Survival and Stationary Distribution in a Stochastic SIS Model

The dynamics of a stochastic SIS epidemic model is investigated. First, we show that the system admits a unique positive global solution starting from the positive initial value. Then, the long-term asymptotic behavior of the model is studied: whenR0≤1, we show how the solution spirals around the disease-free equilibrium of deterministic system under some conditions; whenR0>1, we show that the stochastic model has a stationary distribution under certain parametric restrictions. In particular, we show that random effects may lead the disease to extinction in scenarios where the deterministic model predicts persistence. Finally, numerical simulations are carried out to illustrate the theoretical results.

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On a Stochastic SEIS Model with Treatment Rate of Latent Population

Abstract and Applied Analysis ◽

10.1155/2014/894842 ◽

2014 ◽

Vol 2014 ◽

pp. 1-16

Author(s):

Shujing Gao ◽

Yanfei Dai ◽

Yan Zhang ◽

Yujiang Liu

Keyword(s):

Asymptotic Behavior ◽

Deterministic Model ◽

Reproduction Number ◽

Deterministic System ◽

Treatment Rate ◽

Initial Value ◽

Asymptotic Dynamics ◽

Disease Free ◽

The Basic Reproduction Number

The asymptotic dynamics of a stochastic SEIS epidemic model with treatment rate of latent population is investigated. First, we show that the system provides a unique positive global solution starting from the positive initial value. Then, the long-term asymptotic behavior of the model is studied: ifR0, which is called the basic reproduction number of the corresponding deterministic model, is not more than unity, the solution of the model is oscillating around the disease-free equilibrium of the corresponding deterministic system, whereas ifR0is larger than unity, we show how the solution spirals around the endemic equilibrium of deterministic system under certain parametric restrictions. Finally, numerical simulations are carried out to support our theoretical findings.

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Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function

Mathematical Problems in Engineering ◽

10.1155/2015/745732 ◽

2015 ◽

Vol 2015 ◽

pp. 1-14 ◽

Cited By ~ 1

Author(s):

Yanju Xiao ◽

Weipeng Zhang ◽

Guifeng Deng ◽

Zhehua Liu

Keyword(s):

Sufficient Conditions ◽

Global Dynamics ◽

Sis Model ◽

Delayed Treatment ◽

Sis Epidemic Model ◽

Treatment Function ◽

Lyapunov Coefficient ◽

Saturated Treatment ◽

Theoretical Results ◽

Disease Free

This paper introduces the global dynamics of an SIS model with bilinear incidence rate and saturated treatment function. The treatment function is a continuous and differential function which shows the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. Sufficient conditions for the existence and global asymptotic stability of the disease-free and endemic equilibria are given in this paper. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as subcritical or supercritical. By some complex algebra, the Bogdanov-Takens normal form and the three types of bifurcation curves are derived. Finally, mathematical analysis and numerical simulations are given to support our theoretical results.

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Analysis of a Deterministic and a Stochastic SIS Epidemic Model with Double Epidemic Hypothesis and Specific Functional Response

Discrete Dynamics in Nature and Society ◽

10.1155/2020/5362716 ◽

2020 ◽

Vol 2020 ◽

pp. 1-11

Author(s):

Mouhcine Naim ◽

Fouad Lahmidi

Keyword(s):

Epidemic Model ◽

Deterministic Model ◽

Sufficient Condition ◽

Nonlinear Incidence Rate ◽

Local Asymptotic Stability ◽

Sis Epidemic Model ◽

The Stability ◽

Stochastic Sis Epidemic Model ◽

Disease Free ◽

Sis Epidemic

The purpose of this paper is to investigate the stability of a deterministic and stochastic SIS epidemic model with double epidemic hypothesis and specific nonlinear incidence rate. We prove the local asymptotic stability of the equilibria of the deterministic model. Moreover, by constructing a suitable Lyapunov function, we obtain a sufficient condition for the global stability of the disease-free equilibrium. For the stochastic model, we establish global existence and positivity of the solution. Thereafter, stochastic stability of the disease-free equilibrium in almost sure exponential and pth moment exponential is investigated. Finally, numerical examples are presented.

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Stability of stochastic model for Hepatitis C transmission with an isolation stage

Filomat ◽

10.2298/fil2014795v ◽

2020 ◽

Vol 34 (14) ◽

pp. 4795-4809

Author(s):

Vuk Vujovic ◽

Marija Krstic

Keyword(s):

Hepatitis C ◽

Existence And Uniqueness ◽

Lyapunov Functions ◽

Deterministic Model ◽

Reliable Data ◽

Deterministic System ◽

Stochastic Perturbations ◽

Stochastic Solution ◽

Theoretical Results ◽

Disease Free

In this paper we construct and investigate stability features of two stochastic hepatitis C models with an isolation stage which are obtained by an introduction of stochastic perturbations into the deterministic model for hepatitis C with an isolation stage. One of the stochastic models has only disease- free equilibrium and the other endemic equilibrium state. Aforementioned equilibriums belong to the equilibriums of corresponding deterministic system. For both of models, first of all, we prove the existence and uniqueness of global positive stochastic solution. Thereafter, by using suitable Lyapunov functions, we investigate stability properties of both models. We close the paper with numerical simulation with reliable data of hepatitis C transmission to illustrate our theoretical results.

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Stability of a Nonlinear Stochastic Epidemic Model with Transfer from Infectious to Susceptible

Complexity ◽

10.1155/2020/9614670 ◽

2020 ◽

Vol 2020 ◽

pp. 1-12

Author(s):

Yanmei Wang ◽

Guirong Liu

Keyword(s):

Numerical Simulations ◽

Lyapunov Method ◽

Deterministic Model ◽

Vital Role ◽

Nonlinear Incidence Rate ◽

Sirs Model ◽

Stochastic Epidemic ◽

Almost Surely Exponential Stability ◽

Theoretical Results ◽

Disease Free

We investigate a stochastic SIRS model with transfer from infectious to susceptible and nonlinear incidence rate. First, using stochastic stability theory, we discuss stochastic asymptotic stability of disease-free equilibrium of this model. Moreover, if the transfer rate from infectious to susceptible is sufficiently large, disease goes extinct. Then, we obtain almost surely exponential stability of disease-free equilibrium, which implies that noises can lead to extinction of disease. By the Lyapunov method, we give conditions to ensure that the solution of this model fluctuates around endemic equilibrium of the corresponding deterministic model in average time. Furthermore, numerical simulations show that the fluctuation increases with increase in noise intensity. Finally, these theoretical results are verified by numerical simulations. Hence, noises play a vital role in epidemic transmission. Our results improve and extend previous related results.

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Global Dynamics of Infectious Disease with Arbitrary Distributed Infectious Period on Complex Networks

Discrete Dynamics in Nature and Society ◽

10.1155/2014/161509 ◽

2014 ◽

Vol 2014 ◽

pp. 1-9

Author(s):

Xiaoguang Zhang ◽

Rui Song ◽

Gui-Quan Sun ◽

Zhen Jin

Keyword(s):

Complex Networks ◽

Reproduction Number ◽

Epidemic Models ◽

Infectious Period ◽

Global Dynamics ◽

Globally Asymptotically Stable ◽

Sis Epidemic Model ◽

Different Types ◽

Theoretical Results ◽

Disease Free

Most of the current epidemic models assume that the infectious period follows an exponential distribution. However, due to individual heterogeneity and epidemic diversity, these models fail to describe the distribution of infectious periods precisely. We establish a SIS epidemic model with multistaged progression of infectious periods on complex networks, which can be used to characterize arbitrary distributions of infectious periods of the individuals. By using mathematical analysis, the basic reproduction numberR0for the model is derived. We verify that theR0depends on the average distributions of infection periods for different types of infective individuals, which extend the general theory obtained from the single infectious period epidemic models. It is proved that ifR0<1, then the disease-free equilibrium is globally asymptotically stable; otherwise the unique endemic equilibrium exists such that it is globally asymptotically attractive. Finally numerical simulations hold for the validity of our theoretical results is given.

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The asymptotic behavior of a stochastic multigroup SIS model

International Journal of Biomathematics ◽

10.1142/s1793524518500377 ◽

2018 ◽

Vol 11 (03) ◽

pp. 1850037 ◽

Cited By ~ 3

Author(s):

Chunyan Ji ◽

Daqing Jiang

Keyword(s):

Asymptotic Behavior ◽

Numerical Simulations ◽

Stationary Distribution ◽

Endemic Equilibrium ◽

Time Behavior ◽

Long Time Behavior ◽

Sis Model ◽

Stochastic Perturbations ◽

Long Time ◽

Theoretical Results

In this paper, we explore the long time behavior of a multigroup Susceptible–Infected–Susceptible (SIS) model with stochastic perturbations. The conditions for the disease to die out are obtained. Besides, we also show that the disease is fluctuating around the endemic equilibrium under some conditions. Moreover, there is a stationary distribution under stronger conditions. At last, some numerical simulations are applied to support our theoretical results.

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On the Stochastic Dynamics of a Social Epidemics Model

Discrete Dynamics in Nature and Society ◽

10.1155/2017/6297074 ◽

2017 ◽

Vol 2017 ◽

pp. 1-14

Author(s):

Xun-Yang Wang ◽

Peng-Zhan Zhang ◽

Qing-Shan Yang

Keyword(s):

Stochastic System ◽

Stochastic Dynamics ◽

Deterministic Model ◽

Random Disturbance ◽

Disturbance Intensity ◽

The Stability ◽

The Individual ◽

Theoretical Results ◽

Two Equilibria

Alcohol abuse is a major social problem, which has caused a lot of damages or hidden dangers to the individual and the society. In this paper, with random factors of alcoholism considered in mortality rate of compartment populations, we formulate a stochastic alcoholism model according to compartment theory of infectious disease. Based on this model, we investigate the long-term stochastic dynamics behaviors of two equilibria of the corresponding deterministic model and point out the effect of random disturbance on the stability of the system. We find that when R0≤1, we get the estimation between the trajectory of stochastic system and E0=(Π/μs,0,0,0) in the average in time with respect to the disturbance intensity, while when R0>1, stochastic system is ergodic and has the unique stationary distribution. Finally, we carry out numerical simulations to support the corresponding theoretical results.

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Asymptotic Behavior and Stationary Distribution of a Nonlinear Stochastic Epidemic Model with Relapse and Cure

Journal of Mathematics ◽

10.1155/2020/4307083 ◽

2020 ◽

Vol 2020 ◽

pp. 1-12

Author(s):

Jiying Ma ◽

Qing Yi

Keyword(s):

Stationary Distribution ◽

Stochastic System ◽

Epidemic Model ◽

Basic Reproductive Number ◽

Reproductive Number ◽

Deterministic System ◽

Stochastic Epidemic ◽

Lyapunov Function Method ◽

Deterministic Framework ◽

Theoretical Results

In this paper, by introducing environmental perturbation, we extend an epidemic model with graded cure, relapse, and nonlinear incidence rate from a deterministic framework to a stochastic differential one. The existence and uniqueness of positive solution for the stochastic system is verified. Using the Lyapunov function method, we estimate the distance between stochastic solutions and the corresponding deterministic system in the time mean sense. Under some acceptable conditions, the solution of the stochastic system oscillates in the vicinity of the disease-free equilibrium if the basic reproductive number R0≤1, while the random solution oscillates near the endemic equilibrium, and the system has a unique stationary distribution if R0>1. Moreover, numerical simulation is conducted to support our theoretical results.

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Media effects on the dynamics of a stochastic SIRI epidemic model with relapse and Lévy noise perturbation

International Journal of Biomathematics ◽

10.1142/s1793524519500372 ◽

2019 ◽

Vol 12 (03) ◽

pp. 1950037 ◽

Cited By ~ 2

Author(s):

Badr-Eddine Berrhazi ◽

Mohamed El Fatini ◽

Roger Pettersson ◽

Aziz Laaribi

Keyword(s):

Epidemic Model ◽

Media Coverage ◽

Deterministic Model ◽

Dynamic Properties ◽

Lévy Noise ◽

Noise Perturbation ◽

Global Positive Solution ◽

Levy Noise ◽

Theoretical Results ◽

Disease Free

In this paper, we study the dynamic properties of an SIRI epidemic model incorporating media coverage, and stochastically perturbed by a Lévy noise. We establish the existence of a unique global positive solution. We investigate the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model depending on the basic reproduction number under some noise excitation. Furthermore, we present some numerical simulations to support the theoretical results.

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