### Abstract

Original language | English |
---|---|

Pages (from-to) | 1277-1295 |

Number of pages | 19 |

Journal | Discrete and Continuous Dynamical Systems - Series B |

Volume | 20 |

Issue number | 4 |

Early online date | 28 Feb 2015 |

DOIs | |

Publication status | Published - 2015 |

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### Keywords

- SIRS epidemic model
- SIRS
- Lyapunov function

### Cite this

*Discrete and Continuous Dynamical Systems - Series B*,

*20*(4), 1277-1295. https://doi.org/10.3934/dcdsb.2015.20.1277

}

*Discrete and Continuous Dynamical Systems - Series B*, vol. 20, no. 4, pp. 1277-1295. https://doi.org/10.3934/dcdsb.2015.20.1277

**The threshold of a stochastic SIRS epidemic model in a population with varying size.** / Zhao, Yanan; Jiang, Daqing; Mao, Xuerong; Gray, Alison.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The threshold of a stochastic SIRS epidemic model in a population with varying size

AU - Zhao, Yanan

AU - Jiang, Daqing

AU - Mao, Xuerong

AU - Gray, Alison

PY - 2015

Y1 - 2015

N2 - In this paper, a stochastic susceptible-infected-removed-susceptible (SIRS) epidemic model in a population with varying size is discussed. A new threshold ~R0 is identified which determines the outcome of the disease. When the noise is small, if ~R0 < 1, the infected proportion of the population disappears, so the disease dies out, whereas if ~R0 > 1, the infected proportion persists in the mean and we derive that the disease is endemic. Furthermore, when R0 > 1 and subject to a condition on some of the model parameters, we show that the solution of the stochastic model oscillates around the endemic equilibrium of the corresponding deterministic system with threshold R0, and the intensity of fluctuation is proportional to that of the white noise. On the other hand, when the noise is large, we find that a large noise intensity has the effect of suppressing the epidemic, so that it dies out. These results are illustrated by computer simulations.

AB - In this paper, a stochastic susceptible-infected-removed-susceptible (SIRS) epidemic model in a population with varying size is discussed. A new threshold ~R0 is identified which determines the outcome of the disease. When the noise is small, if ~R0 < 1, the infected proportion of the population disappears, so the disease dies out, whereas if ~R0 > 1, the infected proportion persists in the mean and we derive that the disease is endemic. Furthermore, when R0 > 1 and subject to a condition on some of the model parameters, we show that the solution of the stochastic model oscillates around the endemic equilibrium of the corresponding deterministic system with threshold R0, and the intensity of fluctuation is proportional to that of the white noise. On the other hand, when the noise is large, we find that a large noise intensity has the effect of suppressing the epidemic, so that it dies out. These results are illustrated by computer simulations.

KW - SIRS epidemic model

KW - SIRS

KW - Lyapunov function

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

U2 - 10.3934/dcdsb.2015.20.1277

DO - 10.3934/dcdsb.2015.20.1277

M3 - Article

VL - 20

SP - 1277

EP - 1295

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

SN - 1531-3492

IS - 4

ER -