Dynamical Models of Epidemics: from Black Death to SARS D. Gurarie CWRU 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) 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. Geographic Distribution -1990 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? “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 Daniel Bernoulli 1700-1782 Bernoulli smallpox model (1766) 1) Population cohort of age a, n(a), mortality m(a) 2) Small pox effect Caveat: if inoculation mortality f is included one would need f<.5% for success! 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) Box (compartment) diagrams SI S – Susceptible I – Infectious E – Exposed R - Removed V – Vaccinated … S S SIR Birth I I SEIR R S Death E I V SEIR S E I R V Total population: N = S+I+… recruitment R SIR-type models Ross, Kermak-McKendrick •Population size is large and constant •No birth, death, immigration or emigration •No recovery •No latency •Homogeneous mixing SI b S I SIR with immunity b S m I R Basic Reproduction number: R0=bN/m R0>1 – endemic R0<1 - eradication Residual S(∞)>0 SIR with loss of immunity m b I S R d endemic epidemic Control (i) 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 “Smallpox cohort” SIR (1-n)d lX Z X m Y m+dn m Growth models: variable population N(t) Const recruitment Linear growth due to S (Voltera-Lotka) Linear growth rate due to S,I 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) b= mean number of partners/per I bS/(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!) AIDS for behavioral groups: 6D model Parameters: bh H bh T bt F bf T f H .1 .01 .4 .6 5 .01 Initial state Shom .1 Shet 1 Sfem 1 Ihom Ihet .01 0 Ifem 0 Data (trends) of several African countries 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. Age structured models (smallpox) Continuous population strata n(a,t), age “a”, time “t” Discrete population bins: n=(na) Normal growth Infection Example: 15-bin system with linear growth and structured transmission Age bins: red (young) to blue (old) High survival Low survival 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 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? Some current modeling issues and approaches Spatial/temporal patterns of outbreaks and spread Stochastic modeling Cellular Automata and Agent-Based Models Network Models (STD)