Travelling waves in an infectious disease model with a fixed latent period and a spatio–temporal delay

This paper is concerned with the existence of travelling waves to an infectious disease model with a fixed latent period and a spatio–temporal delay. By analyzing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this model is discussed. By constructing a pair of upper–lower solutions, we use the cross iteration method and the Schauder’s fixed point theorem to prove the existence of a travelling wave solution connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.     

Travelling waves of a delayed SIRS epidemic model with spatial diffusion

This paper is concerned with the existence of travelling waves to an SIRS epidemic model with bilinear incidence rate, spatial diffusion and time delay. By analysing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this system under homogeneous Neumann boundary conditions is discussed. By using the cross iteration method and the Schauder’s fixed point theorem, we reduce the existence of travelling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of a travelling wave solution connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.     

Stability of stochastic SIRS epidemic models with saturated incidence rates and delay

In this article, we consider stochastic susceptible-infected-removed-susceptible (SIRS) epidemic models with saturated incidence rates and delay. We investigate the stochastic stability in probability of the disease-free and endemic equilibria for the stochastic dynamic model with variability in the natural death rate, and the stochastic stability in probability of the endemic equilibrium for the dynamic model when the variability in the environment is proportional to a deviation between the state of the system and the endemic equilibrium. The numerical experiments are provided to support our theoretical results.     

Stochastically perturbed vector-borne disease models with direct transmission

In this paper we study a stochastic epidemic model of vector-borne diseases with direct mode of transmission and its delay modification. More precisely, we extend the deterministic epidemic models by introducing random perturbations around the endemic equilibrium state. By using suitable Lyapunov functions and functionals, we obtain stability conditions for the considered models and study the effect of the delay on the stability of the endemic equilibrium. Finally, numerical simulations for the stochastic model of malaria disease transmission are presented to illustrate our mathematical findings.     

The Asymptotic Behaviour of a Stochastic 3D LANS-α Model

The long-time behaviour of a stochastic 3D LANS-α model on a bounded domain is analysed. First, we reformulate the model as an abstract problem. Next, we establish sufficient conditions ensuring the existence of stationary (steady state) solutions of this abstract nonlinear stochastic evolution equation, and study the stability properties of the model. Finally, we analyse the effects produced by stochastic perturbations in the deterministic version of the system (persistence of exponential stability as well as possible stabilisation effects produced by the noise). The general results are applied to our stochastic LANS-α system throughout the paper.     

Bifurcation and stability of a Mimura–Tsujikawa model with nonlocal delay effect

In this paper, we investigate a Mimura–Tsujikawa model with nonlocal delay effect under the homogeneous Neumann boundary condition. By using Lyapunov–Schmidt reduction, we investigate the existence, multiplicity, stability, and Hopf bifurcation of nontrivial steady‐state solutions bifurcating from the nonzero steady‐state solution. Moreover, we illustrate our general results by applications to models with a one‐dimensional spatial domain. Copyright © 2016 John Wiley & Sons, Ltd.     

具有接种和非局部扩散的流行病模型的行波解（英文）

An epidemic model with vaccination and nonlocal diffusion is proposed, and the existence of traveling wave solutions of this model is studied. By the cross-iteration scheme companied with a pair of upper and lower solutions and Schauder’s fixed point theorem,sufficient conditions are obtained for the existence of a traveling wave solution connecting the disease-free steady state and the endemic steady state.     

Travelling waves of a predator–prey model with a spatiotemporal delay

This paper is concerned with the existence of traveling waves to a predator–prey model with a spatiotemporal delay. By analyzing the corresponding characteristic equations, the local stability of a positive steady state and each of boundary steady states are established, and the existence of Hopf bifurcation at the positive steady state is also discussed. By constructing a pair of upper–lower solutions and by using the cross‐iteration method as well as the Schauder's fixed‐point theorem, the existence of a traveling wave solution connecting the semi‐trivial steady state and the positive steady state is proved. Numerical simulations are carried out to illustrate the main theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Stochastic differential delay equations with jumps,under nonlinear growth condition

In this paper, we investigate the existence and uniqueness of solutions to stochastic differential delay equations under a local Lipschitz condition but without linear growth condition on its coefficients. Moreover, we prove convergence in probability of the Euler–Maruyama approximation as well as of the stochastic theta method approximation to the exact solution.     

Hopf bifurcations in a reaction-diffusion population model with delay effect

A reaction-diffusion population model with a general time-delayed growth rate per capita is considered. The growth rate per capita can be logistic or weak Allee effect type. From a careful analysis of the characteristic equation, the stability of the positive steady state solution and the existence of forward Hopf bifurcation from the positive steady state solution are obtained via the implicit function theorem, where the time delay is used as the bifurcation parameter. The general results are applied to a “food-limited” population model with diffusion and delay effects as well as a weak Allee effect population model.     

Asymptotic Behavior of a Stochastic Predator-prey Model with Beddington-DeAngelis Functional Response and L\'{e}vy Jumps

A stochastic two-prey-one-predator model with Beddington-DeAngelis functional response and L\''{e}vy jumps is proposed and investigated in this paper. First of all, we prove the existence and uniqueness of the global positive solution, and stochastic ultimate boundedness of the solution. Next, under a simple assumption, by using It\^{o} formula and other important inequalities, some sufficient conditions are established to ensure the extinction and persistence in the mean of the system. The results show that neither strong white noise nor L\''{e}vy noise is conducive to the persistence of the population. Finally, the theoretical results are verified by numerical simulations.     

Mathematical Analysis of SIR Epidemic Model with Piecewise Infection Rate and Control Strategies

The limitation of contact between susceptible and infected individuals plays an important role in decreasing the transmission of infectious diseases. Prevention and control strategies contribute to minimizing the transmission rate. In this paper, we propose SIR epidemic model with delayed control strategies, in which delay describes the response and effect time. We study the dynamic properties of the epidemic model from three aspects: steady states, stability and bifurcation. By eliminating the existence of limit cycles, we establish the global stability of the endemic equilibrium, when the delay is ignored. Further, we find that the delayed effect on the infection rate does not affect the stability of the disease-free equilibrium, but it can destabilize the endemic equilibrium and bring Hopf bifurcation. Theoretical results show that the prevention and control strategies can effectively reduce the final number of infected individuals in the population. Numerical results corroborate the theoretical ones.     

Effective construction of Poincar\'e--Bendixson regions

This paper deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincar\''e--Bendixson regions by using transversal curves, that enables us to prove the existence of a limit cycle that has been numerically detected. We apply our results to several known systems, like the Brusselator one or some Li\''{e}nard systems, to prove the existence of the limit cycles and to locate them very precisely in the phase space. Our method, combined with some other classical tools can be applied to obtain sharp bounds for the bifurcation values of a saddle-node bifurcation of limit cycles, as we do for the Rychkov system.     

Stability and traveling waves of an epidemic model with relapse and spatial diffusion

An epidemic model with relapse and spatial diffusion is studied. Such a model is appropriate for tuberculosis, including bovine tuberculosis in cattle and wildlife, and for herpes. By using the linearized method, the local stability of each of feasible steady states to this model is investigated. It is proven that if the basic reproduction number is less than unity, the disease-free steady state is locally asymptotically stable; and if the basic reproduction number is greater than unity, the endemic steady state is locally asymptotically stable. By the cross-iteration scheme companied with a pair of upper and lower solutions and Schauder's fixed point theorem, the existence of a traveling wave solution which connects the two steady states is established. Furthermore, numerical simulations are carried out to complement the main results.     

Analysis of the dynamics of a delayed HIV pathogenesis model

In this paper, considering full Logistic proliferation of CD4⁺ T cells, we study an HIV pathogenesis model with antiretroviral therapy and HIV replication time. We first analyze the existence and stability of the equilibrium, and then investigate the effect of the time delay on the stability of the infected steady state. Sufficient conditions are given to ensure that the infected steady state is asymptotically stable for all delay. Furthermore, we apply the Nyquist criterion to estimate the length of delay for which stability continues to hold, and investigate the existence of Hopf bifurcation by using a delay τ as a bifurcation parameter. Finally, numerical simulations are presented to illustrate the main results.     

A new nonlocal model for the restoration of textured images

In this paper, we focus on the mathematical and numerical study of a new nonlocal reaction-diffusion system for image denoising. This model is motivated by involving the decomposition approach of $H^{-1}$ norm suggested by Meyer [25] which is more appropriate to represent the oscillatory patterns and small details in the textured image. Based on Schaeffer''s fixed point theorem, we prove the existence and uniqueness of solution of the proposed model. To illustrate the efficiency and effectiveness of our model, we test the denoising experimental results as well we compare with some existing models in the literature.     

一类具有感染年龄SIRS模型的分析

根据乙肝的传播特点,通过假设感染率依赖于个体的生理年龄及感染年龄.接种免疫有一定的有效期,建立了具有生理年龄和感染年龄的模型,并研究了地方病平衡点的存在条件.     

Hopf bifurcation in a delayed reaction–diffusion–advection population model

In this paper, we investigate a reaction–diffusion–advection model with time delay effect. The stability/instability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are investigated when the given parameter of the model is near the principle eigenvalue of an elliptic operator. Our results imply that time delay can make the spatially nonhomogeneous positive steady state unstable for a reaction–diffusion–advection model, and the model can exhibit oscillatory pattern through Hopf bifurcation. The effect of advection on Hopf bifurcation values is also considered, and our results suggest that Hopf bifurcation is more likely to occur when the advection rate increases.     

Bifurcation analysis in a diffusive Logistic population model with two delayed density-dependent feedback terms

The present paper is concerned with a diffusive population model of Logistic type with an instantaneous density-dependent term and two delayed density-dependent terms and subject to the zero-Dirichlet boundary condition. By regarding the delay as the bifurcation parameter and analyzing in detail the associated eigenvalue problem, the local asymptotic stability and the existence of Hopf bifurcation for the sufficiently small positive steady state solution are shown. It is found that under the suitable condition, the positive steady state solution of the model will become ultimately unstable after a single stability switch (or change) at a certain critical value of delay through a Hopf bifurcation. However, under the other condition, the positive steady state solution of the model will become ultimately unstable after multiple stability switches at some certain critical values of delay through Hopf bifurcations. In addition, the direction of the above Hopf bifurcations and the stability of the bifurcating periodic solutions are analyzed by means of the center manifold theory and normal form method for partial functional differential equations. Finally, in order to illustrate the correction of the obtained theoretical results, some numerical simulations are also carried out.     

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.