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.     

Traveling waves of a nonlocal diffusion SIRS epidemic model with a class of nonlinear incidence rates and time delay

In this paper, we study the traveling waves of a delayed SIRS epidemic model with nonlocal diffusion and a class of nonlinear incidence rates. When the basic reproduction ratio $\mathscr{R}_0>1$, by using the Schauder''s fixed point theorem associated with upper-lower solutions, we reduce the existence of traveling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of traveling wave solutions connecting the disease-free steady state and the endemic steady state. When $\mathscr{R}_0<1$, the nonexistence of traveling waves is obtained by the comparison principle.     

Traveling waves in a diffusive epidemic model with criss‐cross mechanism

In this paper, we propose a reaction‐diffusion system to describe the spread of infectious diseases within two population groups by self and criss‐cross infection mechanism. Firstly, based on the eigenvalues, we give two methods for the calculation of the critical wave speed c^?. Secondly, by constructing a pair of upper‐lower solutions and using the Schauder fixed‐point theorem, we prove that the system admits positive traveling wave solutions, which connect the initial disease‐free equilibrium at t = ?∞, but the traveling waves need not connect the final disease‐free equilibrium at t = +∞. Hence, we study the asymptotic behaviors of the traveling wave solutions to show that the traveling wave solutions converge to at t = +∞. Finally, by the two‐sided Laplace transform, we establish the nonexistence of traveling waves for the model. The approach in this paper provides an effective method to deal with the existence of traveling wave solutions for the nonmonotone reaction‐diffusion systems consisting of four equations.     

Monotonicity,uniqueness, and stability of traveling waves in a nonlocal reaction‐diffusion system with delay

The purpose of this paper is to study the traveling wave solutions of a nonlocal reaction‐diffusion system with delay arising from the spread of an epidemic by oral‐faecal transmission. Under monostable and quasimonotone it is well known that the system has a minimal wave speed c* of traveling wave fronts. In this paper, we first prove the monotonicity and uniqueness of traveling waves with speed c ?c _?. Then we show that the traveling wave fronts with speed c >c _? are exponentially asymptotically stable.     

Traveling wave fronts in a delayed lattice competitive system

This paper is concerned with the existence, asymptotic behavior, strict monotonicity, and uniqueness of traveling wave fronts connecting two half-positive equilibria in a delayed lattice competitive system. We first prove the existence of traveling wave fronts by constructing upper and lower solutions and Schauder’s fixed point theorem, and then, for sufficiently small intraspecific competitive delays, prove that these traveling wave fronts decay exponentially at both infinities. Furthermore, for system without intraspecific competitive delays, the strict monotonicity and uniqueness of traveling wave fronts are established by means of the sliding method. In addition, we give the exact decay rate of the stronger competitor under some technique conditions by appealing to uniqueness.     

Traveling wavefronts of a delayed lattice reaction-diffusion model

We investigate a system of delayed lattice differential system which is a model of pioneer-climax species distributed on one dimensional discrete space. We show that there exists a constant $c^*&gt;0$, such that the model has traveling wave solutions connecting a boundary equilibrium to a co-existence equilibrium for $c\geq c^*$. We also argue that $c^*$ is the minimal wave speed and the delay is harmless. The Schauder's fixed point theorem combining with upper-lower solution technique is used for showing the existence of wave solution.     

Asymptotic boundary and nonexistence of traveling waves in a discrete diffusive epidemic model

ABSTRACT

In this paper, we establish the asymptotic boundary of traveling waves (first-component) in a discrete diffusive epidemic model, whose existence is well-known. Meanwhile, we derive the nonexistence of traveling wave solutions with non-positive wave speed. We solve two open problems left in two papers [Fu et al., Nonlinear Convex A. 17 (2016), pp. 1739–1751; Wu, J. Differential Equations 262 (2017), pp. 272–282].     

An SIS epidemic model with diffusion

The aim of this paper is to study the dynamics of an SIS epidemic model with diffusion. We first study the well-posedness of the model. And then, by using linearization method and constructing suitable Lyapunov function, we establish the local and global stability of the disease-free equilibrium and the endemic equilibrium, respectively. Furthermore, in view of Schauder fixed point theorem, we show that the model admits traveling wave solutions connecting the disease-free equilibrium and the endemic equilibrium when R_0 1 and c c~*. And also, by virtue of the two-sided Laplace transform, we prove that the model has no traveling wave solution connecting the two equilibria when R_0 1 and c ∈ [0, c~*).     

一类具有扩散项的流行性传染病模型中的行波解

讨论了一类具有扩散项的流行性传染病模型中的行波解的存在性.首先,将对该模型所对应的反应扩散系统的行波解的讨论转化为对二阶常微分系统的上下解的讨论;然后,通过上下解方法建立了这个具有扩散项的传染病模型中行波解的存在性条件,并进一步讨论了扩散因素对行波解的波速的影响,得到被感染人群的流动对病毒的传播有一定的影响.     

On the dynamics of an SEIR epidemic model with a convex incidence rate

An SEIR epidemic model with a nonlinear incidence rate is studied. The incidence is assumed to be a convex function with respect to the infective class of a host population. A bifurcation analysis is performed and conditions ensuring that the system exhibits backward bifurcation are provided. The global dynamics is also studied, through a geometric approach to stability. Numerical simulations are presented to illustrate the results obtained analytically. This research is discussed in the framework of the recent literature on the subject.      

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.     

Analysis and control of an SEIR epidemic system with nonlinear transmission rate

In this paper, the dynamical behaviors of an SEIR epidemic system governed by differential and algebraic equations with seasonal forcing in transmission rate are studied. The cases of only one varying parameter, two varying parameters and three varying parameters are considered to analyze the dynamical behaviors of the system. For the case of one varying parameter, the periodic, chaotic and hyperchaotic dynamical behaviors are investigated via the bifurcation diagrams, Lyapunov exponent spectrum diagram and Poincare section. For the cases of two and three varying parameters, a Lyapunov diagram is applied. A tracking controller is designed to eliminate the hyperchaotic dynamical behavior of the system, such that the disease gradually disappears. In particular, the stability and bifurcation of the system for the case which is the degree of seasonality β₁=0 are considered. Then taking isolation control, the aim of elimination of the disease can be reached. Finally, numerical simulations are given to illustrate the validity of the proposed results.     

Traveling waves for a diffusive SEIR epidemic model with standard incidences

This paper is devoted to the existence of the traveling waves of the equations describing a diffusive susceptible-exposed-infected-recovered(SEIR) model. The existence of traveling waves depends on the basic reproduction rate and the minimal wave speed. We obtain a more precise estimation of the minimal wave speed of the epidemic model, which is of great practical value in the control of serious epidemics. The approach in this paper is to use the Schauder fixed point theorem and the Laplace transform. We also give some numerical results on the minimal wave speed.     

Pulse vaccination on SEIR epidemic model with nonlinear incidence rate

In this paper, we consider an SEIR epidemic model with two time delays and nonlinear incidence rate, and study the dynamical behavior of the model with pulse vaccination. By using the Floquet theorem and comparison theorem, we prove that the infection-free periodic solution is globally attractive when R^*<1, and using a new modelling method, we obtain a sufficient condition for the permanence of the epidemic model with pulse vaccination when R_*>1.     

Traveling waves in a nonlocal dispersal SIR epidemic model

This paper is concerned with traveling wave solutions of a nonlocal dispersal SIR epidemic model. The existence and nonexistence of traveling wave solutions are determined by the basic reproduction number and the minimal wave speed. This threshold dynamics are proved by Schauder’s fixed point theorem and the Laplace transform. The main difficulties are that the semiflow generated by the model does not have the order-preserving property and the solutions lack of regularity.     

Global attractivity for a class of delayed discrete SIRS epidemic models with general nonlinear incidence

This paper deals with global dynamics of a class of delayed discrete susceptible‐infected‐recovered (SIR) compartmental epidemic models with general nonlinear incidence rate and disease‐induced mortality, which are proposed from the Mickens nonstandard discretization of the corresponding delayed continuous epidemic models. By constructing discrete Lyapunov functions, the sufficient conditions for the global attractivity of the disease‐free equilibrium and endemic equilibrium are established. Under some additional assumptions (see (H₃) in Section 3 and (H₄) in Section 4 ), it is shown that the disease‐free equilibrium is globally attractive when basic reproduction number , and when , there is a unique endemic equilibrium, which is globally attractive. Furthermore, some special cases are discussed, and as corollaries, several idiographic results are established. Copyright © 2015 John Wiley & Sons, Ltd.