Global stability of a delayed SIRS epidemic model with a non-monotonic incidence rate

Abstract

Keywords: SIRS epidemic model Nonlinear incidence rate Global asymptotic stability Permanence Monotone iterative technique Lyapunov functional technique In this paper, we investigate a disease transmission model of SIRS type with latent period τ 0 and the specific nonmonotone incidence rate, namely,.For the basic reproduction number R 0 > 1, applying monotone iterative techniques, we establish sufficient conditions for the global asymptotic stability of endemic equilibrium of system which become partial answers to the open problem in [Hai-Feng Huo, ZhanPing Ma, Dynamics of a delayed epidemic model with non-monotonic incidence rate, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 459-468]. Moreover, combining both monotone iterative techniques and the Lyapunov functional techniques to an SIR model by perturbation, we derive another type of sufficient conditions for the global asymptotic stability of the endemic equilibrium.

Stability analysis of a novel epidemics model with vaccination and nonlinear infectious rate

“…In this paper, to overcome the disadvantages above, by considering more complete processes of the epidemic diseases, we first build a novel susceptible-exposed-infective-recovered-partial immunity (SEIRW) model, which incorporates pathogen evolution and human interventions behaviors with vaccines or drugs that alter pathogen genetic diversity, aiming to reflect the transmission dynamics of infectious diseases more realistically. Our model differs from the existing ones in [17][18][19][20][21] in the following respect: (i) our model additionally takes into account the exposed fraction and the partial immunity individuals; (ii) our model is coupled with evolutionary-epidemic strategy and human interference behaviors such as vaccination to a proportion of all newborn children; (iii) in particular, the force of infection of the infectious disease in the existing work is multiplied by a constant such as survival probability while in this paper, it is governed by a nonlinear time-varying growing curve of pathogens, which reflects the viruses' evolution. Subsequently, we transform the SEIRW model into the ordinary differential equation with time-varying perturbation.…”

“…For model (21) to be mathematically tractable and epidemiologically meaningful, it is important to prove that all the state variables are nonnegative and bounded for all time. We prove that all solutions of system (21) with positive initial data will remain positive for all time t P 0.…”

“…. , x 5 (t)] T be any solution of system (21) with the initial functions u 2 C 5 [ À s,0] at t = 0. System (21) can be transformed into the following form…”

“…In [20], Kuniya studied the global asymptotic stability for an age-structured multigroup SIR epidemic model with a discretization approach. Muroya et al [21] investigated the global asymptotic stability of a disease transmission model of SIRS type with latent period and the specific non-monotone incidence rate. The results of these studies, however, do not take into account the whole process of the epidemic diseases such as the exposed fraction and the partial immunity individuals, which are to a great extent owing to deficiency of the genetic variation behavior of the virus and the epidemiology theory.…”

Analysis of a SIQS epidemic model with non-linear incidence rate and application in GMF information dissemination

“…It is wellknown that the rst SIRS epidemic model proposed by Kermack and McKendrick [1] in 1927, and many related models [2][3][4] were studied since then. The incidence rate with general form g(I)S play an important role in epidemic model.…”