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Stochastic epidemic models with varying infectivity and waning immunity

Raphaël Forien, Guodong Pang, Étienne Pardoux, Arsene Brice Zotsa-Ngoufack

TL;DR

This work develops a highly general stochastic epidemic model in which infectivity and immunity are random functions of infection age and are renewed at each new infection. It proves a functional law of large numbers as $N \to \infty$, yielding a deterministic limit given by a two-dimensional Volterra integral system for $(x,y)$ (and corresponding $(\overline{\mathfrak S},\overline{\mathfrak F})$) that generalizes the Kermack–McKendrick PDEs by incorporating random susceptibility and time-varying infectiousness. The endemic-equilibrium analysis reveals a threshold determined by the harmonic mean of the long-time susceptibility, $\mathbb{E}[1/\gamma_*]$: if $R_0 < \mathbb{E}[1/\gamma_*]$ the disease dies out, while if $R_0 > \mathbb{E}[1/\gamma_*]$ a unique endemic equilibrium exists and can be characterized via a fixed-point equation for the limiting infection-pressure, with the endemic infection level given by $\overline{I}_* = (\mathbb{E}[\eta]/R_0)\,\overline{F}_*$. The authors also connect the LLN limit to the classical KMK PDE form under a special SIRS-like parametrization and outline rigorous proofs employing propagation-of-chaos techniques. The results highlight how heterogeneity and gradual waning of immunity shape epidemic thresholds and long-term behavior, with implications for public health strategies in heterogeneous populations.

Abstract

We study an individual-based stochastic epidemic model in which infected individuals become susceptible again following each infection. In contrast to classical compartment models, after each infection, the infectivity is a random function of the time elapsed since one's infection. Similarly, recovered individuals become gradually susceptible after some time according to a random susceptibility function. We study the large population asymptotic behaviour of the model, by proving a functional law of large numbers (FLLN) and investigating the endemic equilibria properties of the limit. The limit depends on the law of the susceptibility random functions but only on the mean infectivity functions. The FLLN is proved by constructing a sequence of i.i.d. auxiliary processes and adapting the approach from the theory of propagation of chaos. The limit is a generalisation of a PDE model introduced by Kermack and McKendrick, and we show how this PDE model can be obtained as a special case of our FLLN limit.% for a particular set of infectivity and susceptibility random functions and initial conditions. For the endemic equilibria, if $ R_0 $ is lower than (or equal to) some threshold, the epidemic does not last forever and eventually disappears from the population, while if $ R_0 $ is larger than this threshold, the epidemic will not disappear and there exists an endemic equilibrium. The value of this threshold turns out to depend on the harmonic mean of the susceptibility a long time after an infection, a fact which was not previously known.

Stochastic epidemic models with varying infectivity and waning immunity

TL;DR

This work develops a highly general stochastic epidemic model in which infectivity and immunity are random functions of infection age and are renewed at each new infection. It proves a functional law of large numbers as , yielding a deterministic limit given by a two-dimensional Volterra integral system for (and corresponding ) that generalizes the Kermack–McKendrick PDEs by incorporating random susceptibility and time-varying infectiousness. The endemic-equilibrium analysis reveals a threshold determined by the harmonic mean of the long-time susceptibility, : if the disease dies out, while if a unique endemic equilibrium exists and can be characterized via a fixed-point equation for the limiting infection-pressure, with the endemic infection level given by . The authors also connect the LLN limit to the classical KMK PDE form under a special SIRS-like parametrization and outline rigorous proofs employing propagation-of-chaos techniques. The results highlight how heterogeneity and gradual waning of immunity shape epidemic thresholds and long-term behavior, with implications for public health strategies in heterogeneous populations.

Abstract

We study an individual-based stochastic epidemic model in which infected individuals become susceptible again following each infection. In contrast to classical compartment models, after each infection, the infectivity is a random function of the time elapsed since one's infection. Similarly, recovered individuals become gradually susceptible after some time according to a random susceptibility function. We study the large population asymptotic behaviour of the model, by proving a functional law of large numbers (FLLN) and investigating the endemic equilibria properties of the limit. The limit depends on the law of the susceptibility random functions but only on the mean infectivity functions. The FLLN is proved by constructing a sequence of i.i.d. auxiliary processes and adapting the approach from the theory of propagation of chaos. The limit is a generalisation of a PDE model introduced by Kermack and McKendrick, and we show how this PDE model can be obtained as a special case of our FLLN limit.% for a particular set of infectivity and susceptibility random functions and initial conditions. For the endemic equilibria, if is lower than (or equal to) some threshold, the epidemic does not last forever and eventually disappears from the population, while if is larger than this threshold, the epidemic will not disappear and there exists an endemic equilibrium. The value of this threshold turns out to depend on the harmonic mean of the susceptibility a long time after an infection, a fact which was not previously known.
Paper Structure (13 sections, 14 theorems, 194 equations, 2 figures)

This paper contains 13 sections, 14 theorems, 194 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model description
  3. Definition of the model
  4. Numbers of infected and uninfected individuals
  5. Functional law of large numbers
  6. The endemic equilibrium
  7. Relating to the Kermack and McKendrick PDEs for the SIRS model
  8. The FLLN limits with a special set of susceptibility/infectivity functions and initial conditions
  9. Connection with the Kermack and McKendrick PDE model
  10. Proofs for the FLLN
  11. Proofs for the endemic equilibrium
  12. Proof of Theorem \ref{['thm-eqlm1']}
  13. Proofs in the case $R_0\geq\mathbb{E}\left[\frac{1}{\gamma_*}\right]$

Key Result

Theorem 3.1

Under Assumption AS-lambda, the set of equations LLN_xy has a unique solution $(\overline{\mathfrak{S}},\overline{\mathfrak{F}})\in D^2(\mathbb{R}_+,\mathbb{R}_+)$. The solution belongs to $C^2(\mathbb{R}_+)$ if $\gamma_0$ has bounded variation and the map $t\mapsto(\mathbb{E}\left[\gamma_0(t)\right

Figures (2)

  • Figure 1: Illustration of a typical realization of the random infectivity and susceptibility functions of an individual from the time of infection to the time of recovery, and then to the time of losing immunity and becoming fully susceptible (or in general, partially susceptible).
  • Figure 2: Illustration of the evolution of an individual's infectivity and susceptibility through time. Each graphic shows the dynamics of an individual's infectivity (blue) and susceptibility (orange). The top graphic corresponds to an individual which is initially susceptible, and the bottom one to an initially infectious individual. Note that, after being reinfected, the second individual remains partially immune even a long time after infection.

Theorems & Definitions (41)

  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 4.1
  • Remark 4.1
  • ...and 31 more