SDE SIS epidemic models

Research output: Contribution to conferenceSpeech

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.
Original languageEnglish
Publication statusUnpublished - 2012
EventWorkshop on Stochastic Modelling in Ecosystems - Glasgow, United Kingdom
Duration: 11 Jun 201212 Jun 2012

Conference

ConferenceWorkshop on Stochastic Modelling in Ecosystems
CountryUnited Kingdom
CityGlasgow
Period11/06/1212/06/12

Fingerprint

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

Keywords

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

Cite this

Gray, A., Greenhalgh, D., Mao, X., & Pan, J. (2012). SDE SIS epidemic models. Workshop on Stochastic Modelling in Ecosystems, Glasgow, United Kingdom.
Gray, Alison ; Greenhalgh, David ; Mao, Xuerong ; Pan, Jiafeng. / SDE SIS epidemic models. Workshop on Stochastic Modelling in Ecosystems, Glasgow, United Kingdom.
@conference{665deb97b85e4e68a9d89c90409d3ab0,
title = "SDE SIS epidemic models",
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 Xuerong Mao and Jiafeng Pan",
year = "2012",
language = "English",
note = "Workshop on Stochastic Modelling in Ecosystems ; Conference date: 11-06-2012 Through 12-06-2012",

}

Gray, A, Greenhalgh, D, Mao, X & Pan, J 2012, 'SDE SIS epidemic models', Workshop on Stochastic Modelling in Ecosystems, Glasgow, United Kingdom, 11/06/12 - 12/06/12.

SDE SIS epidemic models. / Gray, Alison; Greenhalgh, David; Mao, Xuerong; Pan, Jiafeng.

2012. Workshop on Stochastic Modelling in Ecosystems, Glasgow, United Kingdom.

Research output: Contribution to conferenceSpeech

TY - CONF

T1 - SDE SIS epidemic models

AU - Gray, Alison

AU - Greenhalgh, David

AU - Mao, Xuerong

AU - Pan, Jiafeng

PY - 2012

Y1 - 2012

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

M3 - Speech

ER -

Gray A, Greenhalgh D, Mao X, Pan J. SDE SIS epidemic models. 2012. Workshop on Stochastic Modelling in Ecosystems, Glasgow, United Kingdom.