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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$R_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>). 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$R_e^{(k)}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>R</mi> <mi>e</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation> and the fractions infected in the different immunity groups <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\textbf {z}}^{(k)}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="bold">z</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(R_e^{(k)}, z^{(k)})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>R</mi> <mi>e</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <msup> <mi>z</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and the conditional distribution of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$z^{(k)}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>z</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation> given <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$R_e^{(k)}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>R</mi> <mi>e</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation>. The effective reproduction number <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$R_e^{(k)}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>R</mi> <mi>e</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation> is closely related to the initial exponential growth rate <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\rho ^{(k)}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ρ</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation> 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.

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

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2025年11月12日

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Tom Britton,
Andrea Pugliese

摘要

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 $R_0$ $R_{0}$ ). 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 $R_e^{(k)}$ $R_{e}^{(k)}$ and the fractions infected in the different immunity groups ${\textbf {z}}^{(k)}$ $z^{(k)}$ , 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 $(R_e^{(k)}, z^{(k)})$ $(R_{e}^{(k)}, z^{(k)})$ , and the conditional distribution of $z^{(k)}$ $z^{(k)}$ given $R_e^{(k)}$ $R_{e}^{(k)}$ . The effective reproduction number $R_e^{(k)}$ $R_{e}^{(k)}$ is closely related to the initial exponential growth rate $\rho ^{(k)}$ $ρ^{(k)}$ 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.