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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)

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carlos castillo chavez joaquin bustoz jr professor arizona state university

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.htm

Singapore, 08-23-2005

Carlos Castillo-Chavez

Joaquin Bustoz Jr. Professor

Arizona State University

Arizona State University

a tb model with age structure castillo chavez and feng math biosci 1998
A TB model with age-structure(Castillo-Chavez and Feng. Math. Biosci., 1998)

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slide6

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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slide7

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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slide8

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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slide9

Mixing Rules

  • p(t,a,a`) ≥ 0
  • Proportionate mixing:

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slide10

Equations

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slide11

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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slide12

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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slide13

Age Structure Model with vaccination

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age dependent optimal vaccination strategies feng castillo chavez math biosci 1998

Vaccinated

Age-dependent optimal vaccination strategies(Feng, Castillo-Chavez, Math. Biosci., 1998)

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slide15

Basic reproductive Number

(by next generation operator)

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slide16

Stability

There exists an endemic steady state whenever R0()>1.

The infection-free steady state is globally asymptotically stable when R0= R0(0)<1.

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slide17

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

R(y) < R*

Reproductive numbers

Two optimization problems:

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slide20

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 A1and the remaining susceptibles at a later age A2.

. Selected optimal strategy depends on cost function (data).

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generalized household model
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
Transmission Diagram

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key features
Key Features
  • Basic epidemiological unit: cluster (generalized household)
  • Movement of kE2 to I class brings nkE2 to N1population, 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
Basic Cluster Model

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basic reproductive number
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
Diagram of Extended Cluster Model

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slide27
 (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 measuresproportion of time of an “individual spanned within a cluster.

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role of cluster size general model
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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hoppensteadt s theorem 1973

Full system

Hoppensteadt’s 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 0<  <<1.

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bifurcation diagram

1

Bifurcation Diagram

Global bifurcation diagram when 0<<<1 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
Numerical Simulations

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concluding remarks on cluster models
Concluding Remarks on Cluster Models
  • A global forward bifurcation is obtained when  << 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
TB in the US (1953-1999)

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slide37

TB control in the U.S.

CDC’s 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
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
Non-autonomous model (permanent latent class of TB introduced)

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slide40

Effect of HIV

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n t from census data
N(t) from census data

N(t) is from census data and population projection

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results
Results

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results44
Results
  • Left: New case of TB and data (dots)
  • Right: 10% error bound of new cases and data

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regression approach
Regression approach

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A Markov chain model supports the same result

slide46

CONCLUSIONS

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conclusions
Conclusions

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cdc s goal delayed
CDC’s 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
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 tb50
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 tb51
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 Biosciences180: 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-New 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

Models of Dengue Fever and their Public Health Implications

Fabio Sánchez

Ph.D. Candidate

Cornell University

Advisor: Dr. Carlos Castillo-Chavez

Arizona State University

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

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introduction
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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slide55
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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slide56
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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slide58
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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slide59
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
Transmission Cycle

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slide61

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 Ross’s approach (1911)
  • Model incorporates multiple vector densities via its recruitment function

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slide62

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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slide63

Caricature of the Model

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epidemic basic reproductive number r 0
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
Multiple steady states (backward bifurcation)

With control measures

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slide67

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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slide68

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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slide69

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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slide70

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

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

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outline72
Outline
  • Data and the Singapore health system
  • Single outbreak model
  • Results
  • Conclusions

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aedes aegypti
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
Singapore Health System and Data

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singapore health system and data75
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
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
Reported cases from 2001-up to date

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single outbreak model
Single Outbreak Model

VLJ - vectors (mosquitoes)

SEIR - host (humans)

M=V+L+J

N=S+E+I+R

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2001 outbreak
2001 Outbreak

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2001 outbreak80
2001 Outbreak

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2004 outbreak
2004 Outbreak

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2004 outbreak82
2004 Outbreak

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conclusions83
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
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

Arizona State University

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