A multi-season epidemic model with random genetic drift and transmissibility

We consider a model for the spread of an influenza-like disease in which, between seasons, the virus makes a random genetic drift (reducing immunity) and obtains a new random transmissibility (closely related to R₀ R 0 ). Given the immunity status at the start of season k , i.e. the community distribution of years since last infection and their associated immunity levels, the outcome of the epidemic season k , characterized by the effective reproduction number Rₑ⁽ᵏ⁾ R e ( k ) and the fractions infected in the different immunity groups z⁽ᵏ⁾ z ( k ) , is determined by the random genetic drift and transmissibility. It is shown that the community immunity status of consecutive seasons, is an ergodic Markov chain, which converges to a stationary distribution. More analytical progress is made for the case where immunity only lasts for one season: we then characterize the stationary distribution of the community fraction having partial immunity (from being infected last season) as well as the stationary distribution of (Rₑ⁽ᵏ⁾, z⁽ᵏ⁾) ( R e ( k ) , z ( k ) ) , and the conditional distribution of z⁽ᵏ⁾ z ( k ) given Rₑ⁽ᵏ⁾ R e ( k ) . The effective reproduction number Rₑ⁽ᵏ⁾ R e ( k ) is closely related to the initial exponential growth rate ρ ⁽ᵏ⁾ ρ ( k ) of the outbreak, a quantity which can be estimated early in the epidemic season. As a consequence, this conditional distribution may be used for predicting the final size of the epidemic based on its initial growth and immunity status.