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

## TL;DR

The paper introduces a model for influenza-like diseases that incorporates random genetic drift and transmissibility changes between seasons, showing how immunity and outbreak dynamics evolve over time.

## Contribution

The novelty lies in modeling the interplay between random genetic drift, transmissibility, and immunity dynamics as an ergodic Markov chain.

## Key findings

- The community immunity status across seasons forms an ergodic Markov chain converging to a stationary distribution.
- For one-season immunity, the stationary distribution of partial immunity and epidemic parameters is characterized.
- The conditional distribution of epidemic size given initial growth rate can predict outbreak final size.

## Abstract

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 \documentclass[12pt]{minimal}
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\begin{document}$$R_0$$\end{document}R0). 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 \documentclass[12pt]{minimal}
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\begin{document}$$R_e^{(k)}$$\end{document}Re(k) and the fractions infected in the different immunity groups \documentclass[12pt]{minimal}
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\begin{document}$${\textbf {z}}^{(k)}$$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$$(R_e^{(k)}, z^{(k)})$$\end{document}(Re(k),z(k)), and the conditional distribution of \documentclass[12pt]{minimal}
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\begin{document}$$z^{(k)}$$\end{document}z(k) given \documentclass[12pt]{minimal}
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\begin{document}$$R_e^{(k)}$$\end{document}Re(k). The effective reproduction number \documentclass[12pt]{minimal}
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\begin{document}$$R_e^{(k)}$$\end{document}Re(k) is closely related to the initial exponential growth rate \documentclass[12pt]{minimal}
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\begin{document}$$\rho ^{(k)}$$\end{document}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.