Dynamics Models for Tuberculosis Transmission and Control

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

2. Ancient disease • 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 19th century. MTBI Cornell University

3. Transmission Process • 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. MTBI Cornell University

4. Immune System Response Caricature • Bacteria invades lung tissue. • White cells surroundthe 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. MTBI Cornell University

5. Immune System Response Caricature • 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. MTBI Cornell University

6. TB was the main cause of mortality • 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. • TB’s nicknames: White Death, Captain of Death, Time bomb MTBI Cornell University

7. Per Capita Death Rate of TB MTBI Cornell University

8. 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. MTBI Cornell University

9. TB in the US MTBI Cornell University

10. 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. MTBI Cornell University

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

12. Epidemiological Classes MTBI Cornell University

13. Parameters MTBI Cornell University

14. 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) MTBI Cornell University

15. Model Equations MTBI Cornell University

16. Epidemiology(Basic Reproductive Number, R0) 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) MTBI Cornell University

17. Epidemiology(Basic Reproductive Number, R0) 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” MTBI Cornell University

18. R0 • Probability of surviving the latent stage: • Average effective contact rate • Average effective infectious period MTBI Cornell University

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

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

21. Demography and Epidemiology MTBI Cornell University

22. Demography Where MTBI Cornell University

23. Bifurcation Diagram (exponential growth ) MTBI Cornell University

24. Logistic Growth MTBI Cornell University

25. 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 the 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. MTBI Cornell University

26. 1 Bifurcation Diagram MTBI Cornell University

27. 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. MTBI Cornell University

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

29. Fast and Slow TB MTBI Cornell University

30. 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 MTBI Cornell University

31. Variable Latency Period (differentio-integral model) • E0(t): # of individuals in E class at t=0 and still in E class at time t • I0: # of individuals in I class at t=0 and still in I class at time t MTBI Cornell University

32. Exogenous Reinfection MTBI Cornell University

33. Exogenous Reinfection MTBI Cornell University

34. Backward Bifurcation MTBI Cornell University

35. Age Structure Model MTBI Cornell University

36. 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. MTBI Cornell University

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

38. Incidence and Mixing MTBI Cornell University

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

40. Stability There exists an endemic steady state whenever R0()>1. The infection-free steady state is globally asymptotically stable when R0= R0(0)<1. MTBI Cornell University

41. 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. MTBI Cornell University

42. 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. • Optimal strategy depends on data. MTBI Cornell University

43. 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. MTBI Cornell University