Saddle-node bifurcation of limit cycles in an epidemic model with two levels of awareness

TL;DR

This paper investigates bifurcations of limit cycles in an epidemic model with two awareness levels, revealing how nonlinear rates influence disease dynamics and stability, supported by numerical and stochastic analyses.

Contribution

It introduces a novel epidemic model with nonlinear awareness transition rates and analyzes bifurcations and stability conditions using numerical and stochastic methods.

Findings

Limit cycles emerge through saddle-node bifurcations.

Bistability between endemic equilibrium and oscillations.

Stochastic simulations confirm bifurcation scenarios.

Abstract

In this paper we study the appearance of bifurcations of limit cycles in an epidemic model with two types of aware individuals. All the transition rates are constant except for the alerting decay rate of the most aware individuals and the rate of creation of the less aware individuals, which depend on the disease prevalence in a non-linear way. For the ODE model, the numerical computation of the limit cycles and the study of their stability are made by means of the Poincar\'e map. Moreover, sufficient conditions for the existence of an endemic equilibrium are also obtained. These conditions involve a rather natural relationship between the transmissibility of the disease and that of awareness. Finally, stochastic simulations of the model under a very low rate of imported cases are used to confirm the scenarios of bistability (endemic equilibrium and limit cycle) observed in the solutions of the ODE model.