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.     

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.     

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.     

Disease control in a food chain model supplying alternative food

Necessity to find a non-chemical method of disease control is being increasingly felt due to its eco-friendly nature. In this paper the role of alternative food as a disease controller in a disease induced predator–prey system is studied. Stability criteria and the persistence conditions for the system are derived. Bifurcation analysis is done with respect to rate of infection. The main goal of this study is to show the non-trivial consequences of providing alternative food in a disease induced predator–prey system. Numerical simulation results illustrate that there exists a critical infection rate above which disease free system cannot be reached in absence of alternative food whereas supply of suitable alternative food makes the system disease free up to certain infection level. We have computed the disease free regions in various parametric planes. This study is aimed to introduce a new non-chemical method for controlling disease in a predator–prey system.     

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.     

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.     

Hopf bifurcation in an eco-epidemic model

This paper deals with prey-predator system where the prey population is infected with a microparasite. Predator functional response is assumed to be Holling type I. All the feasible equilibrium points of the system are obtained and the condition for the existence of positive (interior) equilibrium point is also determined. The criteria for both local stability and instability involving system parameters are derived. The criterion for existence of Hopf-type small periodic oscillation is shown. The condition for survival of all the population is investigated. We have developed a necessary and sufficient criteria for uniform persistence. Finally, the numerical verifications of analytic results are done.     

Dynamics of a spatiotemporal model on populations in a polluted river

In this paper, we investigate the dynamics for a reaction–diffusion–advection system which models populations in a polluted river. More precisely, we study the stability of steady states, which yields sufficient conditions that lead to population persistence or extinction. Furthermore, some dependence of the stability of the toxicant-only steady state and the population-toxicant coexistence steady state on the model parameters are given.     

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.     

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.     

Threshold of disease transmission in a patch environment

An epidemic model is proposed to describe the dynamics of disease spread between two patches due to population dispersal. It is proved that reproduction number is a threshold of the uniform persistence and disappearance of the disease. It is found that the dispersal rates of susceptible individuals do not influence the persistence and extinction of the disease. Furthermore, if the disease becomes extinct in each patch when the patches are isolated, the disease remains extinct when the population dispersal occurs; if the disease spreads in each patch when the patches are isolated, the disease remains persistent in two patches when the population dispersal occurs; if the disease disappears in one patch and spreads in the other patch when they are isolated, the disease can spread in all the patches or disappear in all the patches if dispersal rates of infectious individuals are suitably chosen. It is shown that an endemic equilibrium is locally stable if susceptible dispersal occurs and infectious dispersal turns off. If susceptible individuals and infectious individuals have the same dispersal rate in each patch, it is shown that the fractions of infectious individuals converge to a unique endemic equilibrium.     

Dynamics of an oncolytic virus model

An ODE model for oncolytic virus dynamics with four parameters is discussed. Although it is hard to obtain the explicit expressions of coordinates of equilibria, we find distribution of equilibria qualitatively, and obtain the exact number of equilibria and their qualitative properties. We further give a complete analysis on local bifurcations such as saddle–node, Bogdanov–Takens, and Hopf bifurcations. Resultant elimination is applied to decompose the semi-algebraic varieties of Lyapunov quantities under the restriction of biological meaning, and we show that the weak focus is of order at most 3.     

Dynamics of a non-autonomous ratio-dependent food chain model

A simple non-autonomous ratio-dependent food chain model is investigated. It is shown that the system is permanence, extinction, ultimate boundedness and globally asymptotic stability under some appropriate conditions. Moreover, by employing Mawhin’s coincidence degree theory, some easily applicable criteria are established for the global existence of positive periodic solution of this model.     

Infinite paths in a Lorentz lattice gas model

We consider infinite paths in an illumination problem on the lattice ℤ², where at each vertex, there is either a two-sided mirror (with probability p≥ 0) or no mirror (with probability 1 −p). The mirrors are independently oriented NE-SW or NW-SE with equal probability. We consider beams of light which are shone from the origin and deflected by the mirrors. The beam of light is either periodic or unbounded. The novel feature of this analysis is that we concentrate on the measure on the space of paths. In particular, under the assumption that the set of unbounded paths has positive measure, we are able to establish a useful ergodic property of the measure. We use this to prove results about the number and geometry of infinite light beams. Extensions to higher dimensions are considered. Received: 14 November 1996 / Revised version: 1 September 1998     

具有分布时滞的互惠系统Hopf分歧分析

考虑了一类具有扩散项和分布时滞的互惠系统,研究了系统Hopf分歧的出现,用中心流行定理证明了周期解的稳定性.     

Persistence of bistable waves in a delayed population model with stage structure on a two-dimensional spatial lattice

In this paper, we study the existence of traveling waves of a delayed population model with age-structure on a 2-dimensional spatial lattice when the maturation time r is relatively small. Under the assumption that the birth function b satisfies the bistable condition without requiring monotonicity, we prove the persistence of traveling wavefronts by means of a perturbation argument based on the existing results on the asymptotic autonomous system and the Fredholm alternative theory.     

Dynamics of an SI epidemic model with external effects in a polluted environment

In a polluted environment, considering the biological population infected with some kinds of diseases and hunted by human beings, we formulate two SI pollution-epidemic models with continuous and impulsive external effects, respectively, and investigate the dynamics of such systems. We assume that only the susceptible population is hunted by human beings. For the continuous system, we obtain sufficient conditions of the ultimate boundedness of solutions and the global asymptotical stability of equilibria. For the impulsive system, by using the comparison theorem and the analysis method, we show that under different conditions the disease-free periodic solution is globally asymptotically attractive, or the system is permanent. Numerical simulations confirm our theoretical results.