Robustness of behaviourally-induced oscillations in epidemic models under a low rate of imported cases

TL;DR

This study investigates how small inflows of imported cases affect the stability of oscillations in epidemic models that include awareness spread, showing that these oscillations are robust and can be predicted by bifurcation analysis.

Contribution

It demonstrates the robustness of oscillations in epidemic models with awareness and imported cases, validated through stochastic simulations and bifurcation analysis.

Findings

Sustained oscillations persist with imported cases across network types.

A sharp transition to high-amplitude periodic outbreaks occurs as alerting rate increases.

Hopf-bifurcation curve accurately predicts the transition to oscillations.

Abstract

This paper is concerned with the robustness of the sustained oscillations predicted by an epidemic ODE model defined on contact networks. The model incorporates the spread of awareness among individuals and, moreover, a small inflow of imported cases. These cases prevent stochastic extinctions when we simulate the epidemics and, hence, they allow to check whether the average dynamics for the fraction of infected individuals are accurately predicted by the ODE model. Stochastic simulations confirm the existence of sustained oscillations for different types of random networks, with a sharp transition from a non-oscillatory asymptotic regime to a periodic one as the alerting rate of susceptible individuals increases from very small values. This abrupt transition to periodic epidemics of high amplitude is quite accurately predicted by the Hopf-bifurcation curve computed from the ODE model using the alerting rate and the infection transmission rate for aware individuals as tuning parameters.