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

We consider a model for an influenza-like disease, in which, between seasons, the virus makes a random genetic drift $\delta$, (reducing immunity by the factor $\delta$) and obtains a new random transmissibility $\tau$ (closely related to $R_0$). Given the immunity status at the start of season $k$: $\textbf{p}^{(k)}$, describing community distribution of years since last infection, and their associated immunity levels $\boldsymbol{\iota}^{(k)}$, the outcome of the epidemic season $k$, characterized by the effective reproduction number $R_e^{(k)}$ and the fractions infected in the different immunity groups $\textbf{z}^{(k)}$, is determined by the random pair $(\delta_k, \tau_k)$. It is shown that the immunity status $(\textbf{p}^{(k)}, \boldsymbol{\iota}^{(k)})$, is an ergodic Markov chain, which converges to a stationary distribution $\bar \pi (\cdot)$. More analytical progress is made for the case where immunity only lasts for one season. We then characterize the stationary distribution of $p_1^{(k)}$, being identical to $z^{(k-1)}$. Further, we also characterize the stationary distribution of $(R_e^{(k)}, z^{(k)})$, and the conditional distribution of $z^{(k)}$ given $R_e^{(k)}$. The effective reproduction number $R_e^{(k)}$ is closely related to the initial exponential growth rate $\rho^{(k)}$ 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.