A stochastic differential equation SIS epidemic model

Research output: Contribution to journalArticle

264 Citations (Scopus)

Abstract

In this paper we extend the classical susceptible-infected-susceptible epidemic model from a deterministic framework to a stochastic one and formulate it as a stochastic differential equation (SDE) for the number of infectious individuals $I(t)$. We then prove that this SDE has a unique global positive solution $I(t)$ and establish conditions for extinction and persistence of $I(t)$. We discuss perturbation by stochastic noise. In the case of persistence we show the existence of a stationary distribution and derive expressions for its mean and variance. The results are illustrated by computer simulations, including two examples based on real-life diseases.
LanguageEnglish
Pages876-902
Number of pages27
JournalSIAM Journal on Applied Mathematics
Volume71
Issue number3
Early online date2 Jun 2011
DOIs
Publication statusPublished - 2011

Fingerprint

SIS Model
Epidemic Model
Persistence
Stochastic Equations
Differential equations
Differential equation
Stationary Distribution
Extinction
Positive Solution
Computer Simulation
Perturbation
Computer simulation
Framework
Life

Keywords

  • susceptible-infected-susceptible model
  • pneumococcus
  • gonorrhea
  • stationary distribution
  • basic reproduction number
  • persistence
  • extinction
  • stochastic differential equations
  • Brownian motion

Cite this

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title = "A stochastic differential equation SIS epidemic model",
abstract = "In this paper we extend the classical susceptible-infected-susceptible epidemic model from a deterministic framework to a stochastic one and formulate it as a stochastic differential equation (SDE) for the number of infectious individuals $I(t)$. We then prove that this SDE has a unique global positive solution $I(t)$ and establish conditions for extinction and persistence of $I(t)$. We discuss perturbation by stochastic noise. In the case of persistence we show the existence of a stationary distribution and derive expressions for its mean and variance. The results are illustrated by computer simulations, including two examples based on real-life diseases.",
keywords = "susceptible-infected-susceptible model, pneumococcus , gonorrhea, stationary distribution, basic reproduction number, persistence, extinction, stochastic differential equations, Brownian motion",
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A stochastic differential equation SIS epidemic model. / Gray, Alison; Greenhalgh, David; Hu, L.; Mao, Xuerong; Pan, Jiafeng.

In: SIAM Journal on Applied Mathematics , Vol. 71, No. 3, 2011, p. 876-902.

Research output: Contribution to journalArticle

TY - JOUR

T1 - A stochastic differential equation SIS epidemic model

AU - Gray, Alison

AU - Greenhalgh, David

AU - Hu, L.

AU - Mao, Xuerong

AU - Pan, Jiafeng

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PY - 2011

Y1 - 2011

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AB - In this paper we extend the classical susceptible-infected-susceptible epidemic model from a deterministic framework to a stochastic one and formulate it as a stochastic differential equation (SDE) for the number of infectious individuals $I(t)$. We then prove that this SDE has a unique global positive solution $I(t)$ and establish conditions for extinction and persistence of $I(t)$. We discuss perturbation by stochastic noise. In the case of persistence we show the existence of a stationary distribution and derive expressions for its mean and variance. The results are illustrated by computer simulations, including two examples based on real-life diseases.

KW - susceptible-infected-susceptible model

KW - pneumococcus

KW - gonorrhea

KW - stationary distribution

KW - basic reproduction number

KW - persistence

KW - extinction

KW - stochastic differential equations

KW - Brownian motion

U2 - 10.1137/10081856X

DO - 10.1137/10081856X

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