Study on a susceptible–infected–vaccinated model with delay and proportional vaccination

A susceptible–infected–vaccinated epidemic model with proportional vaccination and generalized nonlinear rate is formulated and investigated in the paper. We show that the stochastic epidemic model admits a unique and global positive solution with probability one when constructing a proper [Formula: see text]-function therewith. Then a sufficient condition that guarantees the disappearances of diseases is derived when the indicator [Formula: see text]. Further, if [Formula: see text], then we obtain that the solution is weakly permanent with probability one. We also derived the sufficient conditions of the persistence in the mean for the susceptible and infected when another indicator [Formula: see text].

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A hepatitis stochastic epidemic model with acute and chronic stages

Advances in Difference Equations ◽

10.1186/s13662-021-03335-7 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Amir Khan ◽

Ghulam Hussain ◽

Abdullahi Yusuf ◽

Auwalu Hamisu Usman ◽

Usa Wannasingha Humphries

Keyword(s):

Epidemic Model ◽

Transmission Rate ◽

Sufficient Conditions ◽

Environmental Fluctuation ◽

Stochastic Perturbations ◽

Proposed Model ◽

Stochastic Epidemic ◽

Persistence In The Mean ◽

Global Positive Solution ◽

Stochastic Epidemic Model

AbstractThe article is based on the study of hepatitis transmission dynamics using a stochastic epidemic model. We discuss the stochastic perturbations of our proposed model by considering the effect of environmental fluctuation and distribute the transmission rate in the form of white noise. Taking into account the Lyapunov function theory, the uniqueness and existence of the global positive solution are proven. Some sufficient conditions for the extinction and persistence in the mean are established. The numerical simulations are given to verify the main theoretical findings.

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Dynamics of a stochastic rabies epidemic model with markovian switching

International Journal of Biomathematics ◽

10.1142/s1793524521500327 ◽

2021 ◽

pp. 2150032

Author(s):

Hao Peng ◽

Xinhong Zhang ◽

Daqing Jiang

Keyword(s):

Lyapunov Function ◽

White Noise ◽

Numerical Simulations ◽

Positive Solution ◽

Epidemic Model ◽

Sufficient Conditions ◽

Sufficient Condition ◽

Telegraph Noise ◽

Global Positive Solution ◽

Theoretical Results

In this paper, we analyze a stochastic rabies epidemic model which is perturbed by both white noise and telegraph noise. First, we prove the existence of the unique global positive solution. Second, by constructing an appropriate Lyapunov function, we establish a sufficient condition for the existence of a unique ergodic stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for the extinction of diseases. Finally, numerical simulations are introduced to illustrate our theoretical results.

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Dynamics of a stochastic SIQR epidemic model with saturated incidence rate

Filomat ◽

10.2298/fil1815239w ◽

2018 ◽

Vol 32 (15) ◽

pp. 5239-5253 ◽

Cited By ~ 3

Author(s):

Li-Li Wang ◽

Nan-Jing Huang ◽

Donal O’Regan

Keyword(s):

Numerical Simulations ◽

Positive Solution ◽

Incidence Rate ◽

Epidemic Model ◽

Lyapunov Functions ◽

Sufficient Conditions ◽

Saturated Incidence Rate ◽

Saturated Incidence ◽

Global Positive Solution ◽

The Mean

The purpose of this paper is to propose and investigate a stochastic SIQR epidemic model with saturated incidence rate. Firstly, we give some conditions to guarantee the stochastic SIQR epidemic model has a unique global positive solution. Then we verify that the disease in this model will die out exponentially if Rs 0 < 1, while the disease will be persistent in the mean if Rs 0 > 1. Moreover, by constructing suitable Lyapunov functions, we establish some sufficient conditions for the existence of an ergodic stationary distribution for the model. Finally, we provide some numerical simulations to illustrate the analytical results.

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Dynamics of a stochastic predator–prey model with mutual interference

International Journal of Biomathematics ◽

10.1142/s1793524514500260 ◽

2014 ◽

Vol 07 (03) ◽

pp. 1450026 ◽

Cited By ~ 2

Author(s):

Kai Wang ◽

Yanling Zhu

Keyword(s):

Numerical Simulation ◽

Positive Solution ◽

Sufficient Conditions ◽

Stochastic Perturbation ◽

Mutual Interference ◽

Predator Prey ◽

Predator Prey Model ◽

Persistence In The Mean ◽

The Mean ◽

Globally Positive Solution

In this paper, a stochastic predator–prey (PP) model with mutual interference is considered. Some sufficient conditions for the existence of globally positive solution, non-persistence in the mean, weak persistence in the mean, strong persistence in the mean and almost surely extinction of the the model are established. Moreover, the threshold between weak persistence in the mean and almost surely extinction of the prey is obtained. Some examples are given to show the feasibility of the results by numerical simulation. It is significant that such a model is firstly proposed with stochastic perturbation.

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Stochastic dynamics of the transmission of Dengue fever virus between mosquitoes and humans

International Journal of Biomathematics ◽

10.1142/s1793524521500625 ◽

2021 ◽

pp. 2150062

Author(s):

Manjing Guo ◽

Lin Hu ◽

Lin-Fei Nie

Keyword(s):

Dengue Fever ◽

Stochastic Dynamics ◽

Sufficient Conditions ◽

Lyapunov Methods ◽

Differential Model ◽

Persistence In The Mean ◽

Global Positive Solution ◽

The Mean ◽

And Behavior ◽

The Impact

Considering the impact of environmental white noise on the quantity and behavior of vector of disease, a stochastic differential model describing the transmission of Dengue fever between mosquitoes and humans, in this paper, is proposed. By using Lyapunov methods and Itô’s formula, we first prove the existence and uniqueness of a global positive solution for this model. Further, some sufficient conditions for the extinction and persistence in the mean of this stochastic model are obtained by using the techniques of a series of stochastic inequalities. In addition, we also discuss the existence of a unique stationary distribution which leads to the stochastic persistence of this disease. Finally, several numerical simulations are carried to illustrate the main results of this contribution.

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Dynamics of a stochastic SIRS epidemic model with standard incidence under regime switching

International Journal of Biomathematics ◽

10.1142/s1793524521500741 ◽

2021 ◽

pp. 2150074

Author(s):

Jiang Xu ◽

Yinong Wang ◽

Zhongwei Cao

Keyword(s):

Lyapunov Function ◽

Stochastic System ◽

Epidemic Model ◽

Regime Switching ◽

Sufficient Conditions ◽

Sirs Epidemic Model ◽

Persistence In The Mean ◽

The Mean ◽

Stochastic Sirs Epidemic Model ◽

Stochastic Lyapunov Function

The goal of this paper is to introduce and initiate a study of a stochastic SIRS epidemic model with standard incidence which is perturbed by both white and telegraph noises. We first show persistence in the mean and then establish the sufficient conditions for extinction of the disease. Moreover, in the case of persistence, we obtain sufficient conditions for the existence of positive recurrence of the solutions by means of structuring suitable stochastic Lyapunov function with regime switching. Meanwhile, the threshold between persistence in the mean and extinction of the stochastic system is also obtained. Finally, we test our theory conclusion by simulations.

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A stochastic vaccinated epidemic model incorporating Lévy processes with a general awareness-induced incidence

International Journal of Biomathematics ◽

10.1142/s1793524521500443 ◽

2021 ◽

pp. 2150044

Author(s):

Khadija Akdim ◽

Adil Ez-Zetouni ◽

Mehdi Zahid

Keyword(s):

Numerical Simulations ◽

Positive Solution ◽

Dynamic Behavior ◽

Epidemic Model ◽

Equilibrium Points ◽

Sufficient Conditions ◽

Lévy Noise ◽

Global Positive Solution ◽

Levy Noise ◽

Disease Free

In this paper, we investigate a stochastic vaccinated epidemic model with a general awareness-induced incidence perturbed by Lévy noise. First, we show that this model has a unique global positive solution. Therefore, we establish the dynamic behavior of the solution around both disease-free and endemic equilibrium points. Furthermore, when [Formula: see text], we give sufficient conditions for the existence of an ergodic stationary distribution to the model when the jump part in the Lévy noise is null. Finally, we present some examples to illustrate the analytical results by numerical simulations.

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Dynamical Behavior of a Stochastic SIRC Model for Influenza A

Symmetry ◽

10.3390/sym12050745 ◽

2020 ◽

Vol 12 (5) ◽

pp. 745 ◽

Cited By ~ 1

Author(s):

Tongqian Zhang ◽

Tingting Ding ◽

Ning Gao ◽

Yi Song

Keyword(s):

Lyapunov Function ◽

Numerical Simulations ◽

Positive Solution ◽

Epidemic Model ◽

Influenza A ◽

Dynamical Behavior ◽

Sufficient Conditions ◽

Global Positive Solution ◽

Theoretical Results ◽

Stochastic Lyapunov Function

In this paper, a stochastic SIRC epidemic model for Influenza A is proposed and investigated. First, we prove that the system exists a unique global positive solution. Second, the extinction of the disease is explored and the sufficient conditions for extinction of the disease are derived. And then the existence of a unique ergodic stationary distribution of the positive solutions for the system is discussed by constructing stochastic Lyapunov function. Furthermore, numerical simulations are employed to illustrate the theoretical results. Finally, we give some further discussions about the system.

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DYNAMIC ANALYSIS OF AN SIR MODEL WITH STOCHASTIC PERTURBATIONS

International Journal of Biomathematics ◽

10.1142/s1793524513500411 ◽

2013 ◽

Vol 06 (06) ◽

pp. 1350041

Author(s):

ZHENJIE LIU ◽

JINLIANG WANG ◽

YALAN XU ◽

GUOQIANG LI

Keyword(s):

Epidemic Model ◽

Lyapunov Functions ◽

Sir Model ◽

Sir Epidemic Model ◽

Stochastic Perturbations ◽

Sir Epidemic ◽

Stochastic Epidemic ◽

Differential Infectivity ◽

Global Positive Solution ◽

Stochastic Epidemic Model

In this paper, we present a differential infectivity SIR epidemic model with modified saturation incidences and stochastic perturbations. We show that the stochastic epidemic model has a unique global positive solution, and we utilize stochastic Lyapunov functions to show the asymptotic behavior of the solution.

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The rumour process

Journal of Applied Probability ◽

10.1017/s0021900200023081 ◽

1982 ◽

Vol 19 (04) ◽

pp. 759-766

Author(s):

Ross Dunstan

Keyword(s):

Epidemic Model ◽

Asymptotic Form ◽

Deterministic Model ◽

Final Size ◽

Stochastic Epidemic ◽

The Mean ◽

Stochastic Epidemic Model ◽

General Stochastic

The general stochastic epidemic model is used as a model for the spread of rumours. Recursive expressions are found for the mean of the final size of each generation of hearers. Simple expressions are found for the generation size and the asymptotic form of its final size in the deterministic model. An approximating process is presented.

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