1. Jun,2002MTBI Cornell University Carlos Castillo-Chavez Department of Biological Statistics and Computational Biology Department of Theoretical and Applied Mechanics Cornell University, Ithaca, New York, 14853 Dynamics Models for Tuberculosis Transmission and Control

2. Jun,2002MTBI Cornell University TB has a history as long as the human race. TB appears in the history of nearly every culture. TB was probably transferred from animals to humans. TB thrives in dense populations. It was the most important cause of death up to the middle of the 19 th century. Ancient disease

3. Jun,2002MTBI Cornell University Causative agent: Tuberculosis Bacilli (Koch, 1882). Preferred habitat Lung. Main Mode of transmission Host-air-host. Immune Response Immune system tends to respond quickly to initial invasion. Transmission Process

4. Jun,2002MTBI Cornell University 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. Immune System Response Caricature

5. Jun,2002MTBI Cornell University Bacteria cannot cause additional damage as long as confining walls remain unbroken. Most infected individuals never develop active TB (that is, become infectious). Most remain latently-infected for life. Infection progresses and develops into active TB in less than 10% of the cases. Immune System Response Caricature

6. Jun,2002MTBI Cornell University TB was the main cause of mortality Leading cause of death in the past. Accounted for one third of all deaths in the 19 th century. One billion people died of TB during the 19 th and early 20 th centuries. TB’s nicknames: White Death, Captain of Death, Time bomb

7. Jun,2002MTBI Cornell University Per Capita Death Rate of TB

8. Jun,2002MTBI Cornell University Current Situation Two to three million people around the world die of TB each year. Someone is infected with TB every second. One third of the world population is infected with TB ( the prevalence in the US is 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.

9. Jun,2002MTBI Cornell University TB in the US

10. Jun,2002MTBI Cornell 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. Lack of public knowledge about modes of TB transmission and prevention.

11. Jun,2002MTBI Cornell University Earliest Models H.T. Waaler, 1962 C.S. ReVelle, 1967 S. Brogger, 1967 S.H. Ferebee, 1967

12. Jun,2002MTBI Cornell University Epidemiological Classes

13. Jun,2002MTBI Cornell University Parameters

14. Jun,2002MTBI Cornell University 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)

15. Jun,2002MTBI Cornell University Model Equations

16. Jun,2002MTBI Cornell University Epidemiology ( Basic Reproductive Number, R 0 ) The expected number of secondary infections produced by a “typical” infectious individual during his/her entire infectious period when introduced in a population of mostly susceptibles at a demographic steady state. Sir Ronald Ross (1911) Kermack and McKendrick (1927)

17. Jun,2002MTBI Cornell University Epidemiology ( Basic Reproductive Number, R 0 ) Frost (1937) wrote “…it is not necessary that transmission be immediately and completely prevented. It is necessary only that the rate of transmission be held permanently below the level at which a given number of infection spreading (i.e. open) cases succeed in establishing an equivalent number to carry on the succession”

18. Jun,2002MTBI Cornell University R0R0 Probability of surviving the latent stage: Average effective contact rate Average effective infectious period

19. Jun,2002MTBI Cornell University Demography F(N)= , Linear Growth

20. Jun,2002MTBI Cornell University Exponential Growth (Three Thresholds) The Basic Reproductive Number is

21. Jun,2002MTBI Cornell University Demography and Epidemiology

22. Jun,2002MTBI Cornell University Demography Where

23. Jun,2002MTBI Cornell University Bifurcation Diagram ( exponential growth )

24. Jun,2002MTBI Cornell University Logistic Growth

25. Jun,2002MTBI Cornell University Logistic Growth (cont’d) If R 2 * >1 When R 0  1, the disease dies out at an exponential rate. The decay rate is of the order of R 0 – 1. Model is equivalent to a monotone system. A general version of the Poincaré-Bendixson Theorem is used to show that the endemic state (positive equilibrium) is globally stable whenever R 0 >1. When R 0  1, there is no qualitative difference between logistic and exponential growth.

26. Jun,2002MTBI Cornell University Bifurcation Diagram 1

27. Jun,2002MTBI Cornell University Particular Dynamics (R 0 >1 and R 2 * <1) All trajectories approach the origin. Global attraction is verified numerically by randomly choosing 5000 sets of initial conditions.

28. Jun,2002MTBI Cornell University Fast and Slow TB (S. Blower, et al., 1995)

29. Jun,2002MTBI Cornell University Fast and Slow TB

30. Jun,2002MTBI Cornell University Variable Latency Period (Z. Feng, et al,2001) p(s): proportion of infected (noninfectious) individuals who became infective s unit of time ago and who are still infected (non infectious). Number of exposed from 0 to t who are alive and still in the E class Number of those who progress to infectious from 0 to t and who are still alive in I class at time t

31. Jun,2002MTBI Cornell University Variable Latency Period (differentio-integral model) E 0 (t): # of individuals in E class at t=0 and still in E class at time t I 0 : # of individuals in I class at t=0 and still in I class at time t

32. Jun,2002MTBI Cornell University Exogenous Reinfection

33. Jun,2002MTBI Cornell University Exogenous Reinfection

34. Jun,2002MTBI Cornell University Backward Bifurcation

35. Jun,2002MTBI Cornell University Age Structure Model

36. Jun,2002MTBI Cornell University  : 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. Parameters

37. Jun,2002MTBI Cornell University 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. Proportionate mixing: p(t,a,a`)= p(t,a`) Proportionate Mixing

38. Jun,2002MTBI Cornell University Incidence and Mixing

39. Jun,2002MTBI Cornell University Basic reproductive Number (by next generation operator)

40. Jun,2002MTBI Cornell University Stability There exists an endemic steady state whenever R 0 (  )>1. The infection-free steady state is globally asymptotically stable when R 0 = R 0 (0)<1.

41. Jun,2002MTBI Cornell University Optimal Vaccination Strategies Two optimization problems: If the goal is to bring R 0 (  ) 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 R 0 (  ), that is, that minimizes the prevalence.

42. Jun,2002MTBI Cornell 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 A 1 and the remaining susceptibles at a later age A 2. Optimal strategy depends on data.

43. Jun,2002MTBI Cornell University Challenging Questions associated with TB Transmission and Control Impact of immigration. Antibiotic Resistance. Role of public transportation. Globalization—small world dynamics. Time-dependent models. Estimation of parameters and distributions.