Carlos Castillo-Chavez

1
Challenges in Epidemiology The case of
tuberculosis
2005 World Conference on Natural Resource
Modeling Humboldt State University - June 14 -
17 http//www.humboldt.edu/ecomodel/rmaconference
call.htm

• Carlos Castillo-Chavez
• Joaquin Bustoz Jr. Professor
• Arizona State University
• March 25, 2005

2
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.

3
Transmission Process

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

4
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.

5
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.

6
Tuberculosis (TB)
About one out of three persons in the world is
infected with Mycobacterium tuberculosis (TB).
The situation varies considerable from country to
country. In India, for example, about 50 of the
population is infected while in the US the
prevalence is less than 10. There is a vaccine
which is used almost everywhere but in the US.
The effectiveness of the vaccine is questionable.
7
Ro
Number of secondary infections
generated by a typical infectious individual in
a population of mostly susceptibles
Rolt1 No epidemic Rogt1
Epidemic Ro1 Stasis
8
Ro 2
9
Ro 2
10
Ro 2
( End )
11
Ro lt 1
12
Ro lt 1
13
Ro lt 1
14
Ro lt 1
( End )
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Ro
Number of secondary infections
generated by a typical infectious individual in
a population of mostly susceptibles
Rolt1 No epidemic Role of vaccination to reduce
Ro and eliminate the disease. Rogt1 Epidemic
(often leading to and endemic state) Role of
vaccination to reduce Ro but disease still
endemic
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Basic Epidemiological Models SIS
Susceptible - Infected - Susceptible
17
S(t) susceptibles at time t I(t) infected
assumed infectious, at time t N Total population
size (sum of epidemiological classes)
18
SIS - Equations
Parameters
Per-capita death (or birth) rate
Per-capita recovery rate
Transmission coefficient
19
SIS - Model
20
Basic Reproductive Number
average-dead adjusted infectious period
Number of secondary infections generated by a
typical Individual in a population of
susceptible at steady state.
21
SIS - Model
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SIS Transcritical Bifurcation
unstable
23
Harvesting - Models
24
F(I)
0
I
K
Objective
F(I)
minimal disease reproductive window as
increases (temporarily effective vaccine)
0
K
I
25
G(N)
0
N
Objective
G(N)
maximal harvesting rate (Effort E increases)
0
N
26
Harvesting of Populations (Trivial Case One
dimension)
Mean Gain per capita from the harvesting policy
gain per unit harvested
27
Persistent Solutions
Problems
Maximize the mean gain per capita with
the side condition and constant
population size
or
an optimal gain becomes
(trivial result )
Only the constant is fixed since we are
requiring constant population size ( ).
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Model with Age Structure

• n(t,a) Density of individuals with age a at
  time t.

of individuals with ages in (a1 , a2) at time t
Age-specific per-capita death rate Age-specific
per-capita fertility rate
29
Demographic Steady State

• Individuals get older or die
• Some individuals reproduce before they die
• n(t,a) density of individual with age a at time t

n(t,a) satisfies the Lotka-Volterra-Mackendrick
Equation
We assume that the total population density has
reached this demographic steady state.
30
Age Structured Population case

• State space has infinite dimension and fixing
  population size does not determine the harvesting
  rate.
• Under appropriate conditions there is a
  persistent (exponential) solution

where
Lotkas r ( growth exponent) and
The stable age-distribution
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Lotkas characteristic equation
is unique real root.
Once is know then
This is the stable age distribution where
is the number of new borns ( ages in
).
age pyramid growing populations
decaying population
32
Stability of
It is stable in the orbital sense, that is, if a
perturbation is applied then the solution is
reestablished, up to a multiplicative factor.
Assumptions
Population model and optimality problems
considered are homogeneous of degree one. Hence,
we can rescale by setting
33
Age-specific Culling or Harvesting by
can be arbitrarily large. In fact, we allow for
the possibility that
where is a - function peaked
at age

34
Interpretation
When a cohort passes through the age , the
fraction is culled instantaneously.
Net gain from harvesting one individual of age
Total gain per unit of time from applying policy

We require
and, consequently,
It is also the per-capita gain per unit of time
35
Lotka Characteristic Equation when harvesting

and
where
Number of birth per unit of time
Requires
or population goes to extinction without
harvesting
and
population grows
36
Harvesting with maximal intensity
Requirement of constant population size is
satisfied and constant can be
determined
Problem 1
Maximize the gain functional
Subject to the sign condition
for
And the balance condition
37
Example
are positive constant, that is, no age structure
Let be any policy with
Hence, the per-capita gain per unit of time is
, that is, it is independent of choice
of
One could choose . This case, the
Poisson case ( constants) is in fact
atypical.
38
Problem 1 Take the transformation
Transformation takes the rate into a hazard
function .
Or after the inversion
39
The transition from
puts constraints on
for
If
then
since
Define the Kernel Functions
40
Gain functional and constraint
condition
Can be rewritten as
and
Functional and are linear in
and
41
Problem 1
Maximize under conditions
Result ( Hadeler and Muller, 2005 )
In a generic situation optimal policies harvest
either one age class only or at most two ages
classes
Proof ( Lagrange functions - looking for maximum)
42
Kuhn-Tucker conditions for this problem are
(and formally
)
Two cases I)
maximum at single age
II)
need not vanish
Maximum ( generic situation can be assumed at two
points )
43
Per Capita Death Rate of TB
44
Incidence of TB since 1850
45
Basic Epidemiological Models SIS and SIR
Susceptible - Infected - Recovered
46
S(t) susceptible at time t I(t) infected
assumed infectious at time t R(t) recovered,
permanently immune N Total population size
(SIR)
47
SIR - Equations
Parameters
Per-capita death (or birth) rate
Per-capita recovery rate
Transmission coefficient
48
SIR - Model (Invasion)
49
Establishment of a Critical Mass of
Infectives!Ro gt1 implies growth while Rolt1
extinction.
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Phase Portraits
51
SIR Transcritical Bifurcation
unstable
52
Bifurcation Diagram
53
SIR Model with Age Structure

• s(t,a) Density of susceptible individuals
  with age a at time t.
• i(t,a) Density of infectious individuals
  with age a at time t.
• r(t,a) Density of recovered individuals
  with age a at time t.

of susceptible individuals with ages in (a1 ,
a2) at time t
of infectious individuals with ages in (a1 ,
a2) at time t
of recovered individuals with ages in (a1 , a2)
at time t
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Parameters

• ? recruitment/birth rate.
• ?(a) age-specific probability of becoming
  infected.
• c(a) age-specific per-capita contact rate.
• ?(a) age-specific per-capita mortality rate.
• ?(a) age-specific per-capita recovery rate.

55
Mixing
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 .
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Mixing Rules

• p(t,a,a) 0
• Proportionate mixing

57
Equations
58
Demographic Steady State
n(t,a) density of individual with age a at time t
n(t,a) satisfies the Mackendrick Equation
We assume that the total population density has
reached this demographic steady state.
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Parameters

• ? 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.

60
Age Structure Model with vaccination
61
Incidence and Mixing
62
F(I)
0
I
K
Objective
F(I)
minimal disease reproductive window
0
K
I
63
Basic reproductive Number (by next generation
operator)
64
Stability
There exists an endemic steady state whenever
R0(?)gt1. The infection-free steady state is
globally asymptotically stable when R0 R0(0)lt1.
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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.

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Optimal Strategies

• Oneage strategy vaccinate the susceptible
  population at exactly age A.
• Twoage strategy vaccinate part of the
  susceptible population at exactly age A1 and the
  remaining susceptibles at a later age A2.
• . Selected optimal strategy depends on cost
  function (data).

67
Some References

• On the role of variables latent periods in
  mathematical models for tuberculosis.(Castillo-Cha
  vez, Feng, and Huang) J. Dynamical and Diff. Eq.,
  (2001)
• A model for TB with exogenous re-infection
  (Castillo-Chavez, Feng, and Capurro) J.
  Theoretical Population Biology (2000)
• Global stability of an age-structure model for TB
  and its applications to optimal vaccination
  strategies (Castillo-Chavez, Feng) J. Math.
  Biosci. (1998)
• Role of public transportation (Castillo-Chavez,
  Capurro, Velasco-Hernanderz, and Zellner)
  Aportaciones Matematicas, Serie Communications
  (1998)

68
Some References

• Long-term TB evolution (Aparicio, Capurro, and
  Castillo-Chavez) (IMA Volume 125)
• Transmission and dynamics of tuberculosis on
  general households, J. of Theoretical Biology
  (2000).
• Markers of disease evolution the case of
  tuberculosis, Journal of Theoretical Biology,
  215 227-237, March 2002 (Aparicio, Capurro and
  Castillo-Chavez)
• Tuberculosis Models with Fast and Slow Dynamics
  The Role of Close and Casual Contacts,
  Mathematical Biosciences 180 187-205, December
  2002(Song, Castillo-Chavez and Aparicio).
• Dynamical Models of Tuberculosis and
  applications, Journal of Mathematical Biosciences
  and Engineering, 1(2) 361-404, 2004
  (Castillo-Chavez, and Song).