Yongli Cai

A stochastic SIRS epidemic model with nonlinear incidence rate

A stochastic SIRS model with ratio-dependent incidence rate is developed.The global dynamics of the deterministic model is shown.The stochastic dynamics of the SDE model is given.The existence of a unique stationary distribution of the SDE model is displayed. In this paper, we investigate the global dynamics of a general SIRS epidemic model with a ratio-dependent incidence rate and its corresponding stochastic differential equation version. For the deterministic model, we show that the basic reproduction number R0 determines whether there is an endemic outbreak or not: if R0 1, the disease persists. For the stochastic model, we show that its related reproduction number R0S can determine whether there is a unique disease-free stationary distribution or a unique endemic stationary distribution. In addition, we provide analytic results regarding the stochastic boundedness and permanence/extinction. One of the most interesting findings is that random fluctuations introduced in our stochastic model can suppress disease outbreak, which can provide us some useful control strategies to regulate disease dynamics.

Spatiotemporal complexity in a predator--prey model with weak Allee effects

In this article, we study the rich dynamics of a diffusive predator-prey system with Allee effects in the prey growth. Our model assumes a prey-dependent Holling type-II functional response and a density dependent death rate for predator. We investigate the dissipation and persistence property, the stability of nonnegative and positive constant steady state of the model, as well as the existence of Hopf bifurcation at the positive constant solution. In addition, we provide results on the existence and non-existence of positive non-constant solutions of the model. We also demonstrate the Turing instability under some conditions, and find that our model exhibits a diffusion-controlled formation growth of spots, stripes, and holes pattern replication via numerical simulations. One of the most interesting findings is that Turing instability in the model is induced by the density dependent death rate in predator.

Allee-Effect-Induced Instability in a Reaction-Diffusion Predator-Prey Model

We investigate the spatiotemporal dynamics induced by Allee effect in a reaction-diffusion predator-prey model. In the case without Allee effect, there is nonexistence of diffusion-driven instability for the model. And in the case with Allee effect, the positive equilibrium may be unstable under certain conditions. This instability is induced by Allee effect and diffusion together. Furthermore, via numerical simulations, the model dynamics exhibits both Allee effect and diffusion controlled pattern formation growth to holes, stripes-holes mixture, stripes, stripes-spots mixture, and spots replication, which shows that the dynamics of the model with Allee effect is not simple, but rich and complex.

Dynamical Analysis of a Delayed Reaction-Diffusion Predator-Prey System

This work deals with the analysis of a delayed diffusive predator-prey system under Neumann boundary conditions. The dynamics are investigated in terms of the stability of the nonnegative equilibria and the existence of Hopf bifurcation by analyzing the characteristic equations. The direction of Hopf bifurcation and the stability of bifurcating periodic solution are also discussed by employing the normal form theory and the center manifold reduction. Furthermore, we prove that the positive equilibrium is asymptotically stable when the delay is less than a certain critical value and unstable when the delay is greater than the critical value.

Spatiotemporal Dynamics in a Reaction–Diffusion Epidemic Model with a Time-Delay in Transmission

In this paper, we investigate the effects of time-delay and diffusion on the disease dynamics in an epidemic model analytically and numerically. We give the conditions of Hopf and Turing bifurcations in a spatial domain. From the results of mathematical analysis and numerical simulations, we find that for unequal diffusive coefficients, time-delay and diffusion may induce that Turing instability results in stationary Turing patterns, Hopf instability results in spiral wave patterns, and Hopf–Turing instability results in chaotic wave patterns. Our results well extend the findings of spatiotemporal dynamics in the delayed reaction–diffusion epidemic model, and show that time-delay has a strong impact on the pattern formation of the reaction–diffusion epidemic model.

Stationary Patterns of a Cross-Diffusion Epidemic Model

We investigate the complex dynamics of cross-diffusion epidemic model. We first give the conditions of the local and global stability of the nonnegative constant steady states, which indicates that the basic reproduction number determines whether there is an endemic outbreak or not. Furthermore, we prove the existence and nonexistence of the positive nonconstant steady states, which guarantees the existence of the stationary patterns.

Global stability of the steady states of an epidemic model incorporating intervention strategies

In this paper, we investigate the global stability of the steady states of a general reaction-diffusion epidemiological model with infection force under intervention strategies in a spatially heterogeneous environment. We prove that the reproductoin number R0 can be played an essential role in determining whether the disease will extinct or persist: if R0< 1, there is a unique disease-free equilibrium which is globally asymptotically stable; and if R0 >1, there exists a unique endemic equilibrium which is globally asymptotically stable. Furthermore, we study the relation between R0 with the diffusion and spatial heterogeneity and find that, it seems very necessary to create a low-risk habitat for the population to effectively control the spread of the epidemic disease. This may provide some potential applications in disease control.