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

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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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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

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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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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

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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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Vaccinated

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()>1.

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

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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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R(y) < R*

Reproductive numbers

Two optimization problems:

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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
• 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 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

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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 measuresproportion 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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Full system

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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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

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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)

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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

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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Effect of HIV

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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

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A Markov chain model supports the same result

CONCLUSIONS

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Conclusions

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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
• 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 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

Fabio Sánchez

Ph.D. Candidate

Cornell University

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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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• 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 Ross’s 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)

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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With control measures

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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

VLJ - vectors (mosquitoes)

SEIR - host (humans)

M=V+L+J

N=S+E+I+R

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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:

David Murillo (ASU)

Karen Hurman (N.C. State)

Gerardo Chowell-Puente (LANL)

Ministry of Health of Singapore

Prof. Laura Harrington (Cornell)