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Tutorials 3: Epidemiological Mathematical Modeling, The Case of Tuberculosis. 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 7/18/2015 Arizona State University Primary Collaborators: Juan Aparicio (Universidad Metropolitana, Puerto Rico) Angel Capurro (Universidad de Belgrano, Argentina, deceased) Zhilan Feng (Purdue University) Wenzhang Huang (University of Alabama) Baojung Song (Montclair State University) 7/18/2015 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. 7/18/2015 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: 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 . 7/18/2015 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 7/18/2015 Arizona State University Outline Brief Introduction to TB Long-term TB evolution Dynamical models for TB transmission The impact of social networks – cluster models A control strategy of TB for the U.S.: TB and HIV 7/18/2015 Arizona State University Long History of Prevalence • TB has a long history. • TB transferred from animal-populations. • Huge prevalence. • It was a one of the most fatal diseases. 7/18/2015 Arizona State University Transmission Process • Pathogen? Tuberculosis Bacilli (Koch, 1882). • Where? Lung. • How? Host-air-host • Immunity? Immune system responds quickly 7/18/2015 Arizona State University Immune System Response • Bacteria invades lung tissue • White cells surround the invaders and try to destroy them. • Body builds a wall of cells and fibers around the bacteria to confine them, forming a small hard lump. 7/18/2015 Arizona State University Immune System Response • Bacteria cannot cause more damage as long as the confining walls remain unbroken. • Most infected individuals never progress to active TB. • Most remain latently-infected for life. • Infection progresses and develops into active TB in less than 10% of the cases. 7/18/2015 Arizona State University Current Situations • Two million people around the world die of TB each year. • Every second someone is infected with TB today. • One third of the world population is infected with TB (the prevalence in the US around 10-15% ). • Twenty three countries in South East Asia and Sub Saharan Africa account for 80% total cases around the world. • 70% untreated actively infected individuals die. 7/18/2015 Arizona State University Reasons for TB Persistence • Co-infection with HIV/AIDS (10% who are HIV positive are also TB infected) • Multi-drug resistance is mostly due to incomplete treatment • Immigration accounts for 40% or more of all new recent cases. 7/18/2015 Arizona State University Basic Model Framework B' ( N , T , I ) F (N ) S S B( N , S , I ) E kE I E r1 I ( d) I r 2E • • • • 7/18/2015 N=S+E+I+T, Total population F(N): Birth and immigration rate B(N,S,I): Transmission rate (incidence) B`(N,S,I): Transmission rate (incidence) Arizona State University T T Model Equations dS F(N) CS I I, N dt dE CS I ( kr )E'CT I , 2 N N dt dI kE( d r )E, 1 dt dT r Er I ' CT I T, 1 N dt 2 N SEI T 7/18/2015 Arizona State University R0 R0 C k r1 d r2 k Probability of surviving to infectious stage: Average successful contact rate Average infectious period 7/18/2015 Arizona State University C 1 r1 d k r2 k Phase Portraits 7/18/2015 Arizona State University Bifurcation Diagram I* Global T ranscritical Bifurcation 1 7/18/2015 Arizona State University R0 Fast and Slow TB (S. Blower, et al., 1995) pSI S (1 p)SI E S 7/18/2015 kE E Arizona State University I ( d)I Fast and Slow TB dS SI S , dt dE (1 p) SI kE E, dt dI p SI kE dI I. dt 7/18/2015 Arizona State University What is the role of long and variable latent periods? (Feng, Huang and Castillo-Chavez. JDDE, 2001) 7/18/2015 Arizona State University A one-strain TB model with a distributed period of latency Assumption Let p(s) represents the fraction of individuals who are still in the latent class at infection age s, and Then, the number of latent individuals at time t is: and the number of infectious individuals at time t is: 7/18/2015 Arizona State University The model 7/18/2015 Arizona State University The reproductive number Result: The qualitative behavior is similar to that of the ODE model. Q: What happens if we incorporate resistant strains? 7/18/2015 Arizona State University What is the role of long and variable latent periods? (Feng, Hunag and Castillo-Chavez, JDDE, 2001) A one-strain TB model 1/k is the latency period 7/18/2015 Arizona State University Bifurcation Diagram I* Global T ranscritical Bifurcation 1 7/18/2015 Arizona State University R0 A TB model with exogenous reinfection (Feng, Castillo-Chavez and Capurro. TPB, 2000) 7/18/2015 Arizona State University Exogenous Reinfection I p cS EN S S cS I N kE E E I rI ( d )I cT I N 7/18/2015 Arizona State University T T The model 7/18/2015 Arizona State University Basic reproductive number is Note: R0 does not depend on p. A backward bifurcation occurs at some pc (i.e., E* exists for R0 < 1) Backward bifurcation 7/18/2015 Number of infectives I vs. time Arizona State University Backward Bifurcation 7/18/2015 Arizona State University Dynamics depends on initial values 7/18/2015 Arizona State University A two-strain TB model (Castillo-Chavez and Feng, JMB, 1997) Drug sensitive strain TB - Treatment for active TB: 12 months - Treatment for latent TB: 9 months - DOTS (directly observed therapy strategy) - In the US bout 22% of patients currently fail to complete their treatment within a 12-month period and in some areas the failure rate reaches 55% (CDC, 1991) Multi-drug resistant strain TB - Infection by direct contact - Infection due to incomplete treatment of sensitive TB - Patients may die shortly after being diagnosed - Expensive treatment 7/18/2015 Arizona State University A diagram for two-strain TB transmission 1 S ’’ L1 * 2 qr2 L2 +d1 k1 I1 pr2 r1 (1-(p+q))r2 T * K2 +d2 7/18/2015 I2 r2 is the treatment rate for individuals with active TB q is the fraction of treatment failure Arizona State University 7/18/2015 Arizona State University The two-strain TB model r2 is the treatment rate for individuals with active TB q is the fraction of treatment failure 7/18/2015 Arizona State University Reproductive numbers For the drug-sensitive strain: For the drug-resistant strain: 7/18/2015 Arizona State University Equilibria and stability Resistant TB only Coexistence There are four possible equilibrium points: E1 : disease-free equilibrium (always exists) E2 : boundary equilibrium with L2 = I2 = 0 (R1 > 1; q = 0) Sensitive TB only q=0 E3 : interior equilibrium with I1 > 0 and I2 > 0 (conditional) E4 : boundary equilibrium with L1 = I1 = 0 (R2 > 1) Resistant TB only Stability dependent on R1 and R2 Coexistence 7/18/2015 Arizona State University q>0 Bifurcation diagram Resistant TB only TB-free Resistant TB only Coexistence Sensitive TB only Fraction of infections vs time q=0 7/18/2015 q >0 Arizona State University Contour plot of the fraction of resistant TB, J/N, vs treatment rate r2 and fraction of treatment failure q 7/18/2015 Arizona State University Optimal control strategies of TB through treatment of sensitive TB Jung, E., Lenhart, S. and Feng, Z. (2002), Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems “Case holding", which refers to activities and techniques used to ensure regularity of drug intake for a duration adequate to achieve a cure “Case finding", which refers to the identification (through screening, for example) of individuals latently infected with sensitive TB who are at high risk of developing the disease and who may benefit from preventive intervention These preventive treatments will reduce the incidence (new cases per unit of time) of drug sensitive TB and hence indirectly reduce the incidence of drug resistant TB 7/18/2015 Arizona State University A diagram for two-strains TB transmission with controls 1 S * 2 L2 ’’ +d1 r1u1 k1 L1 (1-u2)pr2 (1-u2) qr2 I1 * K2 +d2 7/18/2015 I2 Arizona State University T (1-(1-u2)(p+q))r2 The two-strain system with time-dependent controls (Jung, Lenhart and Feng. DCDSB, 2002) u1(t): Effort to identify and treat typical TB individuals 1-u2(t): Effort to prevent failure of treatment of active TB 0 < u1(t), u2(t) <1 are Lebesgue integrable functions 7/18/2015 Arizona State University Objective functional B1 and B2 are balancing cost factors. We need to find an optimal control pair, u1 and u2, such that where ai, bi are fixed positive constants, and tf is the final time. 7/18/2015 Arizona State University 7/18/2015 Arizona State University Numerical Method: An iteration method Jung, E., Lenhart, S. and Feng, Z. (2002), Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems 1. Guess the value of the control over the simulated time. 2. Solve the state system forward in time using the Runge-Kutta scheme. 3. Solve the adjoint system backward in time using the Runge-Kutta scheme using the solution of the state equations from 2. 4. Update the control by using a convex combination of the previous control and the value from the characterization. 5. Repeat the these process of until the difference of values of unknowns at the present iteration and the previous iteration becomes negligibly small. 7/18/2015 Arizona State University Optimal control strategies Jung, E., Lenhart, S. and Feng, Z. (2002), Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems u2(t) Control TB cases (L2+I2)/N 7/18/2015 u1(t) without control With control Arizona State University Controls for various population sizes Jung, E., Lenhart, S. and Feng, Z. (2002), Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems u1(t) u2(t) 7/18/2015 Arizona State University Demography dN F(N) N dI, dt dE C(N E I) I ( kr )E, 2 N dt dI kE( d r )I. 1 dt F(N)=, a constant F(N)rN, Exponential Growth N F(N)rN1 , Logistic Growth K Results: More than one Threshold Possible 7/18/2015 Arizona State University Bifurcation Diagram--Not Complete or Correct Picture I* Global T ranscritical Bifurcation 1 7/18/2015 Arizona State University R0 Demography and Epidemiology R0 C k r1 d r2 k 7/18/2015 Arizona State University Demography Where: r R2 * du d (mr nr c(mr k )2 4d (C d )(kC mr nr )c (d (mr nr ) C)(mr k ) * u 2d (C d )(kC mr nr ) 7/18/2015 Arizona State University Bifurcation Diagram (exponential growth ) R1 r N 0 I N u * N 0 ( R2 1) I ( R2 1) 1 I N I 0 I N I 0 7/18/2015 0 R0 1 Arizona State University Logistic Growth βC k R0 μ r1 d μ r2 k R* 2 7/18/2015 r R0 1 k μ d μ d k r1 R0 Arizona State University Logistic Growth (cont’d) If R2* >1 • When R0 1, the disease dies out at an exponential rate. The decay rate is of the order of R0 – 1. • Model is equivalent to a monotone system. A general version of Poincaré-Bendixson Theorem is used to show that the endemic state (positive equilibrium) is globally stable whenever R0 >1. • When R0 1, there is no qualitative difference between logistic and exponential growth. 7/18/2015 Arizona State University Bifurcation Diagram I* Global T ranscritical Bifurcation 1 7/18/2015 Arizona State University R0 Particular Dynamics (R0 >1 and * R2 <1) All trajectories approach the origin. Global attraction is verified numerically by randomly choosing 5000 sets of initial conditions. 7/18/2015 Arizona State University Particular Dynamics (R0 >1 and R2* <1) All trajectories approach the origin. Global attraction is verified numerically by randomly choosing 5000 sets of initial conditions. 7/18/2015 Arizona State University Conclusions on Density-dependent Demography • Most relevant population growth patterns handled with the examples. • Qualitatively all demographic patterns have the same impact on TB dynamics. • In the case R0<1, both exponential growth and logistic grow lead to the exponential decay of TB cases at the rate of R0-1. • When parameters are in a particular region, theoretically model predicts that TB could regulate the entire population. •However, today, real parameters are unlikely to fall in that region. 7/18/2015 Arizona State University A fatal disease • Leading cause of death in the past, accounted for one third of all deaths in the 19th century. • One billion people died of TB during the 19th and early 20th centuries. 7/18/2015 Arizona State University Per Capita Death Rate of TB 7/18/2015 Arizona State University Non Autonomous Model Here, N(t) is a known function of t or it comes from data (time series). The death rates are known functions of time, too. 7/18/2015 Arizona State University Births and immigration adjusted to fit data 7/18/2015 Arizona State University Life Expectancy in Years 7/18/2015 Arizona State University Incidence = k E 7/18/2015 Arizona State University Incidence of TB since 1850 7/18/2015 Arizona State University Conclusions • Contact rates increased--people move massively to cities • Life span increased in part because of reduce impact of TB-induced mortality • Prevalence of TB high • Progression must have slow down dramatically 7/18/2015 Arizona State University