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