Stability of a stochastic SIR system (original) (raw)
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Mathematics
So called SIR epidemic model with distributed delay and stochastic perturbations is considered. It is shown, that the known sufficient conditions of stability in probability of the equilibria of this model, formulated immediately in the terms of the system parameters, can be improved by virtue of the method of Lyapunov functionals construction and the method of Linear Matrix Inequalities (LMIs). It is also shown, that stability can be investigated immediately via numerical simulation of a solution of the considered model.
Dynamics for a stochastic delayed SIRS epidemic model
Nonlinear Analysis: Modelling and Control, 2020
In this paper, we consider a stochastic delayed SIRS epidemic model with seasonal variation. Firstly, we prove that the system is mathematically and biologically well-posed by showing the global existence, positivity and stochastically ultimate boundneness of the solution. Secondly, some sufficient conditions on the permanence and extinction of the positive solutions with probability one are presented. Thirdly, we show that the solution of the system is asymptotical around of the disease-free periodic solution and the intensity of the oscillation depends of the intensity of the noise. Lastly, the existence of stochastic nontrivial periodic solution for the system is obtained.
The stability of an SIR epidemic model with time delays
Mathematical Biosciences and Engineering, 2005
In this paper, an SIR epidemic model for the spread of an infectious disease transmitted by direct contact among humans and vectors (mosquitoes ) which have an incubation time to become infectious is formulated. It is shown that a disease-free equilibrium point is globally stable if no endemic equilibrium point exists. Further, the endemic equilibrium point (if it exists) is globally stable with respect to a "weak delay". Some known results are generalized.
Random Operators and Stochastic Equations, 2018
We treat a delayed SIR (susceptible, infected, recovered) epidemic model with a saturated incidence rate and its perturbation through the contact rate using a white noise. We start with a deterministic model and then add a perturbation on the contact rate using a white noise to obtain a stochastic model. We prove the existence and uniqueness of the global positive solution for both deterministic and stochastic delayed differential equations. Under suitable conditions on the parameters, we study the global asymptotic stability of the disease-free equilibrium of the deterministic model and the almost sure stability of the disease-free equilibrium of the stochastic model.
Dynamical behaviors of a stochastic SIRS epidemic model
Journal of Mathematical and Computational Science
In this paper, we study the dynamical behavior of a stochastic SIRS epidemic model with specific nonlinear incidence rate and vaccination. We show the existence and positivity of the solution of the SIRS stochastic differential equation. We defined a number R and we prove the disease free equilibrium is almost sure exponentially stable if R < 1. We studying the behavior around the endemic equilibrium E*. Numerical simulations presented our theoretical results.
Deterministic and stochastic stability of an SIRS epidemic model with a saturated incidence rate
Random Operators and Stochastic Equations, 2017
In this paper, we formulate an epidemic model for the spread of an infectious disease in a population of varying size. The total population is divided into three distinct epidemiological subclass of individuals (susceptible, infectious and recovered) and we study a deterministic and stochastic models with saturated incidence rate. The stochastic model is obtained by incorporating a random noise into the deterministic model. In the deterministic case, we briefly discuss the global asymptotic stability of the disease free equilibrium by using a Lyapunov function. For the stochastic version, we study the global existence and positivity of the solution. Under suitable conditions on the intensity of the white noise perturbation, we prove that there are a
Stability Analysis of a Stochastic SIR Epidemic Model with Specific Nonlinear Incidence Rate
International Journal of Stochastic Analysis, 2013
We investigate a stochastic SIR epidemic model with specific nonlinear incidence rate. The stochastic model is derived from the deterministic epidemic model by introducing random perturbations around the endemic equilibrium state. The effect of random perturbations on the stability behavior of endemic equilibrium is discussed. Finally, numerical simulations are presented to illustrate our theoretical results.
Dynamical Behavior of a Stochastic SIRS Epidemic Model
Mathematical Modelling of Natural Phenomena, 2015
In this paper we study the Kernack-MacKendrick model under telegraph noise. The telegraph noise switches at random between two SIRS models. We give out conditions for the persistence of the disease and the stability of a disease free equilibrium. We show that the asymptotic behavior highly depends on the value of a threshold λ which is calculated from the intensities of switching between environmental states, the total size of the population as well as the parameters of both SIRS systems. According to the value of λ, the system can globally tend towards an endemic state or a disease free state. The aim of this work is also to describe completely the ω-limit set of all positive solutions to the model. Moreover, the attraction of the ω-limit set and the stationary distribution of solutions will be shown.
Analysis of a stochastic SIRS epidemic model with specific functional response
Applied Mathematical Sciences, 2016
A new stochastic SIRS epidemic model with specific functional response is proposed and analyzed. First, we show that the model is biologically well-posed by proving the global existence, positivity and boundedness of solutions. Moreover, sufficient conditions for the extinction and persistence of the disease are also obtained. In the end, some numerical simulations are presented to illustrate our analytical results.