# American Institute of Mathematical Sciences

September  2019, 24(9): 4827-4849. doi: 10.3934/dcdsb.2019033

## Analysis of a stochastic SIRS model with interval parameters

 1 School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, China 2 Department of Mathematics, University of Florida, Gainesville, FL 32611, USA 3 School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, China

* Corresponding author: Qimin Zhang

Received  February 2018 Revised  September 2018 Published  September 2019 Early access  February 2019

Many studies of mathematical epidemiology assume that model parameters are precisely known. However, they can be imprecise due to various uncertainties. Deterministic epidemic models are also subjected to stochastic perturbations. In this paper, we analyze a stochastic SIRS model that includes interval parameters and environmental noises. We define the stochastic basic reproduction number, which is shown to govern disease extinction or persistence. When it is less than one, the disease is predicted to die out with probability one. When it is greater than one, the model admits a stationary distribution. Thus, larger stochastic noises (resulting in a smaller stochastic basic reproduction number) are able to suppress the emergence of disease outbreaks. Using numerical simulations, we also investigate the influence of parameter imprecision and susceptible response to the disease information that may change individual behavior and protect the susceptible from infection. These parameters can greatly affect the long-term behavior of the system, highlighting the importance of incorporating parameter imprecision into epidemic models and the role of information intervention in the control of infectious diseases.

Citation: Kangbo Bao, Libin Rong, Qimin Zhang. Analysis of a stochastic SIRS model with interval parameters. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4827-4849. doi: 10.3934/dcdsb.2019033
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##### References:
The path of $S(t)$, $I(t)$, $R(t)$ and the histogram of the probability density function of $I(150)$ assuming p = 0.1, $(S_0,I_0,R_0,Z_0) = (479.0,20.0,1.0,10.0)$ under different noise intensities
The path of $S(t)$, $I(t)$, $R(t)$ assuming p = 0.1, $(S_0,I_0,R_0,Z_0) = (479.0,20.0,1.0,10.0)$ with different noise intensities
Variation of $\mathscr{R}_0$ and $\mathscr{R}_s$ as $p$ varies
The path of $S(t)$, $I(t)$, $R(t)$ with initial $(S_0,I_0,R_0,Z_0) = (479.0,20.0,1.0,10.0)$ for p = 0.2, p = 0.4 and p = 0.6, respectively
The path of $I(t)$ with initial $(S_0,I_0,R_0,Z_0) = (479.0,20.0,1.0,10.0)$ under different noise intensities and imprecise parameter p
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