A stochastic differential equation SIS epidemic model with two independent Brownian motions

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2 Citations (Scopus)

Abstract

In this paper, we introduce two perturbations in the classical deterministic susceptible–infected–susceptible epidemic model. Greenhalgh and Gray [4] in 2011 use a perturbation on β in SIS model. Based on their previous work, we consider another perturbation on the parameter μ+ γ and formulate the original model as a stochastic differential equation (SDE) with two independent Brownian Motions for the number of infected population. We then prove that our Model has a unique and bounded global solution I ( t ) . Also we establish conditions for extinction and persistence of the infected population I ( t ) . Under the conditions of persistence, we show that there is a unique stationary distribution and derive its mean and variance. Computer simulations illustrate our results and provide evidence to back up our theory.
Original languageEnglish
Pages (from-to)1536-1550
Number of pages15
JournalJournal of Mathematical Analysis and Applications
Volume474
Issue number2
Early online date13 Feb 2019
DOIs
Publication statusPublished - 15 Jun 2019

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SIS Model
Brownian movement
Epidemic Model
Stochastic Equations
Brownian motion
Differential equations
Differential equation
Perturbation
Persistence
Deterministic Model
Stationary Distribution
Global Solution
Extinction
Computer Simulation
Model
Computer simulation

Keywords

  • SIS model
  • independent Brownian motion
  • extinction
  • persistence
  • stationary distribution

Cite this

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title = "A stochastic differential equation SIS epidemic model with two independent Brownian motions",
abstract = "In this paper, we introduce two perturbations in the classical deterministic susceptible–infected–susceptible epidemic model. Greenhalgh and Gray [4] in 2011 use a perturbation on β in SIS model. Based on their previous work, we consider another perturbation on the parameter μ+ γ and formulate the original model as a stochastic differential equation (SDE) with two independent Brownian Motions for the number of infected population. We then prove that our Model has a unique and bounded global solution I ( t ) . Also we establish conditions for extinction and persistence of the infected population I ( t ) . Under the conditions of persistence, we show that there is a unique stationary distribution and derive its mean and variance. Computer simulations illustrate our results and provide evidence to back up our theory.",
keywords = "SIS model, independent Brownian motion, extinction, persistence, stationary distribution",
author = "Siyang Cai and Yongmei Cai and Xuerong Mao",
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T1 - A stochastic differential equation SIS epidemic model with two independent Brownian motions

AU - Cai, Siyang

AU - Cai, Yongmei

AU - Mao, Xuerong

PY - 2019/6/15

Y1 - 2019/6/15

N2 - In this paper, we introduce two perturbations in the classical deterministic susceptible–infected–susceptible epidemic model. Greenhalgh and Gray [4] in 2011 use a perturbation on β in SIS model. Based on their previous work, we consider another perturbation on the parameter μ+ γ and formulate the original model as a stochastic differential equation (SDE) with two independent Brownian Motions for the number of infected population. We then prove that our Model has a unique and bounded global solution I ( t ) . Also we establish conditions for extinction and persistence of the infected population I ( t ) . Under the conditions of persistence, we show that there is a unique stationary distribution and derive its mean and variance. Computer simulations illustrate our results and provide evidence to back up our theory.

AB - In this paper, we introduce two perturbations in the classical deterministic susceptible–infected–susceptible epidemic model. Greenhalgh and Gray [4] in 2011 use a perturbation on β in SIS model. Based on their previous work, we consider another perturbation on the parameter μ+ γ and formulate the original model as a stochastic differential equation (SDE) with two independent Brownian Motions for the number of infected population. We then prove that our Model has a unique and bounded global solution I ( t ) . Also we establish conditions for extinction and persistence of the infected population I ( t ) . Under the conditions of persistence, we show that there is a unique stationary distribution and derive its mean and variance. Computer simulations illustrate our results and provide evidence to back up our theory.

KW - SIS model

KW - independent Brownian motion

KW - extinction

KW - persistence

KW - stationary distribution

UR - https://www.journals.elsevier.com/journal-of-mathematical-analysis-and-applications

U2 - 10.1016/j.jmaa.2019.02.039

DO - 10.1016/j.jmaa.2019.02.039

M3 - Article

VL - 474

SP - 1536

EP - 1550

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

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ER -