Global Stability of a Delayed SIRI Epidemic Model with Nonlinear Incidence

Abstract

In this paper we propose the global dynamics of an SIRI epidemic model with latency and a general nonlinear incidence function. The model is based on the susceptible-infective-recovered (SIR) compartmental structure with relapse (SIRI). Sufficient conditions for the global stability of equilibria (the disease-free equilibrium and the endemic equilibrium) are obtained by means of Lyapunov-LaSalle theorem. Also some numerical simulations are given to illustrate this result.

Global dynamics of an epidemiological model with age of infection and disease relapse

“…Hence, if β(a) and γ (a) are constants, then system (2) reduces to system (1). By choosing appropriate kernel functions, system (2) also contains infectious disease models with time delay, and the global stability result in this work provides the global dynamics for these delayed epidemic models.…”

“…In system (1), the population is divided into three compartments depending on disease status, where S(t) represents the number of individuals who are not previously exposed to the virus at time t, I(t) represents the number of infected individuals who are infectious and are able to spread the disease by contact with susceptible individuals at time t, and R(t) represents the number of recovered individuals who are previously infected with the virus but not currently shedding virus (latent) at time t. In system (1), it is assumed that the population is homogeneous mixing with constant size. The parameter A is the constant birth rate, μ is the death rate, β > 0 is the contact rate, that is, the average number of effective contacts of an infective individual per unit time, and γ > 0 is the rate at which infective individuals recover (latent).…”

“…In [22], the basic reproduction number was identified, and it was shown to be a sharp threshold to determine whether or not the disease dies out or approaches an endemic value. In [20], Moreira and Wang extended system (1) to include more general incidence functions, and by using an elementary analysis of Liénard's equation and Lyapunov's direct method, and sufficient conditions were established for the global asymptotic stability of the disease-free and endemic equilibria.…”

“…There are growing interests in infectious disease models with disease relapse (see, e.g. [1,11,24,25,27,28,30]). In [25], van den Driessche et al formulated and analysed a model including a general exposed distribution and the possibility of relapse in which a constant exposed period was assumed, for the spread of bovine TB (Mycobacterium bovis) in a cattle herd.…”

“…We note that in model (1), infectious individuals are assumed to be equally infectious during their periodic infectivity. However, laboratory studies suggest that the infectivity of infectious individuals be different at the differential age of infection [33].…”