# The Model

In order to model the 2009 Swine Flu epidemic across Birmingham we adapted a stochastic SIR model based of the works of Jewell, Keeling and Roberts:

**The Model:**

The probability of moving from the S to S or S to I state in a week’s time step is modelled by a time inhomogenous Poisson process across the week which incorporates the schools sizes, absence number and school distances. The rate function for this process is given by:

where: is the number of pupils in school i, is the absence number of pupils in school j at time t-1 and

is a spatial kernel given by: .

and are parameters to fit which relate to the geographical and infectious nature of the disease respectively.

Thus the probability of school i moving from S to I at time step t is given by:

(ie the school has encountered at least one other infectious school in this time step).

Thus the probability of school i moving from S to S is given by:

The recovery time is initially modelled at a geometric random variable with parameter which is to be determined from the data.

**Extensions:**

In order to better encapsulate the epidemic dynamics we also studied the following extensions to the model:

External Pressure:

Here we add another parameter to try and account for infections occurring outside of school mixing:

**Negative Binomial Infectious Period:**

Here we distribute the infectious period as a Negative Binomial with parameters and .

Finally we considered a five parameter model encompassing the external pressure and negative binomial infectious period.