Multistability Switches and Codimension-2 Bifurcation in an SIRS(Z) Model with Two Delays

Abstract

In an [Formula: see text] model, the combined effect of two delays, incubation delay and information delay, is investigated on disease dynamics. Here [Formula: see text] is information density. A unique disease-free equilibrium is obtained which is locally stable for basic reproduction number ([Formula: see text]) below one and unstable for [Formula: see text] above one for all delays. The unique endemic equilibrium, which exists for [Formula: see text] exhibits stability switches at the critical values of delays regardless of its initial stability (i.e. either stable or unstable without delay). We observe that both delays have a significant impact on stability switching. If one delay destabilizes the endemic equilibrium, another delay may restore it, and vice versa. In our analysis, we fix one delay in specific ranges while varying and showing the effect of the other delay. We note that if the first delay is changed, the impact of the second delay changes as well. We explicitly show this result for all ranges of the first delay. We show that in the presence of multiple codimension-1 Hopf bifurcations, endemic equilibrium switches stability multiple times, which we prove both analytically and numerically. When delays are varied in the presence of a codimension-2 Hopf–Hopf bifurcation, endemic equilibrium either switches stability from unstable to stable to unstable or generates different frequency oscillations around it. We investigated this result numerically. The double frequencies oscillations are observed as a result of this codimension-2 bifurcation. We also find a region in two-parametric plane to show various dynamical properties varied from stability to double frequency oscillation.