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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi> <mml:mn>0</mml:mn></mml:msub> </mml:math> ). 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>R</mml:mi> <mml:mi>e</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:msubsup> </mml:math> and the fractions infected in the different immunity groups <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>z</mml:mi></mml:mrow> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:msup> </mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:msubsup><mml:mi>R</mml:mi> <mml:mi>e</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:msubsup> <mml:mo>,</mml:mo> <mml:msup><mml:mi>z</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:msup> <mml:mo>)</mml:mo></mml:mrow> </mml:math> , and the conditional distribution of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>z</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:msup> </mml:math> given <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>R</mml:mi> <mml:mi>e</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:msubsup> </mml:math> . The effective reproduction number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>R</mml:mi> <mml:mi>e</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:msubsup> </mml:math> is closely related to the initial exponential growth rate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>ρ</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:msup> </mml:math> 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.