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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 7/17/2015 Arizona State University A TB model with age-structure (Castillo-Chavez and Feng. Math. Biosci., 1998) 7/17/2015 Arizona State University 7/17/2015 Arizona State University 7/17/2015 Arizona State University 7/17/2015 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. a2 # of susceptible individuals with ages in (a1 , a2) s ( t , a ) da : a1 at time t a2 i(t, a)da : # of infectious individuals with ages a1 in (a1 , a2) at time t a2 r (t, a)da : # of recovered individuals with ages in (a1 , a2) a1 at time t 7/17/2015 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. 7/17/2015 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 . 7/17/2015 Arizona State University Mixing Rules p(t,a,a`) ≥ 0 1. 2. 3. p(t,a,a')da'1 0 c(a) p(t,a,a')n(t,a) c(a') p(t,a',a)n(t,a') 4. Proportionate mixing: p(t,a,a') p(t,a') c(a')n(t,a') c(u)n(t,u)du 0 7/17/2015 Arizona State University Equations dt s(t , a) (a)c(a) B(t ) s (t , a) (a)s(t , a), da i(t , a) (a)c(a) B(t ) s (t , a) ( (a) (a))i(t , a), dt da r (t , a) (a)i(t , a) - (a)r (t , a). dt da B(t) i(t,a') p(t,a,a')da' 0 n(t, a') p(t,a,a') c(a')n(t,a') c(a)n(t,a)da 0 7/17/2015 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 dt n(t, a) (a)n(t, a), da n(t, a) e 0a (a)da , as t We assume that the total population density has reached this demographic steady state. 7/17/2015 Arizona State University Parameters • • • • • • • • 7/17/2015 : 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. Arizona State University Age Structure Model with vaccination dt dt dt s(t,a) (a)c(a)B(t) s(t,a) ((a) (a))s(t,a), da v(t,a) (a)s(t,a) (a)v(t,a) (a)B(t)v(t,a), da l(t,a) (a)c(a)B(t) s(t,a) (a)c(a)B(t) j(t,a) da (a)B(t)v(t,a) (k (a))l(t,a dt dt 7/17/2015 i(t,a) kl(t,a) (r (a))i(t,a), da j(t,a) ri(t,a) (a)c(a)B(t) j(t,a) da Arizona State University - (a) j(t,a). Age-dependent optimal vaccination strategies (Feng, Castillo-Chavez, Math. Biosci., 1998) Vaccinated 7/17/2015 Arizona State University Basic reproductive Number (by next generation operator) ( k) ( r) k ( )dd R0() p( )()c()e e rk 00 (a)F (a)(1F (a)) a F (a)exp( (b)db) denotes the probability that a 0 susceptible individual has not been vaccinated at age a. 7/17/2015 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. 7/17/2015 Arizona State University Optimal Vaccination Strategies Two optimization problems: 1. 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). 2. If the budget is fixed (cost) find a vaccination strategy (a) that minimizes R0(), that is, that minimizes the prevalence. 7/17/2015 Arizona State University Reproductive numbers Two optimization problems: 7/17/2015 R()< R* Arizona State University One-age and two-age vaccination strategies 7/17/2015 Arizona State University Optimal Strategies 1. One–age strategy: vaccinate the susceptible population at exactly age A. 2. Two–age strategy: vaccinate part of the susceptible population at exactly age A1 and the remaining susceptibles at a later age A2. 3. . Selected optimal strategy depends on cost function (data). 7/17/2015 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. 7/17/2015 Arizona State University Transmission Diagram knE2 NE22 knE2 S 1 E2 S2 S1 S2 7/17/2015 knE 2 kE2 I E1 E2 S1 E1 In I Arizona State University S2 N2 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 nI back to N2 population, where nI (S1 /N1)= S1 go to S2 and nI (E1 /N1)= E1 go to E2 7/17/2015 Arizona State University Basic Cluster Model 7/17/2015 Arizona State University Basic Reproductive Number R0c n k Q0 f k Where: Q0 n is the expected number of infections produced by one infectious individual within his/her cluster. f k k denotes the fraction that survives over the latency period. 7/17/2015 Arizona State University Diagram of Extended Cluster Model 7/17/2015 Arizona State University (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 measures proportion of time of an “individual spanned within a cluster. 7/17/2015 Arizona State University 7/17/2015 Arizona State University 7/17/2015 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 p K n* 1 1 K p L L Thecluster size n* is optimal as it maximizes the relative impact of within to between cluster transmission. 7/17/2015 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. 7/17/2015 Arizona State University Bifurcation Diagram I* Global T ranscritical Bifurcation 1 R0 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). 7/17/2015 Arizona State University Numerical Simulations 7/17/2015 Arizona State University Concluding Remarks on Cluster Models 1. 2. 3. 7/17/2015 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 Arizona State University TB in the US (1953-1999) 7/17/2015 Arizona State University 7/17/2015 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? 7/17/2015 Arizona State University Model Construction dN F (N ) dI dt Since d has been approximately equal to zero over the past 50 years in the US, we only consider dN F (N ). dt Hence, N can be computed independently of TB. 7/17/2015 Arizona State University Non-autonomous model (permanent latent class of TB introduced) dL1 (N (t) L L I ) I ( (t) A(t) p k r )L , 1 2 1 1 dt N (t) dL2 pL ( (t) r )L , 1 2 2 dt dI (k A(t))L ( (t) d (t) r )I . 1 3 dt N(t), (t), A(t), d (t) are known. 7/17/2015 Arizona State University Effect of HIV A(t) 3 (t 1983) 1 Exp2 (t 1983) , if t1983; 0, 7/17/2015 Arizona State University Otherwise. Parameter estimation and simulation setup 7/17/2015 Parameter Estimation 0.22 c 10 k 0.001 r1 0.05 r2 0.05 r3 0.65 p 0.1 Arizona State University N(t) from census data N(t) is from census data and population projection 7/17/2015 Initial I(0) L1(0) L2(0) Values 874230 106 106 Arizona State University Results 7/17/2015 Arizona State University Results Left: New case of TB and data (dots) Right: 10% error bound of new cases and data 7/17/2015 Arizona State University Regression approach Regression Equation : Log Y 11.3970 0.0597 X 0.0006 X 2 7/17/2015 A Markov chainArizona modelState supports the same result University CONCLUSIONS 7/17/2015 Arizona State University Conclusions 7/17/2015 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. 7/17/2015 Arizona State University Our work on TB Aparicio, J., A. Capurro and C. Castillo-Chavez, “On the long-term dynamics and reemergence of tuberculosis.” In: Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, IMA Volume 125, 351-360, Springer-Veralg, BerlinHeidelberg-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. 7/17/2015 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: 209225, 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 . 7/17/2015 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: 187205, 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 7/17/2015 Arizona State University Models of Dengue Fever and their Public Health Implications Fabio Sánchez Ph.D. Candidate Cornell University Advisor: Dr. Carlos Castillo-Chavez 7/17/2015 Arizona State University Outline Introduction Single strain model Two-strain model with collective behavior change Single outbreak model Conclusions 7/17/2015 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. 7/17/2015 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) 7/17/2015 Arizona State University Four antigenically distinct serotypes (DEN1, 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 7/17/2015 Arizona State University 7/17/2015 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. 7/17/2015 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 7/17/2015 Arizona State University Transmission Cycle 7/17/2015 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 7/17/2015 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 7/17/2015 Arizona State University Caricature of the Model 7/17/2015 Arizona State University Epidemic basic reproductive number, R0 The average number of secondary cases of a disease caused by a “typical” infectious individual. R0 7/17/2015 m ( ) Arizona State University Multiple steady states (backward bifurcation) With control measures 7/17/2015 Arizona State University Change in Host Behavior and its Impact on the Co-evolution of Dengue 7/17/2015 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. 7/17/2015 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 i 1 1 R 0 i i (m i ) ( i ) m Where, i m i - represents the proportion of mosquitoes that make it from the latent stage to the infectious stage 1 ( i ) - represents the average time of the host spent in the infectious stage 1 m 7/17/2015 - represents the average life-span of the mosquito Arizona State University Regions of Stability of Endemic Equilibria From the stability analysis of the endemic equilibria, the following necessary condition arose ( )( k 2 1 R 1 k k Rk 2 k k Rk2 k 2 Ri < 2 (m k ) R k 1 k 1 R 2 1 ( ) 1 k k k kk ( k 1 k ) ( k ) ( ) ) ( ( ) which defines the regions illustrated above. 7/17/2015 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. 7/17/2015 Arizona State University A Comparison Study of the 2001 and 2004 Dengue Fever Outbreaks in Singapore 7/17/2015 Arizona State University Outline Data and the Singapore health system Single outbreak model Results Conclusions 7/17/2015 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 7/17/2015 Arizona State University Singapore Health System and Data 7/17/2015 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. 7/17/2015 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 7/17/2015 Arizona State University Reported cases from 2001-up to date DF+DHF 450 400 confirmed cases 350 300 250 200 150 100 50 0 1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196 209 222 week 7/17/2015 Arizona State University Single Outbreak Model VLJ - vectors (mosquitoes) SEIR - host (humans) dS J S , dt M dE J S E, dt M dI E I, dt dR I. dt N=S+E+I+R 7/17/2015 dV I V ( d ), dt N dL I V ( d ), dt N dJ L ( d ). dt M=V+L+J Arizona State University 2001 Outbreak 7/17/2015 Arizona State University 2001 Outbreak 7/17/2015 Arizona State University 2004 Outbreak 7/17/2015 Arizona State University 2004 Outbreak 7/17/2015 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 7/17/2015 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 7/17/2015 Arizona State University