A discrete-time susceptible-infectious-recovered-susceptible model for the analysis of influenza data.

We develop a discrete time compartmental model to describe the spread of seasonal influenza virus. As time and disease state variables are assumed to be discrete, this model is considered to be a discrete time, stochastic, Susceptible-Infectious-Recovered-Susceptible (DT-SIRS) model, where weekly counts of disease are assumed to follow a Poisson distribution. We allow the disease transmission rate to also vary over time, and the disease can only be reintroduced after extinction if there is a contact with infected individuals from other host populations. To capture the variability of influenza activities from one season to the next, we define the seasonality with a 4-week period effect that may change over years. We examine three different transmission rates and compare their performance to that of existing approaches. Even though there is limited information for susceptible and recovered individuals, we demonstrate that the simple models for transmission rates effectively capture the behaviour of the disease dynamics. We use a Bayesian approach for inference. The framework is applied in an analysis of the temporal spread of influenza in the province of Manitoba, Canada, 2012-2015.