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March  2017, 22(2): 247-266. doi: 10.3934/dcdsb.2017013

Dynamics of a nonlocal SIS epidemic model with free boundary

Jia-Feng Cao , Wan-Tong Li , and  Fei-Ying Yang

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

* Corresponding author

Received  February 2016 Revised  March 2016 Published  December 2016

This paper is concerned with the spreading or vanishing of a epidemic disease which is characterized by a diffusion SIS model with nonlocal incidence rate and double free boundaries. We get the full information about the sufficient conditions that ensure the disease spreading or vanishing, which exhibits a detailed description of the communicable mechanism of the disease. Our results imply that the nonlocal interaction may enhance the spread of the disease.

Keywords: SIS model, reaction-diffusion, free boundary, spreading-vanishing dichotomy, nonlocal incidence rate.
Mathematics Subject Classification: 35K57, 35R20, 92D2.
Citation: Jia-Feng Cao, Wan-Tong Li, Fei-Ying Yang. Dynamics of a nonlocal SIS epidemic model with free boundary. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 247-266. doi: 10.3934/dcdsb.2017013
References:
[1]

S. Ai and R. Albashaireh, Traveling waves in spatial SIRS models, J. Dynam. Differential Equations, 26 (2014), 143-164. doi: 10.1007/s10884-014-9348-3. Google Scholar

[2]

L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst., 21 (2008), 1-20. doi: 10.3934/dcds.2008.21.1. Google Scholar

[3]

D. Aronson, The asymptotic speed of propagation of a simple epidemic, Research Notes in Math., 14 (1977), 1-23, Pitman, London. Google Scholar

[4]

R. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations Wiley Series in Mathematical and Computational Biology, 2003.Google Scholar

[5]

J. Cao, W. T. Li, J. Wang and F. Yang, A free boundary problem of a diffusive SIRS model with nonlinear incidence, submitted, 2015.Google Scholar

[6]

X. Chen and A. Friedman, A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800. doi: 10.1137/S0036141099351693. Google Scholar

[7]

X. Chen and A. Friedman, A free boundary problem for an elliptic-hyperbolic system: an application to tumor growth, SIAM J. Math. Anal., 35 (2003), 974-986. doi: 10.1137/S0036141002418388. Google Scholar

[8]

Y. Du and Z. Guo, Spreading-Vanishing dichotomy in a diffusive logistic model with a free boundary Ⅱ, J. Differential Equations, 250 (2011), 4336-4366. doi: 10.1016/j.jde.2011.02.011. Google Scholar

[9]

Y. Du, Z. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142. doi: 10.1016/j.jfa.2013.07.016. Google Scholar

[10]

Y. Du and Z. Lin, Spreading-Vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. doi: 10.1137/090771089. Google Scholar

[11]

Y. Du and Z. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132. doi: 10.3934/dcdsb.2014.19.3105. Google Scholar

[12]

Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724. doi: 10.4171/JEMS/568. Google Scholar

[13]

Y. Du, H. Matsuzawa and M. Zhou, Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46 (2014), 375-396. doi: 10.1137/130908063. Google Scholar

[14]

Y. Du and X. Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 279-305. doi: 10.1016/j.anihpc.2013.11.004. Google Scholar

[15]

A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 459-482. doi: 10.1017/S0308210507000455. Google Scholar

[16]

T. Faria, W. Huang and J. Wu, Travelling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. Lond. Ser. A, 462 (2006), 229-261. doi: 10.1098/rspa.2005.1554. Google Scholar

[17]

J. Ge, K. Kim, Z. Lin and H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509. doi: 10.1016/j.jde.2015.06.035. Google Scholar

[18]

J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system, J. Dynam. Differential Equations, 24 (2012), 873-895. doi: 10.1007/s10884-012-9267-0. Google Scholar

[19]

J. S. Guo and C. H. Wu, Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 28 (2015), 1-27. doi: 10.1088/0951-7715/28/1/1. Google Scholar

[20]

W. Huang, M. Han and K. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Math. Biosci. Eng., 7 (2010), 51-66. doi: 10.3934/mbe.2010.7.51. Google Scholar

[21]

G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207. doi: 10.1007/s11538-009-9487-6. Google Scholar

[22]

H. Huang and M. Wang, The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2039-2050. doi: 10.3934/dcdsb.2015.20.2039. Google Scholar

[23]

D. Kendall, Some problems in theory of dams, J. Roy. Statist. Soc. Ser. B., 19 (1957), 207-212. Google Scholar

[24]

D. Kendall, Mathematical Models of the Spread of Infection Mathematics and Computer Science in Biology and Medicine. H. M. Stationary Off, London, 1965.Google Scholar

[25]

A. Inkyung and Z. Lin, The spreading fronts of an infective environment in a manenvironment-man epidemic model, arXiv: 1408.6326.Google Scholar

[26]

K. Kim, Z. Lin and Q. Zhang, An SIR epidemic model with free boundary, Nonlinear Anal. Real World Appl., 14 (2013), 1992-2001. doi: 10.1016/j.nonrwa.2013.02.003. Google Scholar

[27]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type Academic Press, New York, London, 1968.Google Scholar

[28]

C. Lei, Z. Lin and Q. Zhang, The spreading front of invasive species in favorable habitat or unfavorable habitat, J. Differential Equations, 257 (2014), 145-166. doi: 10.1016/j.jde.2014.03.015. Google Scholar

[29]

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

[30]

Z. Lin, Y. Zhao and P. Zhou, The infected frontier in an SEIR epidemic model with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2355-2376. doi: 10.3934/dcdsb.2013.18.2355. Google Scholar

[31]

Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, Tutor. Math. Biosci. Ⅳ: Evol. Ecol. , Springer, 1922 (2008), 171{205.Google Scholar

[32]

D. Mollison, Possible velocities for a simple epidemic, Adv. Appl. Probab., 4 (1972), 233-257. doi: 10.2307/1425997. Google Scholar

[33]

D. Mottoni, P. Orlandi and A. Tesei, Asymptotic behavior for a system describing epidemics with migration and spatial spread of infection, Nonlinear Anal., 3 (1979), 663-675. doi: 10.1016/0362-546X(79)90095-6. Google Scholar

[34]

R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst., 33 (2013), 2007-2031. doi: 10.3934/dcds.2013.33.2007. Google Scholar

[35]

L. I. Rubinstein, The Stefan Problem American Mathematical Society, Providence, RI, 1971.Google Scholar

[36]

J. Wang and L. Zhang, Invasion by an inferior or superior competitor: A diffusive competition model with a free boundary in a heterogeneous environment, J. Math. Anal. Appl., 423 (2015), 377-398. doi: 10.1016/j.jmaa.2014.09.055. Google Scholar

[37]

M. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394. doi: 10.1016/j.jde.2014.02.013. Google Scholar

[38]

M. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258 (2015), 1252-1266. doi: 10.1016/j.jde.2014.10.022. Google Scholar

[39]

M. Wang and J. Zhao, Free boundary problem for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26 (2014), 655-672. doi: 10.1007/s10884-014-9363-4. Google Scholar

[40]

M. Wang and J. Zhao, A free boundary problem for a predator-prey model with double free boundaries, J. Dynam. Differential Equations, (2015), 1-23. doi: 10.1007/s10884-015-9503-5. Google Scholar

[41]

M. Wang, A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment, J. Funct. Anal., 270 (2016), 483-508. doi: 10.1016/j.jfa.2015.10.014. Google Scholar

[42]

Z. C. Wang and J. Wu, Traveling waves of a diffusive Kermack-McKendrick epidemic model with nonlocal delayed transmission, Proc. R. Soc. Lond. Ser. A, 466 (2010), 237-261. doi: 10.1098/rspa.2009.0377. Google Scholar

[43]

C. H. Wu, Spreading speed and traveling waves for a two-species weak competition system with free boundary, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2441-2455. doi: 10.3934/dcdsb.2013.18.2441. Google Scholar

[44]

J. Yang and B. Lou, Traveling wave solutions of competitive models with free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 817-826. doi: 10.3934/dcdsb.2014.19.817. Google Scholar

[45]

J. Zhao and M. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Anal. Real World Appl., 16 (2014), 250-263. doi: 10.1016/j.nonrwa.2013.10.003. Google Scholar

[46]

P. Zhou and D. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014), 1927-1954. doi: 10.1016/j.jde.2013.12.008. Google Scholar

show all references

References:
[1]

S. Ai and R. Albashaireh, Traveling waves in spatial SIRS models, J. Dynam. Differential Equations, 26 (2014), 143-164. doi: 10.1007/s10884-014-9348-3. Google Scholar

[2]

L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst., 21 (2008), 1-20. doi: 10.3934/dcds.2008.21.1. Google Scholar

[3]

D. Aronson, The asymptotic speed of propagation of a simple epidemic, Research Notes in Math., 14 (1977), 1-23, Pitman, London. Google Scholar

[4]

R. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations Wiley Series in Mathematical and Computational Biology, 2003.Google Scholar

[5]

J. Cao, W. T. Li, J. Wang and F. Yang, A free boundary problem of a diffusive SIRS model with nonlinear incidence, submitted, 2015.Google Scholar

[6]

X. Chen and A. Friedman, A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800. doi: 10.1137/S0036141099351693. Google Scholar

[7]

X. Chen and A. Friedman, A free boundary problem for an elliptic-hyperbolic system: an application to tumor growth, SIAM J. Math. Anal., 35 (2003), 974-986. doi: 10.1137/S0036141002418388. Google Scholar

[8]

Y. Du and Z. Guo, Spreading-Vanishing dichotomy in a diffusive logistic model with a free boundary Ⅱ, J. Differential Equations, 250 (2011), 4336-4366. doi: 10.1016/j.jde.2011.02.011. Google Scholar

[9]

Y. Du, Z. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142. doi: 10.1016/j.jfa.2013.07.016. Google Scholar

[10]

Y. Du and Z. Lin, Spreading-Vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. doi: 10.1137/090771089. Google Scholar

[11]

Y. Du and Z. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132. doi: 10.3934/dcdsb.2014.19.3105. Google Scholar

[12]

Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724. doi: 10.4171/JEMS/568. Google Scholar

[13]

Y. Du, H. Matsuzawa and M. Zhou, Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46 (2014), 375-396. doi: 10.1137/130908063. Google Scholar

[14]

Y. Du and X. Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 279-305. doi: 10.1016/j.anihpc.2013.11.004. Google Scholar

[15]

A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 459-482. doi: 10.1017/S0308210507000455. Google Scholar

[16]

T. Faria, W. Huang and J. Wu, Travelling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. Lond. Ser. A, 462 (2006), 229-261. doi: 10.1098/rspa.2005.1554. Google Scholar

[17]

J. Ge, K. Kim, Z. Lin and H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509. doi: 10.1016/j.jde.2015.06.035. Google Scholar

[18]

J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system, J. Dynam. Differential Equations, 24 (2012), 873-895. doi: 10.1007/s10884-012-9267-0. Google Scholar

[19]

J. S. Guo and C. H. Wu, Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 28 (2015), 1-27. doi: 10.1088/0951-7715/28/1/1. Google Scholar

[20]

W. Huang, M. Han and K. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Math. Biosci. Eng., 7 (2010), 51-66. doi: 10.3934/mbe.2010.7.51. Google Scholar

[21]

G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207. doi: 10.1007/s11538-009-9487-6. Google Scholar

[22]

H. Huang and M. Wang, The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2039-2050. doi: 10.3934/dcdsb.2015.20.2039. Google Scholar

[23]

D. Kendall, Some problems in theory of dams, J. Roy. Statist. Soc. Ser. B., 19 (1957), 207-212. Google Scholar

[24]

D. Kendall, Mathematical Models of the Spread of Infection Mathematics and Computer Science in Biology and Medicine. H. M. Stationary Off, London, 1965.Google Scholar

[25]

A. Inkyung and Z. Lin, The spreading fronts of an infective environment in a manenvironment-man epidemic model, arXiv: 1408.6326.Google Scholar

[26]

K. Kim, Z. Lin and Q. Zhang, An SIR epidemic model with free boundary, Nonlinear Anal. Real World Appl., 14 (2013), 1992-2001. doi: 10.1016/j.nonrwa.2013.02.003. Google Scholar

[27]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type Academic Press, New York, London, 1968.Google Scholar

[28]

C. Lei, Z. Lin and Q. Zhang, The spreading front of invasive species in favorable habitat or unfavorable habitat, J. Differential Equations, 257 (2014), 145-166. doi: 10.1016/j.jde.2014.03.015. Google Scholar

[29]

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

[30]

Z. Lin, Y. Zhao and P. Zhou, The infected frontier in an SEIR epidemic model with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2355-2376. doi: 10.3934/dcdsb.2013.18.2355. Google Scholar

[31]

Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, Tutor. Math. Biosci. Ⅳ: Evol. Ecol. , Springer, 1922 (2008), 171{205.Google Scholar

[32]

D. Mollison, Possible velocities for a simple epidemic, Adv. Appl. Probab., 4 (1972), 233-257. doi: 10.2307/1425997. Google Scholar

[33]

D. Mottoni, P. Orlandi and A. Tesei, Asymptotic behavior for a system describing epidemics with migration and spatial spread of infection, Nonlinear Anal., 3 (1979), 663-675. doi: 10.1016/0362-546X(79)90095-6. Google Scholar

[34]

R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst., 33 (2013), 2007-2031. doi: 10.3934/dcds.2013.33.2007. Google Scholar

[35]

L. I. Rubinstein, The Stefan Problem American Mathematical Society, Providence, RI, 1971.Google Scholar

[36]

J. Wang and L. Zhang, Invasion by an inferior or superior competitor: A diffusive competition model with a free boundary in a heterogeneous environment, J. Math. Anal. Appl., 423 (2015), 377-398. doi: 10.1016/j.jmaa.2014.09.055. Google Scholar

[37]

M. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394. doi: 10.1016/j.jde.2014.02.013. Google Scholar

[38]

M. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258 (2015), 1252-1266. doi: 10.1016/j.jde.2014.10.022. Google Scholar

[39]

M. Wang and J. Zhao, Free boundary problem for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26 (2014), 655-672. doi: 10.1007/s10884-014-9363-4. Google Scholar

[40]

M. Wang and J. Zhao, A free boundary problem for a predator-prey model with double free boundaries, J. Dynam. Differential Equations, (2015), 1-23. doi: 10.1007/s10884-015-9503-5. Google Scholar

[41]

M. Wang, A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment, J. Funct. Anal., 270 (2016), 483-508. doi: 10.1016/j.jfa.2015.10.014. Google Scholar

[42]

Z. C. Wang and J. Wu, Traveling waves of a diffusive Kermack-McKendrick epidemic model with nonlocal delayed transmission, Proc. R. Soc. Lond. Ser. A, 466 (2010), 237-261. doi: 10.1098/rspa.2009.0377. Google Scholar

[43]

C. H. Wu, Spreading speed and traveling waves for a two-species weak competition system with free boundary, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2441-2455. doi: 10.3934/dcdsb.2013.18.2441. Google Scholar

[44]

J. Yang and B. Lou, Traveling wave solutions of competitive models with free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 817-826. doi: 10.3934/dcdsb.2014.19.817. Google Scholar

[45]

J. Zhao and M. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Anal. Real World Appl., 16 (2014), 250-263. doi: 10.1016/j.nonrwa.2013.10.003. Google Scholar

[46]

P. Zhou and D. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014), 1927-1954. doi: 10.1016/j.jde.2013.12.008. Google Scholar

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