SIR Epidemic Models CS 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
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dR/dt = aI
a is recovery rate
if illness (infectious period) lasts two days, then a = 0.5/day
The # 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
Only 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
Exposed (E) population is not yet infected
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
The set of susceptible neighbors < N
a = 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
Von Neumann 4 cell neighborhood
S -> I if Heads
chose 1 neighbor by rolling dye
I -> R if Heads
R -> R
A: Initial State 3 I, others S (1< r < 2)
Add simultaneous updates, birth, immigration…
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