Global stability for epidemic model with constant latency and infectious periods

Abstract

In recent years many delay epidemiological models have been proposed to study at which stage of the epidemics the delays can destabilize the disease free equilibrium, or the endemic equilibrium, giving rise to stability switches. One of these models is the SEIR model with constant latency time and infectious periods [2], for which the authors have proved that the two delays are harmless in inducing stability switches. However, it is left open the problem of the global asymptotic stability of the endemic equilibrium whenever it exists. Even the Lyapunov functions approach, recently proposed by Huang and Takeuchi to study many delay epidemiological models, fails to work on this model. In this paper, an age-infection model is presented for the delay SEIR epidemic model, such that the properties of global asymptotic stability of the equilibria of the age-infection model imply the same properties for the original delay-differential epidemic model. By introducing suitable Lyapunov functions to study the global stability of the disease free equilibrium (when R0 ≤ 1) and of the endemic equilibria (whenever R0 > 1) of the age-infection model, we can infer the corresponding global properties for the equilibria of the delay SEIR model in [2], thus proving that the endemic equilibrium in [2] is globally asymptotically stable whenever it exists.

Dynamics of an age-of-infection cholera model

“…When choosing kernel functions for differential infectivity in direct and indirect transmission, model (2.1) contains several multi-stage ODE models, such as the staged progression model in [21] and the mass-action model in [18]. In [10,16], delay models are regarded as special cases of age-of-infection models with only direct transmission. By taking appropriate kernel functions, our model (2.1) also contains delay cholera models, and our global stability result (Theorem 3.1) provides the global dynamics for these delay models.…”

Global stability properties of age‐dependent epidemic models with varying rates of recurrence

“…In recent years, Lyapunov's second method has been a popular technique to study global stability of epidemiological models. A Volterra‐type Lyapunov function, V ( X ) = X − 1− ln X , it has been used in to prove global stability of several epidemiological models. Recent novel applications of Volterra‐type functions in multistage epidemic models were obtained by Guo, Li and Shuai .…”

“…Huang and coauthors solved the global stability problem for basic age‐structured model of viral infections using the Volterra‐type functions. In , they reformulated a delay SIR epidemic model as an age‐of‐infection model; they have used the Lyapunov's method and Volterra‐type functions to establish a similar global stability threshold. This function is used also in a class of host–vector models with age‐dependent .…”

“…We will analyse the global stability of the equilibria in our structured models by means of suitable Lyapunov functions and the LaSalle's invariance principle for semiflows. The global Lyapunov functions are motivated by the papers .…”