Behavior-induced oscillations in epidemic outbreaks with distributed memory: beyond the linear chain trick using numerical methods

TL;DR

The paper investigates how behavioral adaptation driven by distributed memory of past infections can induce sustained epidemic waves, even when susceptibles are effectively nondepleting. It extends the incidence-based information index framework by incorporating gamma-distributed memory kernels with general shape $\alpha$ and rate $\mu$, and employs a pseudospectral discretization to transform the infinite-delay renewal equation into a finite-dimensional ODE system amenable to continuation analysis. Analytic results for equilibria show the disease-free state is historically stable for $R_0<1$, while a positive established equilibrium $E_+$ exists when $R_0>1$ and its stability depends on the memory kernel via $\widehat{K}(\lambda)$; Hopf bifurcations are possible when memory delays are nontrivial. Numerically, larger memory concentration (higher $\alpha$) and longer average delays promote oscillations, and a nonzero minimal contact rate $\beta_1$ can stabilize the system, with Hopf branches analyzed using MatCont on the approximating ODE system. The work demonstrates how distributed-memory effects shape epidemic waves and provides a practical framework to analyze complex memory kernels beyond the linear-chain trick, with implications for understanding and controlling behavior-driven outbreaks.

Abstract

We considered a model for an infectious disease outbreak, when the depletion of susceptible individuals is negligible, and assumed that individuals adapt their behavior according to the information they receive about new cases. In line with the information index approach, we supposed that individuals react to past information according to a memory kernel that is continuously distributed in the past. We analyzed equilibria and their stability, with analytical results for selected cases. Thanks to the recently developed pseudospectral approximation of delay equations, we studied numerically the long-term dynamics of the model for memory kernels defined by gamma distributions with a general non-integer shape parameter, extending the analysis beyond what is allowed by the linear chain trick. In agreement with previous studies, we showed that behavior adaptation alone can cause sustained waves of infections even in an outbreak scenario, and notably in the absence of other processes like demographic turnover, seasonality, or waning immunity. Our analysis gives a more general insight into how the period and peak of epidemic waves depend on the shape of the memory kernel and how the level of minimal contact impacts the stability of the behavior-induced positive equilibrium.

Figures (7)

• Figure 1: Stable and unstable regions of the EE for \ref{['model']} with a gamma-distributed kernel and $f$ defined as in \ref{['f1']} in the $(\beta_0,\tau)$ plane, for $\alpha = 2$ (left) and $\alpha = 3$ (right). Blue: analytic curve. Red: obtained with MatCont using $N=20$, $\rho=0.5\alpha/\tau$, and $r=1$. The dashed line delimits the region of existence of the EE (DFE is LAS to the left of the dashed line).
• Figure 2: Stable and unstable regions of the EE for \ref{['model']} with a gamma-distributed kernel and $f$ defined as in \ref{['f1']}. Left: regions in the $(\beta_0,\alpha)$ plane, for $\tau = 14$. Right: regions in the $(\tau,\alpha)$ plane, for $\beta_0=5$. Curves were obtained with MatCont using $N=20$, $\rho=0.5\alpha/\tau$, and $r=1$ (left) and $N=30$, $\rho=0.5\alpha/\tau$, and $r=0.25$ (right).
• Figure 3: Left: bifurcation diagram for \ref{['model']} with a gamma-distributed kernel and $f$ defined as in \ref{['f1']} with $\alpha$ as the bifurcation parameter, for $\beta_0=3$ and $\tau = 14$. Right: profiles of the periodic solutions for the values $\alpha = 4.8$ (red), $\alpha = 5.4$ (blue), and $\alpha = 6$ (orange). Curves were obtained with MatCont using $N=20$, $\rho=0.5\alpha/\tau$, and $r=0.25$.
• Figure 4: Left: bifurcation diagram for \ref{['model']} with a gamma-distributed kernel and $f$ defined as in \ref{['f1']} with $\alpha$ as the bifurcation parameter, for $\beta_0=5$ and $\tau = 14$. Right: profiles of the periodic solutions for the values $\alpha = 4.6$ (red), $\alpha = 5$ (blue), and $\alpha = 5.5$ (orange). Curves were obtained with MatCont using $N=20$, $\rho=0.5\alpha/\tau$, and $r=0.25$.
• Figure 5: Hopf curves for \ref{['model']} with a gamma-distributed kernel and $f$ defined as in \ref{['f_mincontact']}, for $\tau = 14$. Left: curves in the $(\beta_0,\alpha)$ plane with $\beta_1 = 0.05$ (red) and $\beta_1 = 0.08$ (blue). In both cases, the stable region is the one below the corresponding curve. Right: curve in the $(\beta_0,\beta_1)$ plane for $\alpha = 10.5$. Curves were obtained with MatCont using $N=20$ and $r=1$.

Theorems & Definitions (1)

• Proposition 3.1