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.     

Analysis of a Kind of Stochastic Dynamics Model with Nonlinear Function

In this paper, we establish stochastic differential equations on the basis of a nonlinear deterministic model and study the global dynamics. For the deterministic model, we show that the basic reproduction number $\Re _0$ determines whether there is an endemic outbreak or not: if $\Re _0< 1$, the disease dies out; while if $\Re _0> 1$, the disease persists. For the stochastic model, we provide analytic results regarding the stochastic boundedness, perturbation, permanence and extinction. Finally, some numerical examples are carried out to confirm the analytical results. One of the most interesting findings is that stochastic fluctuations introduced in our stochastic model can suppress disease outbreak, which can provide us some useful control strategies to regulate disease dynamics.     

一类具有Logistic增长和病程的SIR模型

研究具有Logistic增长和病程的SIR流行病模型.运用微分、积分方程理论,得到再生数R0<1时,无病平衡点E0是全局渐近稳定的;而当R0>1时,地方病平衡点E*是局部渐近稳定的.     

Stationary Distribution and Extinction of Stochastic Beddington-DeAngelis Predator-prey Model with Distributed Delay

In this paper, the formats of Julia sets for a class of nonlinear complex dynamic systems with variable coefficients were studied under certain conditions. For the complex dynamic systems in piecewise cases, we proposed some methods to control the forms of their Julia sets and stable domains analytically. What’s more, we illustrated that our methods worked well by computational simulations. Our work provides a better understanding about how to control the Julia sets of certain complex dynamic systems.     

Hybrid stochastic epidemic SIR models with hidden states

This paper focuses on realistic hybrid SIR models that take into account stochasticity. The proposed systems are applicable to most incidence rates that are used in the literature including the bilinear incidence rate, the Beddington–DeAngelis incidence rate, and a Holling type II functional response. Given that many diseases can lead to asymptomatic infections, this paper looks at a system of stochastic differential equations that also includes a class of hidden state individuals, for which the infection status is unknown. Assuming that the direct observation of the percentage of hidden state individuals being infected, $\alpha \left(t\right)$, is not given and only a noise-corrupted observation process is available. Using nonlinear filtering techniques in conjunction with an invasion type analysis, this paper shows that the long-term behavior of the disease is governed by a threshold $\lambda \in \mathbb{R}$ that depends on the model parameters. It turns out that if $\lambda <0$ the number $I\left(t\right)$ of infected individuals converges to zero exponentially fast (extinction). However, if $\lambda >0$, the infection is endemic and the system is persistent. We showcase our theorems by applying them in some illuminating examples.     

具有标准发生率的随机SIR传染病模型的建立及平稳分布与疾病灭绝研究

研究了一类具有标准发生率以及考虑随机扰动与系统变量成正比的随机SIR传染病模型.首先,对于任意的正的初值,系统存在唯一的全局正解以及通过构造合适的随机李雅普诺夫函数,得到了模型遍历平稳分布存在的充分条件.其次,给出了疾病灭绝的充分条件,并与模型遍历平稳分布存在的充分条件作对比,得出了在特定条件下随机SIR模型的阈值.最...     

Dynamics of a Diffusive SIR Epidemic Model with Time Delay

This paper is devoted to a reaction-diffusion system for a SIR epidemic model with time delay and incidence rate. Firstly, the nonnegativity and boundedness of solutions determined by nonnegative initial values are obtained. Secondly, the existence and local stability of the disease-free equilibrium as well as the endemic equilibrium are investigated by analyzing the characteristic equations. Finally, the global asymptotical stability are obtained via Lyapunov functionals.     

一类时滞SIR传染病模型的稳定性与Hopf分岔分析

研究了一类具有时滞及非线性发生率的SIR传染病模型．首先利用特征值理论分析了地方病平衡点的稳定性,并以时滞为分岔参数,给出了Hopf分岔存在的条件．然后,应用规范型和中心流形定理给出了关于Hopf分岔周期解的稳定性及分岔方向的计算公式．最后,用Matlab软件进行了数值模拟．     

The Dynamics of Stochastic Predator-prey Models with Non-constant Mortality Rate and General Nonlinear Functional Response

In this paper, we investigate the dynamics of stochastic predator- prey models with non-constant mortality rate and general nonlinear functional response. For the stochastic system, we firstly prove the existence of the global unique positive solution. Secondly, we establish sufficient conditions for the extinction and persistence in the mean of autonomous stochastic model and obtain a critical value between them. Then by constructing a appropriate Lyapunov function, we prove that there exists a unique stationary distribution and it has ergodicity in the case of persistence. Finally, numerical simulations are introduced to illustrate our theoretical results.     

一类具有logistic增长和饱和发生率的SIRI传染病模型的全局动力学性态

研究一类具有logistic增长、潜伏期时滞、饱和发生率和疾病复发的传染病动力学模型.通过计算得到了疾病的基本再生数;通过分析相应特征方程根的分布,讨论了可行平衡点的局部稳定性和Hopf分支的存在性;通过构造Lyapunov泛函,得到了保证地方病平衡点全局渐近稳定的充分条件.     

随机扰动下的增长模型

以确定性的Solow增长模型为背景,文章将随机扰动引入其中.因此模型中的要素方程从ODE过渡到It SDE,并对其作某些定性的分析.进而运用二次Liapunov函数等方法对随机Solow模型的零解的稳定性及其平稳分布进行了有益的探索,得出了一些有意义的结果.文章的主要结论是:随机干扰将实质性地改变由确定性增长模型所描述的经济增长路径.     

一类具有非线性发生率的SIR传染病模型的全局稳定性

研究一类具有非线性发生率的SIR传染病模型.应用微分方程定性理论分别得到了该系统无病平衡点、地方病平衡点全局渐近稳定的充分条件,并进行了数值模拟.     

一类具有常数输入的随机SIR流行病模型的定性分析

研究了一类易感者和恢复者具有常数输入的随机SIR传染病模型.利用停时理论及Lyapunov分析方法,证明了该随机模型正解的全局存在唯一性和有界性,讨论了随机模型的解在相应确定模型的无病平衡点和地方病平衡点附近的振荡行为以及得到了随机模型的解的平均持久和疾病灭绝的充分条件.最后,通过数值模拟验证和理论推导的一致性.     

Analysis of a Kind of Quitting Smoking Model with Beddington-DeAngelis Function

In this paper, we discuss the dynamics of quitting smoking with Beddington-Deangelis function. Firstly, the basic reproduction number of the model is obtained by establishing the basic reproduction matrix. Then, by using the Routh-Hurwitz criterion and Lyapunov functionals and LaSalle’s Invariant Principle and the second additive compound matrix, local and global dynamics of the model are analyzed. Based on the partial rank correlation coefficients (PRCCs), we discuss some biological implications and focus on the impact of some key model parameters. Finally, the numerical simulations show the theoretical analysis more intuitively, and we give some strategies to control the spread of smokers.     

Dynamics of a stochastic density dependent predator-prey system with Beddington-DeAngelis functional response

In this paper, we discuss a stochastic density dependent predator-prey system with Beddington-DeAngelis functional response. First, we show that this system has a unique positive solution as this is essential in any population dynamics model. Then, we investigate the asymptotic behavior of this system. When the white noise is small, the stochastic system imitates the corresponding deterministic system. Either there is a stationary distribution, or the predator population will die out. While if the white noise is large, besides the extinction of the predator population, both species in the system may also die out, which does not happen in the deterministic system. Finally, simulations are carried out to conform to our results.     

带随机延滞的ARCH模型的几何遍历性

本文对带随机延滞的ARCH模型进行了分析,并得到该模型伴随几何遍历的一个判别准则.     

非线性发病率随机流行病模型的动力学行为

研究了一类具有非线性发病率的随机SEIR传染病模型的绝灭性及平稳分布问题,通过构造合适的Lyapunov函数及控制噪声强度,在适当的条件下,得到模型的全局解存在唯一、指数稳定,且解具有平稳分布及遍历性.利用线性化及Fourier变换,证明了解渐近服从四维正态分布,并给出均值及方差矩阵的表达式.数值模拟验证了我们所得的主要结果.     

一类带随机延滞的时间序列模型的遍历性

本利用马氏化方法和一般状态空间马氏链的基本理论研究了一类带随机延滞的时间序列模型的遍历性，得到了该模型伴随几何遍历的一个判别准则．     

具有Logistic增长和年龄结构的SIS模型

讨论具有Logistic增长和年龄结构的SIS流行病模型.运用微分、积分方程理论,得到了当再生数R0<1时,无病平衡点E0是全局渐近稳定的;当R0>1时,地方病平衡点E*是局部渐近稳定的.