsaatvdadt

atsaaat stBacaatsdadt

).,()(- ),()()()(),(),(

),,())((),(),(

, ),()()()(),(

),,()()(),()(),()(),(

),,())()(( ),()()()(),(

Age Structure Model


• : recruitment rate.• (a): age-specific probability of becoming infected.• c(a): age-specific per-capita contact rate.• (a); age-specific per-capita mortality rate.• k: progression rate from infected to infectious.• r: treatment rate.• : reduction proportion due to prior exposure to TB.• : reduction proportion due to vaccination.

Parameters


• p(t,a,a`): probability that an individual of age a has contact with an individual of age a` given that it has a contact with a member of the population .

• Proportionate mixing: p(t,a,a`)= p(t,a`)

Proportionate Mixing


),(),(),(),(),(),(

.)(),0( ),(),0( ),(),0( ),(),0( ),(),0(

:valuesBoundary .0)0,()0,()0,()0,( ,)0,(

: valuesInitial

mixing random ,),()(

)',()'()',(

')',,()',()',()(

00

000

0

0

atjatiatlatvatsatn

ajajaiaialalavavasas

tjtitltvts

duutnuc

atnacatp

daaatpatnatitB

Incidence and Mixing


Basic reproductive Number(by next generation operator)

.ageat natedbeen vaccinot has individual esusceptibl

ay that probabilit thedenotes ))(exp()(

))(1()()(

)()()()()(

0

)()(

0 00

a

dbbaF

aFaFa

ddeecpkr

kR

a

rk


Stability

There exists an endemic steady state whenever R0()>1.

The infection-free steady state is globally asymptotically stable when R0= R0(0)<1.


Optimal Vaccination Strategies

Two optimization problems:

If the goal is to bring R0() to pre-assigned value then find the vaccination strategy (a) that minimizes the total cost associated with this goal (reduced prevalence to a target level). If the budget is fixed (cost) find a vaccination strategy (a) that minimizes R0(), that is, that minimizes the prevalence.


Optimal Strategies

• One–age strategy: vaccinate the susceptible population

at exactly age A.• Two–age strategy: vaccinate part of the susceptible population at exactly age A1 and the remaining susceptibles at a later age A2.

Optimal strategy depends on data.


Challenging Questions associated with TB Transmission and Control

• Impact of immigration.• Antibiotic Resistance.• Role of public transportation.• Globalization—small world dynamics.• Time-dependent models.• Estimation of parameters and distributions.

Documents

Jun,2002MTBI Cornell University Carlos Castillo-Chavez Department of Biological Statistics and Computational Biology Department of Theoretical and Applied