SEIR models with host heterogeneity: theoretical aspects and applications to seasonal influenza dynamics

Abstract

Population heterogeneity is a key factor in epidemic dynamics, influencing both transmission and final epidemic size. While heterogeneity is often modelled through age structure, spatial location, or contact patterns, differences in host susceptibility have recently gained attention, particularly during the COVID-19 pandemic. Building on the framework of Diekmann and Inaba (Journal of Mathematical Biology, 2023), we focus on the special case of SEIR epidemic models, assuming that at the epidemic start there is no pre-existing immunity. Under two distinct assumptions linking susceptibility and infectiousness, one obtains a closed system of 3 ODEs, which can be easily simulated and for which some analytical results are obtained. In particular, we proved that heterogeneity in susceptibility reduces the epidemic final size compared to homogeneous models with the same basic reproduction number $R_0$. We specialised in the case where susceptibility is distributed according to a gamma or extended Beta distribution, showing how the epidemic final size depends on the variance of the distribution. In the case of a gamma-distributed susceptibility, the resulting model consists of a system of ODEs with just one parameter more than the classical SEIR model; this makes it practical for fitting epidemic data. We illustrate its use by fitting data on seasonal influenza in Italy, and comparing the results to those obtained with simple SEIR models with pre-existing immunity.