# American Institute of Mathematical Sciences

September  2015, 20(7): 2039-2050. doi: 10.3934/dcdsb.2015.20.2039

## The reaction-diffusion system for an SIR epidemic model with a free boundary

 1 Department of Mathematics, Harbin Institute of Technology, Harbin 150080, China 2 Natural Science Research Center, Harbin Institute of Technology, Harbin 150080

Received  September 2014 Revised  March 2015 Published  July 2015

The reaction-diffusion system for an $SIR$ epidemic model with a free boundary is studied. This model describes a transmission of diseases. The existence, uniqueness and estimates of the global solution are discussed first. Then some sufficient conditions for the disease vanishing are given. With the help of investigating the long time behavior of solution to the initial and boundary value problem in half space, the long time behavior of the susceptible population $S$ is obtained for the disease vanishing case.
Citation: Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039
##### References:
 [1] N. F. Britton, Essential Mathematical Biology,, Springer Undergraduate Mathematics Series, (2003). doi: 10.1007/978-1-4471-0049-2. Google Scholar [2] V. Capasso, Mathematical Structures of Epidemic Systems,, Lecture Notes in Biomath., (1993). doi: 10.1007/978-3-540-70514-7. Google Scholar [3] J. Crank, Free and Moving Boundary Problems,, Oxford Science Publications, (1984). Google Scholar [4] Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary,, SIAM J. Math. Anal., 42 (2010), 377. doi: 10.1137/090771089. Google Scholar [5] Y. H. Du and Z. G. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competition,, Discrete Cont. Dyn. Syst. Ser. B, 19 (2014), 3105. doi: 10.3934/dcdsb.2014.19.3105. Google Scholar [6] J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system,, J. Dyn. Diff. Equa., 24 (2012), 873. doi: 10.1007/s10884-012-9267-0. Google Scholar [7] Y. Kaneko, Spreading and vanishing behaviors for radially symmetric solutions of free boundary problems for reaction-diffusion equations,, Nonlinear Anal.: Real World Appl., 18 (2014), 121. doi: 10.1016/j.nonrwa.2014.01.008. Google Scholar [8] Y. Kaneko and Y. Yamada, A free boundary problem for a reaction diffusion equation appearing in ecology,, Advan. Math. Sci. Appl., 21 (2011), 467. Google Scholar [9] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics,, Proceedings of the Royal Society of London Series, 115 (1972), 700. Google Scholar [10] K. I. Kim, Z. G. Lin and Q. Y. Zhang, An SIR epidemic model with free boundary,, Nonlinear Anal.: Real World Appl., 14 (2013), 1992. doi: 10.1016/j.nonrwa.2013.02.003. Google Scholar [11] Z. G. Lin, Y. N. Zhao and P. Zhou, The infected frontier in an SEIR epidemic model with infinite delay,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2355. doi: 10.3934/dcdsb.2013.18.2355. Google Scholar [12] J. D. Murray, Mathematical Biology. I. An Introduction,, $3^{rd}$ edition, (2002). Google Scholar [13] L. I. Rubenstein, The Stefan Problem,, Translations of Mathematical Monographs, (1971). Google Scholar [14] M. X. Wang, On some free boundary problems of the prey-predator model,, J. Differential Equations, 256 (2014), 3365. doi: 10.1016/j.jde.2014.02.013. Google Scholar [15] M. X. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient,, J. Differential Equations, 258 (2015), 1252. doi: 10.1016/j.jde.2014.10.022. Google Scholar [16] M. X. Wang, Spreading and vanishing in the diffusive prey-predator model with a free boundary,, Commun. Nonlinear Sci. Numer. Simulat., 23 (2015), 311. doi: 10.1016/j.cnsns.2014.11.016. Google Scholar [17] M. X. Wang and Y. Zhang, Two kinds of free boundary problems for the diffusive prey-predator model,, Nonlinear Anal.: Real World Appl., 24 (2015), 73. doi: 10.1016/j.nonrwa.2015.01.004. Google Scholar [18] M. X. Wang and J. F. Zhao, Free boundary problems for a Lotka-Volterra competition system,, J. Dyn. Diff. Equat., 26 (2014), 655. doi: 10.1007/s10884-014-9363-4. Google Scholar [19] M. X. Wang and J. F. Zhao, A free boundary problem for a predator-prey model with double free boundaries, preprint,, , (). Google Scholar [20] J. F. Zhao and M. X. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment,, Nonlinear Anal.: Real World Appl., 16 (2014), 250. doi: 10.1016/j.nonrwa.2013.10.003. Google Scholar

show all references

##### References:
 [1] N. F. Britton, Essential Mathematical Biology,, Springer Undergraduate Mathematics Series, (2003). doi: 10.1007/978-1-4471-0049-2. Google Scholar [2] V. Capasso, Mathematical Structures of Epidemic Systems,, Lecture Notes in Biomath., (1993). doi: 10.1007/978-3-540-70514-7. Google Scholar [3] J. Crank, Free and Moving Boundary Problems,, Oxford Science Publications, (1984). Google Scholar [4] Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary,, SIAM J. Math. Anal., 42 (2010), 377. doi: 10.1137/090771089. Google Scholar [5] Y. H. Du and Z. G. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competition,, Discrete Cont. Dyn. Syst. Ser. B, 19 (2014), 3105. doi: 10.3934/dcdsb.2014.19.3105. Google Scholar [6] J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system,, J. Dyn. Diff. Equa., 24 (2012), 873. doi: 10.1007/s10884-012-9267-0. Google Scholar [7] Y. Kaneko, Spreading and vanishing behaviors for radially symmetric solutions of free boundary problems for reaction-diffusion equations,, Nonlinear Anal.: Real World Appl., 18 (2014), 121. doi: 10.1016/j.nonrwa.2014.01.008. Google Scholar [8] Y. Kaneko and Y. Yamada, A free boundary problem for a reaction diffusion equation appearing in ecology,, Advan. Math. Sci. Appl., 21 (2011), 467. Google Scholar [9] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics,, Proceedings of the Royal Society of London Series, 115 (1972), 700. Google Scholar [10] K. I. Kim, Z. G. Lin and Q. Y. Zhang, An SIR epidemic model with free boundary,, Nonlinear Anal.: Real World Appl., 14 (2013), 1992. doi: 10.1016/j.nonrwa.2013.02.003. Google Scholar [11] Z. G. Lin, Y. N. Zhao and P. Zhou, The infected frontier in an SEIR epidemic model with infinite delay,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2355. doi: 10.3934/dcdsb.2013.18.2355. Google Scholar [12] J. D. Murray, Mathematical Biology. I. An Introduction,, $3^{rd}$ edition, (2002). Google Scholar [13] L. I. Rubenstein, The Stefan Problem,, Translations of Mathematical Monographs, (1971). Google Scholar [14] M. X. Wang, On some free boundary problems of the prey-predator model,, J. Differential Equations, 256 (2014), 3365. doi: 10.1016/j.jde.2014.02.013. Google Scholar [15] M. X. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient,, J. Differential Equations, 258 (2015), 1252. doi: 10.1016/j.jde.2014.10.022. Google Scholar [16] M. X. Wang, Spreading and vanishing in the diffusive prey-predator model with a free boundary,, Commun. Nonlinear Sci. Numer. Simulat., 23 (2015), 311. doi: 10.1016/j.cnsns.2014.11.016. Google Scholar [17] M. X. Wang and Y. Zhang, Two kinds of free boundary problems for the diffusive prey-predator model,, Nonlinear Anal.: Real World Appl., 24 (2015), 73. doi: 10.1016/j.nonrwa.2015.01.004. Google Scholar [18] M. X. Wang and J. F. Zhao, Free boundary problems for a Lotka-Volterra competition system,, J. Dyn. Diff. Equat., 26 (2014), 655. doi: 10.1007/s10884-014-9363-4. Google Scholar [19] M. X. Wang and J. F. Zhao, A free boundary problem for a predator-prey model with double free boundaries, preprint,, , (). Google Scholar [20] J. F. Zhao and M. X. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment,, Nonlinear Anal.: Real World Appl., 16 (2014), 250. doi: 10.1016/j.nonrwa.2013.10.003. Google Scholar
 [1] Keng Deng. Asymptotic behavior of an SIR reaction-diffusion model with a linear source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5945-5957. doi: 10.3934/dcdsb.2019114 [2] Lu Yang, Meihua Yang. Long-time behavior of stochastic reaction-diffusion equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2627-2650. doi: 10.3934/dcdsb.2017102 [3] Jia-Feng Cao, Wan-Tong Li, Meng Zhao. On a free boundary problem for a nonlocal reaction-diffusion model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4117-4139. doi: 10.3934/dcdsb.2018128 [4] 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 [5] Shin-Ichiro Ei, Toshio Ishimoto. Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems. Networks & Heterogeneous Media, 2013, 8 (1) : 191-209. doi: 10.3934/nhm.2013.8.191 [6] Nguyen Huu Du, Nguyen Thanh Dieu. Long-time behavior of an SIR model with perturbed disease transmission coefficient. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3429-3440. doi: 10.3934/dcdsb.2016105 [7] E. C.M. Crooks, E. N. Dancer, Danielle Hilhorst. Fast reaction limit and long time behavior for a competition-diffusion system with Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 39-44. doi: 10.3934/dcdsb.2007.8.39 [8] Ana Carpio, Gema Duro. Explosive behavior in spatially discrete reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 693-711. doi: 10.3934/dcdsb.2009.12.693 [9] Keng Deng, Yixiang Wu. Asymptotic behavior for a reaction-diffusion population model with delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 385-395. doi: 10.3934/dcdsb.2015.20.385 [10] Yuncheng You. Asymptotic dynamics of reversible cubic autocatalytic reaction-diffusion systems. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1415-1445. doi: 10.3934/cpaa.2011.10.1415 [11] Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189 [12] Wei Feng, Xin Lu. Global periodicity in a class of reaction-diffusion systems with time delays. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 69-78. doi: 10.3934/dcdsb.2003.3.69 [13] Wenzhang Huang, Maoan Han, Kaiyu Liu. Dynamics of an SIS reaction-diffusion epidemic model for disease transmission. Mathematical Biosciences & Engineering, 2010, 7 (1) : 51-66. doi: 10.3934/mbe.2010.7.51 [14] Liang Zhang, Zhi-Cheng Wang. Threshold dynamics of a reaction-diffusion epidemic model with stage structure. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3797-3820. doi: 10.3934/dcdsb.2017191 [15] Hongyan Zhang, Siyu Liu, Yue Zhang. Dynamics and spatiotemporal pattern formations of a homogeneous reaction-diffusion Thomas model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1149-1164. doi: 10.3934/dcdss.2017062 [16] Xiaoyan Zhang, Yuxiang Zhang. Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2625-2640. doi: 10.3934/dcdsb.2018124 [17] Xin Li, Xingfu Zou. On a reaction-diffusion model for sterile insect release method with release on the boundary. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2509-2522. doi: 10.3934/dcdsb.2012.17.2509 [18] Irena Lasiecka, To Fu Ma, Rodrigo Nunes Monteiro. Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1037-1072. doi: 10.3934/dcdsb.2018141 [19] Dingshi Li, Kening Lu, Bixiang Wang, Xiaohu Wang. Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 187-208. doi: 10.3934/dcds.2018009 [20] Peter V. Gordon, Cyrill B. Muratov. Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks & Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767

2018 Impact Factor: 1.008