1、10/10/2018,Arizona State University,Carlos Castillo-Chavez Joaquin Bustoz Jr. Professor Arizona State University,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
2、 Institute for Mathematical Sciences, National University of Singapore and Regional Emerging Diseases Intervention (REDI) Centre, Singaporehttp:/www.ims.nus.edu.sg/Programs/infectiousdiseases/index.htmSingapore, 08-23-2005,10/10/2018,Arizona State University,Primary Collaborators: Juan Aparicio (Uni
3、versidad Metropolitana, Puerto Rico) Angel Capurro (Universidad de Belgrano, Argentina, deceased) Zhilan Feng (Purdue University) Wenzhang Huang (University of Alabama) Baojung Song (Montclair State University),10/10/2018,Arizona State University,Our work on TB,Aparicio, J., A. Capurro and C. Castil
4、lo-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
5、, Denise Kirschner and Abdul-Aziz Yakubu, 2002Aparicio J., A. Capurro and C. Castillo-Chavez, “Transmission and Dynamics of Tuberculosis on Generalized Households” Journal of Theoretical Biology 206, 327-341, 2000Aparicio, J., A. Capurro and C. Castillo-Chavez, Markers of disease evolution: the case
6、 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
7、 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, 2002Berezovsky, F., G. Karev, B. Song, and C. Castillo-Chavez, Simple Models with Surprised Dynamics, Journal of Mathematical Bi
8、osciences 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.,10/10/2018,Arizona State University,Our work on TB,Castillo-Chavez, C., A. Capurro, M. Zellner and J. X. Velasco-Hernandez, “El transporte public
9、o 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
10、. 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 f
11、or TB and its applications to optimal vaccination strategies,” Mathematical Biosciences, 151,135-154, 1998Feng, 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 v
12、ariable latent periods in mathematical models for tuberculosis, Journal of Dynamics and Differential Equations .,10/10/2018,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,
13、 Mathematical Biosciences 180: 187-205, December 2002Song, 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-
14、294, Springer-Veralg, Berlin-Heidelberg-New York. Edited by Carlos Castillo-Chavez with Pauline van den Driessche, Denise Kirschner and Abdul-Aziz Yakubu, 2002,10/10/2018,Arizona State University,Outline,Brief Introduction to TB Long-term TB evolution Dynamical models for TB transmission The impact
15、of social networks cluster models A control strategy of TB for the U.S.: TB and HIV,10/10/2018,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.,10/10/2018,Arizona State Universit
16、y,Pathogen?Tuberculosis Bacilli (Koch, 1882). Where? Lung. How?Host-air-host Immunity?Immune system responds quickly,Transmission Process,10/10/2018,Arizona State University,Bacteria invades lung tissue White cells surround the invaders and try to destroy them. Body builds a wall of cells and fibers
17、 around the bacteria to confine them, forming a small hard lump.,Immune System Response,10/10/2018,Arizona State University,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.
18、 Infection progresses and develops into active TB in less than 10% of the cases.,Immune System Response,10/10/2018,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
19、 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.,10/10/2018,Arizona State University,Reasons for TB Persistence,Co-infection wit
20、h 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.,10/10/2018,Arizona State University,Basic Model Framework,N=S+E+I+T, Total populationF(N): Birth and immigration rateB(
21、N,S,I): Transmission rate (incidence)B(N,S,I): Transmission rate (incidence),10/10/2018,Arizona State University,Model Equations,10/10/2018,Arizona State University,R0,Probability of surviving to infectious stage:Average successful contact rateAverage infectious period,10/10/2018,Arizona State Unive
22、rsity,Phase Portraits,10/10/2018,Arizona State University,Bifurcation Diagram,10/10/2018,Arizona State University,Fast and Slow TB (S. Blower, et al., 1995),10/10/2018,Arizona State University,Fast and Slow TB,10/10/2018,Arizona State University,What is the role of long and variable latent periods?
23、(Feng, Huang and Castillo-Chavez. JDDE, 2001),10/10/2018,Arizona State University,A one-strain TB model with a distributed period of latency,AssumptionLet p(s) represents the fraction of individuals who are still in the latent class at infection age s, andThen, the number of latent individuals at ti
24、me t is:and the number of infectious individuals at time t is:,10/10/2018,Arizona State University,The model,10/10/2018,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?,10/10/201
25、8,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,10/10/2018,Arizona State University,Bifurcation Diagram,10/10/2018,Arizona State University,A TB model with exogenous reinfec
26、tion (Feng, Castillo-Chavez and Capurro. TPB, 2000),10/10/2018,Arizona State University,Exogenous Reinfection,E,10/10/2018,Arizona State University,The model,10/10/2018,Arizona State University,Basic reproductive number isNote: R0 does not depend on p. A backward bifurcation occurs at some pc (i.e.,
27、 E* exists for R0 1),Backward bifurcation,Number of infectives I vs. time,10/10/2018,Arizona State University,Backward Bifurcation,10/10/2018,Arizona State University,Dynamics depends on initial values,10/10/2018,Arizona State University,A two-strain TB model (Castillo-Chavez and Feng, JMB, 1997),Dr
28、ug 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
29、-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,10/10/2018,Arizona State University,A diagram for two-strain TB transmission,S,L1,I1,T,L2,I2,+d1,+d2,k1,pr2,(1-(p+q)r2,q
30、r2,K2,r1,1,2,*,*,r2 is the treatment rate for individuals with active TB q is the fraction of treatment failure,10/10/2018,Arizona State University,10/10/2018,Arizona State University,The two-strain TB model,r2 is the treatment rate for individuals with active TB q is the fraction of treatment failu
31、re,10/10/2018,Arizona State University,Reproductive numbers,For the drug-sensitive strain:For the drug-resistant strain:,10/10/2018,Arizona State University,Equilibria and stability,There are four possible equilibrium points:E1 : disease-free equilibrium (always exists)E2 : boundary equilibrium with
32、 L2 = I2 = 0 (R1 1; q = 0)E3 : interior equilibrium with I1 0 and I2 0 (conditional)E4 : boundary equilibrium with L1 = I1 = 0 (R2 1)Stability dependent on R1 and R2,Bifurcation diagram,10/10/2018,Arizona State University,q 0,Fraction of infections vs time,10/10/2018,Arizona State University,Contour
33、 plot of the fraction of resistant TB, J/N, vs treatment rate r2 and fraction of treatment failure q,10/10/2018,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 tubercu
34、losis 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 latent
35、ly infected with sensitive TB who are at high risk of developing the disease and who may benefit from preventive interventionThese 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,10/10/20
36、18,Arizona State University,A diagram for two-strains TB transmission with controls,S,L1,I1,T,L2,I2,+d1,+d2,k1,(1-u2)pr2,(1-(1-u2)(p+q)r2,(1-u2) qr2,K2,r1u1,1,2,*,*,10/10/2018,Arizona State University,u1(t): Effort to identify and treat typical TB individuals 1-u2(t): Effort to prevent failure of tr
37、eatment of active TB 0 u1(t), u2(t) 1 are Lebesgue integrable functions,The two-strain system with time-dependent controls (Jung, Lenhart and Feng. DCDSB, 2002),10/10/2018,Arizona State University,Objective functional,B1 and B2 are balancing cost factors. We need to find an optimal control pair, u1
38、and u2, such thatwhereai, bi are fixed positive constants, and tf is the final time.,10/10/2018,Arizona State University,10/10/2018,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 mod
39、el, Discrete and Continuous Dynamical Systems,Guess the value of the control over the simulated time.Solve the state system forward in time using the Runge-Kutta scheme.Solve the adjoint system backward in time using the Runge-Kutta scheme using the solution of the state equations from 2. Update the
40、 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.,10/10/2018,Arizona State University,Optimal
41、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,u1(t),u2(t),without control,With control,TB cases (L2+I2)/N,Control,10/10/2018,Arizona State University,Controls for various popul
42、ation sizes Jung, E., Lenhart, S. and Feng, Z. (2002), Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems,10/10/2018,Arizona State University,Demography,F(N)=, a constant,Results: More than one Threshold Possible,10/10/2018,Arizona State Unive
43、rsity,Bifurcation Diagram-Not Complete or Correct Picture,10/10/2018,Arizona State University,Demography and Epidemiology,10/10/2018,Arizona State University,Demography,Where:,10/10/2018,Arizona State University,Bifurcation Diagram (exponential growth ),10/10/2018,Arizona State University,Logistic G
44、rowth,10/10/2018,Arizona State University,Logistic Growth (contd),If R2* 1When 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 Poincar-Bendixson Theorem is used to show that the endemic state (p
45、ositive equilibrium) is globally stable whenever R0 1. When R0 1, there is no qualitative difference between logistic and exponential growth.,10/10/2018,Arizona State University,Bifurcation Diagram,10/10/2018,Arizona State University,Particular Dynamics (R0 1 and R2* 1),All trajectories approach the
46、 origin. Global attraction is verified numerically by randomly choosing 5000 sets of initial conditions.,10/10/2018,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
47、conditions.,10/10/2018,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 R01, both exponential growth and logistic grow lead t
48、o 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.,10/10/2018,Arizona State University,A fatal disease,Leadi
49、ng 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.,10/10/2018,Arizona State University,Per Capita Death Rate of TB,10/10/2018,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.,10/10/2018,Arizona State University,Births and immigration adjusted to fit data,
