Genuine and spurious bistability in a simple epidemic model with waning immunity

TL;DR

This work analyzes an infection-age-structured SIS model with gamma-distributed infection durations and waning immunity, showing that the shape of the duration distribution can destabilize the endemic equilibrium and even produce genuine bistability with a stable periodic orbit. It develops three finite-dimensional approaches (Erlang and hypoexponential via ODE reductions, and a pseudospectral discretization) to approximate the infinite-dimensional model and perform bifurcation analyses. The results demonstrate genuine bistability and oscillations driven by infection-age structure, while revealing that common Erlang-based approximations can yield spurious bistability absent in the underlying gamma model. The study highlights the critical influence of distributional shape on epidemic dynamics and provides robust numerical tools to study such effects, with implications for how compartmental reductions may mislead public-health inferences.

Abstract

We study an infection-age structured epidemic model in which both the infectivity and the rate of loss of immunity depend on the time-since-infection. The model can be equivalently viewed as a nonlinear renewal equation for the incidence of infection or as a partial differential equation for the density of infected individuals. We explicitly consider gamma, rather than Erlang, distributed durations of infection using a combination of ODE approximations and numerical bifurcation methods. We show that the shape of this distribution strongly influences stability of the endemic equilibrium, even when the basic reproduction number $R_0$ and the mean duration of infectiousness are fixed. Moreover, we establish the existence of regions of bistability, where a stable endemic equilibrium coexists with a stable periodic orbit. To our knowledge, this provides the first example of bistability in infection-age structured models with waning immunity alone. Finally, we show how common compartmental modelling approaches, which impose implicit assumptions on the distribution of the duration of infection, can lead to spurious dynamical outcomes. Taken together, our analysis underscores the crucial role of distributional structure in epidemic modelling and provides new insights into the rich dynamics of infection-age structured SIS/SIRS models.

Figures (5)

• Figure 1: A model schematic of the SIS model structured by infection-age. Susceptible individuals are denoted by $S(t)$ and are infected with force of infection $\Lambda(t)$. Newly infected individuals have infection-age $a=0$ and enter at the left-hand boundary of the age-structured infected population. As these infected individuals progress through infection, their infectivity, and corresponding contribution to the force of infection, evolves according to $\beta(a)$. Infected individuals return to the susceptible class at infection-age dependent rate $h(a),$ which is the hazard rate corresponding to the random variable $\mathbb{A}$ that defines the duration of infection and corresponds to the immunity waning rate. Individuals can remain in the infectious class but no longer contribute to the force of infection if $\beta(a)$ has compact support.
• Figure 2: Stability of the EE in two-parameter planes In all cases, the EE loses stability through a Hopf bifurcation. In Panel A, $\tau=5$ and $j=20$ are fixed. In Panel B, we hold $R_0=2.2$ and $k_d=2$ fixed. In Panel C, we hold $R_0=2.2$ and $\tau=5$ fixed. The grey dashed lines correspond to parameter choices for the one-parameter continuations.
• Figure 3: One-parameter bifurcation diagram of the EE $I^*$ as a function of $R_0$ In both cases, the EE becomes stable following a transcritical bifurcation of the DFE at $R_0 =1$ and loses stability through a supercritical Hopf bifurcation. In Panel B, the EE undergoes a subcritical Hopf bifurcation which leads to the existence of an unstable periodic orbit and a region of bistability. In Panels A and B, we consider $\tau=5$ and $j=20$, while $k_d=2$ and Panel A and $k_d=5$ in Panel B. These parameter choices correspond to the Panel A of Figure \ref{['fig:planes']}.
• Figure 4: Genuine bistability in the simple epidemic model. In both Panel A and B, we take $R_0 =3.75, \tau=5$, and $j=20$. Panel A shows a one-parameter bifurcation diagram with $k_d$ as bifurcation parameter. The EE loses stability through a supercritical Hopf bifurcation which leads to a stable periodic orbit. This stable periodic orbit loses stability in a limit point of cycles bifurcation which results in a region of bistability of the EE and the stable periodic orbit. Panel B illustrates this bistability via time simulations of the simple epidemic model from two different initial conditions. Here, as $j = 20 \in \mathbb{N},$ the Erlang approximation is exact.
• Figure 5: Spurious bistability in the simple epidemic model. In both Panel A and B, we take $R_0 = 2.2, \tau=5$, and $k_d = 4.2$. Panel A shows a one-parameter bifurcation diagram with $j$ as bifurcation parameter. The EE loses stability through a supercritical Hopf bifurcation which leads to a stable periodic orbit at $j^* = 8.31$. Panel B shows two time simulations of this simple epidemic model using the Erlang and hypoexponential approximation to approximate the nonlinear RE with $j = 8.32$. The Erlang approximation predicts spurious convergence to the EE while the hypoexponential approximation accurately captures the stable periodic orbit.

Theorems & Definitions (1)

• Theorem 3.1