jun,2002mtbi cornell university carlos castillo-chavez department of biological statistics and...
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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
Jun,2002 MTBI Cornell University
• 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
Jun,2002 MTBI Cornell University
• 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
Jun,2002 MTBI Cornell University
• 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
Jun,2002 MTBI Cornell University
• 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
Jun,2002 MTBI Cornell University
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
Jun,2002 MTBI Cornell University
Per Capita Death Rate of TB
Jun,2002 MTBI Cornell University
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.
Jun,2002 MTBI Cornell University
TB in the US
Jun,2002 MTBI Cornell University
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.
Jun,2002 MTBI Cornell University
Earliest Models
• H.T. Waaler, 1962
• C.S. ReVelle, 1967
• S. Brogger, 1967
• S.H. Ferebee, 1967
Jun,2002 MTBI Cornell University
Epidemiological Classes
Jun,2002 MTBI Cornell University
Parameters
Jun,2002 MTBI Cornell University
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
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),,(' ITNB
Jun,2002 MTBI Cornell University
Model Equations
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Jun,2002 MTBI Cornell University
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)
Jun,2002 MTBI Cornell University
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”
Jun,2002 MTBI Cornell University
R0
• Probability of surviving the latent stage:
• Average effective contact rate
• Average effective infectious period
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dr 1
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krk
drC
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Jun,2002 MTBI Cornell University
Demography
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Growth lExponentia ,)(
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F(N)=, Linear Growth
Jun,2002 MTBI Cornell University
Exponential Growth(Three Thresholds)
The Basic Reproductive Number is
krk
drC
210R
Jun,2002 MTBI Cornell University
Demography and Epidemiology
krrk
drrC
211R
Jun,2002 MTBI Cornell University
Demography
Where
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))((2))()(())((42)((*
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u
Jun,2002 MTBI Cornell University
1R
Bifurcation Diagram(exponential growth )
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Jun,2002 MTBI Cornell University
Logistic Growth
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rR*
Jun,2002 MTBI Cornell University
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.
Jun,2002 MTBI Cornell University
Bifurcation Diagram
0R
*I
n Bifurcatio
calTranscriti Global
1
Jun,2002 MTBI Cornell University
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.
Jun,2002 MTBI Cornell University
Fast and Slow TB
(S. Blower, et al., 1995)
S
IE kE
Id)( E
SSIp )1(
SIp
Jun,2002 MTBI Cornell University
.
,)1(
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Fast and Slow TB
Jun,2002 MTBI Cornell University
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
Jun,2002 MTBI Cornell University
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
Jun,2002 MTBI Cornell University
Exogenous Reinfection
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c NISp
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NIcT
Jun,2002 MTBI Cornell University
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Exogenous Reinfection
Jun,2002 MTBI Cornell University
Backward Bifurcation
Jun,2002 MTBI Cornell University
atjaat jtBacaatriatjdadt
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),,())((),(),(
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Age Structure Model
Jun,2002 MTBI Cornell University
• : 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
Jun,2002 MTBI Cornell University
• 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
Jun,2002 MTBI Cornell University
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Incidence and Mixing
Jun,2002 MTBI Cornell University
Basic reproductive Number(by next generation operator)
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ay that probabilit thedenotes ))(exp()(
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)()()()()(
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rk
Jun,2002 MTBI Cornell University
Stability
There exists an endemic steady state whenever R0()>1.
The infection-free steady state is globally asymptotically stable when R0= R0(0)<1.
Jun,2002 MTBI Cornell University
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.
Jun,2002 MTBI Cornell University
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.
Jun,2002 MTBI Cornell University
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.