# American Institute of Mathematical Sciences

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November  2015, 20(9): 2933-2947. doi: 10.3934/dcdsb.2015.20.2933

## Persistence-time estimation for some stochastic SIS epidemic models

 1 Department of Applied Mathematics and Statistics and Operations Research, University of the Basque Country UPV/EHU, Spain, Spain 2 Department of Di erential Equations and Numerical Analysis, University of Sevilla, Spain

Received  July 2014 Revised  July 2015 Published  September 2015

In this paper, we study two stochastic SIS epidemic models: the first one with a constant population size, and the second one with a death factor. We analyze persistence and extinction behaviors for these models. The persistence time depends on the initial population size and satisfies a stationary backward Kolmogorov differential equation, which is a linear second-order partial differential equation with variable degenerate coefficients. We solve this equation numerically using a classical finite element method. We give computational evidence that the importance of understanding the dynamics of both the deterministic and the stochastic epidemic models is due to the numerical approximations to the mean persistence time. This can give more information about the model and may perhaps explain strange behaviors, such as the differences between the deterministic model and the stochastic one for long times.
Citation: Francisco de la Hoz, Anna Doubova, Fernando Vadillo. Persistence-time estimation for some stochastic SIS epidemic models. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2933-2947. doi: 10.3934/dcdsb.2015.20.2933
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