Dynamical Models of Epidemics: from Black Death to SARS

1. Dynamical Models of Epidemics: from Black Death to SARS D. Gurarie CWRU

2. History Epidemics in History – Plague in 14th Century Europe killed 25 million – Aztecs lost half of 3.5 million to smallpox – 20 million people in influenza epidemic of 1919 Diseases at Present – 1 million deaths per year due to malaria – 1 million deaths per year due to measles – 2 million deaths per year due to tuberculosis– 3 million deaths per year due to HIV – Billions infected with these diseases History of Epidemiology . Hippocrates's On the Epidemics (circa 400 BC) . John Graunt's Natural and Political Observations made upon the Bills of Mortality (1662) . Louis Pasteur and Robert Koch (middle 1800's) History of Mathematical Epidemiology . Daniel Bernoulli studied the effect of vaccination with cow pox on life expectancy (1760) . Ross's Simple Epidemic Model (1911) . Kermack and McKendrick's General Epidemic Model (1927)

3. Geographic Distribution -1990 Schistosomiasis • Chronic parasitic trematode infection • 200-300 million people worldwide • Significant morbidity (esp. anemia) • Premature mortality • Life-cycle is complex, requiring species-specific intermediate snail host • Optimal control strategies have not been established.

4. Smallpox: XVIII century • Known facts: • Short duration (10 days), high mortality (75%) • Life-long immunity for survivors • Prevention: immunity by inoculation (??) • Problem: could public health (life expectancy) be improved by inoculation? Daniel Bernoulli 1700-1782 “I simply wish that, in a matter which so closely concerns the well-being of mankind, no decision shall be made without all the knowledge which a little analysis and calculation can provide.” Daniel Bernoulli, on smallpox inoculation, 1766

5. Bernoulli smallpox model (1766) 1) Population cohort of age a, n(a), mortality m(a) 2) Small pox effect Caveat: if inoculation mortalityf is included one would needf<.5% for success!

6. Modeling issues and strategies • State variables for host/parasite • “mean” or “distributed” (deterministic/stochastic) • Prevalence or level/intensity • Disease stages (latent,…) • Susceptibility and infectiousness • Transmission • Homogeneous (uniformly mixed populations): “mass action” • Heterogeneous: age/gender/ behavioral strata, spatially structured contacts • Environmental factors • Multi-host systems, parasites with complex life cycles, …. • Goals of epidemic modeling • Prediction • Risk assessment • Control (intervention, prevention)

7. Box (compartment) diagrams S I Birth Death SI S I R SIR recruitment SEIR SEIR S S E E I I R R V V S – Susceptible E – Exposed I – Infectious R - Removed V – Vaccinated … Total population: N = S+I+…

8. S I b SIR with immunity S I R b m Residual S(∞)>0 SIR-type modelsRoss, Kermak-McKendrick • Population size is large and constant • No birth, death, immigration or emigration • No recovery • No latency • Homogeneous mixing SI Basic Reproduction number: R0=bN/m R0>1 – endemic R0<1 - eradication

9. I S R m b d endemic epidemic SIR with loss of immunity • Control • R0=“transmissiom”x”pop. density”/”recovery”<1. Hence critical density N>m/b • to sustain endemic level • (ii) Vaccination removes a fraction of N from transmission cycle: so eradication • (equilibrium I<0) requires (1-1/R0) fraction of N vaccinated

10. (1-n)d l X X Z Y m m+dn m “Smallpox cohort” SIR

11. Growth models: variable population N(t) Const recruitment Linear growth due to S (Voltera-Lotka) Linear growth rate due to S,I

12. HIV/AIDS and STD • Variable population N=S+I • Natural growth a for S • Mortality m=10/year for I • Transmission: b S I/(S+I) • = mean number of partners/per I • S/(S+I) probability of infecting S (S-fraction of N) Typical collapse Conclusion: Transm. treatment Treatment w/o prevention of spread can only increase g (collapse!)

13. AIDS for behavioral groups: 6D model Parameters: Initial state

14. Data (trends) of several African countries

15. Heterogeneous transmission for distributed populations • SIR type are only conceptual models • Idealize transmissions and individual characteristics (susceptibilities) • Real epidemics requires heterogeneous models: • age structure • spatial/behavioral heterogeneity, etc.

16. Age structured models (smallpox) Continuous population strata n(a,t), age “a”, time “t” Discrete population bins: n=(na) Normal growth Infection

17. Example: 15-bin system with linear growth and structured transmission Age bins: red (young) to blue (old) High survival Low survival

18. Fisher’s Equation (1937) Original motivation: spread of a genetype in a population) • Infection: S(x,t), I(x,t)– (distributed) susceptibles and infectives • Population density is constant N • No birth or death • No recovery or latent period • Only local infection • Infection rate is proportional to the number of infectives • Individuals disperse diffusively with constant D

19. Solutions: propagating density waves Spreading wave in uniform medium with const pop. density Spreading wave with variable pop. density (red) • Problems: • Equilibrium, Basic Reproduction Number? • Speed of propagation (traveling waves)? • Parameters for control, prevention?

20. Some current modeling issues and approaches • Spatial/temporal patterns of outbreaks and spread • Stochastic modeling • Cellular Automata and Agent-Based Models • Network Models (STD)