Stability and bifurcations in an epidemic model with varying immunity period

TL;DR

This paper introduces a delay differential equation epidemic model with varying immunity, analyzing stability, bifurcations, and dynamical regimes through analytical proofs and numerical tools.

Contribution

It develops a new epidemic model incorporating distributed delays for immunity and provides comprehensive stability and bifurcation analysis.

Findings

Disease-free equilibrium is globally stable.

Endemic equilibrium can be stable or unstable depending on parameters.

Bifurcation analysis reveals transitions between dynamical regimes.

Abstract

An epidemic model with distributed time delay is derived to describe the dynamics of infectious diseases with varying immunity. It is shown that solutions are always positive, and the model has at most two steady states: disease-free and endemic. It is proved that the disease-free equilibrium is locally and globally asymptotically stable. When an endemic equilibrium exists, it is possible to analytically prove its local and global stability using Lyapunov functionals. Bifurcation analysis is performed using DDE-BIFTOOL and traceDDE to investigate different dynamical regimes in the model using numerical continuation for different values of system parameters and different integral kernels.