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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.htm
Singapore, 08-23-2005
Carlos Castillo-Chavez
Joaquin Bustoz Jr. Professor
Arizona State University
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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)
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Our work on TB




Aparicio, J., A. Capurro and C. Castillo-Chavez, “On the long-term dynamics and reemergence of tuberculosis.” In: Mathematical Approaches for Emerging and Reemerging
Infectious Diseases: An Introduction, IMA Volume 125, 351-360, Springer-Veralg, BerlinHeidelberg-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
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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: 187205, 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
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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
B' ( N , T , I )
F (N )
S
S
B( N , S , I )
E
kE
I
E
r1 I
(  d ) I
r 2E
•
•
•
•
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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)
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T
T
Model Equations
dS F(N)  CS I  I,
N
dt
dE   CS I ( kr )E'CT I ,
2
N
N
dt
dI kE( dr )E,
1
dt
dT r Er I  ' CT I  T,
1
N
dt 2
N SEI T

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R0
R0 





C





k
 r1 d  r2 k






Probability of surviving to infectious stage:
Average successful contact rate
Average infectious period
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C
1
  r1  d
k
  r2  k
Phase Portraits
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Bifurcation Diagram
I*
Global Transcriti cal
Bifurcatio n
1
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R0
Fast and Slow TB
(S. Blower, et al., 1995)
pSI
 S
(1  p)SI
S
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kE
E
E
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I
(  d)I
Fast and Slow TB
dS     SI  S ,
dt
dE  (1 p) SI  kE  E,
dt
dI  p SI  kE  dI  I.
dt
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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
I*
Global Transcriti cal
Bifurcatio n
1
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R0
A TB model with exogenous reinfection
(Feng, Castillo-Chavez and Capurro. TPB, 2000)
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Exogenous Reinfection
I
p cS
EN
cS I
 S
N
S
kE
E
E
I
rI
(   d) I
cT I
N
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T
T
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 < 1)
Backward bifurcation
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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




1
S
’’
L1
*
2
qr2
L2
+d1
k1

I1
pr2
r1
(1-(p+q))r2
T
*
K2
+d2
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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 Resistant TB only
Coexistence
There are four possible equilibrium points:
E1 : disease-free equilibrium (always exists)
E2 : boundary equilibrium with L2 = I2 = 0 (R1 > 1; q = 0)
Sensitive TB only
q=0
E3 : interior equilibrium with I1 > 0 and I2 > 0 (conditional)
E4 : boundary equilibrium with L1 = I1 = 0 (R2 > 1)
Resistant TB only
Stability dependent on R1 and R2
Coexistence
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q>0
Bifurcation diagram
Resistant TB only
TB-free
Resistant TB only
Coexistence
Sensitive TB only
Fraction of infections vs time
q=0
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q >0
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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 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
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A diagram for two-strains TB transmission with controls




1
S
*
2
L2
’’
+d1

r1u1
k1
L1
(1-u2)pr2
(1-u2) qr2
I1
*
K2
+d2
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I2
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T
(1-(1-u2)(p+q))r2
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 < u1(t), u2(t) <1 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
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.
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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)
Control
TB cases
(L2+I2)/N
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u1(t)
without control
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
u1(t)
u2(t)
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Demography
dN F(N)N dI,
dt
dE   C(N E I) I ( kr )E,
2
N
dt
dI kE( dr )I.
1
dt

F(N)=, a constant
F(N)rN, Exponential Growth


N

F(N)rN 1 , Logistic Growth
K 

Results: More than one Threshold Possible

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Bifurcation Diagram--Not
Complete or Correct Picture
I*
Global Transcriti cal
Bifurcatio n
1
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R0
Demography and Epidemiology
R0 





C





k
 r1 d  r2 k

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




Demography
Where:
r


R2  *
du
2  4d (C  d )(kC  mr nr )c  (d (mr  nr )  C )(mr  k )
d
(
m

n


c
(
m

k
)
r
r
r
u* 
2d (C  d )(kC  mr nr )
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Bifurcation Diagram
(exponential
growth )
R1
r
N 0
I
N
u
*
 N  0 ( R2  1)

 I   ( R2  1)
1
I
N
I 
0
I
N
I 0
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0
R0
1
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Logistic Growth




βC
k



R0 


μ r1  d  μ r2  k 






R* 
2
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r
R0 1
k
μ d
μ  d  k  r1 R0
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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.
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Bifurcation Diagram
I*
Global Transcriti cal
Bifurcatio n
1
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R0
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.
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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.
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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 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.
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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
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