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

Alessia Andò; Simone De Reggi; Francesca Scarabel; Rossana Vermiglio; Jianhong Wu; Alessia Andò; Simone De Reggi; Francesca Scarabel; Rossana Vermiglio; Jianhong Wu

doi:10.3934/mbe.2026004

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 1: 76-96. doi: 10.3934/mbe.2026004

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

1.
Department of Mathematics, Computer Science and Physics, University of Udine, Via delle Scienze 206, 33100 Udine, Italy
2.
Department of Mathematics, University of Trento, Via Sommarive 14, 38123 Povo (Trento), Italy
3.
School of Mathematics, University of Leeds, Woodhouse, Leeds LS2 9JT, United Kingdom
4.
CDLab – Computational Dynamics Laboratory, University of Udine, 33100 Udine, Italy
5.
Laboratory for Industrial and Applied Mathematics, Y-EMERGE, York University, Toronto M3J 1P3, Canada
6.
Fields-CQAM Laboratory of Mathematics for Public Health, York University, Toronto M3J 1P3, Canada

Received: 29 July 2025 Revised: 16 October 2025 Accepted: 18 November 2025 Published: 28 November 2025

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.
- incidence-based social distancing,
- behavioral epidemiology,
- pseudospectral approximation,
- MatCont,
- periodic solutions,
- stability,
- infectious disease model,
- linear chain trick
Citation: Alessia Andò, Simone De Reggi, Francesca Scarabel, Rossana Vermiglio, Jianhong Wu. Behavior-induced oscillations in epidemic outbreaks with distributed memory: Beyond the linear chain trick using numerical methods[J]. Mathematical Biosciences and Engineering, 2026, 23(1): 76-96. doi: 10.3934/mbe.2026004

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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.

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