SIR Epidemic ModelsCS 390/590

Fall 2009

Why SIR models?

• Infectious diseases kill millions of people world-wide– Malaria, TB, HIV/AIDS

• New diseases emerge suddenly and spread quickly– SARS, West Nile Virus, HIV, Avian Influenza…

• Effective and fast control measures are needed• Models allow you to predict (estimate) when you don’t KNOW

– What are the costs and benefits of different control strategies?– When should there be quarantines?– Who should receive vaccinations?– When should wildlife or domestic animals be killed?– Which human populations are most vulnerable?– How many people are likely to be infected? To get sick? To die?

• Epidemic Spread as a CAS– How much can scientists hope to predict?

Simple SIR Model

• Susceptibles (S) have no immunity from the disease.

• Infecteds (I) have the disease and can spread it to others.

• Recovereds (R) have recovered from the disease and are immune to

further infection.

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SIR Differential Equation Model

• Change in Recovered PopulationdR/dt = aIa is recovery rateif illness (infectious period) lasts two days, then a = 0.5/day

• Change in the Susceptible PopulationThe # of possible contacts that spread infection is SI. Without births or immigration, S can only decrease dS/dt =−rSI, r is the transmission constant (infection rate) r is % of the possible contacts that result in the disease being spread r represents disease infectiousness & population interactions

• Change in Infected PopulationOnly susceptibles become infected, and all infecteds recover. What I gains comes from what S has lost; and what I loses, R gains. dI/dt =−dS/dt −dR/dt = rSI-aI

Lipsitch (et al 2003) SARS SEIR Model predicts the effect of quarantine

Susceptible-Exposeds-Infecteds-Recovereds Exposed (E) population is not yet infected

Simplifying assumptions 1. There are no births.

2. The only deaths are because of SARS.

3. The number of contacts of an infected individual with a susceptible person is constant and does not depend on the population density.

4. For susceptible individuals with exposure to the disease, the quarantine proportion (q) is the same for non-infected as for infected people.

5. Quarantine and isolation are completely effective. Someone who has the disease and is in

quarantine or isolation cannot spread the disease. Transmission requires an S to contact an IU

Reproductive Rate of the Pathogen

• The reproductive number R0 is the expected number of secondary cases from each infectious case

– R0 > 1 leads to exponential growth – At stage n of the spread, R0

n new people get infected

• One infectious person eventually gives rise to R0 =kb / (v +m +w) secondary infectious cases of SARS.

– Rate of spread from IU to S• b:probability that contact between IU and S spreads SARS • k: contacts per day between IU and S

– Rate of leaving IU (without entering S)• m—per capita death rate (from SARS)• v—per capita recovery rate

• w—probability per day of IU transferred to ID

• With quarantine R0 = (1 −q) kb / (v +m +w)

Metrics of the epidemic

• The reproductive number R0 is the expected number of secondary cases from each infectious case

– R0 > 1 leads to exponential growth – At stage n of the spread, R0

n new people get infected

• R0 determines

• The duration of spread before– The pathogen disappears (before infecting everyone)– All susceptibles die or become immune

• Note many individuals are NOT susceptible to many diseases

• The total number infected, recovered (immune) and dead

Spatial (CA) SIR models

• Why does space matter?

• In SIR, R0 = r/a (infection rate/recovery rate)

• Differential Eq model assumes an infinite number of susceptible neighbors

• CA assumes a finite set of neighbors N

The set of susceptible neighbors < N– the # of susceptibles decreases when an infection occurs

– You can’t infect the neighbor who infected you

– Some of your neighbors have already been infected by the neighbor who infected you

SIR CA exerciseODE modela = 1/2 (you are infectious for 1 time step = 1 day)

r = 1/2, R0 = 1 (on the cusp of an epidemic)--First, contacting only 1 person per time step

CA Simulation Von Neumann 4 cell neighborhoodS -> I if Heads

chose 1 neighbor by rolling dyeI -> R if HeadsR -> R

A: Initial State 3 I, others S (1< r < 2)• Hand raised = Infected• Arms crossed = Recovered• Arms at sides = Susceptible

Add simultaneous updates, birth, immigration…

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Background readingReading for Wednesday • Keeling & Grenfell, “Individual Based

Perspectives on R0”, Journal of Theoretical Biology v. 203 (2000)

• Angela B. Shiflet and George W. Shiflet: Introduction to Computational Science

Module 6