Stochastic SIR model with jumps

People have always attached importance to the prevention and the control of the epidemic disease. The study of the epidemic model provides us a powerful tool. Unfortunately the previous model cannot be applied to massive diseases, such as avian influenza. Therefore we need to revise the model. In this paper, we take the lead in using the stochastic differential equation with jumps to study the asymptotic behavior of the stochastic SIR model.     

Mathematical Analysis for an Age-Structured Heroin Epidemic Model

In this paper, an age-structured heroin epidemic model, where the susceptibility of individuals and the relapse of heroin users in treatment are described by two age-dependent variables, is formulated and analyzed. The basic reproduction ratio of the model is derived and proved to be a threshold condition, which completely determines the global behaviors of the model. The asymptotic smoothness of the semiflow generated by the family of solutions, uniform persistence and existence of an interior global attractor have been presented for establishing and defining a Lyapunov functional on this attractor. Some control strategies of heroin and two special cases of the model formulation are addressed.

     

Dynamic behaviour for a nonautonomous heroin epidemic model with time delay

In this paper, we have modified the White and Comiskey heroin epidemic model (White and Comiskey in Math. Biosci. 208:312–324, 2007) into a nonautonomous heroin epidemic model with distributed time delay. We have introduced some new threshold values R _* and R ^* and further obtained that the heroin-using career will be permanent when R _*>1 and the heroin-using career will be going to extinct when R ^*<1. Using the method of Lyapunov functional, some sufficient conditions are derived for the global asymptotic stability of the system. The aim of this modification is to identify the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.     

On a discrete risk model with two-sided jumps

In this paper, we consider a discrete renewal risk model with phase-type interarrival times and two-sided jumps. In this model, downward jumps represent claim loss, while upward jumps are also allowed to represent random gains. Assume that the downward jumps have an arbitrary probability function and the upward jumps have a rational probability generating function. We study the (Gerber-Shiu) discounted penalty function. The generating function, the recursive formula as well as an explicit expression for the discounted penalty function are obtained.     

A note on global stability for a heroin epidemic model with distributed delay

By using the direct Lyapunov method and constructing appropriate Lyapunov functional, the global stability for the heroin epidemic model with distributed delay is investigated. It is shown that the disease endemic equilibrium of the system is globally asymptotically stable whenever it exists. This improves the related results presented in [J. Liu, T. Zhang, Global behavior of a heroin epidemic model with distributed delay, Appl. Math. Lett. 24 (2011) 1685–1692].     

Global dynamical analysis of a heroin epidemic model on complex networks

In this paper, a heroin epidemic model on complex networks is proposed. By the next generation matrix, the basic reproduction number $R_0$ is obtained. If $R_0<1$, then the drug-free equilibrium is globally asymptotically stable. If $R_0>1$, there is an unique endemic equilibrium and it is also globally asymptotically stable. Our results show that if the degree of the network is large enough, the drug transmission always spreads. Sensitivity analysis of the basic reproduction number with the various parameters in the model are carried out to verify the important effects for control the drug transmission. Some simulations illustrate our theoretical results     

Representable Lexicographic Products

A linear ordering is said to be representable if it can be order-embedded into the reals. Representable linear orderings have been characterized as those which are separable in the order topology and have at most countably many jumps. We use this characterization to study the representability of a lexicographic product of linear orderings. First we count the jumps in a lexicographic product in terms of the number of jumps in its factors. Then we relate the separability of a lexicographic product to properties of its factors, and derive a classification of representable lexicographic products. This revised version was published online in September 2006 with corrections to the Cover Date.     

Threshold dynamics of the stochastic epidemic model with jump-diffusion infection force

This paper formulates a stochastic SIR epidemic model by supposing that the infection force is perturbed by Brown motion and L\''{e}vy jumps. The globally positive and bounded solution is proved firstly by constructing the suitable Lyapunov function. Then, a stochastic basic reproduction number $R_0^{L}$ is derived, which is less than that for the deterministic model and the stochastic model driven by Brown motion. Analytical results show that the disease will die out if $R_0^{L}<1$, and $R_0^{L}>1$ is the necessary and sufficient condition for persistence of the disease. Theoretical results and numerical simulations indicate that the effects of L\''{e}vy jumps may lead to extinction of the disease while the deterministic model and the stochastic model driven by Brown motion both predict persistence. Additionally, the method developed in this paper can be used to investigate a class of related stochastic models driven by L\''{e}vy noise.     

Dynamical behavior of a hybrid switching SIS epidemic model with vaccination and Lévy jumps

In this paper, the dynamical behavior of a hybrid switching SIS epidemic model with vaccination and Lévy jumps is considered. Besides a standard geometric Brownian motion, another two driving processes are taken into account: a stationary Poisson point process and a continuous time finite-state Markov chain. Firstly, we establish sufficient conditions for persistence in the mean of the disease. Then we obtain sufficient conditions for extinction of the disease. In addition, we also establish sufficient conditions for the existence of positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching.     

一类含时滞SIS流行病模型的全局稳定性

该文研究了一类含有限分布时滞的SIS流行病模型, 利用李亚普诺夫泛函的方法,得到了地方病平衡点和无病平衡点全局稳定的充要条件. 揭示了时滞对平衡点稳定性的影响 .      

Numerical simulations and modeling for stochastic biological systems with jumps

This paper gives a numerical method to simulate sample paths for stochastic differential equations (SDEs) driven by Poisson random measures. It provides us a new approach to simulate systems with jumps from a different angle. The driving Poisson random measures are assumed to be generated by stationary Poisson point processes instead of Lévy processes. Methods provided in this paper can be used to simulate SDEs with Lévy noise approximately. The simulation is divided into two parts: the part of jumping integration is based on definition without approximation while the continuous part is based on some classical approaches. Biological explanations for stochastic integrations with jumps are motivated by several numerical simulations. How to model biological systems with jumps is showed in this paper. Moreover, method of choosing integrands and stationary Poisson point processes in jumping integrations for biological models are obtained. In addition, results are illustrated through some examples and numerical simulations. For some examples, earthquake is chose as a jumping source which causes jumps on the size of biological population.     

The gerber-shiu expected discounted penalty function for Lévy insurance risk processes

In this paper,we study a general Lévy risk process with positive and negative jumps.A renewal equation and an infinite series expression are obtained for the expected discounted penalty function of this risk model.We also examine some asymptotic behaviors for the ruin probability as the initial capital tends to infinity.     

Wave Propagation in an Elastic Medium Intersected by Systems of Parallel Fractures

An elastic anisotropic medium intersected by systems of parallel fractures is investigated. Every fracture is considered as a plane boundary with jumps of displacements and stresses, and these jumps are linear functions of displacements and stresses averaged on the boundary. For this medium, an effective model is constructed by the method of matrix averaging. The equations of this model describe wave propagation in the given medium and are more complicated than the equations of elasticity theory. In particular cases, the equations obtained are converted to the equations of elastic media. On the basis of the equations of the effective model, expressions for the densities of the kinetic and potential energies are derived, and conditions of absoption in the medium are established. Bibliography: 15 titles.     

Permanence of a discrete SIRS epidemic model with time delays

In this paper, we consider the permanence of a discrete SIRS epidemic model with time delays. This model is constructed from the discretization by the Euler method. Applying the technique to prove the existence of an eventual lower bound in a continuous epidemic model, a sufficient condition for the permanence of the above discrete SIRS epidemic model is obtained.     

On the threshold dividend strategy for a generalized jump-diffusion risk model

In this paper, we generalize the Cramér-Lundberg risk model perturbed by diffusion to incorporate jumps due to surplus fluctuation and to relax the positive loading condition. Assuming that the surplus process has exponential upward and arbitrary downward jumps, we analyze the expected discounted penalty (EDP) function of Gerber and Shiu (1998) under the threshold dividend strategy. An integral equation for the EDP function is derived using the Wiener-Hopf factorization. As a result, an explicit analytical expression is obtained for the EDP function by solving the integral equation. Finally, phase-type downward jumps are considered and a matrix representation of the EDP function is presented.     

一类具有阶段结构和接种的SIR传染病模型

根据不同程度的感染者有不同的传染率,建立了一个具有阶段结构和双线性传染率的S IR流行病模型,得到了模型的阈值参数R0,证明了模型平衡点的全局性态完全由R0的值确定.并进行了数值模拟.     

Optimal control of the genital herpes epidemic

A spatial stochastic model to study the optimal control of the epidemic is introduced. The equilibrium states of the epidemic model are found. The stability and instability in linear approximation of this model are investigated. The optimal control of the unstable equilibrium states is studied. The control functions are obtained from the conditions that ensure the optimal stabilization of these states. Graphical and numerical simulation of the obtained results are presented.     

Study a class of Hilfer fractional stochastic integrodifferential equations with Poisson jumps

Abstract

In this article, we derive the sufficient conditions for the existence of mild solutions of Hilfer fractional stochastic integrodifferential equations with nonlocal conditions and Poisson jumps in Hilbert spaces. Results will be obtained in the pth mean square sense by using the fractional calculus, semigroup theory and stochastic analysis techniques. The article generalizes many of the existing results in the literature in terms of (1) Riemann–Liouville and Caputo derivatives are the special cases. (2) In the sense of pth mean square norm. (3) Stochastic integrodifferential with nonlocal conditions and Poisson jumps. A numerical example is provided to validate the obtained theoretical results.     

股票价格序列中随机跳跃存在性的一种检验方法

提出股票价格序列跳跃的一种检验方法.假设价格具有连续样本路径,建立一个关于股票价格样本观察的统计量,利用中心极限定理求得该统计量的极限分布为正态分布,这样,当该统计量超出基于极限分布算出的临界水平时,可以拒绝原假设,认为样本中存在跳跃.用此方法来应用于中国股市沪深股票指数,得到了中国股市存在随机跳跃的直接证据.提出的跳跃检验方法无需对连续部分的波动率形式作过多的假设,克服了波动率模型对检验准确性的影响.结果对金融资产的定价、投资和风险管理都具有积极的意义.