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

1. Tutorials 1: Epidemiological Mathematical Modeling Applications in Homeland Security. 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 ASU/SUMS/MTBI/SFI

2. Bioterrorism The possibility of bioterrorist acts stresses the need for the development of theoretical and practical mathematical frameworks to systemically test our efforts to anticipate, prevent and respond to acts of destabilization in a global community ASU/SUMS/MTBI/SFI

3. From defense threat reduction agency Homeland Security Telecom Pharmaceuticals Ports & Airports Buildings Response Attribution Food Water Supply Urban Treatment and Consequence Management Roads & Transport Electric Power Detection Interdiction Warning ASU/SUMS/MTBI/SFI

4. From defense threat reduction agency From defense threat reduction agency Food Safety Medical Surveillance Animal/Plant Health Other Public Health Choke Points Urban Monitoring Characterization Metros Data Mining, Fusion, and Management State and Local Governments Emergency Management Tools Federal Response Plan ASU/SUMS/MTBI/SFI Toxic Industrials

5. Ricardo Oliva: Ricardo Oliva: Research Areas • Biosurveillance; • Agroterrorism; • Bioterror response logistics; • Deliberate release of biological agents; • Impact assessment at all levels; • Causes: spread of fanatic behaviors. ASU/SUMS/MTBI/SFI

6. Modeling Challenges &Mathematical ApproachesFrom a “classical” perspective to a global scale • Deterministic • Stochastic • Computational • Agent Based Models ASU/SUMS/MTBI/SFI

7. Some theoretical/modeling challenges • Individual and Agent Based Models--what can they do? • Mean Field or Deterministic Approaches--how do we average? • Space? Physical or sociological? • Classical approaches (PDEs, meta-population models) or network/graph theoretic approaches • Large scale simulations--how much detail? ASU/SUMS/MTBI/SFI

8. Ecological/Epidemiological view point • Invasion • Persistence • Co-existence • Evolution • Co-evolution • Control ASU/SUMS/MTBI/SFI

9. Epidemiological/Control Units • Cell • Individuals • Houses/Farms • Generalized households • Communities • Cities/countries ASU/SUMS/MTBI/SFI

10. Temporal Scales • Single outbreaks • Long-term dynamics • Evolutionary behavior ASU/SUMS/MTBI/SFI

11. Social Complexity • Spatial distribution • Population structure • Social Dynamics • Population Mobility • Demography--Immigration • Social hierarchies • Economic systems/structures ASU/SUMS/MTBI/SFI

12. Links/Topology/Networks • Local transportation network • Global transportation network • Migration • Topology (social and physical) • Geography--borders. ASU/SUMS/MTBI/SFI

13. Control/Economics/Logistics • Vaccination/Education • Alternative public health approaches • Cost, cost & cost • Public health infrastructure • Response time ASU/SUMS/MTBI/SFI

14. Critical Response Time in FMD epidemics A. L. Rivas, S. Tennenbaum, C. Castillo-Chávez et al.{American Journal of Veterinary Research}(Canadian Journal of Veterinary Research)

15. It is critical to determine the time needed and available to implement a successful intervention.

16. : 1-5 cases (1- 7 days post-onset) 1-5 cases (8-14 days post-onset) 3 2 1 The context--Foot and Mouth Disease BRAZIL ARGENT .INA ATLANTIC OCEAN

17. “exponential”growth Daily cases in the first month of the epidemic Number of daily cases

18. The Basic Reproductive Number R0 R0is the average number of secondary cases generated by an infectious unit when it is introduced into a susceptible population (at demographic steady state) of the same units. If R0 >1 then an epidemic is expected to occur--number of infected units increases If R0 < 1 then the number of secondary infections is not enough to sustain an apidemic. The goal of public health interventions is to reduce R0 to a number below 1. However, timing is an issue! How fast do we need to respond? ASU/SUMS/MTBI/SFI

19. Estimated CRTs for implementing intervention(s) resulting in R_o <= 1 (successful intervention) 3.0 days 2.6 days 1.4 days

20. Epidemic Models ASU/SUMS/MTBI/SFI

21. Basic Epidemiological Models: SIR Susceptible - Infected - Recovered ASU/SUMS/MTBI/SFI

22. R S I S(t): susceptible at time t I(t): infected assumed infectious at time t R(t): recovered, permanently immune N: Total population size (S+I+R) ASU/SUMS/MTBI/SFI

23. SIR - Equations Parameters ASU/SUMS/MTBI/SFI

24. SIR - Model (Invasion) ASU/SUMS/MTBI/SFI

25. Ro “Number of secondary infections generated by a “typical” infectious individual in a population of mostly susceptibles at a demographic steady state Ro<1 No epidemic Ro>1 Epidemic ASU/SUMS/MTBI/SFI

26. Establishment of a Critical Mass of Infectives!Ro >1 implies growth while Ro<1 extinction. ASU/SUMS/MTBI/SFI

27. Phase Portraits ASU/SUMS/MTBI/SFI

28. SIR Transcritical Bifurcation unstable ASU/SUMS/MTBI/SFI

29. Deliberate Release of Biological Agents ASU/SUMS/MTBI/SFI

30. Effects of Behavioral Changes in a Smallpox Attack Model Impact of behavioral changes on response logistics and public policy (appeared in Mathematical Biosciences, 05) Sara Del Valle1,2 Herbert Hethcote2, Carlos Castillo-Chavez1,3, Mac Hyman1 1Los Alamos National Laboratory 2University of Iowa 3Cornell University ASU/SUMS/MTBI/SFI

31. MODEL • All individuals are susceptible • The population is divided into two groups: normally active and less active • No vital dynamics included (single outbreak) • Disease progression: Exposed (latent) and Infectious • News of a smallpox outbreak leads to the implementation of the following interventions: • Quarantine • Isolation • Vaccination (ring and mass vaccination) • Behavioral changes (3 levels: high, medium & low) ASU/SUMS/MTBI/SFI

32. The Model The subscript refers to normally active (n) or less active (l): Susceptibles (S), Exposed (E), Infectious (I), Vaccinated (V), Quarantined (Q), Isolated (W), Recovered (R), Dead (D) Sn En In R Q W V S E I Sl El Il D ASU/SUMS/MTBI/SFI

33. The Model • The behavioral change rates are modeled by a non-negative, bounded, monotone increasing function i(for i =S, E, I) given by with ASU/SUMS/MTBI/SFI

34. Numerical Simulations ASU/SUMS/MTBI/SFI

35. Numerical Simulations ASU/SUMS/MTBI/SFI

36. Conclusions • Behavioral changes play a key role. • Integrated control policies are most effective: behavioral changes and vaccination have a huge impact. • Delays are bad. ASU/SUMS/MTBI/SFI

37. Mass Transportation and Epidemics

38. "An Epidemic Model with Virtual Mass Transportation" ASU/SUMS/MTBI/SFI

39. Mass Transportation Systems/HUBSBaojun SongJuan ZhangCarlos Castillo-Chavez ASU/SUMS/MTBI/SFI

41. NSU NSU SU SU Subway SU SU NSU NSU Subway Transportation Model ASU/SUMS/MTBI/SFI

42. Vaccination Strategies • Vaccinate civilian health-care and public health workers • Ring vaccination (Trace vaccination) • Mass vaccination • Mass vaccination if ring vaccination fails • Integrated approaches likely to be most effective

43. Assumptions • The population is divided into N neighborhoods; • Epidemiologically each individual is in one of four status: susceptible, exposed, infectious, and recovered; • A person is either a subway user or not • A ``vaccinated” class is included--everybody who is successfully vaccinated is sent to the recovered class

44. Proportionate mixing K subpopulations with densities N1(t), N2(t), …, Nk(t) at time t. cl : the average number of contacts per individual, per unit time among members of the lth subgroup.   Pij : the probability that an i-group individual has a contact with a j-group individual given that it had a contact with somebody.

45. Proportionate mixing (Mixing Axioms) (1) Pij >0 (2) (3) ci Ni Pij = cj Nj Pji Then is the only separable solution satisfying (1) , (2), and (3).

46. Definitions the mixing probability between non-subway users from neighborhood i given that they mixed. the mixing probability of non-subway and subway users from neighborhood i, given that they mixed. the mixing probability of subway and non-subway users from neighborhood i, given that they mixed. the mixing probability between subway users from neighborhood i, given that they mixed. the mixing probability between subway users from neighborhoods i and j, given that they mixed. the mixing probability between non-subway users from neighborhoods i and j, given that they mixed. the mixing probability between non-subway user from neighborhood i and subway users from neighborhood j, given that they mixed.

47. Formulae of Mixing Probabilities (depends on activity level and allocated time)

48. Identities of Mixing Probabilities

49. State Variables • i index of neighborhood • Wi number of individuals of susceptibles of SU in neighborhood i • Xi number of individuals of exposed of SU in neighborhood i • Yi number of individuals of infectious of SU in neighborhood i • Zi number of individuals of recovered of SU in neighborhood i • Si number of individuals of susceptibles of NSU in neighborhood i • Ei number of individuals of exposed of NSU in neighborhood i • Ii number of individuals of infectious of NSU in neighborhood i • Ri number of individuals of recovered of NSU in neighborhood i

50. Smallpox Model for NSU in neighborhood i Si Ei Ii Ai Ri