# American Institute of Mathematical Sciences

April  2019, 24(4): 1627-1652. doi: 10.3934/dcdsb.2018223

## A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients

 College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China

* Corresponding author: S. Guo

Received  February 2018 Published  June 2018

Fund Project: The second author is supported by NSF of China (Grants No. 11671123).

This paper is devoted to a spatial heterogeneous SIS model with the infected group equipped with a free boundary. Our main aim is to determine whether the disease is spreading forever or extinct eventually, and to illustrate, under the nonhomogeneous spatial environment, free boundaries can have a large influence on the infected behavior at the large time. For this purpose, we first introduce a basic reproduction number and then establish a spreading-vanishing dichotomy. Then by investigating the effect of the diffusion rate, initial domain and spreading speed on the asymptotic behavior of the infected group, we establish some sufficient conditions and even necessary and sufficient conditions for disease spreading or vanishing.

Citation: Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223
##### References:
 [1] L. J. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete and Continuous Dynamical Systems, 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.  Google Scholar [2] R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics, Journal of Mathematical Biology, 29 (1991), 315-338.  doi: 10.1007/BF00167155.  Google Scholar [3] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar [4] J.-F. Cao, W.-T. Li and F.-Y. Yang, Dynamics of a nonlocal SIS epidemic model with free boundary, Discrete & Continuous Dynamical Systems-Series B, 22 (2017), 247-266.  doi: 10.3934/dcdsb.2017013.  Google Scholar [5] X. Chen and A. Friedman, A free boundary problem arising in a model of wound healing, SIAM Journal on Mathematical Analysis, 32 (2000), 778-800.  doi: 10.1137/S0036141099351693.  Google Scholar [6] Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations, vol. 2, World Scientific, 2006. doi: 10.1142/9789812774446.  Google Scholar [7] Y. Du and Z. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, Ⅱ, Journal of Differential Equations, 250 (2011), 4336-4366.  doi: 10.1016/j.jde.2011.02.011.  Google Scholar [8] Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM Journal on Mathematical Analysis, 42 (2010), 377-405.  doi: 10.1137/090771089.  Google Scholar [9] Y. Du and L. Liu, Remarks on the uniqueness problem for the logistic equation on the entire space, Bulletin of the Australian Mathematical Society, 73 (2006), 129-137.  doi: 10.1017/S0004972700038685.  Google Scholar [10] Y. Du and L. Ma, Logistic type equations on $\mathbb{R}^n$ by a squeezing method involving boundary blow-up solutions, Journal of the London Mathematical Society, 64 (2001), 107-124.  doi: 10.1017/S0024610701002289.  Google Scholar [11] W.-E. Fitzgibbon and M. Langlais, Simple models for the transmission of microparasites between host populations living on noncoincident spatial domains, in Structured Population Models in Biology and Epidemiology, Springer, 1936 (2008), 115–164. doi: 10.1007/978-3-540-78273-5_3.  Google Scholar [12] J. Ge, K. I. Kim, Z. Lin and H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, Journal of Differential Equations, 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035.  Google Scholar [13] S. Guo, Spatio-temporal patterns in a diffusive model with non-local delay effect, IMA Journal of Applied Mathematics, 82 (2017), 864-908.  doi: 10.1093/imamat/hxx018.  Google Scholar [14] S. Guo and S. Yan, Hopf bifurcation in a diffusive Lotka-Volterra type system with nonlocal delay effect, Journal of Differential Equations, 260 (2016), 781-817.  doi: 10.1016/j.jde.2015.09.031.  Google Scholar [15] X. He and W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system Ⅰ: Heterogeneity vs. homogeneity, Journal of Differential Equations, 254 (2013), 528-546.  doi: 10.1016/j.jde.2012.08.032.  Google Scholar [16] V. Hutson, Y. Lou and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, Journal of Differential Equations, 185 (2002), 97-136.  doi: 10.1006/jdeq.2001.4157.  Google Scholar [17] O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type, vol. 23, American Mathematical Soc., 1968.  Google Scholar [18] H. Li and S. Guo, Dynamics of a SIRC epidemiological model, Electron. J. Differential Equations, 2017 (2007), Paper No. 121, 18 pp.  Google Scholar [19] Z. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.  doi: 10.1088/0951-7715/20/8/004.  Google Scholar [20] J. L. Lockwood, M. F. Hoopes and M. P. Marchetti, Invasion Ecology, John Wiley & Sons, 2013. Google Scholar [21] Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, Journal of Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010.  Google Scholar [22] D. Mollison, Epidemic models: Their structure and relation to data, vol. 5, Biometrics Cambridge, 52 (1996), P778. doi: 10.2307/2532920.  Google Scholar [23] J. D. Murray, Mathematical Biology. Ⅱ, Spatial Models and Biomedical Applications, Interdisciplinary Applied Mathematics, Vol. 18, Springer-Verlag, New York, 2003.  Google Scholar [24] W.-M. Ni, The Mathematics of Diffusion, SIAM, 2011. doi: 10.1137/1.9781611971972.  Google Scholar [25] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in Mathematics for Life Science and Medicine, Springer, 2007, 97–122.  Google Scholar [26] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, UK, 1997. Google Scholar [27] M. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient, Journal of Differential Equations, 258 (2015), 1252-1266.  doi: 10.1016/j.jde.2014.10.022.  Google Scholar [28] M. Wang and J. Zhao, Free boundary problems for a Lotka-Volterra competition system, Journal of Dynamics and Differential Equations, 26 (2014), 655-672.  doi: 10.1007/s10884-014-9363-4.  Google Scholar [29] Y. Wu and X. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, Journal of Differential Equations, 261 (2016), 4424-4447.  doi: 10.1016/j.jde.2016.06.028.  Google Scholar [30] S. Yan and S. Guo, Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity, Discrete & Continuous Dynamical Systems-B, 23 (2018), 1559-1579.  doi: 10.3934/dcdsb.2018059.  Google Scholar [31] X. Zhong, S. Guo and M. Peng, Stability of stochastic SIRS epidemic models with saturated incidence rates and delay, Stochastic Analysis and Applications, 35 (2017), 1-26.  doi: 10.1080/07362994.2016.1244644.  Google Scholar

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##### References:
 [1] L. J. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete and Continuous Dynamical Systems, 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.  Google Scholar [2] R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics, Journal of Mathematical Biology, 29 (1991), 315-338.  doi: 10.1007/BF00167155.  Google Scholar [3] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar [4] J.-F. Cao, W.-T. Li and F.-Y. Yang, Dynamics of a nonlocal SIS epidemic model with free boundary, Discrete & Continuous Dynamical Systems-Series B, 22 (2017), 247-266.  doi: 10.3934/dcdsb.2017013.  Google Scholar [5] X. Chen and A. Friedman, A free boundary problem arising in a model of wound healing, SIAM Journal on Mathematical Analysis, 32 (2000), 778-800.  doi: 10.1137/S0036141099351693.  Google Scholar [6] Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations, vol. 2, World Scientific, 2006. doi: 10.1142/9789812774446.  Google Scholar [7] Y. Du and Z. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, Ⅱ, Journal of Differential Equations, 250 (2011), 4336-4366.  doi: 10.1016/j.jde.2011.02.011.  Google Scholar [8] Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM Journal on Mathematical Analysis, 42 (2010), 377-405.  doi: 10.1137/090771089.  Google Scholar [9] Y. Du and L. Liu, Remarks on the uniqueness problem for the logistic equation on the entire space, Bulletin of the Australian Mathematical Society, 73 (2006), 129-137.  doi: 10.1017/S0004972700038685.  Google Scholar [10] Y. Du and L. Ma, Logistic type equations on $\mathbb{R}^n$ by a squeezing method involving boundary blow-up solutions, Journal of the London Mathematical Society, 64 (2001), 107-124.  doi: 10.1017/S0024610701002289.  Google Scholar [11] W.-E. Fitzgibbon and M. Langlais, Simple models for the transmission of microparasites between host populations living on noncoincident spatial domains, in Structured Population Models in Biology and Epidemiology, Springer, 1936 (2008), 115–164. doi: 10.1007/978-3-540-78273-5_3.  Google Scholar [12] J. Ge, K. I. Kim, Z. Lin and H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, Journal of Differential Equations, 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035.  Google Scholar [13] S. Guo, Spatio-temporal patterns in a diffusive model with non-local delay effect, IMA Journal of Applied Mathematics, 82 (2017), 864-908.  doi: 10.1093/imamat/hxx018.  Google Scholar [14] S. Guo and S. Yan, Hopf bifurcation in a diffusive Lotka-Volterra type system with nonlocal delay effect, Journal of Differential Equations, 260 (2016), 781-817.  doi: 10.1016/j.jde.2015.09.031.  Google Scholar [15] X. He and W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system Ⅰ: Heterogeneity vs. homogeneity, Journal of Differential Equations, 254 (2013), 528-546.  doi: 10.1016/j.jde.2012.08.032.  Google Scholar [16] V. Hutson, Y. Lou and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, Journal of Differential Equations, 185 (2002), 97-136.  doi: 10.1006/jdeq.2001.4157.  Google Scholar [17] O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type, vol. 23, American Mathematical Soc., 1968.  Google Scholar [18] H. Li and S. Guo, Dynamics of a SIRC epidemiological model, Electron. J. Differential Equations, 2017 (2007), Paper No. 121, 18 pp.  Google Scholar [19] Z. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.  doi: 10.1088/0951-7715/20/8/004.  Google Scholar [20] J. L. Lockwood, M. F. Hoopes and M. P. Marchetti, Invasion Ecology, John Wiley & Sons, 2013. Google Scholar [21] Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, Journal of Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010.  Google Scholar [22] D. Mollison, Epidemic models: Their structure and relation to data, vol. 5, Biometrics Cambridge, 52 (1996), P778. doi: 10.2307/2532920.  Google Scholar [23] J. D. Murray, Mathematical Biology. Ⅱ, Spatial Models and Biomedical Applications, Interdisciplinary Applied Mathematics, Vol. 18, Springer-Verlag, New York, 2003.  Google Scholar [24] W.-M. Ni, The Mathematics of Diffusion, SIAM, 2011. doi: 10.1137/1.9781611971972.  Google Scholar [25] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in Mathematics for Life Science and Medicine, Springer, 2007, 97–122.  Google Scholar [26] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, UK, 1997. Google Scholar [27] M. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient, Journal of Differential Equations, 258 (2015), 1252-1266.  doi: 10.1016/j.jde.2014.10.022.  Google Scholar [28] M. Wang and J. Zhao, Free boundary problems for a Lotka-Volterra competition system, Journal of Dynamics and Differential Equations, 26 (2014), 655-672.  doi: 10.1007/s10884-014-9363-4.  Google Scholar [29] Y. Wu and X. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, Journal of Differential Equations, 261 (2016), 4424-4447.  doi: 10.1016/j.jde.2016.06.028.  Google Scholar [30] S. Yan and S. Guo, Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity, Discrete & Continuous Dynamical Systems-B, 23 (2018), 1559-1579.  doi: 10.3934/dcdsb.2018059.  Google Scholar [31] X. Zhong, S. Guo and M. Peng, Stability of stochastic SIRS epidemic models with saturated incidence rates and delay, Stochastic Analysis and Applications, 35 (2017), 1-26.  doi: 10.1080/07362994.2016.1244644.  Google Scholar
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