# 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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##### References:
 [1] Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020196 [2] Hal L. Smith, Horst R. Thieme. Persistence and global stability for a class of discrete time structured population models. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4627-4646. doi: 10.3934/dcds.2013.33.4627 [3] Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59 [4] Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020222 [5] Kun Wang, Yinnian He, Yueqiang Shang. Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 665-684. doi: 10.3934/dcdsb.2010.13.665 [6] Hao Wang, Wei Yang, Yunqing Huang. An adaptive edge finite element method for the Maxwell's equations in metamaterials. Electronic Research Archive, 2020, 28 (2) : 961-976. doi: 10.3934/era.2020051 [7] Alex Bihlo, James Jackaman, Francis Valiquette. On the development of symmetry-preserving finite element schemes for ordinary differential equations. Journal of Computational Dynamics, 2020, 7 (2) : 339-368. doi: 10.3934/jcd.2020014 [8] Gonzalo Galiano, Julián Velasco. Finite element approximation of a population spatial adaptation model. Mathematical Biosciences & Engineering, 2013, 10 (3) : 637-647. doi: 10.3934/mbe.2013.10.637 [9] Daniele Boffi, Lucia Gastaldi. Discrete models for fluid-structure interactions: The finite element Immersed Boundary Method. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 89-107. doi: 10.3934/dcdss.2016.9.89 [10] Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic & Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639 [11] Can Huang, Zhimin Zhang. The spectral collocation method for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 667-679. doi: 10.3934/dcdsb.2013.18.667 [12] P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020178 [13] Cornel M. Murea, H. G. E. Hentschel. A finite element method for growth in biological development. Mathematical Biosciences & Engineering, 2007, 4 (2) : 339-353. doi: 10.3934/mbe.2007.4.339 [14] Meng Zhao, Aijie Cheng, Hong Wang. A preconditioned fast Hermite finite element method for space-fractional diffusion equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3529-3545. doi: 10.3934/dcdsb.2017178 [15] Changling Xu, Tianliang Hou. Superclose analysis of a two-grid finite element scheme for semilinear parabolic integro-differential equations. Electronic Research Archive, 2020, 28 (2) : 897-910. doi: 10.3934/era.2020047 [16] Min Yang, Guanggan Chen. Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1565-1581. doi: 10.3934/dcdsb.2019240 [17] Viorel Barbu. Existence for nonlinear finite dimensional stochastic differential equations of subgradient type. Mathematical Control & Related Fields, 2018, 8 (3&4) : 501-508. doi: 10.3934/mcrf.2018020 [18] Jianguo Huang, Sen Lin. A $C^0P_2$ time-stepping virtual element method for linear wave equations on polygonal meshes. Electronic Research Archive, 2020, 28 (2) : 911-933. doi: 10.3934/era.2020048 [19] Yayun Zheng, Xu Sun. Governing equations for Probability densities of stochastic differential equations with discrete time delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3615-3628. doi: 10.3934/dcdsb.2017182 [20] Weiyin Fei, Liangjian Hu, Xuerong Mao, Dengfeng Xia. Advances in the truncated Euler–Maruyama method for stochastic differential delay equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2081-2100. doi: 10.3934/cpaa.2020092

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