1. SIR Epidemics
박상훈

2. 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. a r

3. 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?

4. The SIR Epidemic Model First studied, Kermack & McKendrick, 1927.
Consider a disease spread by contact with infected individuals.
Individuals recover from the disease and gain further immunity from it.
S = fraction of susceptibles in a population
I = fraction of infecteds in a population
R = fraction of recovereds in a population

5. The SIR Epidemic Model
Differential equations (involving the variables S, I, and R and their rates of change with respect to time t) are
An equivalent compartment diagram is

6. The SIR Epidemic Model
Non-dimensionalise equations

7. Basic Reproductive ratio
The average number of individuals directly infected by an infectious case during his/her entire infectious period
In a population
if R0 > 1 : epidemic
if R0 = 1 : endemic stage
if R0 < 1 : sucessful control of infection
If population is completely susceptible
measles : R0 = 15-20
smallpox : R0 = 3 – 5

8. Threshold for SIR Epidemic
initial per capita growth rate if the infectives in dimensional terms
Separable in (u,v,w)-space

9. Threshold for SIR Epidemic
they tend to limits,

10. Threshold for SIR Epidemic
If R0>1, the total number infected by the epidemic is Nw1,
Where w1 is the unique positive root of Equation
We can also analyse the time course of the epidemic.
There is no closed form solution of this equation, and a numerical solution is often required, but analytical progress may be made for a small epidemic