Setting Up Epidemic SIR Model

Question:

Model a daycare that measles spreads when a sick child comes in. In this daycare, we have 100 susceptible children, and there is a recruitment rate of 5 new kids per year. The rate at which measles is transmitted from an infected child to a susceptible child is assumed to be 0.0005 per year. Also assume that infected children will recover from the disease in 2 weeks.

Info:

So I know

$\displaystyle \frac {dS}{dt} = -\beta S I$

$\displaystyle \frac {dI}{dt}= \beta S I + \gamma I$

$\displaystyle \frac {dR}{dt}= \gamma I$

Attempt:

So I would assume $\displaystyle \beta = 0.0005$

And I think $\displaystyle S = 100 + 5t - \gamma I$

How do I calculate $\displaystyle \gamma$. I know they recover in 14 days. Would it be 14/365?

What else am I missing?

Any help is certainly appreciated to help me understand a little better. I have tried to read wikipedia. Thanks

Re: Setting Up Epidemic SIR Model

Hmm. I'm not sure I would agree with your model as yet. Start with English first, and then translate to differential equations. Let S be the number of susceptibles, I be the number of infectees, and R be the number of recoverees. (Coining words there, I know.) Then I think you have the following:

dS/dt = recruitment rate minus infection rate (since infected children are not counted as susceptible anymore).

dI/dt = rate due to interactions between susceptibles and infectees minus recovery rate.

dR/dt = proportional to the number of infectees

So from this, what do you think a reasonable model would be? What are the initial conditions?

Re: Setting Up Epidemic SIR Model

Quote:

Originally Posted by

**Len** How do I calculate $\displaystyle \gamma$. I know they recover in 14 days. Would it be 14/365?

Wikipedia glosses $\displaystyle 1 / \gamma$ as average recovery period. (So flip your fraction.)