# 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:
$\lambda_i (t) = N_i \beta \sum_{j \in I(t-1)} {A_{j,t-1} K(d_{ij})}$
where: $N_i$ is the number of pupils in school i, $A_{j,t-1}$ is the absence number of pupils in school j at time t-1 and
$K: \mathbb{R}^+ \to (0,1)$
is a spatial kernel given by: $\exp(-\alpha x)$.
$\alpha$ and $\beta$ 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:
$P_i(t) = 1 - \exp(-\lambda_i(t))$
(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:
$\exp(-\lambda_i(t))$
The recovery time is initially modelled at a geometric random variable with parameter $\gamma$ 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 $\delta$ to try and account for infections occurring outside of school mixing:

$\lambda_i (t) = (N_i \beta \sum_{j \in I(t-1)} {A_{j,t-1} K(d_{ij})}) + \delta N_i$

Negative Binomial Infectious Period:

Here we distribute the infectious period as a Negative Binomial with parameters $\gamma$ and $r$.

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

The Data Set and Thresholding