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

Tamás Tekeli, Andrea Pugliese, Cinzia Soresina

TL;DR

The paper analyzes SEIR dynamics with host susceptibility heterogeneity, deriving two formulations that link susceptibility and infectiousness under different assumptions. Using a gamma-distributed susceptibility as a pivotal case, the authors obtain explicit final-size relations and demonstrate that greater heterogeneity reduces the epidemic attack size relative to a homogeneous population with the same $R_0$, while correlated susceptibility and infectiousness further dampen spread. They extend the framework to Beta distributions and provide numerical illustrations showing that the final size depends mainly on the variance of susceptibility rather than distribution shape. Applying the model to Italian seasonal influenza data, they find that a Gamma-heterogeneity model can fit seasons without requiring implausible pre-existing immunity, offering a practical parameterization via a single variance-controlling parameter $p$. The work highlights the importance of incorporating host heterogeneity into forecasting and public health planning, while acknowledging simplifying assumptions and outlining directions for including age structure, vaccination, and waning immunity.

Abstract

Population heterogeneity is a key factor in epidemic dynamics, influencing both transmission and final epidemic size. While heterogeneity is often modeled 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-models, which are widely used for influenza and other respiratory infections. We derive the model equations under two distinct assumptions linking susceptibility and infectiousness. Analytical results show that heterogeneity in susceptibility reduces the epidemic final size compared to homogeneous models with the same basic reproduction number $\Ro$. In the case of gamma-distributed susceptibility, we obtain stronger results on the epidemic final size. The resulting model captures population heterogeneity through a single parameter, which makes it practical for fitting epidemic data. We illustrate its use by applying it to seasonal influenza in Italy.

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

TL;DR

The paper analyzes SEIR dynamics with host susceptibility heterogeneity, deriving two formulations that link susceptibility and infectiousness under different assumptions. Using a gamma-distributed susceptibility as a pivotal case, the authors obtain explicit final-size relations and demonstrate that greater heterogeneity reduces the epidemic attack size relative to a homogeneous population with the same , while correlated susceptibility and infectiousness further dampen spread. They extend the framework to Beta distributions and provide numerical illustrations showing that the final size depends mainly on the variance of susceptibility rather than distribution shape. Applying the model to Italian seasonal influenza data, they find that a Gamma-heterogeneity model can fit seasons without requiring implausible pre-existing immunity, offering a practical parameterization via a single variance-controlling parameter . The work highlights the importance of incorporating host heterogeneity into forecasting and public health planning, while acknowledging simplifying assumptions and outlining directions for including age structure, vaccination, and waning immunity.

Abstract

Population heterogeneity is a key factor in epidemic dynamics, influencing both transmission and final epidemic size. While heterogeneity is often modeled 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-models, which are widely used for influenza and other respiratory infections. We derive the model equations under two distinct assumptions linking susceptibility and infectiousness. Analytical results show that heterogeneity in susceptibility reduces the epidemic final size compared to homogeneous models with the same basic reproduction number . In the case of gamma-distributed susceptibility, we obtain stronger results on the epidemic final size. The resulting model captures population heterogeneity through a single parameter, which makes it practical for fitting epidemic data. We illustrate its use by applying it to seasonal influenza in Italy.

Paper Structure

This paper contains 16 sections, 5 theorems, 68 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model construction
  3. Basic properties
  4. The system with specific distributions of susceptibility
  5. The case of gamma-distributed susceptibility
  6. Beta-distributed susceptibility
  7. Numerical illustrations
  8. Final size equation for the Gamma distribution
  9. The case of independent infectiousness and susceptibility
  10. The case of correlated susceptibility and infectiousness
  11. Model fit to data on seasonal influenza in Italy
  12. Data description
  13. Fitting methods
  14. Results
  15. Discussion
  16. ...and 1 more sections

Key Result

Theorem 3.1

Equation finalsize_gen admits a unique solution $\bar{s}_\infty \in (0,1)$ if $\mathcal{R}_0 > 1$, no solutions in $(0,1)$ if $\mathcal{R}_0 \le 1$. The total susceptible fraction $\bar{S}_\infty = \Phi_0(\bar{s}_\infty)$ is larger than the final susceptible fraction $S_\infty$ found in the homogene

Figures (4)

  • Figure 1: Plots of Gamma (left) and Beta (right) susceptibility distributions used in the simulations in Figure \ref{['fig1']}. Gamma distributions have variance $1/p$, where $p$ is specified in the legend. For the Beta distributions, the values of $a$ and $b$ are also specified in the legend, while $L$ is obtained from \ref{['L_beta']}; it can be verified that all Beta distributions have variance 1/3.
  • Figure 2: Plots showing proportion of $E(t) + I(t)$ (above) and $R(t)$ (below) for differently distributed susceptibility over time (all with $\mathcal{R}_0 = 2$). On the left panels, plots are shown with infectiousness independent of susceptibility; on the right panels, one can see plots with directly correlated infectiousness and susceptibility. The black curve represents the homogeneous SEIR model, coloured solid curves represent Gamma distributions with different variance (see legend), coloured dashed curves represent Beta-distributed heterogeneity, with the same variance $(Var(X)=1/3)$ as the light-blue, solid curve.
  • Figure 3: Susceptible fraction at the end of an epidemic for model \ref{['sys_bar']}, plotted as a function of the variance. Curves are shown for the Gamma distribution and the Beta distributions with either the parameter choices in \ref{['choice_beta1']} or \ref{['choice_beta2']}. In all cases, $\mathcal{R}_0=1.5$.
  • Figure 4: Comparisons between the weekly infections estimated from the homogeneous or heterogeneous SEIR models (dashed lines) and the estimated cases from the data using formula \ref{['casi']} (dots) in the seasons 2016-17, 2019-2020, and 2023-24

Theorems & Definitions (10)

  • Theorem 3.1
  • Theorem 5.1
  • Theorem 5.2
  • Proposition 5.3
  • proof : Proof of Theorem \ref{['theo3.1']}
  • proof : Proof of Theorem \ref{['theor_ind']}
  • proof : Proof of Theorem \ref{['theor_corr']}
  • Lemma A.1
  • proof
  • proof : Proof of Proposition \ref{['prop_compare']}