A stochastic susceptible-infective-susceptible (SIS) epidemic model is considere

Question

A stochastic susceptible-infective-susceptible (SIS) epidemic model is considered, which consists of susceptible S(t) and infected I(t) populations. The susceptible becomes infected, recovers and becomes susceptible again. The stochastic version of the model is given by [4, 67] dS(t) = (-alpha S(t)I(t)/N + beta I (t)) dt + 1/Squareroot 2 Squareroot alpha S(t)I(t)/N + beta I(t)(dW_1 - dW_2) dI(t) = (alpha S(t)I(t)/N - beta I(t)) dt + 1/Squareroot 2 Squareroot S(t)I(t)/N + beta I(t) (-dw_1 + dw_2) where S(0) + I(0) = S(t) + I(t) = Total population N (constant), alpha is the rate at which susceptible becomes infected, beta is the rate at which infected individuals after recovery become susceptible again and W_1 and W_2 are two Wiener processes. Taking alpha = 0.04, beta = 0.01, S(0) = 950, I(0) = 50 and time period = [0, 100], solve the system numerically, compare it with the deterministic solution and comment on the result.

Explanation / Answer

Highlights

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A stochastic susceptible–infected–removed model is formulated when population size varies with time.

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Extinction of solution is derived when intensities of white noises are controlled by parameters of model.

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Persistence, stationary distribution and ergodicity are followed by careful computation via constructing suitable functions.

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Three examples and numerical simulations are separately carried out to check the validity of the main results.

Abstract

In this paper, the dynamics of a stochastic susceptible–infected–removed model in a population with varying size is investigated. We firstly show that the stochastic epidemic model has a unique global positive solution with any positive initial value. Then we verify that random perturbations lead to extinction when some conditions are being valid. Moreover, we prove that the solution of the stochastic epidemic model is persistent in the mean by building up a suitable Lyapunov function and using generalized Itô’s formula. Further, the stochastic epidemic model admits a stationary distribution around the endemic equilibrium when parameters satisfy some sufficient conditions. To end this contribution and to check the validity of the main results, numerical simulations are separately carried out to illustrate these results.

Keywords

Varying population size

Stochastic SIR model

Extinction

Persistence in the mean

Stationary distribution

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