scholarly journals Threshold Analysis and Stationary Distribution of a Stochastic Model with Relapse and Temporary Immunity

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 331 ◽  
Author(s):  
Peng Liu ◽  
Xinzhu Meng ◽  
Haokun Qi
Keyword(s):  
Stochastic Model ◽  
Stationary Distribution ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Markov Semigroup ◽  
Threshold Analysis ◽  
Numerical Examples ◽  
Stochastic Properties ◽  
Symmetric Method ◽  
Temporary Immunity

In this paper, a stochastic model with relapse and temporary immunity is formulated. The main purpose of this model is to investigate the stochastic properties. For two incidence rate terms, we apply the ideas of a symmetric method to obtain the results. First, by constructing suitable stochastic Lyapunov functions, we establish sufficient conditions for the extinction and persistence of this system. Then, we investigate the existence of a stationary distribution for this model by employing the theory of an integral Markov semigroup. Finally, the numerical examples are presented to illustrate the analytical findings.

scholarly journals Correction: Liu, P., et al. Threshold Analysis and Stationary Distribution of a Stochastic Model with Relapse and Temporary Immunity. Symmetry 2020, 12, 331

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 629
Author(s):  
Peng Liu ◽  
Xinzhu Meng ◽  
Haokun Qi
Keyword(s):  
Stochastic Model ◽  
Stationary Distribution ◽  
Threshold Analysis ◽  
Temporary Immunity

The authors wish to make the following corrections and explanations to this paper [...]

The stationary distribution and ergodicity of a stochastic mutualism model

2018 ◽  
Vol 68 (3) ◽  
pp. 685-690 ◽  
Author(s):  
Jingliang Lv ◽  
Sirun Liu ◽  
Heng Liu
Keyword(s):  
Stochastic Model ◽  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Ergodic Property ◽  
Model Simulations ◽  
Analytical Results ◽  
Stationary Distribution And Ergodicity ◽  
Mutualism Model ◽  
Mutualism System

Abstract This paper is concerned with a stochastic mutualism system with toxicant substances and saturation terms. We obtain the sufficient conditions for the existence of a unique stationary distribution to the equation and it has an ergodic property. It is interesting and surprising that toxicant substances have no effect on the stationary distribution of the stochastic model. Simulations are also carried out to confirm our analytical results.

scholarly journals Dynamics of a stochastic HIV/AIDS model with treatment under regime switching

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Miaomiao Gao ◽  
Daqing Jiang ◽  
Tasawar Hayat ◽  
Ahmed Alsaedi ◽  
Bashir Ahmad
Keyword(s):  
Stationary Distribution ◽  
Regime Switching ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Aids Model ◽  
Telegraph Noise ◽  
Spread Dynamics ◽  
Global Positive Solution ◽  
Theoretical Results ◽  
Hiv Aids

<p style='text-indent:20px;'>This paper focuses on the spread dynamics of an HIV/AIDS model with multiple stages of infection and treatment, which is disturbed by both white noise and telegraph noise. Switching between different environmental states is governed by Markov chain. Firstly, we prove the existence and uniqueness of the global positive solution. Then we investigate the existence of a unique ergodic stationary distribution by constructing suitable Lyapunov functions with regime switching. Furthermore, sufficient conditions for extinction of the disease are derived. The conditions presented for the existence of stationary distribution improve and generalize the previous results. Finally, numerical examples are given to illustrate our theoretical results.</p>

scholarly journals Dynamic Analysis of a Stochastic Rumor Propagation Model with Regime Switching

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3277
Author(s):  
Fangju Jia ◽  
Chunzheng Cao
Keyword(s):  
Dynamic Analysis ◽  
Numerical Simulations ◽  
Stationary Distribution ◽  
Regime Switching ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Propagation Model ◽  
Rumor Propagation ◽  
White Noises

We study the rumor propagation model with regime switching considering both colored and white noises. Firstly, by constructing suitable Lyapunov functions, the sufficient conditions for ergodic stationary distribution and extinction are obtained. Then we obtain the threshold Rs which guarantees the extinction and the existence of the stationary distribution of the rumor. Finally, numerical simulations are performed to verify our model. The results indicated that there is a unique ergodic stationary distribution when Rs>1. The rumor becomes extinct exponentially with probability one when Rs<1.

Stationary distribution and extinction for a stochastic two-compartment model of B-cell chronic lymphocytic leukemia

2021 ◽  
pp. 2150065
Author(s):  
Miaomiao Gao ◽  
Daqing Jiang ◽  
Xiangdan Wen
Keyword(s):  
Chronic Lymphocytic Leukemia ◽  
Stationary Distribution ◽  
Lyapunov Functions ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Compartment Model ◽  
Lymphocytic Leukemia ◽  
Two Compartment Model ◽  
Theoretical Results ◽  
Cell Chronic Lymphocytic Leukemia

In this paper, we study the dynamical behavior of a stochastic two-compartment model of [Formula: see text]-cell chronic lymphocytic leukemia, which is perturbed by white noise. Firstly, by constructing suitable Lyapunov functions, we establish sufficient conditions for the existence of a unique ergodic stationary distribution. Then, conditions for extinction of the disease are derived. Furthermore, numerical simulations are presented for supporting the theoretical results. Our results show that large noise intensity may contribute to extinction of the disease.

scholarly journals Common Weak Linear Copositive Lyapunov Functions for Positive Switched Linear Systems

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Yuangong Sun ◽  
Zhaorong Wu ◽  
Fanwei Meng
Keyword(s):  
Stability Analysis ◽  
Complex Systems ◽  
Linear Systems ◽  
Algebraic Structure ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Switched Linear Systems ◽  
Numerical Examples ◽  
The Stability ◽  
Necessary And Sufficient

Lyapunov functions play a key role in the stability analysis of complex systems. In this paper, we study the existence of a class of common weak linear copositive Lyapunov functions (CWCLFs) for positive switched linear systems (PSLSs) which generalize the conventional common linear copositive Lyapunov functions (CLCLFs) and can be used as handy tool to deal with the stability of PSLSs not covered by CLCLFs. We not only establish necessary and sufficient conditions for the existence of CWCLFs but also clearly describe the algebraic structure of all CWCLFs. Numerical examples are also given to demonstrate the effectiveness of the obtained results.

scholarly journals Stationary distribution of a stochastic SIRD epidemic model of Ebola with double saturated incidence rates and vaccination

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Panpan Wang ◽  
Jianwen Jia
Keyword(s):  
Stochastic Model ◽  
Numerical Simulations ◽  
Positive Solution ◽  
Stationary Distribution ◽  
Epidemic Model ◽  
Existence And Uniqueness ◽  
Lyapunov Functions ◽  
Incidence Rates ◽  
Saturated Incidence ◽  
Global Positive Solution

Abstract In this paper, a stochastic SIRD model of Ebola with double saturated incidence rates and vaccination is considered. Firstly, the existence and uniqueness of a global positive solution are obtained. Secondly, by constructing suitable Lyapunov functions and using Khasminskii’s theory, we show that the stochastic model has a unique stationary distribution. Moreover, the extinction of the disease is also analyzed. Finally, numerical simulations are carried out to portray the analytical results.

scholarly journals Ergodic Stationary Distribution of Hepatitis C Virus Model Incorporating Two Treatment Effects

2021 ◽  
Author(s):  
Auwal Abdullahi
Keyword(s):  
Hepatitis C ◽  
Stationary Distribution ◽  
Treatment Effects ◽  
Disease Model ◽  
Sufficient Conditions ◽  
Treatment Rate ◽  
Lyapunov Approach ◽  
Infectious Disease Model ◽  
Virus Model ◽  
Stochastic Properties

In this paper, the dynamics of Hepatitis C infectious disease model with two treatment effects are studied through the Ito Stochastic Differential Equations (SDEs). While the first treatment rate reduces the reproduction of virion, the other mitigates the new infections. Though the deterministic behaviour of the model has been extensively studied, little is known about its stochastic properties. Thus, we examine sufficient conditions for the existence and uniqueness of the ergodic stationary distribution of the model via stochastic Lyapunov approach. The existence of a unique positive solution is also studied. The numerical simulations of the SDE model are performed through the Euler-Maruyama method and compared with their deterministic counterparts. The results obtained by SDEs are found to conform to those reported through their deterministic analogues.

scholarly journals Long-time behaviors of two stochastic mussel-algae models

2021 ◽  
Vol 18 (6) ◽  
pp. 8392-8414
Author(s):  
Dengxia Zhou ◽  
◽  
Meng Liu ◽  
Ke Qi ◽  
Zhijun Liu ◽  
...  
Keyword(s):  
Periodic Solution ◽  
Stationary Distribution ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
The Other ◽  
Environmental Perturbations ◽  
Periodic Model ◽  
Long Time ◽  
The Mean

<abstract><p>In this paper, we develop two stochastic mussel-algae models: one is autonomous and the other is periodic. For the autonomous model, we provide sufficient conditions for the extinction, nonpersistent in the mean and weak persistence, and demonstrate that the model possesses a unique ergodic stationary distribution by constructing some suitable Lyapunov functions. For the periodic model, we testify that it has a periodic solution. The theoretical findings are also applied to practice to dissect the effects of environmental perturbations on the growth of mussel.</p></abstract>

scholarly journals Stochastic analysis of GSB process

2014 ◽  
Vol 95 (109) ◽  
pp. 149-159 ◽  
Author(s):  
Vladica Stojanovic ◽  
Biljana Popovic ◽  
Predrag Popovic
Keyword(s):  
Time Series ◽  
Stochastic Model ◽  
Stochastic Analysis ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Correlation Structure ◽  
Necessary And Sufficient ◽  
Stochastic Properties ◽  
Standard Investigation ◽  
Threshold Regime

We present a modification (and partly a generalization) of STOPBREAK process, which is the stochastic model of time series with permanent, emphatic fluctuations. The threshold regime of the process is obtained by using, so called, noise indicator. Now, the model, named the General Split- BREAK (GSB) process, is investigated in terms of its basic stochastic properties. We analyze some necessary and sufficient conditions of the existence of stationary GSB process, and we describe its correlation structure. Also, we define the sequence of the increments of the GSB process, named Split-MA process. Besides the standard investigation of stochastic properties of this process, we also give the conditions of its invertibility.

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