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

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### Models of Dengue Fever and their Public Health Implications

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

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)

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- 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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- : 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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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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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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- : 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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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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One-age and two-age vaccination strategies

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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 nI back to N2 population, where nI (S1 /N1)= S1 go to S2 and nI (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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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

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

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Parameter estimation and simulation setup

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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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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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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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- 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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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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Epidemic basic reproductive number,R0

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

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Change in Host Behavior and its Impact on the Co-evolution of Dengue

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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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- 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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- 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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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:

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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