An SIR epidemic model with partial temporary immunity modeled with delay

Abstract

The SIR epidemic model for disease dynamics considers recovered individuals to be permanently immune, while the SIS epidemic model considers recovered individuals to be immediately resusceptible. We study the case of temporary immunity in an SIR-based model with delayed coupling between the susceptible and removed classes, which results in a coupled set of delay differential equations. We find conditions for which the endemic steady state becomes unstable to periodic outbreaks. We then use analytical and numerical bifurcation analysis to describe how the severity and period of the outbreaks depend on the model parameters.

Synchronous versus asynchronous oscillations for antigenically varyingPlasmodium falciparumwith host immune response

“…In terms of the original parameters, the value of the delay for an Hopf bifurcation is given by Equation (29). For a long-lived variant-specific IR relative to the cross-reactive IR, and all other processes being relatively equal, then the efficacy of the variantspecific IR E z will be much larger than that of the cross-reactive IR, E w .…”

“…We introduce a new slow time scale s = η 2 t and assume state variables depend on t and s, where oscillatory behaviour occurs in time scale t and slow evolution of the amplitude of oscillations occurs in the long time scale s. With the new slow time scale, the delay term in Equation (7) expands as (see [21,29] for a discussion of the validity of this expansion)…”

Delay-periodic solutions and their stability using averaging in delay-differential equations, with applications

“…which has applications in lasers [7][8][9][10][11], population epidemics [6,12], and malaria infection [13,14]. More generally, Eqs.…”

“…A third approach is to allow the delay to remain in the leading order problem but to take advantage of solution properties such as pulsations that allow for the construction of solutions other than by direct methods. For example, in [5,6] the authors use ideas from matched asymptotic expansions to patch together a complete solution based on approximate solutions constructed over different intervals of time.…”

Global Hopf bifurcation analysis of an susceptible-infective-removed epidemic model incorporating media coverage with time delay

“…For instance, it can change the stability of equilibrium and thus lead to periodic solutions by Hopf bifurcation [6,[10][11][12]14,17,18,21,[27][28][29][30].…”