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 11/9/2018 Arizona State University

Arizona State University A TB model with age-structure (Castillo-Chavez and Feng. Math. Biosci., 1998) 11/9/2018 Arizona State University

Arizona State University 11/9/2018 Arizona State University

Arizona State University 11/9/2018 Arizona State University

Arizona State University 11/9/2018 Arizona State University

Arizona State University 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 11/9/2018 Arizona State University

Arizona State University 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. 11/9/2018 Arizona State University

Arizona State University 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 . 11/9/2018 Arizona State University

Arizona State University Mixing Rules p(t,a,a`) ≥ 0 Proportionate mixing: 11/9/2018 Arizona State University

Arizona State University Equations 11/9/2018 Arizona State University

Arizona State University 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. 11/9/2018 Arizona State University

Arizona State University 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. 11/9/2018 Arizona State University

Arizona State University Age Structure Model with vaccination 11/9/2018 Arizona State University

Arizona State University Age-dependent optimal vaccination strategies (Feng, Castillo-Chavez, Math. Biosci., 1998) Vaccinated 11/9/2018 Arizona State University

Basic reproductive Number (by next generation operator) 11/9/2018 Arizona State University

Arizona State University Stability There exists an endemic steady state whenever R0()>1. The infection-free steady state is globally asymptotically stable when R0= R0(0)<1. 11/9/2018 Arizona State University

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. 11/9/2018 Arizona State University

Reproductive numbers Two optimization problems: R(y) < R* 11/9/2018 Arizona State University

One-age and two-age vaccination strategies 11/9/2018 Arizona State University

Arizona State University 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). 11/9/2018 Arizona State University

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. 11/9/2018 Arizona State University

Arizona State University     Transmission Diagram 11/9/2018 Arizona State University

Arizona State University 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 11/9/2018 Arizona State University

Arizona State University Basic Cluster Model 11/9/2018 Arizona State University

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. 11/9/2018 Arizona State University

Diagram of Extended Cluster Model 11/9/2018 Arizona State University

Arizona State University 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. 11/9/2018 Arizona State University

Arizona State University 11/9/2018 Arizona State University

Arizona State University 11/9/2018 Arizona State University

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. 11/9/2018 Arizona State University

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. 11/9/2018 Arizona State University

Arizona State University 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). 11/9/2018 Arizona State University

Numerical Simulations 11/9/2018 Arizona State University

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 11/9/2018 Arizona State University

Arizona State University TB in the US (1953-1999) Europe may include TB educational facilities in high school curriculum (Porter & Grange, 2000) 11/9/2018 Arizona State University

Arizona State University 11/9/2018 Arizona State University

Arizona State University 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? 11/9/2018 Arizona State University

Arizona State University 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. 11/9/2018 Arizona State University

Non-autonomous model (permanent latent class of TB introduced) 11/9/2018 Arizona State University

Arizona State University Effect of HIV 11/9/2018 Arizona State University

Parameter estimation and simulation setup  0.22 c 10 k 0.001 r1 0.05 r2 r3 0.65 p 0.1 11/9/2018 Arizona State University

Arizona State University N(t) from census data N(t) is from census data and population projection Initial I(0) L1(0) L2(0) Values 874230 106 11/9/2018 Arizona State University

Arizona State University Results 11/9/2018 Arizona State University

Arizona State University Results Left: New case of TB and data (dots) Right: 10% error bound of new cases and data 11/9/2018 Arizona State University

Regression approach A Markov chain model supports the same result 11/9/2018 Arizona State University A Markov chain model supports the same result

Arizona State University CONCLUSIONS 11/9/2018 Arizona State University

Arizona State University Conclusions 11/9/2018 Arizona State University

Arizona State University 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. 11/9/2018 Arizona State University

Arizona State University 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. 11/9/2018 Arizona State University

Arizona State University 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 . 11/9/2018 Arizona State University

Arizona State University 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-New York. Edited by Carlos Castillo-Chavez with Pauline van den Driessche, Denise Kirschner and Abdul-Aziz Yakubu, 2002 11/9/2018 Arizona State University

Models of Dengue Fever and their Public Health Implications Fabio Sánchez Ph.D. Candidate Cornell University Advisor: Dr. Carlos Castillo-Chavez 11/9/2018 Arizona State University

Arizona State University Outline Introduction Single strain model Two-strain model with collective behavior change Single outbreak model Conclusions 11/9/2018 Arizona State University

Arizona State University 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 11/9/2018 Arizona State University

Arizona State University 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) 11/9/2018 Arizona State University

Arizona State University 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. 213-261. 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). 11/9/2018 Arizona State University

Arizona State University 11/9/2018 Arizona State University

Arizona State University 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. 11/9/2018 Arizona State University

Arizona State University 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 11/9/2018 Arizona State University

Arizona State University Transmission Cycle 11/9/2018 Arizona State University

Arizona State University 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 11/9/2018 Arizona State University

Arizona State University 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 11/9/2018 Arizona State University

Arizona State University Caricature of the Model 11/9/2018 Arizona State University

Epidemic basic reproductive number, R0 The average number of secondary cases of a disease caused by a “typical” infectious individual. 11/9/2018 Arizona State University

Multiple steady states (backward bifurcation) With control measures 11/9/2018 Arizona State University

Change in Host Behavior and its Impact on the Co-evolution of Dengue 11/9/2018 Arizona State University

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. 11/9/2018 Arizona State University

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 11/9/2018 Arizona State University

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. 11/9/2018 Arizona State University

Arizona State University 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. 11/9/2018 Arizona State University

Arizona State University A Comparison Study of the 2001 and 2004 Dengue Fever Outbreaks in Singapore 11/9/2018 Arizona State University

Arizona State University Outline Data and the Singapore health system Single outbreak model Results Conclusions 11/9/2018 Arizona State University

Arizona State University Aedes aegypti Has adapted well to humans Mostly found in urban areas Eggs can last up to a year in dry land 11/9/2018 Arizona State University

Singapore Health System and Data 11/9/2018 Arizona State University

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. 11/9/2018 Arizona State University

Arizona State University 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 11/9/2018 Arizona State University

Reported cases from 2001-up to date 11/9/2018 Arizona State University

Arizona State University Single Outbreak Model SEIR - host (humans) VLJ - vectors (mosquitoes) M=V+L+J N=S+E+I+R 11/9/2018 Arizona State University

Arizona State University 2001 Outbreak 11/9/2018 Arizona State University

Arizona State University 2001 Outbreak 11/9/2018 Arizona State University

Arizona State University 2004 Outbreak 11/9/2018 Arizona State University

Arizona State University 2004 Outbreak 11/9/2018 Arizona State University

Arizona State University 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 11/9/2018 Arizona State University

Arizona State University 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 11/9/2018 Arizona State University