×

Dynamics of a nonlocal dispersal SIS epidemic model with Neumann boundary conditions. (English) Zbl 1416.35285

Summary: In this paper we study a nonlocal dispersal susceptible-infected-susceptible (SIS) epidemic model with Neumann boundary condition, where the spatial movement of individuals is described by a nonlocal (convolution) diffusion operator, the transmission rate and recovery rate are spatially heterogeneous, and the total population number is constant. We first define the basic reproduction number \(R_0\) and discuss the existence, uniqueness and stability of steady states of the nonlocal dispersal SIS epidemic model in terms of \(R_0\). Then we consider the impacts of the large diffusion rates of the susceptible and infectious populations on the persistence and extinction of the disease. The obtained results indicate that the nonlocal movement of the susceptible or infectious individuals will enhance the persistence of the infectious disease. In particular, our analytical results suggest that the spatial heterogeneity tends to boost the spread of the infectious disease.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
35B40 Asymptotic behavior of solutions to PDEs
45A05 Linear integral equations
45F05 Systems of nonsingular linear integral equations
47G20 Integro-differential operators
92D30 Epidemiology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Allen, L. J.S.; Bolker, B. M.; Lou, Y.; Nevai, A. L., Asymptotic profiles of the steady states for an SIS epidemic patch model, SIAM J. Appl. Math., 67, 1283-1309 (2007) · Zbl 1121.92054
[2] Allen, L. J.S.; Bolker, B. M.; Lou, Y.; Nevai, A. L., Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst., 21, 1-20 (2008) · Zbl 1146.92028
[3] Allen, L. J.S.; Lou, Y.; Nevai, A. L., Spatial patterns in a discrete-time SIS patch model, J. Math. Biol., 58, 339-375 (2009) · Zbl 1162.92033
[4] Andreu-Vaillo, F.; Mazón, J. M.; Rossi, J. D.; Toledo-Melero, J. J., Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, vol. 165 (2010), Amer. Math. Soc.: Amer. Math. Soc. Providence, Rhode Island · Zbl 1214.45002
[5] Aronson, D. G., The asymptotic speed of propagation of a simple epidemic, (Fitzgibbon, W. E.; Walker, H. F., Nonlinear Diffusion. Nonlinear Diffusion, Res. Notes Math., vol. 14 (1977), Pitman: Pitman London), 1-23 · Zbl 0361.35011
[6] Berestycki, H.; Hamel, F.; Roques, L., Analysis of the periodically fragmented environment model. I. Species persistence, J. Math. Biol., 51, 75-113 (2005) · Zbl 1066.92047
[7] Bates, P., On some nonlocal evolution equations arising in materials science, (Brunner, H.; Zhao, X. Q.; Zou, X., Nonlinear Dynamics and Evolution Equations. Nonlinear Dynamics and Evolution Equations, Fields Inst. Commun., vol. 48 (2006), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 13-52 · Zbl 1101.35073
[8] Bates, P.; Fife, P. C.; Ren, X.; Wang, X., Traveling waves in a convolution model for phase transition, Arch. Ration. Mech. Anal., 138, 105-136 (1997) · Zbl 0889.45012
[9] Bates, P.; Zhao, G., Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332, 428-440 (2007) · Zbl 1114.35017
[10] Chasseigne, E.; Chaves, M.; Rossi, J. D., Asymptotic behavior for nonlocal diffusion equation, J. Math. Pures Appl., 86, 271-291 (2006) · Zbl 1126.35081
[11] Cortázar, C.; Coville, J.; Elgueta, M.; Martínez, S., A nonlocal inhomogeneous dispersal process, J. Differential Equations, 241, 332-358 (2007) · Zbl 1127.45003
[12] Coville, J., On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249, 2921-2953 (2010) · Zbl 1218.45002
[13] Coville, J.; Dávila, J.; Martínez, S., Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39, 1693-1709 (2008) · Zbl 1161.45003
[14] Coville, J.; Dávila, J.; Martínez, S., Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30, 179-223 (2013) · Zbl 1288.45007
[15] Cui, R.; Lam, K. Y.; Lou, Y., Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J. Differential Equations, 263, 2343-2373 (2017) · Zbl 1388.35086
[16] Cui, R.; Lou, Y., A spatial SIS model in advective heterogeneous environments, J. Differential Equations, 261, 3305-3343 (2016) · Zbl 1342.92231
[17] Diekmann, O.; Heesterbeek, J. A.P.; Metz, J. A.J., On the definition and the computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28, 365-382 (1990) · Zbl 0726.92018
[18] Fife, P., Some nonclassical trends in parabolic and parabolic-like evolutions, (Kirkilionis, M.; Krömker, S.; Rannacher, R.; Tomi, F., Trends in Nonlinear Analysis (2003), Springer: Springer Berlin), 153-191 · Zbl 1072.35005
[19] García-Melián, J.; Rossi, J. D., On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations, 246, 21-38 (2009) · Zbl 1162.35055
[20] García-Melián, J.; Rossi, J. D., Maximum and antimaximum principles for some nonlocal diffusion operators, Nonlinear Anal., 71, 6116-6121 (2009) · Zbl 1185.45005
[21] García-Melián, J.; Rossi, J. D., A logistic equation with refuge and nonlocal diffusion, Commun. Pure Appl. Anal., 8, 2037-2053 (2009) · Zbl 1180.45002
[22] Heffernan, J. M.; Smith, R. J.; Wahl, L. M., Perspectives on the basic reproductive ratio, J. R. Soc. Interface, 2, 281-293 (2005)
[23] Huang, W.; Han, M.; Liu, K., Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Math. Biosci. Eng., 7, 51-66 (2010) · Zbl 1190.35120
[24] Hutson, V.; Martinez, S.; Mischaikow, K.; Vickers, G. T., The evolution of dispersal, J. Math. Biol., 47, 483-517 (2003) · Zbl 1052.92042
[25] Kao, C. Y.; Lou, Y.; Shen, W., Random dispersal vs non-local dispersal, Discrete Contin. Dyn. Syst., 26, 551-596 (2010) · Zbl 1187.35127
[26] Kendall, D. G., Discussion of ‘Measles periodicity and community size’ by M.S. Bartlett, J. R. Stat. Soc., A, 120, 64-76 (1977)
[27] Kendall, D. G., Mathematical models of the spread of infection, (Mathematics and Computer Science in Biology and Medicine (1965)), 213-225
[28] Li, H.; Peng, R.; Wang, F. B., Varying total population enhances disease persistence: qualitative analysis on a diffusive SIS epidemic model, J. Differential Equations, 262, 885-913 (2017) · Zbl 1355.35107
[29] Li, W. T.; Sun, Y. J.; Wang, Z. C., Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11, 2302-2313 (2010) · Zbl 1196.35015
[30] Li, W. T.; Zhang, Li; Zhang, G. B., Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35, 1531-1560 (2015) · Zbl 1319.35088
[31] Mollison, D., Possible velocities for a simple epidemic, Adv. in Appl. Probab., 4, 233-257 (1972) · Zbl 0251.92012
[32] Pan, S.; Li, W. T.; Lin, G., Travelling wave fronts in nonlocal reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60, 377-392 (2009) · Zbl 1178.35206
[33] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0516.47023
[34] Peng, R., Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model, I, J. Differential Equations, 247, 1096-1119 (2009) · Zbl 1165.92035
[35] Peng, R.; Liu, S., Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Anal., 71, 239-247 (2009) · Zbl 1162.92037
[36] Peng, R.; Zhao, X. Q., A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25, 1451-1471 (2012) · Zbl 1250.35172
[37] Peng, R.; Yi, F., Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: effects of epidemic risk and population movement, Phys. D, 259, 8-25 (2013) · Zbl 1321.92076
[38] Rass, L.; Radcliffe, J., Spatial Deterministic Epidemics, Math. Surveys Monogr., vol. 102 (2003), Amer. Math. Soc.: Amer. Math. Soc. Providence, PI · Zbl 1018.92028
[39] Ruan, S., Spatial-temporal dynamics in nonlocal epidemiological models, (Takeuchi, Y.; Sato, K.; Iwasa, Y., Mathematics for Life Science and Medicine (2007), Springer-Verlag: Springer-Verlag Berlin), 99-122
[40] Shen, W.; Zhang, A., Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 15, 747-795 (2010) · Zbl 1196.45002
[41] Shen, W.; Zhang, A., Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. Amer. Math. Soc., 140, 1681-1696 (2012) · Zbl 1243.45008
[42] Sun, J. W.; Li, W. T.; Yang, F. Y., Approximate the Fokker-Planck equation by a class of nonlocal dispersal problems, Nonlinear Anal., 74, 3501-3509 (2011) · Zbl 1219.35317
[43] Sun, J. W.; Yang, F. Y.; Li, W. T., A nonlocal dispersal equation arising from a selection-migration model in genetics, J. Differential Equations, 257, 1372-1402 (2014) · Zbl 1320.35060
[44] Sun, J. W.; Li, W. T.; Wang, Z. C., A nonlocal dispersal logistic model with spatial degeneracy, Discrete Contin. Dyn. Syst., 35, 3217-3238 (2015) · Zbl 1386.35220
[45] Sun, Y. J.; Li, W. T.; Wang, Z. C., Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differential Equations, 251, 551-581 (2011) · Zbl 1228.35017
[46] Thieme, H. R., Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70, 188-211 (2009) · Zbl 1191.47089
[47] van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, John A. Jacquez memorial volume, Math. Biosci., 180, 29-48 (2002) · Zbl 1015.92036
[48] Wang, W.; Zhao, X., A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71, 147-168 (2011) · Zbl 1228.35118
[49] Wang, W.; Zhao, X., Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11, 1652-1673 (2012) · Zbl 1259.35120
[50] Wang, X., Metastability and stability of patterns in a convolution model for phase transitions, J. Differential Equations, 183, 434-461 (2002) · Zbl 1011.35073
[51] Wu, Y.; Zou, X., Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J. Differential Equations, 261, 4424-4447 (2016) · Zbl 1346.35199
[52] Yang, F. Y.; Li, W. T., Dynamics of a nonlocal dispersal SIS epidemic model, Commun. Pure Appl. Anal., 16, 781-797 (2017) · Zbl 1364.35390
[53] Yang, F. Y.; Li, W. T.; Sun, J. W., Principal eigenvalues for some nonlocal eigenvalue problems and applications, Discrete Contin. Dyn. Syst., 36, 4027-4049 (2016) · Zbl 1386.35286
[54] Zhang, G. B.; Li, W. T.; Sun, Y. J., Asymptotic behavior for nonlocal dispersal equations, Nonlinear Anal., 72, 4466-4474 (2010) · Zbl 1191.35065
[55] Zhao, G.; Ruan, S., Spatial and temporal dynamics of a nonlocal viral infection model, SIAM J. Appl. Math., 78, 1954-1980 (2018) · Zbl 1410.35261
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.