Carlos Castillo-Chavez

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Tutorials 4 Epidemiological Mathematical
Modeling, The Cases of Tuberculosis and Dengue.
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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A TB model with age-structure(Castillo-Chavez
and Feng. Math. Biosci., 1998)
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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.

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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

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Equations
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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.

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Age Structure Model with vaccination
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Age-dependent optimal vaccination strategies
(Feng, Castillo-Chavez, Math. Biosci., 1998)
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Basic reproductive Number (by next generation
operator)
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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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Reproductive numbers
Two optimization problems
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One-age and two-age vaccination strategies
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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).

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Generalized Household Model

• Incorporates contact type (close vs. casual) and
  focus on close and prolonged contacts.
• Generalized households become the basic
  epidemiological unit rather than individuals.
• Use epidemiological time-scales in model
  development and analysis.

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Transmission Diagram
                         
 
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Key Features

• Basic epidemiological unit cluster (generalized
  household)
• Movement of kE2 to I class brings nkE2 to N1
  population, where by assumptions nkE2(S2 /N2) go
  to S1 and nkE2(E2/N2) go to E1
• Conversely, recovery of ??I infectious bring n?I
  back to N2 population, where n?I (S1 /N1) ?? S1
  go to S2 and n?I (E1 /N1) ?? E1 go to E2

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Basic Cluster Model
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Basic Reproductive Number
Where
is the expected number of infections produced by
one infectious individual within his/her
cluster. denotes the fraction that survives over
the latency period.
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Diagram of Extended Cluster Model
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? (n)
Both close casual contacts are included in the
extended model. The risk of infection per
susceptible, ? , is assumed to be a nonlinear
function of the average cluster size n. The
constant p measures proportion of time of an
individual spanned within a cluster.
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Role of Cluster Size (General Model)
E(n) denotes the ratio of within cluster to
between cluster transmission. E(n) increases and
reaches its maximum value at The cluster size
n is optimal as it maximizes the relative impact
of within to between cluster transmission.
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Hoppensteadts Theorem (1973)
Reduced system
where x ? Rm, y ? Rn and ? is a positive real
parameter near zero (small parameter). Five
conditions must be satisfied (not listed here).
If the reduced system has a globally
asymptotically stable equilibrium, then the full
system has a g.a.s. equilibrium whenever 0lt ?
ltlt1.
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Bifurcation Diagram

• Global bifurcation diagram when 0lt?ltlt1 where ?
  denotes
• the ratio between rate of progression to active
  TB and the
• average life-span of the host (approximately).

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Numerical Simulations
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Concluding Remarks on Cluster Models

• A global forward bifurcation is obtained when ?
  ltlt 1
• E(n) measures the relative impact of close
  versus casual contacts can be defined. It defines
  optimal cluster size (size that maximizes
  transmission).
• Method can be used to study other transmission
  diseases with distinct time scales such as
  influenza

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TB in the US (1953-1999)
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TB control in the U.S.

• CDCs goal
• 3.5 cases per 100,000 by 2000
• One case per million by 2010.
• Can CDC meet this goal?

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Model Construction
Since d has been approximately equal to zero over
the past 50 years in the US, we only consider
Hence, N can be computed independently of TB.
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Non-autonomous model (permanent latent class of
TB introduced)
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Effect of HIV
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Parameter estimation and simulation setup
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N(t) from census data
N(t) is from census data and population projection
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Results
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Results

• Left New case of TB and data (dots)
• Right 10 error bound of new cases and data

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Regression approach
A Markov chain model supports the same result
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CONCLUSIONS
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Conclusions
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CDCs Goal Delayed

• Impact of HIV.
• Lower curve does not include HIV impact
• Upper curve represents the case rate when HIV is
  included
• Both are the same before 1983. Dots represent
  real data.

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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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Models of Dengue Fever and their Public Health
Implications

• Fabio Sánchez
• Ph.D. Candidate
• Cornell University
• Advisor Dr. Carlos Castillo-Chavez

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Outline

• Introduction
• Single strain model
• Two-strain model with collective behavior change
• Single outbreak model
• Conclusions

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Introduction

• Mosquito transmitted disease
• 50 to 100 million reported cases every year
• Nearly 2.5 billion people at risk around the
  world (mostly in the tropics)
• Human generated breeding sites are a major
  problem.

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• Dengue hemorrhagic fever (worst case of the
  disease)
• About 1/4 to 1/2 million cases per year with a
  fatality ratio of 5 (most of fatalities occur in
  children)

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• Four antigenically distinct serotypes (DEN-1,
  DEN-2, DEN-3 and DEN-4)
• Permanent immunity but no cross immunity
• After infection with a particular strain there is
  at most 90 days of partial immunity to other
  strains

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• There is geographic strain variability.
• Each region with strain i, does not have all the
  variants of strain i.
• Geographic spread of new variants of existing
  local strains poses new challenges in a globally
  connected society.

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• Aedes aegypti (principal vector)
• viable eggs can survive without water for a long
  time (approximately one year)
• adults can live 20 to 30 days on average.
• only females take blood meals
• latency period of approximately 10 days later (on
  the average).
• Aedes albopictus a.k.a. the Asian tiger mosquito

- can also transmit dengue
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Transmission Cycle
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The Model

• Coupled nonlinear ode system
• Includes the immature (egg/larvae) vector stage
• Incorporates a general recruitment function for
  the immature stage of the vector
• SIR model for the host (human) system--following
  Rosss approach (1911)
• Model incorporates multiple vector densities via
  its recruitment function

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State Variables

• Vector State Variables
• E? viable eggs (were used as the larvae/egg
  stage)
• V? adult mosquitoes
• J? infected adult mosquitoes
• Host State Variables (Humans)
• S? susceptible hosts
• I? infected hosts
• R? recovered hosts

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Caricature of the Model
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Epidemic basic reproductive number, R0

• The average number of secondary cases of a
  disease caused by a typical infectious
  individual.

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Multiple steady states (backward bifurcation)
With control measures
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Change in Host Behavior and its Impact on the
Co-evolution of Dengue
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Introduction to the Model
Host System
Vector System

• Our model expands on the work of Esteva and
  Vargas, by incorporating a behavioral change
  class in the host system and a latent stage in
  the vector system.

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Basic Reproductive Number, R0

• The Basic Reproductive number represents the
  number of secondary infections caused by a
  typical infectious individual
• Calculated using the Next Generation Operator
  approach

Where,
- represents the proportion of mosquitoes that
make it from the latent stage to the infectious
stage
- represents the average time of the host spent
in the infectious stage
- represents the average life-span of the
mosquito
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Regions of Stability of Endemic Equilibria
From the stability analysis of the endemic
equilibria, the following necessary condition
arose
which defines the regions illustrated above.
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Conclusions

• A model for the transmission dynamics of two
  strains of dengue was formulated and analyzed
  with the incorporation of a behavioral change
  class.
• Behavioral change impacts the disease dynamics.
• Results support the necessity of the behavioral
  change class to model the transmission dynamics
  of dengue.

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A Comparison Study of the 2001 and 2004 Dengue
Fever Outbreaks in Singapore
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Outline

• Data and the Singapore health system
• Single outbreak model
• Results
• Conclusions

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Aedes aegypti

• Has adapted well to humans
• Mostly found in urban areas
• Eggs can last up to a year in dry land

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Singapore Health System and Data
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Singapore Health System and Data

• Prevention and Control
• The National Environment Agency carries out
  entomological investigation around the residence
  and/or workplace of notified cases, particularly
  if these cases form a cluster where they are
  within 200 meters of each other. They also carry
  out epidemic vector control measures in outbreak
  areas and areas of high Aedes breeding habitats.

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Preventive Measures

• Clustering of cases by place and time
• Intensified control actions are implemented in
  these cluster areas
• Surveillance control programs
• Vector control
• Larval source reduction (search-and-destroy)
• Health education
• House to house visits by health officers
• Dengue Prevention Volunteer Groups (National
  Environment Agency)
• Law enforcement
• Large fines for repeat offenders

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Reported cases from 2001-up to date
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Single Outbreak Model

• SEIR - host (humans)

VLJ - vectors (mosquitoes)
MVLJ
NSEIR
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2001 Outbreak
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2001 Outbreak
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2004 Outbreak
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2004 Outbreak
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Conclusions

• Monitoring of particular strains may help prevent
  future outbreaks
• Elimination of breeding sites is an important
  factor, however low mosquito densities are
  capable of producing large outbreaks
• Having a well-structured public health system
  helps but other approaches of prevention are
  needed
• Transient (tourists) populations could possibly
  trigger large outbreaks
• By introduction of a new strain
• Large pool of susceptible increases the
  probability of transmission

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Acknowledgements

• Collaborators
• Chad Gonsalez (ASU)
• David Murillo (ASU)
• Karen Hurman (N.C. State)
• Gerardo Chowell-Puente (LANL)
• Ministry of Health of Singapore
• Prof. Laura Harrington (Cornell)
• Advisor Dr. Carlos Castillo-Chavez