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

Filomat ◽  
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.

scholarly journals Survival and Stationary Distribution in a Stochastic SIS Model

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Yanli Zhou ◽  
Weiguo Zhang ◽  
Sanling Yuan
Keyword(s):  
Stationary Distribution ◽  
Deterministic Model ◽  
Deterministic System ◽  
Initial Value ◽  
Sis Model ◽  
Sis Epidemic Model ◽  
Stochastic Sis Epidemic Model ◽  
Theoretical Results ◽  
Disease Free

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.

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

Complexity ◽  
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.

scholarly journals Evolutionary Dynamics of Stochastic SEIR Models with Migration and Human Awareness in Complex Networks

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Yue Zhang ◽  
Yuxuan Li
Keyword(s):  
Complex Networks ◽  
Numerical Experiments ◽  
Evolutionary Dynamics ◽  
Sufficient Conditions ◽  
Deterministic System ◽  
Stochastic Solution ◽  
Epidemic Dynamic ◽  
Theoretical Results ◽  
Disease Persistence ◽  
Human Awareness

In this paper, a stochastic SEIR (Susceptible-Exposed-Infected-Removed) epidemic dynamic model with migration and human awareness in complex networks is constructed. The awareness is described by an exponential function. The existence of global positive solutions for the stochastic system in complex networks is obtained. The sufficient conditions are presented for the extinction and persistence of the disease. Under the conditions of disease persistence, the distance between the stochastic solution and the local disease equilibrium of the corresponding deterministic system is estimated in the time sense. Some numerical experiments are also presented to illustrate the theoretical results. Although the awareness introduced in the model cannot affect the extinction of the disease, the scale of the disease will eventually decrease as human awareness increases.

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

2019 ◽  
Vol 12 (03) ◽  
pp. 1950037 ◽  
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.

scholarly journals The Behavior of an SVIR Epidemic Model with Stochastic Perturbation

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Yanan Zhao ◽  
Daqing Jiang
Keyword(s):  
Epidemic Model ◽  
Lyapunov Functions ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Stochastic Perturbation ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Time Average ◽  
The Difference ◽  
Disease Free

We discuss a stochastic SIR epidemic model with vaccination. We investigate the asymptotic behavior according to the perturbation and the reproduction numberR0. We deduce the globally asymptotic stability of the disease-free equilibrium whenR0≤ 1and the perturbation is small, which means that the disease will die out. WhenR0>1, we derive that the disease will prevail, which is measured through the difference between the solution and the endemic equilibrium of the deterministic model in time average. The key to our analysis is choosing appropriate Lyapunov functions.

scholarly journals On a Stochastic SEIS Model with Treatment Rate of Latent Population

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.

The asymptotic behavior and ergodicity of stochastically perturbed SVIR epidemic model

2016 ◽  
Vol 09 (03) ◽  
pp. 1650042 ◽  
Author(s):  
Yanan Zhao ◽  
Daqing Jiang
Keyword(s):  
Epidemic Model ◽  
Lyapunov Functions ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Dynamical Behavior ◽  
Standard Technique ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Number Formula ◽  
Disease Free

In this paper, we introduce stochasticity into an SIR epidemic model with vaccination. The stochasticity in the model is a standard technique in stochastic population modeling. When the perturbations are small, by the method of stochastic Lyapunov functions, we carry out a detailed analysis on the dynamical behavior of the stochastic model regarding of the basic reproduction number [Formula: see text]. If [Formula: see text], the solution of the model is oscillating around a steady state, which is the disease-free equilibrium of the corresponding deterministic model. If [Formula: see text], there is a stationary distribution and the solution has the ergodic property, which means that the disease will prevail.

scholarly journals Survival and ergodicity of a stochastic Holling-III predator–prey model with Markovian switching in an impulsive polluted environment

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Wen Qin ◽  
Hanjun Zhang ◽  
Qingsong He
Keyword(s):  
Existence And Uniqueness ◽  
Lyapunov Functions ◽  
Colored Noise ◽  
Polluted Environment ◽  
Existence And Uniqueness Theorem ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Theoretical Results

AbstractBased on the effects of white noise and colored noise, we propose a stochastic Holling-III predator–prey model in an impulsive polluted environment. Firstly, we prove an existence and uniqueness theorem of the presented model. Secondly, we establish sufficient criteria of extinction, nonpersistence in mean, and weak persistence in mean for both prey and predator species. Thirdly, with the aid of Lyapunov functions, we prove that this system is ergodic and has a unique stationary distribution under certain conditions. Finally, we verify the theoretical results by performing some numerical simulations.

scholarly journals Global dynamics of a novel deterministic and stochastic SIR epidemic model with vertical transmission and media coverage

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaodong Wang ◽  
Chunxia Wang ◽  
Kai Wang
Keyword(s):  
Vertical Transmission ◽  
Epidemic Model ◽  
Media Coverage ◽  
Existence And Uniqueness ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Global Dynamics ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Stochastic Sir

AbstractIn this paper, we study a novel deterministic and stochastic SIR epidemic model with vertical transmission and media coverage. For the deterministic model, we give the basic reproduction number $R_{0}$ R 0 which determines the extinction or prevalence of the disease. In addition, for the stochastic model, we prove existence and uniqueness of the positive solution, and extinction and persistence in mean. Furthermore, we give numerical simulations to verify our results.

scholarly journals The Global Behavior of a Periodic Epidemic Model with Travel between Patches

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Luosheng Wen ◽  
Bin Long ◽  
Xin Liang ◽  
Fengling Zeng
Keyword(s):  
Epidemic Model ◽  
Transmission Rate ◽  
Global Behavior ◽  
Uniform Persistence ◽  
Basic Reproduction Ratio ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Periodic Transmission ◽  
Theoretical Results ◽  
Disease Free

We establish an SIS (susceptible-infected-susceptible) epidemic model, in which the travel between patches and the periodic transmission rate are considered. As an example, the global behavior of the model with two patches is investigated. We present the expression of basic reproduction ratioR0and two theorems on the global behavior: ifR0< 1 the disease-free periodic solution is globally asymptotically stable and ifR0> 1, then it is unstable; ifR0> 1, the disease is uniform persistence. Finally, two numerical examples are given to clarify the theoretical results.

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