Diffusion-induced spatio-temporal oscillations in an epidemic model with two delays

TL;DR

This paper analyzes a diffusive epidemic model with delays, revealing how bifurcations lead to stable oscillations with various spatial patterns, and provides explicit conditions for their existence.

Contribution

It introduces a detailed bifurcation analysis of a delayed, diffusive epidemic model, including explicit formulas for bifurcation criticality and conditions for spatially heterogeneous oscillations.

Findings

Hopf bifurcation causes destabilization of the endemic equilibrium.

Stable oscillations can be spatially homogeneous or heterogeneous depending on parameters.

Explicit conditions for the existence of different oscillation patterns are derived.

Abstract

We investigate a diffusive, stage-structured epidemic model with the maturation delay and freely-moving delay. Choosing delays and diffusive rates as bifurcation parameters, the only possible way to destabilize the endemic equilibrium is through Hopf bifurcation. The normal forms of Hopf bifurcations on the center manifold are calculated, and explicit formulae determining the criticality of bifurcations are derived. There are two different kinds of stable oscillations near the first bifurcation: on one hand, we theoretically prove that when the diffusion rate of infected immature individuals is sufficiently small or sufficiently large, the first branch of Hopf bifurcating solutions is always spatially homogeneous; on the other, fixing this diffusion rate at an appropriate size, stable oscillations with different spatial profiles are observed, and the conditions to guarantee the existence of such solutions are given. These phenomena are investigated by calculating the corresponding eigenfunction of the Laplacian at the first Hopf bifurcation point.