Global dynamics of an epidemic model with relapse and nonlinear incidence

On the basis of a basic SIR epidemic model, we propose and study an epidemic model with nonlinear incidence. The model also incorporates many features of the recovered such as relapse and with/without immunity. A threshold dynamics is established, which is completely determined by the basic reproduction number. The global stability of the disease‐free equilibrium is proved by means of the fluctuation lemma. To prove the global stability of the endemic equilibrium, we need some novel techniques including the transformation of variables, the construction of a new type of Lyapunov functions, and the seeking of an appropriate positively invariant set of the model.     

Global stability of a discrete multigroup SIR model with nonlinear incidence rate

In this paper, by applying nonstandard finite difference scheme, we propose a discrete multigroup Susceptible‐Infective‐Removed (SIR) model with nonlinear incidence rate. Using Lyapunov functions, it is shown that the global dynamics of this model are completely determined by the basic reproduction number . If , then the disease‐free equilibrium is globally asymptotically stable; if , then there exists a unique endemic equilibrium and it is globally asymptotically stable. Example and numerical simulations are presented to illustrate the results. Copyright © 2017 John Wiley & Sons, Ltd.     

Global dynamics of a multistage SIR model with distributed delays and nonlinear incidence rate

In this paper, a multistage susceptible‐infectious‐recovered model with distributed delays and nonlinear incidence rate is investigated, which extends the model considered by Guo et al. [H. Guo, M. Y. Li and Z. Shuai, Global dynamics of a general class of multistage models for infectious diseases, SIAM J. Appl. Math., 72 (2012), 261–279]. Under some appropriate and realistic conditions, the global dynamics is completely determined by the basic reproduction number R₀. If R₀≤1, then the infection‐free equilibrium is globally asymptotically stable and the disease dies out in all stages. If R₀>1, then a unique endemic equilibrium exists, and it is globally asymptotically stable, and hence the disease persists in all stages. The results are proved by utilizing the theory of non‐negative matrices, Lyapunov functionals, and the graph‐theoretical approach. Copyright © 2016 John Wiley & Sons, Ltd.     

Global analysis for an epidemical model of vector-borne plant viruses with disease resistance and nonlinear incidence

Vector-borne disease models play an important role in understanding the mechanism of plant disease transmission. In this paper, we study a vector-borne model with plant disease resistance, disease exposed period and nonlinear incidence. We compute the basic reproduction number, determine the implicit locations of equilibria and then investigate their global stability by generalizing a classic geometric approach to higher dimensional systems. Higher dimensions cause greater difficulties such as the construction of the transformation matrix and the estimate of the $Lozinski\tilde{\iota}$ measure in this geometric approach. For a complete control of vector-borne diseases, a quantitative way is provided by the given expression of the basic reproduction number, from which we need not only increasing plant disease resistance but also decreasing the contact rate between infected plants and susceptible vectors instead of a single one of them.     

Global asymptotical properties for a diffused HBV infection model with CTL immune response and nonlinear incidence

This article proposes a diffused hepatitis B virus (HBV) model with CTL immune response and nonlinear incidence for the control of viral infections. By means of different Lyapunov functions, the global asymptotical properties of the viral-free equilibrium and immune-free equilibrium of the model are obtained. Global stability of the positive equilibrium of the model is also considered. The results show that the free diffusion of the virus has no effect on the global stability of such HBV infection problem with Neumann homogeneous boundary conditions.     

Traveling waves in an SEIR epidemic model with a general nonlinear incidence rate

ABSTRACT

We study the traveling waves of reaction-diffusion equations for a diffusive SEIR model with a general nonlinear incidence. The existence of traveling waves is determined by the basic reproduction number of the corresponding ordinary differential equations and the minimal wave speed. Its proof is showed by introducing an auxiliary system, applying Schauder fixed point theorem and then a limiting argument. The non-existence proof is obtained by two-sided Laplace transform when the speed is less than the critical velocity. Finally, we present some examples to support our theoretical results.     

Modeling diseases with latency and nonlinear incidence rates: global dynamics of a multi‐group model

In this paper, we perform global stability analysis of a multi‐group SEIR epidemic model in which we can consider the heterogeneity of host population and the effects of latency and nonlinear incidence rates. For a simpler version that assumes an identical natural death rate for all groups, and with a gamma distribution for the latency, the basic reproduction number is defined by the theory of the next generation operator and proved to be a sharp threshold determining whether or not disease spread. Under certain assumptions, the disease‐free equilibrium is globally asymptotically stable if R₀≤1 and there exists a unique endemic equilibrium which is globally asymptotically stable if R₀>1. The proofs of global stability of equilibria exploit a matrix‐theoretic method using Perron eigenvetor, a graph‐theoretic method based on Kirchhoff's matrix tree theorem and Lyapunov functionals. Copyright © 2015 John Wiley & Sons, Ltd.     

具有常数迁入率和非线性传染率βI~pS~q的SI模型分析

对一种具有种群动力和非线性传染率的传染病模型进行了研究,建立了具有常数迁入率和非线性传染率βI~pS~q的SI模型.与以往的具有非线性传染率的传染病模型相比,这种模型引入了种群动力,也就是种群的总数不再为常数,因此,该类模型更精确地描述了传染病传播的规律.还讨论了模型的正不变集,运用微分方程稳定性理论分析了模型平衡点的存在性及稳定性,得出了疾病消除平衡点和地方病平衡点的全局渐进稳定的充分条件.进一步的,得出了在某些参数范围内会出现Hopf分支现象,并对上述模型进行了生物学讨论.     

Global dynamics for an age-structured epidemic model with media impact and incomplete vaccination

Some studies show that the efficacy of vaccines may decline and the vaccine-induced immunity wanes with time elapsed since vaccination (e.g., BCG, HBV vaccines). Meanwhile, the use of media can effectively contain the emerging epidemics. In our work, an SVEIR epidemic model with media impact, age-dependent vaccination and latency is proposed, where the efficacy of vaccines depends on the time since vaccination. Uniform persistence and the sharp threshold dynamics are established, that is, the vaccination reproduction number completely determines the global dynamics of the model system. Our main results established can be extended to the SVEIR model with the general nonlinear infectious force function of monotonicity and concavity.     

Global analysis of a multi-group animal epidemic model with indirect infection and time delay

The transmission mechanism of some animal diseases is complex because of the multiple transmission pathways and multiple-group interactions, which lead to the limited understanding of the dynamics of these diseases transmission. In this paper, a delay multi-group dynamic model is proposed in which time delay is caused by the latency of infection. Under the biologically motivated assumptions, the basic reproduction number $R_0$ is derived and then the global stability of the disease-free equilibrium and the endemic equilibrium is analyzed by Lyapunov functionals and a graph-theoretic approach as for time delay. The results show the global properties of equilibria only depend on the basic reproductive number $R_0$: the disease-free equilibrium is globally asymptotically stable if $R_0\leq 1$; if $R_0>1$, the endemic equilibrium exists and is globally asymptotically stable, which implies time delay span has no effect on the stability of equilibria. Finally, some specific examples are taken to illustrate the utilization of the results and then numerical simulations are used for further discussion. The numerical results show time delay model may experience periodic oscillation behaviors, implying that the spread of animal diseases depends largely on the prevention and control strategies of all sub-populations.     

Effect of discretization on dynamical behavior of SEIR and SIR models with nonlinear incidence

In this paper, using the forward Euler and backward Euler methods, we present four discrete epidemic models with the nonlinear incidence rate. We discuss the effect of two discretizations on the stability of the endemic equilibrium for these models. Numerical simulations are performed to illustrate our analytic results.     

A class of Lyapunov functions and the global stability of some epidemic models with nonlinear incidence

In this paper, by investigating an SIR epidemic model with nonlinear incidence, we present a new technique for proving the global stability of the endemic equilibrium, which consists of introducing a variable transformation and constructing a more general Lyapunov function. For the model we obtain the following results. The disease-free equilibrium is globally stable in the feasible region as the basic reproduction number is less than or equal to unity, and the endemic equilibrium is globally stable in the feasible region as the basic reproduction number is greater than unity.The generality of the technique is illustrated by considering certain nonlinear incidences and SIS and SIRS epidemic models.     

Global dynamics of a Vector‐Borne disease model with two delays and nonlinear transmission rate

In this paper, we investigate a Vector‐Borne disease model with nonlinear incidence rate and 2 delays: One is the incubation period in the vectors and the other is the incubation period in the host. Under the biologically motivated assumptions, we show that the global dynamics are completely determined by the basic reproduction number R₀. The disease‐free equilibrium is globally asymptotically stable if R₀≤1; when R₀>1, the system is uniformly persistent, and there exists a unique endemic equilibrium that is globally asymptotically. Numerical simulations are conducted to illustrate the theoretical results.     

TWO DIFFERENTIAL INFECTIVITY EPIDEMIC MODELS WITH NONLINEAR INCIDENCE RATE

This paper considers two differential infectivity（DI） epidemic models with a nonlinear incidence rate and constant or varying population size. The models exhibits two equilibria, namely., a disease-free equilibrium O and a unique endemic equilibrium. If the basic reproductive number σ is below unity,O is globally stable and the disease always dies out. If σ〉1, O is unstable and the sufficient conditions for global stability of endemic equilibrium are derived. Moreover,when σ〈 1 ,the local or global asymptotical stability of endemic equilibrium for DI model with constant population size in n-dimensional or two-dimensional space is obtained.     

Epidemic transmission on SEIR stochastic models with nonlinear incidence rate

Our interest is to quantify the spread of an infective process with latency period and generic incidence rate that takes place in a finite and homogeneous population. Within a stochastic framework, two random variables are defined to describe the variations of the number of secondary cases produced by an index case inside of a closed population. Computational algorithms are presented in order to characterize both random variables. Finally, theoretical and algorithmic results are illustrated by several numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Global dynamics of an age‐structured virus model with saturation effects

An infection‐age virus dynamics model for human immunodeficiency virus (or hepatitis B virus) infections with saturation effects of infection rate and immune response is investigated in this paper. It is shown that the global dynamics of the model is completely determined by two critical values R ₀, the basic reproductive number for viral infection, and R ₁, the viral reproductive number at the immune‐free infection steady state (R ₁<R ₀). If R ₀<1, the uninfected steady state E ⁰ is globally asymptotically stable; if R ₀>1 > R ₁, the immune‐free infected steady state E ^? is globally asymptotically stable; while if R ₁>1, the antibody immune infected steady state is globally asymptotically stable. Moreover, our results show that ignoring the saturation effects of antibody immune response or infection rate will result in an overestimate of the antibody immune reproductive number. Copyright © 2016 John Wiley & Sons, Ltd.