May  2017, 16(3): 781-798. doi: 10.3934/cpaa.2017037

Dynamics of a nonlocal dispersal SIS epidemic model

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People's Republic of China

wtli@lzu.edu.cn (W.-T. Li)(Corresponding author)

Received  March 2016 Revised  January 2017 Published  February 2017

This paper is concerned with a nonlocal dispersal susceptible-infected-susceptible (SIS) epidemic model with Dirichlet boundary condition, where the rates of disease transmission and recovery are assumed to be spatially heterogeneous. We introduce a basic reproduction number $R_0$ and establish threshold-type results on the global dynamic in terms of R0. More specifically, we show that if the basic reproduction number is less than one, then the disease will be extinct, and if the basic reproduction number is larger than one, then the disease will persist. Particularly, our results imply that the nonlocal dispersal of the infected individuals may suppress the spread of the disease even though in a high-risk domain.

Citation: Fei-Ying Yang, Wan-Tong Li. Dynamics of a nonlocal dispersal SIS epidemic model. Communications on Pure & Applied Analysis, 2017, 16 (3) : 781-798. doi: 10.3934/cpaa.2017037
References:
[1]

L. J. S. AllenB. M. BolkerY. 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

[2]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, AMS, Providence, Rhode Island, 2010. doi: 10.1090/surv/165. Google Scholar

[3]

P. Bates and G. Zhao, Existence, uniquenss, and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428-440. doi: 10.1016/j.jmaa.2006.09.007. Google Scholar

[4]

H. BerestyckiF. Hamel and L. Roques, Analysis of the periodically fragmented environment model. Ⅰ. Species persistence, J. Math. Biol., 51 (2005), 75-113. doi: 10.1007/s00285-004-0313-3. Google Scholar

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J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953. doi: 10.1016/j.jde.2010.07.003. Google Scholar

[6]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. Google Scholar

[7]

J. Garcĺa-Melián and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations, 246 (2009), 21-38. doi: 10.1016/j.jde.2008.04.015. Google Scholar

[8]

J. Garcĺa-Melián and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion, Commun. Pure Appl. Anal., 8 (2009), 2037-2053. doi: 10.3934/cpaa.2009.8.2037. Google Scholar

[9]

V. HutsonS. MartinezK. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517. doi: 10.1007/s00285-003-0210-1. Google Scholar

[10]

W. HuangM. 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

[11]

C. Y. KaoY. Lou and W. Shen, Random dispersal vs non-local dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596. doi: 10.3934/dcds.2010.26.551. Google Scholar

[12]

C. Y. KaoY. Lou and W. Shen, Evolution of mixed dispersal in periodic environments, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047-2072. doi: 10.3934/dcdsb.2012.17.2047. Google Scholar

[13]

W. T. LiY. J. Sun and Z. C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313. doi: 10.1016/j.nonrwa.2009.07.005. Google Scholar

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J. D. Murray, Mathematical Biology, Ⅱ, Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.Google Scholar

[15]

S. PanW. T. Li and G. Lin, Travelling wave fronts in nonlocal reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392. doi: 10.1007/s00033-007-7005-y. Google Scholar

[16]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag New York Berlin Heidelberg Tokyo, 1983.Google Scholar

[17]

R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Anal., 71 (2009), 239-247. doi: 10.1016/j.na.2008.10.043. Google Scholar

[18]

R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model.Ⅰ, J. Differential Equations, 247 (2009), 1096-1119. doi: 10.1016/j.jde.2009.05.002. Google Scholar

[19]

R. Peng and X. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471. doi: 10.1088/0951-7715/25/5/1451. Google Scholar

[20]

R. Peng and F. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reactiondiffusion model: effects of epidemic risk and population movement, Phys. D, 259 (2013), 8-25. doi: 10.1016/j.physd.2013.05.006. Google Scholar

[21]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 249 (2010), 747-795. doi: 10.1016/j.jde.2010.04.012. Google Scholar

[22]

W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. Amer. Math. Soc., 140 (2012), 1681-1696. doi: 10.1090/S0002-9939-2011-11011-6. Google Scholar

[23]

N. Rawal and W. Shen, Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications, J. Dynam. Diff. Eqns., 24 (2012), 927-954. doi: 10.1007/s10884-012-9276-z. Google Scholar

[24]

J. W. SunF. Y. Yang and W. T. Li, A nonlocal dispersal equation arising from a selectionmigration model in genetics, J. Differential Equations, 257 (2014), 1372-1402. doi: 10.1016/j.jde.2014.05.005. Google Scholar

[25]

J. W. SunW. T. Li and Z. C. Wang, A nonlocal dispersal logistic model with spatial degeneracy, Discrete Contin. Dyn. Syst., 35 (2015), 3217-3238. doi: 10.3934/dcds.2015.35.3217. Google Scholar

[26]

Y. J. SunW. T. Li and Z. C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differential Equations, 251 (2011), 551-581. doi: 10.1016/j.jde.2011.04.020. Google Scholar

[27]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211. doi: 10.1137/080732870. Google Scholar

[28]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, John A. Jacquez memorial volume. Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[29]

X. Wang, Metastability and stability of patterns in a convolution model for phase transitions, J. Differential Equations, 183 (2002), 434-461. doi: 10.1006/jdeq.2001.4129. Google Scholar

[30]

W. Wang and X. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168. doi: 10.1137/090775890. Google Scholar

[31]

W. Wang and X. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673. doi: 10.1137/120872942. Google Scholar

[32]

F. Y. YangY. LiW. T. Li and Z. C. Wang, Traveling waves in a nonlocal dispersal KermackMcKendrick epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1969-1993. doi: 10.3934/dcdsb.2013.18.1969. Google Scholar

[33]

F. Y. YangW. T. Li and Z. C. Wang, Traveling waves in a nonlocal dispersal SIR epidemic model, Nonlinear Anal. Real World Appl., 23 (2015), 129-147. doi: 10.1016/j.nonrwa.2014.12.001. Google Scholar

[34]

L. Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neural networks, J. Differential Equations, 197 (2004), 162-196. doi: 10.1016/S0022-0396(03)00170-0. Google Scholar

[35]

G. B. ZhangW. T. Li and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure, Math. Comput. Modelling, 49 (2009), 1021-1029. doi: 10.1016/j.mcm.2008.09.007. Google Scholar

show all references

References:
[1]

L. J. S. AllenB. M. BolkerY. 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

[2]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, AMS, Providence, Rhode Island, 2010. doi: 10.1090/surv/165. Google Scholar

[3]

P. Bates and G. Zhao, Existence, uniquenss, and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428-440. doi: 10.1016/j.jmaa.2006.09.007. Google Scholar

[4]

H. BerestyckiF. Hamel and L. Roques, Analysis of the periodically fragmented environment model. Ⅰ. Species persistence, J. Math. Biol., 51 (2005), 75-113. doi: 10.1007/s00285-004-0313-3. Google Scholar

[5]

J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953. doi: 10.1016/j.jde.2010.07.003. Google Scholar

[6]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. Google Scholar

[7]

J. Garcĺa-Melián and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations, 246 (2009), 21-38. doi: 10.1016/j.jde.2008.04.015. Google Scholar

[8]

J. Garcĺa-Melián and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion, Commun. Pure Appl. Anal., 8 (2009), 2037-2053. doi: 10.3934/cpaa.2009.8.2037. Google Scholar

[9]

V. HutsonS. MartinezK. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517. doi: 10.1007/s00285-003-0210-1. Google Scholar

[10]

W. HuangM. 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

[11]

C. Y. KaoY. Lou and W. Shen, Random dispersal vs non-local dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596. doi: 10.3934/dcds.2010.26.551. Google Scholar

[12]

C. Y. KaoY. Lou and W. Shen, Evolution of mixed dispersal in periodic environments, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047-2072. doi: 10.3934/dcdsb.2012.17.2047. Google Scholar

[13]

W. T. LiY. J. Sun and Z. C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313. doi: 10.1016/j.nonrwa.2009.07.005. Google Scholar

[14]

J. D. Murray, Mathematical Biology, Ⅱ, Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.Google Scholar

[15]

S. PanW. T. Li and G. Lin, Travelling wave fronts in nonlocal reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392. doi: 10.1007/s00033-007-7005-y. Google Scholar

[16]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag New York Berlin Heidelberg Tokyo, 1983.Google Scholar

[17]

R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Anal., 71 (2009), 239-247. doi: 10.1016/j.na.2008.10.043. Google Scholar

[18]

R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model.Ⅰ, J. Differential Equations, 247 (2009), 1096-1119. doi: 10.1016/j.jde.2009.05.002. Google Scholar

[19]

R. Peng and X. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471. doi: 10.1088/0951-7715/25/5/1451. Google Scholar

[20]

R. Peng and F. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reactiondiffusion model: effects of epidemic risk and population movement, Phys. D, 259 (2013), 8-25. doi: 10.1016/j.physd.2013.05.006. Google Scholar

[21]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 249 (2010), 747-795. doi: 10.1016/j.jde.2010.04.012. Google Scholar

[22]

W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. Amer. Math. Soc., 140 (2012), 1681-1696. doi: 10.1090/S0002-9939-2011-11011-6. Google Scholar

[23]

N. Rawal and W. Shen, Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications, J. Dynam. Diff. Eqns., 24 (2012), 927-954. doi: 10.1007/s10884-012-9276-z. Google Scholar

[24]

J. W. SunF. Y. Yang and W. T. Li, A nonlocal dispersal equation arising from a selectionmigration model in genetics, J. Differential Equations, 257 (2014), 1372-1402. doi: 10.1016/j.jde.2014.05.005. Google Scholar

[25]

J. W. SunW. T. Li and Z. C. Wang, A nonlocal dispersal logistic model with spatial degeneracy, Discrete Contin. Dyn. Syst., 35 (2015), 3217-3238. doi: 10.3934/dcds.2015.35.3217. Google Scholar

[26]

Y. J. SunW. T. Li and Z. C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differential Equations, 251 (2011), 551-581. doi: 10.1016/j.jde.2011.04.020. Google Scholar

[27]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211. doi: 10.1137/080732870. Google Scholar

[28]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, John A. Jacquez memorial volume. Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[29]

X. Wang, Metastability and stability of patterns in a convolution model for phase transitions, J. Differential Equations, 183 (2002), 434-461. doi: 10.1006/jdeq.2001.4129. Google Scholar

[30]

W. Wang and X. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168. doi: 10.1137/090775890. Google Scholar

[31]

W. Wang and X. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673. doi: 10.1137/120872942. Google Scholar

[32]

F. Y. YangY. LiW. T. Li and Z. C. Wang, Traveling waves in a nonlocal dispersal KermackMcKendrick epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1969-1993. doi: 10.3934/dcdsb.2013.18.1969. Google Scholar

[33]

F. Y. YangW. T. Li and Z. C. Wang, Traveling waves in a nonlocal dispersal SIR epidemic model, Nonlinear Anal. Real World Appl., 23 (2015), 129-147. doi: 10.1016/j.nonrwa.2014.12.001. Google Scholar

[34]

L. Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neural networks, J. Differential Equations, 197 (2004), 162-196. doi: 10.1016/S0022-0396(03)00170-0. Google Scholar

[35]

G. B. ZhangW. T. Li and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure, Math. Comput. Modelling, 49 (2009), 1021-1029. doi: 10.1016/j.mcm.2008.09.007. Google Scholar

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