Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model

Global stability analysis of a delayed susceptible–infected–susceptible epidemic model

“…Our model is based on the assumption that the disease transmission occurs only when susceptible and infected individuals contact each other at the same time, namely, any susceptible individuals at present time cannot contact the infected individuals at the past time. This key assumption makes our model different from those considered in previous literature [3,4,6,9] in the sense that the incidence rate in our equation of susceptible individuals does not contain any delay. Thanks to this new assumption, we are able to establish a threshold theorem of global stability without any technical condition, while in the literature (see [3,4] for example), an additional assumption of sufficiently small probability of immunity lost is required to obtain global stability of endemic equilibrium.…”

“…This key assumption makes our model different from those considered in previous literature [3,4,6,9] in the sense that the incidence rate in our equation of susceptible individuals does not contain any delay. Thanks to this new assumption, we are able to establish a threshold theorem of global stability without any technical condition, while in the literature (see [3,4] for example), an additional assumption of sufficiently small probability of immunity lost is required to obtain global stability of endemic equilibrium. It is also noted that by choosing special distribution function p(τ ) and using a standard linear chain trick [8, p. 96], our model reduces to the SEIS model with multiple latent classes considered in [2], and our result coincides with that obtained in [2].…”

“…p(τ ) ≥ 0 with τ ∈ [0, ∞) is the probability density function of transmission delay, μ ≥ 0 corresponds to the death rate during latent period, γ ≥ 0 is the recovery rate of infected individuals, and q ∈ [0, 1] denotes the probability of immunity lost. We remark that the delayed nonlinear incidence rates adopted in our model are different from those in [3,4,6,9]. In those models, the incidence rates take the same form βS(t)I(t − τ ) in both equations of S (t) and I (t).…”

“…We shall construct two suitable Lyapunov functionals which are modified from those in [3,4,6,7,9]. Our result is more decent in the sense that no additional condition is required to obtain global stability of endemic equilibrium; while in the literature (cf.…”

“…Our result is more decent in the sense that no additional condition is required to obtain global stability of endemic equilibrium; while in the literature (cf. [3,4]), a technical assumption that the probability of immunity lost should be small is needed. Our main results are given in Section 2.…”