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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
Singapore,
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)
Vaccinated
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15. 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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17. 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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18. Reproductive numbers Two optimization problems: R(y) < R* 11/9/2018
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19. One-age and two-age vaccination strategies
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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 A1 and the remaining susceptibles at a later age A2.
. Selected optimal strategy depends on cost function (data).
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21. 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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25. 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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26. Diagram of Extended Cluster Model
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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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30. 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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31. Hoppensteadt’s Theorem (1973)
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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Bifurcation Diagram 1
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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33. Numerical Simulations
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34. 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 ( )
Europe may include TB educational facilities in high school curriculum (Porter & Grange, 2000)
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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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39. Non-autonomous model (permanent latent class of TB introduced)
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Effect of HIV
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41. Parameter estimation and simulation setup
0.22 c 10 k
0.001 r1
0.05 r2 r3
0.65 p
0.1
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N(t) from census data
N(t) is from census data and population projection
Initial
I(0)
L1(0)
L2(0)
Values
874230
106
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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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45. Regression approach A Markov chain model supports the same result
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A Markov chain model supports the same result

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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 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, , 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, , 2000
Aparicio, J., A. Capurro and C. Castillo-Chavez, Markers of disease evolution: the case of tuberculosis, Journal of Theoretical Biology, 215: , 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, , 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): , 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: , 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, , 1998.
Castillo-Chavez,C and B. Song: Dynamical Models of Tuberculosis and applications, Journal of Mathematical Biosciences and Engineering, 1(2): , 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, , 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: , 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, , 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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52. 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.
Infection with one type does not typically provide immunity to a second type
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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
Gubler, D.J.: Dengue. In: Monath T.P. (ed.) The arbovirus: Epidemiology and Ecology. Vol II, pp CRC press, Florida, USA, (1986).
Mendez Galvan, J.F., Castellanos, R.M., Manual para la vigilancia epidemiologica del dengue. Secretaria de Salud, Mexico, D.F., (1994).
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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)
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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64. Epidemic basic reproductive number, R0
The average number of secondary cases of a disease caused by a “typical” infectious individual.
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65. Multiple steady states (backward bifurcation)
With control measures
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66. Change in Host Behavior and its Impact on the Co-evolution of Dengue
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67. 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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68. 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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69. 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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74. Singapore Health System and Data
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75. 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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77. Reported cases from 2001-up to date
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Single Outbreak Model
SEIR - host (humans)
VLJ - vectors (mosquitoes)
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
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