Carlos Castillo-Chavez Joaquin Bustoz Jr. Professor Arizona State University

1. 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 Arizona State University

2. 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) Arizona State University

3. 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

4. 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

5. 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 Biosciences180: 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 Arizona State University

6. 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 Arizona State University

7. 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. Arizona State University

8. Transmission Process • Pathogen? Tuberculosis Bacilli (Koch, 1882). • Where? Lung. • How? Host-air-host • Immunity? Immune system responds quickly Arizona State University

9. 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. Arizona State University

10. 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. Arizona State University

11. 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. Arizona State University

12. 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. Arizona State University

13. Basic Model Framework • 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

14. Model Equations Arizona State University

15. R0 • Probability of surviving to infectious stage: • Average successful contact rate • Average infectious period Arizona State University

16. Phase Portraits Arizona State University

17. 1 Bifurcation Diagram Arizona State University

18. Fast and Slow TB (S. Blower, et al., 1995) Arizona State University

19. Fast and Slow TB Arizona State University

20. What is the role of long and variable latent periods?(Feng, Huang and Castillo-Chavez. JDDE, 2001) Arizona State University

21. 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: Arizona State University

22. The model Arizona State University

23. The reproductive number Result: The qualitative behavior is similar to that of the ODE model. Q: What happens if we incorporate resistant strains? Arizona State University

24. 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 Arizona State University

25. 1 Bifurcation Diagram Arizona State University

26. A TB model with exogenous reinfection(Feng, Castillo-Chavez and Capurro. TPB, 2000) Arizona State University

27. Exogenous Reinfection E Arizona State University

28. The model Arizona State University

29. Basic reproductive number is Note: R0 does not depend on p. A backward bifurcationoccurs at some pc(i.e., E* exists for R0< 1) Backward bifurcation Number of infectives I vs. time Arizona State University

30. Backward Bifurcation Arizona State University

31. Dynamics depends on initial values Arizona State University

32. 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 Arizona State University

33. A diagram for two-strain TB transmission   +d1  ’’  1 r1 k1 I1 L1 T S pr2 (1-(p+q))r2 * 2  qr2 * L2 K2 +d2 I2 r2is the treatment rate for individuals with active TB q is the fraction of treatment failure Arizona State University

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35. The two-strain TB model r2 is the treatment rate for individuals with active TB q is the fraction of treatment failure Arizona State University

36. Reproductive numbers For the drug-sensitive strain: For the drug-resistant strain: Arizona State University

37. Resistant TB only Coexistence Sensitive TB only q=0 Resistant TB only Coexistence q>0 Equilibria and stability There are four possible equilibrium points: E1 : disease-free equilibrium (always exists) E2 : boundary equilibrium with 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 Arizona State University Bifurcation diagram

38. Resistant TB only Coexistence TB-free Resistant TB only Sensitive TB only q = 0 Fraction of infections vs time q >0 Arizona State University

39. Contour plot of the fraction of resistant TB, J/N, vs treatment rate r2 and fraction of treatment failure q Arizona State University

40. Optimal control strategies of TB through treatment of sensitive TBJung, 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 Arizona State University

41. A diagram for two-strains TB transmission with controls   +d1  ’’ r1u1  1 k1 I1 L1 T S (1-u2)pr2 * 2  (1-(1-u2)(p+q))r2 (1-u2) qr2 * L2 K2 +d2 I2 Arizona State University

42. 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 Arizona State University

43. Objective functional B1 and B2 are balancing cost factors. We need to find an optimal control pair, u1 and u2, such that where ai, biare fixed positive constants, and tf is the final time. Arizona State University

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45. Numerical Method: An iteration methodJung, E., Lenhart, S. and Feng, Z. (2002), Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems • Guess the value of the control over the simulated time. • Solve the state systemforward in time using the Runge-Kutta scheme. • Solve the adjoint systembackward in time using the Runge-Kutta scheme using the solution of the state equations from 2. • 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. Arizona State University

46. 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) u1(t) Control without control TB cases (L2+I2)/N With control Arizona State University

47. u1(t) u2(t) Controls for various population sizesJung, E., Lenhart, S. and Feng, Z. (2002), Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems Arizona State University

48. Demography F(N)=, a constant Results: More than one Threshold Possible Arizona State University

49. 1 Bifurcation Diagram--Not Complete or Correct Picture Arizona State University

50. Demography and Epidemiology Arizona State University