Views

Download Presentation
## Carlos Castillo-Chavez Joaquin Bustoz Jr. Teacher Arizona State College

Download Now

**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) 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 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. 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 . Arizona State University**Mixing Rules**• p(t,a,a`) ≥ 0 • Proportionate mixing: Arizona State University**Equations**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. 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. Arizona State University**Age Structure Model with vaccination**Arizona State University**Vaccinated**Age-dependent optimal vaccination strategies(Feng, Castillo-Chavez, Math. Biosci., 1998) Arizona State University**Basic reproductive Number**(by next generation operator) 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. 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. Arizona State University**R(y) < R***Reproductive numbers Two optimization problems: Arizona State University**One-age and two-age vaccination strategies**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 A1and the remaining susceptibles at a later age A2. . Selected optimal strategy depends on cost function (data). 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. Arizona State University**Transmission Diagram**Arizona State University**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 Arizona State University**Basic Cluster Model**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. Arizona State University**Diagram of Extended Cluster Model**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 measuresproportion of time of an “individual spanned within a cluster. 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. Arizona State University**Full system**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. Arizona State University**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). Arizona State University**Numerical Simulations**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 Arizona State University**TB in the US (1953-1999)**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? 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. Arizona State University**Non-autonomous model (permanent latent class of TB**introduced) Arizona State University**Effect of HIV**Arizona State University**Parameter estimation and simulation setup**Arizona State University**N(t) from census data**N(t) is from census data and population projection Arizona State University**Results**Arizona State University**Results**• Left: New case of TB and data (dots) • Right: 10% error bound of new cases and data Arizona State University**Regression approach**Arizona State University A Markov chain model supports the same result**CONCLUSIONS**Arizona State University**Conclusions**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. 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. 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 . Arizona State University