# 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.
Language English 876-902 27 SIAM Journal on Applied Mathematics 71 3 2 Jun 2011 10.1137/10081856X Published - 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

@article{95f7d688ba5b45988cf18bc360a46621,
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",
author = "Alison Gray and David Greenhalgh and L. Hu and Xuerong Mao and Jiafeng Pan",
note = "Changed last author",
year = "2011",
doi = "10.1137/10081856X",
language = "English",
volume = "71",
pages = "876--902",
journal = "SIAM Journal on Applied Mathematics",
issn = "0036-1399",
number = "3",

}

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

N1 - Changed last author

PY - 2011

Y1 - 2011

N2 - 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.

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

M3 - Article

VL - 71

SP - 876

EP - 902

JO - SIAM Journal on Applied Mathematics

T2 - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 3

ER -