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

TL;DR

The paper develops a stochastic, multi-season epidemic framework for influenza-like disease with yearly random genetic drift $δ_k$ and random transmissibility $τ_k$. Immunity wanes across $r$ years, yielding a Markov-chain description of the population's immunity state that converges to a stationary distribution $\bar{π}$. In the analytically tractable case $r=2$ (immunity lasting one season), the authors derive explicit joint and conditional distributions for the effective reproduction number $R_e^{(k)}$ and the season final size $z^{(k)}$, and they outline the transition structure and stationary behavior. Numerical illustrations demonstrate how $R_e$ and $z$ relate to the initial growth and prior immunity, and show how longer-lasting immunity (larger $r$) reduces attack sizes. The framework offers a basis for predicting seasonal outcomes and for fitting drift/transmissibility distributions to data, with extensions to multiple strains and population heterogeneity.

Abstract

We consider a model for an influenza-like disease, in which, between seasons, the virus makes a random genetic drift $δ$, (reducing immunity by the factor $δ$) and obtains a new random transmissibility $τ$ (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 $\boldsymbolι^{(k)}$, 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 $(δ_k, τ_k)$. It is shown that the immunity status $(\textbf{p}^{(k)}, \boldsymbolι^{(k)})$, is an ergodic Markov chain, which converges to a stationary distribution $\bar π(\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 $ρ^{(k)}$ 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.

Figures (7)

• Figure 1: 200 draws of the random variables $(\tau,\delta)$ for each case described in Table \ref{['tab:pairs']}
• Figure 2: Examples of the bivariate distributions of $(z^{(k)},R_e^{(k})$ given $p^k = z^{(k-1)} = p$ computed using Proposition \ref{['prop_biv']} for $p=0.1$ (left panel) or $p=0.5$ (right panel) in case 1 of Table \ref{['tab:pairs']}.
• Figure 3: The distributions of $z^{(k)}$ given $p^{(k)}=z^{(k-1)}$ (solid curves), and given the pair $(R_e^{(k)},p^{(k)})$ (dashed curves). The red curves are computed with $p^{(k)}=p=0.1$, the blue ones with $p=0.5$. The conditional distributions are based on $R_e^{(k)}=1.6$ The four panels correspond to the four distributions of the pairs $(\delta_k,\tau_k)$ as shown in Table \ref{['tab:pairs']}.
• Figure 4: Estimates of the stationary distribution (black curve) of the attack ratio $z^{(k)}$, and of the conditional distributions of $z^{(k)}$ given $R_e^{(k)}=R_e$ (red and blue curves); values used for $R_e$ in the legend. The four panels correspond to the four distributions of the pairs $(\delta_k,\tau_k)$ as shown in Table \ref{['tab:pairs']}.
• Figure 5: Bivariate graphs of $(R_e^{(k)},z^{(k)})$ found along the simulations of the model with $r=2$. The curves are the functions $R_e = -\log(1-z)/z$ above which the bivariate distributioin always lies, as shown in Section \ref{['Sec-r=2']}.

Theorems & Definitions (8)

• Proposition 2.1
• proof
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof