Demographic stochasticity in the SDE SIS epidemic model

David Greenhalgh; Yanfeng Liang; Xuerong Mao

doi:10.3934/dcdsb.2015.20.2859

Demographic stochasticity in the SDE SIS epidemic model

David Greenhalgh, Yanfeng Liang, Xuerong Mao

Mathematics And Statistics

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Abstract

In this paper we discuss the stochastic differential equation (SDE) susceptible- infected-susceptible (SIS) epidemic model with demographic stochasticity. First we prove that the SDE has a unique nonnegative solution which is bounded above. Then we give conditions needed for the solution to become extinct. Next we use the Feller test to calculate the respective probabilities of the solution first hitting zero or the upper limit. We confirm our theoretical results with numerical simulations and then give simulations with realistic parameter values for two example diseases: gonorrhea and pneumococcus.

Original language	English
Pages (from-to)	2859-2884
Number of pages	26
Journal	Discrete and Continuous Dynamical Systems - Series B
Volume	20
Issue number	9
DOIs	https://doi.org/10.3934/dcdsb.2015.20.2859
Publication status	Published - 30 Sept 2015

Keywords

SIS epidemic model
Brownian motion
stochastic differential equations
feller test
extinction
demographic stochasticity

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10.3934/dcdsb.2015.20.2859

Greenhalgh-etal-DCDS-2015-Demographic-stochasticity-in-the-SDE-DIDAccepted author manuscript, 6.78 MB

Cite this

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title = "Demographic stochasticity in the SDE SIS epidemic model",

abstract = "In this paper we discuss the stochastic differential equation (SDE) susceptible- infected-susceptible (SIS) epidemic model with demographic stochasticity. First we prove that the SDE has a unique nonnegative solution which is bounded above. Then we give conditions needed for the solution to become extinct. Next we use the Feller test to calculate the respective probabilities of the solution first hitting zero or the upper limit. We confirm our theoretical results with numerical simulations and then give simulations with realistic parameter values for two example diseases: gonorrhea and pneumococcus.",

keywords = "SIS epidemic model, Brownian motion, stochastic differential equations, feller test, extinction, demographic stochasticity",

author = "David Greenhalgh and Yanfeng Liang and Xuerong Mao",

note = "This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Discrete and Continuous Dynamical Systems - Series B following peer review. The definitive publisher-authenticated version Greenhalgh, D., Liang, Y., & Mao, X. (2015). Demographic stochasticity in the SDE SIS epidemic model. Discrete and Continuous Dynamical Systems - Series B, 20(9), 2859-2884. 10.3934/dcdsb.2015.20.2859 is available online at: http://dx.doi.org/10.3934/dcdsb.2015.20.2859",

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T1 - Demographic stochasticity in the SDE SIS epidemic model

AU - Greenhalgh, David

AU - Liang, Yanfeng

AU - Mao, Xuerong

N1 - This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Discrete and Continuous Dynamical Systems - Series B following peer review. The definitive publisher-authenticated version Greenhalgh, D., Liang, Y., & Mao, X. (2015). Demographic stochasticity in the SDE SIS epidemic model. Discrete and Continuous Dynamical Systems - Series B, 20(9), 2859-2884. 10.3934/dcdsb.2015.20.2859 is available online at: http://dx.doi.org/10.3934/dcdsb.2015.20.2859

PY - 2015/9/30

Y1 - 2015/9/30

N2 - In this paper we discuss the stochastic differential equation (SDE) susceptible- infected-susceptible (SIS) epidemic model with demographic stochasticity. First we prove that the SDE has a unique nonnegative solution which is bounded above. Then we give conditions needed for the solution to become extinct. Next we use the Feller test to calculate the respective probabilities of the solution first hitting zero or the upper limit. We confirm our theoretical results with numerical simulations and then give simulations with realistic parameter values for two example diseases: gonorrhea and pneumococcus.

AB - In this paper we discuss the stochastic differential equation (SDE) susceptible- infected-susceptible (SIS) epidemic model with demographic stochasticity. First we prove that the SDE has a unique nonnegative solution which is bounded above. Then we give conditions needed for the solution to become extinct. Next we use the Feller test to calculate the respective probabilities of the solution first hitting zero or the upper limit. We confirm our theoretical results with numerical simulations and then give simulations with realistic parameter values for two example diseases: gonorrhea and pneumococcus.

KW - SIS epidemic model

KW - Brownian motion

KW - stochastic differential equations

KW - feller test

KW - extinction

KW - demographic stochasticity

UR - http://aimsciences.org/journals/home.jsp?journalID=2

U2 - 10.3934/dcdsb.2015.20.2859

DO - 10.3934/dcdsb.2015.20.2859

M3 - Article

SN - 1531-3492

VL - 20

SP - 2859

EP - 2884

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

IS - 9

ER -

Demographic stochasticity in the SDE SIS epidemic model

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Demographic stochasticity in the SDE SIS epidemic model

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