Carlos Castillo-Chavez

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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.h
tm Singapore, 08-23-2005

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

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Primary CollaboratorsJuan 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)
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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.

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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 .

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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-N
  ew York. Edited by Carlos Castillo-Chavez with
  Pauline van den Driessche, Denise Kirschner and
  Abdul-Aziz Yakubu, 2002

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Outline

• Brief Introduction to TB
• Long-term TB evolution
• Dynamical models for TB transmission
• The impact of social networks cluster models
• A control strategy of TB for the U.S. TB and HIV

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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.

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Transmission Process

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

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Immune System Response

• 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.

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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.

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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.

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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.

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Basic Model Framework

• NSEIT, Total population
• F(N) Birth and immigration rate
• B(N,S,I) Transmission rate (incidence)
• B(N,S,I) Transmission rate (incidence)

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Model Equations
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R0

• Probability of surviving to infectious stage
• Average successful contact rate
• Average infectious period

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Phase Portraits
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Bifurcation Diagram
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Fast and Slow TB (S. Blower, et al., 1995)
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Fast and Slow TB
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What is the role of long and variable latent
periods?(Feng, Huang and Castillo-Chavez. JDDE,
2001)
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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

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The model
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The reproductive number
Result The qualitative behavior is similar to
that of the ODE model. Q What happens if we
incorporate resistant strains?
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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
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Bifurcation Diagram
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A TB model with exogenous reinfection(Feng,
Castillo-Chavez and Capurro. TPB, 2000)
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Exogenous Reinfection
E
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The model
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• Basic reproductive number is
• Note R0 does not depend on p.
• A backward bifurcation occurs at some pc (i.e.,
  E exists for R0 lt 1)

Backward bifurcation
Number of infectives I vs. time
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Backward Bifurcation
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Dynamics depends on initial values
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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

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A diagram for two-strain TB transmission
?
?
?d1
?
?
?
?1
r1
k1
I1
L1
T
S
pr2
(1-(pq))r2
?
?2
?
qr2
?
L2
K2
?d2
I2
r2 is the treatment rate for individuals with
active TB q is the fraction of treatment failure
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The two-strain TB model
r2 is the treatment rate for individuals with
active TB q is the fraction of treatment
failure
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Reproductive numbers

• For the drug-sensitive strain
• For the drug-resistant strain

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Equilibria and stability

• There are four possible equilibrium points
• E1 disease-free equilibrium (always exists)
• E2 boundary equilibrium with L2 I2 0 (R1 gt
  1 q 0)
• E3 interior equilibrium with I1 gt 0 and I2 gt 0
  (conditional)
• E4 boundary equilibrium with L1 I1 0 (R2 gt
  1)
• Stability dependent on R1 and R2

Bifurcation diagram
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Fraction of infections vs time
q gt0
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Contour plot of the fraction of resistant TB,
J/N, vs treatment rate r2 and fraction of
treatment failure q

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Optimal control strategies of TB through
treatment of sensitive TBJung, 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

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A diagram for two-strains TB transmission with
controls
?
?
?d1
?
?
r1u1
?
?1
k1
I1
L1
T
S
(1-u2)pr2
?
?2
?
(1-(1-u2)(pq))r2
(1-u2) qr2
?
L2
K2
?d2
I2
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The two-strain system with time-dependent
controls(Jung, Lenhart and Feng. DCDSB, 2002)

• u1(t) Effort to identify and treat typical TB
  individuals
• 1-u2(t) Effort to prevent failure of treatment
  of active TB
• 0 lt u1(t), u2(t) lt1 are Lebesgue integrable
  functions

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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.

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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 Continuous Dynamical Systems

• Guess the value of the control over the simulated
  time.
• Solve the state system forward in time using the
  Runge-Kutta scheme.
• Solve the adjoint system backward in time using
  the Runge-Kutta scheme using the solution of the
  state equations from 2.
• 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.

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Optimal control strategies Jung, E., Lenhart, S.
and Feng, Z. (2002), Optimal control of
treatments in a two-strain tuberculosis model,
Discrete and Continuous Dynamical Systems
u2(t)
u1(t)
Control
without control
TB cases (L2I2)/N
With control
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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 Continuous
Dynamical Systems
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Demography
F(N)?, a constant
Results More than one Threshold Possible
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Bifurcation Diagram--Not Complete or Correct
Picture
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Demography and Epidemiology
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Demography
Where
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  Bifurcation Diagram (exponential growth )
 
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Logistic Growth
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Logistic Growth (contd)

• If R2 gt1
• 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 gt1.
• When R0 ? 1, there is no qualitative difference
  between logistic and exponential growth.

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Bifurcation Diagram
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Particular Dynamics(R0 gt1 and R2 lt1)
All trajectories approach the origin. Global
attraction is verified numerically by randomly
choosing 5000 sets of initial conditions.
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Particular Dynamics(R0 gt1 and R2 lt1)
All trajectories approach the origin. Global
attraction is verified numerically by randomly
choosing 5000 sets of initial conditions.
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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 R0lt1, 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.

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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.

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Per Capita Death Rate of TB
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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.
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Births and immigration adjusted to fit data
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Life Expectancy in Years
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Incidence k E
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Incidence of TB since 1850
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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