Threshold behaviour of a stochastic SIR model

In this paper, we investigate the threshold behaviour of a susceptible-infected-recovered (SIR) epidemic model with stochastic perturbation. When the noise is small, we show that the threshold determines the extinction and persistence of the epidemic. Compared with the corresponding deterministic system, this value is affected by white noise, which is less than the basic reproduction number of the deterministic system. On the other hand, we obtain that the large noise will also suppress the epidemic to prevail, which never happens in the deterministic system. These results are illustrated by computer simulations.     

The survival analysis for a single-species population model in a polluted environment

This paper concentrates on studying the long-term behavior of a single-species population living in a polluted environment. A new mathematical model is derived assuming that a born organism takes with it a quantity of internal toxicant, and the amount of toxicant stored in each living organism which dies is drifted into the environment. Sufficient criteria for uniform persistence, weak persistence in the mean or extinction of the population are obtained. Also we find some sufficient conditions, depending on the parameters of the model and the clean up rate, under which the population will be persistent.     

Survival analysis of a cooperation system with random perturbations in a polluted environment

Taking white noises, Markovian switchings and Lévy jump noises into account, a stochastic cooperation system of two species in a polluted environment is developed and analyzed. Persistence–extinction thresholds are obtained for each population. The results reveal that white noises, Markovian switchings and Lévy jumps have sufficient effect to the persistence and extinction of the species.     

A periodic epidemic model in a patchy environment

An epidemic model in a patchy environment with periodic coefficients is investigated in this paper. By employing the persistence theory, we establish a threshold between the extinction and the uniform persistence of the disease. Further, we obtain the conditions under which the positive periodic solution is globally asymptotically stable. At last, we present two examples and numerical simulations.     

Threshold dynamics for a cholera epidemic model with periodic transmission rate

A cholera epidemic model with periodic transmission rate is presented. The basic reproduction number is defined. It is shown that the disease-free equilibrium is globally asymptotically stable and the cholera eventually disappears if the basic reproduction number is less than one. And if the basic reproduction number is greater than one, there exists a positive periodic solution which is globally asymptotically stable. Numerical simulations are provided to illustrate analytical results.     

一类Lotka-Volterra捕食-竞争扩散系统的概周期解

研究了一类带扩散项的n种群Lotka-volterra非自治捕食-竞争系统，应用Liapunov泛函方法得到系统持久生存和存在唯一全局渐近稳定正概周期解的新的充分条件，并举例说明定理的应用．     

An epidemic model of a vector-borne disease with direct transmission and time delay

This paper considers an epidemic model of a vector-borne disease which has direct mode of transmission in addition to the vector-mediated transmission. The incidence term is assumed to be of the bilinear mass-action form. We include both a baseline ODE version of the model, and, a differential-delay model with a discrete time delay. The ODE model shows that the dynamics is completely determined by the basic reproduction number R₀. If R₀?1, the disease-free equilibrium is globally stable and the disease dies out. If R₀>1, a unique endemic equilibrium exists and is locally asymptotically stable in the interior of the feasible region. The delay in the differential-delay model accounts for the incubation time the vectors need to become infectious. We study the effect of that delay on the stability of the equilibria. We show that the introduction of a time delay in the host-to-vector transmission term can destabilize the system and periodic solutions can arise through Hopf bifurcation.     

Persistence and Extinction in Two Species Models with Impulse

Both uniform persistence and global extinction are established for a two species predatorprey and competition system with impulse by appealing to theories of abstract persistence, asymptotically autonomous semiflows, and the comparison theorem.     

Dynamics of a lattice gas model

Lotka–Volterra equations (LVEs) for mutualisms predict that when mutualistic effects between species are strong, population sizes of the species increase infinitely, which is the so-called divergence problem. Although many models have been established to avoid the problem, most of them are rather complicated. This paper considers a mutualism model of two species, which is derived from reactions on lattice and has a form similar to that of LVEs. Population sizes in the model will not increase infinitely since there is interspecific competition for sites on the lattice. Global dynamics of the model demonstrate essential features of mutualisms and basic mechanisms by which the mutualisms can lead to persistence/extinction of mutualists. Our analysis not only confirms typical dynamics obtained by numerical simulations in a previous work, but also exhibits a new one. Saddle-node bifurcation, transcritical bifurcation and pitchfork bifurcation in the system are demonstrated, while a relationship between saddle-node bifurcation and pitchfork bifurcation in the model is displayed. Numerical simulations validate and extend our conclusions.     

Boundedness,persistence and extinction of a stochastic non-autonomous logistic system with time delays

This paper investigates the stochastic non-autonomous logistic system with time delays. Under two simple assumptions on the environmental noise, it is shown that the stochastic system has a unique global positive solution, and this positive solution is asymptotically bounded. The conditions for extinction, weak persistence of solutions are also obtained by the exponential martingale inequality. Finally, a numerical example is provided to illustrate our results.     

Dynamics of a Deterministic and Stochastic Susceptible-exposed-infectious-recovered Epidemic Model

We investigate a susceptible-exposed-infectious-recovered (SEIR) epidemic model with asymptomatic infective individuals. First, we formulate a deterministic model, and give the basic reproduction number $\mathcal{R}_{0}$. We show that the disease is persistent, if $\mathcal{R}_{0}>1$, and it is extinct, if $\mathcal{R}_{0}<1$. Then, we formulate a stochastic version of the deterministic model. By constructing suitable stochastic Lyapunov functions, we establish sufficient criteria for the extinction and the existence of ergodic stationary distribution to the model. As a case, we study the COVID-19 transmission in Wuhan, China, and perform some sensitivity analysis. Our numerical simulations are carried out to illustrate the analytic results.     

Stability of disease free sets in epidemic models

In this paper the attractivity properties of disease free subsets are considered in the context of disease transmission models. Sufficient conditions are derived for the existence of stable disease free subsets in a general compartmental disease transmission model. The conditions are stated in terms of the system linearized along the trajectories limited to a subset of disease free states. The proof is in the framework of the classical direct method of Lyapunov. As illustrations of the result a multigroup SIRS vaccination model and a Lotka–Volterra system with prey epidemic interaction are presented.     

The permanence and extinction of a nonlinear growth rate single-species non-autonomous dispersal models with time delays

In this paper, we consider the effect of diffusion on the permanence and extinction of a non-autonomous nonlinear growth rate single-species dispersal model with time delays. Firstly, the sufficient conditions of the permanence and extinction of the species are established, which shows if the growth rate and dispersal coefficients is suitable, the species is permanent, on the contrary, it is extinction. Secondly, an interesting result is established, that is, if only the species in some patches even in one patch is permanent, then it is also permanent in other patches. Finally, some examples together with their numerical simulations show the feasibility of our main results.     

Optimal control of a vector borne disease with horizontal transmission

The paper presents the optimal control applied to a vector borne disease with direct transmission in host population. First, we show the existence of the control problem and then use both analytical and numerical techniques to investigate that there are cost effective control efforts for prevention of direct and indirect transmission of disease. In order to do this three control functions are used, one for vector-reduction strategies and the other two for personal (human) protection and blood screening, respectively. We completely characterize the optimal control and compute the numerical solution of the optimality system using an iterative method.     

一类污染环境下具有脉冲输入的竞争培养模型的定性分析

本文研究了污染环境下具有脉冲输入的竞争培养模型.利用乘子理论和小振幅扰动法,我们得到了种群灭绝周期解全局渐近稳定的充分条件,同时还得到了种群持久的条件.我们的结果表明环境污染能最终导致种群灭绝.     

Persistence and extinction in stochastic non-autonomous logistic systems

This paper studies two widely used stochastic non-autonomous logistic models. For the first system, sufficient conditions for extinction, non-persistence in the mean, weak persistence and stochastic permanence are established. The critical number between weak persistence and extinction is obtained. For the second system, sufficient criteria for extinction, non-persistence in the mean, weak persistence in the mean, strong persistence in the mean and stochastic permanence are established. The critical number between weak persistence in the mean and extinction is obtained. It should be pointed out that this research is systematical and complete. In fact, the behaviors of the two models in every coefficient cases are cleared up by the results obtained in this paper.     

The threshold of a stochastic SIRS epidemic model with saturated incidence

In this paper, we investigate the dynamics of a stochastic SIRS epidemic model with saturated incidence. When the noise is small, we obtain a threshold of the stochastic system which determines the extinction and persistence of the epidemic. Besides, we find that large noise will suppress the epidemic from prevailing.     

Nontrivial periodic solution of a stochastic non-autonomous model with biodegradation of microcystins

In this paper, a stochastic non-autonomous model with biodegradation of microcystins is considered. Firstly, the threshold of the model that determines whether the microcystin-degrading bacteria extinct is obtained. Then, we investigate the persistence of species: it shows that there exists a unique nontrivial positive periodic solution. Finally, we conclude our study by a short discussion.