Download SIR models and CAs

SIR Epidemic Models
CS 423/523
Spring 2012
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.
r
a
SIR Differential Equation Model
• Change in Recovered Population
dR/dt = aI
a is recovery rate
if illness (infectious period) lasts two days, then a = 0.5/day
• Change in the Susceptible Population
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
• Change in Infected Population
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
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, R0n new people get infected
• One infectious person eventually gives rise to
R0 =r/a (infection rate/recovery rate) secondary infectious cases
• R0 is the key metric of an epidemic
– R0 > 1 leads to exponential growth and epidemic spread
– R0 < 1 the epidemic will not take off
– At stage n of the spread, R0n new people get infected
• R0 determines
– The duration of spread before
• The pathogen disappears OR
• All susceptible individuals die or become immune
– The total number of infections from an epidemic
Spatial (CA) SIR models
• Why does space matter?
• In ODEs (diff eqs), R0 = r/a (infection rate/recovery rate)
– Assumes an infinite number of susceptible neighbors
– Assumes the population is well mixed
• 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 exercise
Initial condition: 1 I, all others S
• Hand raised = Infected
• Arms crossed = Recovered
• Arms at sides = Susceptible
1st Run: an ODE
An I infects 1 S each time step (r = 1)
Choose who to infect at random
An I stays I for 1 time step (a = 1)
R0 = 1 (on the cusp of an epidemic)
Next:
2nd run: Make this a CA—choose who to infect from your 4 neighbors
3rd run: Decrease a (recovery rate) to ½ so it takes 2 time steps to recover
4th run: Reset a = 1 and decrease infection rate r = ½
Next: Choose a set of parameters that causes an epidemic (all cells become I) iff
>=10% of cells are initially infected