Dynamics and asymptotic profiles of a nonlocal dispersal SIS epidemic model with bilinear incidence and Neumann boundary conditions. (English) Zbl 1495.35030
Summary: This paper is concerned with a nonlocal (convolution) dispersal susceptible-infected-susceptible (SIS) epidemic model with bilinear incidence and Neumann boundary conditions. First we establish the existence and uniqueness of stationary solutions by reducing the system to a single equation. Then we study the asymptotic profiles of the endemic steady states for large and small diffusion rates to illustrate the persistence or extinction of the infectious disease. The lack of regularity of the endemic steady state makes it more difficult to obtain the limit function of the sequence of endemic steady states. We also observe the concentration phenomenon which occurs when the diffusion rate of the infected individuals tends to zero. Our analytical results demonstrate that limiting the movement of susceptible individuals is not effective in eliminating the infectious disease unless the total population size is relatively small.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K57 | Reaction-diffusion equations |
| 35R09 | Integro-partial differential equations |
| 92D30 | Epidemiology |
Keywords:
nonlocal dispersal; susceptible-infected-susceptible (SIS) epidemic model; endemic steady state; asymptotic profilesReferences:
| [1] | 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 |
| [2] | Andreu-Vaillo, F.; Mazón, J. M.; Rossi, J. D.; Toledo-Melero, J., Nonlocal Diffusion Problems, Mathematical Surveys and Monographs (2010), AMS: AMS Providence, Rhode Island · Zbl 1214.45002 |
| [3] | Bailey, N. T.J., Spatial models in the epidemiology of infectious diseases, (Jäger, W.; Rost, H.; Tautu, P., Biological Growth and Spread. Biological Growth and Spread, Lecture Notes in Biomathematics, vol. 38 (1980), Springer: Springer Berlin), 233-261 · Zbl 0452.92022 |
| [4] | Bartlett, M. S., Deterministic and stochastic models for recurrent epidemics, (Neymann, J., Contributions to Biology and Problems of Health. Contributions to Biology and Problems of Health, Proc. 3rd Berkeley Symp. Math. Stat. and Prob., vol. 4 (1956), University of California Press: University of California Press Berkeley), 81-109 · Zbl 0070.15004 |
| [5] | 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 |
| [6] | Beardmore, I.; Beardmore, R., The global structure of a spatial model of infections disease, Proc. R. Soc. Lond. A, 459, 1427-1448 (2003) · Zbl 1064.35069 |
| [7] | Busenberg, S. N.; Travis, C. C., Epidemic models with spatial spread due to population migration, J. Math. Biol., 16, 181-198 (1983) · Zbl 0522.92020 |
| [8] | Capasso, V., Global solution for a diffusive nonlinear deterministic epidemic model, SIAM J. Appl. Math., 35, 274-284 (1978) · Zbl 0415.92018 |
| [9] | Castellano, K.; Salako, R. B., On the effect of lowering population’s movement to control the spread of an infectious disease, J. Differ. Equ., 316, 1-27 (2022) · Zbl 1486.92210 |
| [10] | Coville, J., On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differ. Equ., 249, 2921-2953 (2010) · Zbl 1218.45002 |
| [11] | Cui, R., Asymptotic profiles of the endemic steady state of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate, Discrete Contin. Dyn. Syst., Ser. B, 26, 2997-3022 (2021) · Zbl 1471.35275 |
| [12] | Cui, R.; Lam, K. Y.; Lou, Y., Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J. Differ. Equ., 263, 2343-2373 (2017) · Zbl 1388.35086 |
| [13] | Cui, R.; Li, H.; Peng, R.; Zhou, M., Concentration behavior of endemic steady state for a reaction-diffusion-advection SIS epidemic model with mass action infection mechanism, Calc. Var. Partial Differ. Equ., 60, 184 (2021) · Zbl 1480.35380 |
| [14] | Cui, R.; Lou, Y., A spatial SIS model in advective heterogeneous environments, J. Differ. Equ., 261, 3305-3343 (2016) · Zbl 1342.92231 |
| [15] | Deng, K.; Wu, Y., Dynamics of a susceptible-infected-susceptible epidemic reaction-diffusion model, Proc. R. Soc. Edinb., Sect. A, 146, 929-946 (2016) · Zbl 1353.92094 |
| [16] | Deng, K.; Wu, Y., Corrigendum: Dynamics of a susceptible-infected-susceptible epidemic reaction-diffusion model, Proc. R. Soc. Edinb. · Zbl 1353.92094 |
| [17] | de Monttoni, P.; Orlandi, E.; Tesei, A., Asymptotic behavior for a system describing epidemics with migration and spatial spread of infection, Nonlinear Anal., 3, 663-675 (1979) · Zbl 0416.35009 |
| [18] | Fife, P., Some nonclassical trends in parabolic and parabolic-like evolutions, (Trends in Nonlinear Analysis (2003), Springer: Springer Berlin), 153-191 · Zbl 1072.35005 |
| [19] | Fitzgibbon, W. E.; Langlais, M., Simple models for the transmission of microparasites between host populations living on non coincident spatial domain, (Magal, P.; Ruan, S., Structured Population Models in Biology and Epidemiology. Structured Population Models in Biology and Epidemiology, Lecture Notes in Math., vol. 1936 (2008), Springer-Verlag: Springer-Verlag Berlin), 115-164 · Zbl 1138.92029 |
| [20] | 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 |
| [21] | Kao, C.-Y.; Lou, Y.; Shen, W., Random dispersal vs non-local dispersal, Discrete Contin. Dyn. Syst., 26, 551-596 (2010) · Zbl 1187.35127 |
| [22] | Kendall, D. G., Deterministic and stochastic epidemics in closed populations, (Neymann, J., Contributions to Biology and Problems of Health. Contributions to Biology and Problems of Health, Proc. 3rd Berkeley Symp. Math. Stat. and Prob., vol. 4 (1956), University of California Press: University of California Press Berkeley), 149-165 · Zbl 0070.15101 |
| [23] | Kendall, D. G., Mathematical models of the spread of infection, (Mathematics and Computer Science in Biology and Medicine (1965)), 213-225 |
| [24] | Kermack, W. O.; McKendrick, A. G., A contribution to the mathematical theory of epidemics, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 115, 700-721 (1927) · JFM 53.0517.01 |
| [25] | Kuniya, T.; Wang, J., Global dynamics of an SIR epidemic model with nonlocal diffusion, Nonlinear Anal., Real World Appl., 43, 262-282 (2018) · Zbl 1392.92102 |
| [26] | Kuperman, M. N.; Wio, H. S., Front propagation in epidemiological models with spatial dependence, Physica A, 272, 206-222 (1999) |
| [27] | Kuto, K.; Matsuzawa, H.; Peng, R., Concentration profile of endemic steady state of a reaction-diffusion-advection SIS epidemic model, Calc. Var. Partial Differ. Equ., 56, 112 (2017) · Zbl 1444.35102 |
| [28] | Li, H.; Peng, R.; Wang, Z., On a diffusive susceptible-infected-susceptible epidemic model with mass action mechanism and birth-death effect: analysis, simulations, and comparison with other mechanisms, SIAM J. Appl. Math., 78, 2129-2153 (2018) · Zbl 1393.35102 |
| [29] | Lei, C.; Xiong, J.; Zhou, X., Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment, Discrete Contin. Dyn. Syst., Ser. B, 25, 81-98 (2020) · Zbl 1425.92188 |
| [30] | Mollison, D., The rate of spatial propagation of simple epidemic, (Neymann, J., Contributions to Probability Theory. Contributions to Probability Theory, Proc. 6th Berkeley Symp. Math. Stat. and Prob., vol. 3 (1972), University of California Press: University of California Press Berkeley), 579-614 · Zbl 0266.92008 |
| [31] | Murray, J. D., Mathematical Biology, II, Spatial Models and Biomedical Applications, Interdisciplinary Applied Mathematics, vol. 18 (2003), Springer-Verlag: Springer-Verlag New York · Zbl 1006.92002 |
| [32] | Noble, J. V., Geographic and temporal development of plagues, Nature, 250, 726-729 (1974) |
| [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. Differ. Equ., 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.; Yi, F., Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: effects of epidemic risk and population movement, Physica D, 259, 8-25 (2013) · Zbl 1321.92076 |
| [37] | Rass, L.; Radcliffe, J., Spatial Deterministic Epidemics, Math. Surveys Monogr., vol. 102 (2003), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1018.92028 |
| [38] | 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 |
| [39] | Ruan, S.; Wu, J., Modeling spatial spread of communicable diseases involving animal hosts, (Cantrell, S.; Cosner, C.; Ruan, S., Spatial Ecology (2009), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, FL), 293-316 · Zbl 1183.92074 |
| [40] | Shen, W.; Zhang, A., Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differ. Equ., 249, 747-795 (2010) · Zbl 1196.45002 |
| [41] | 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 |
| [42] | Sun, J.-W.; Yang, F.-Y.; Li, W.-T., A nonlocal dispersal equation arising from a selection-migration model in genetics, J. Differ. Equ., 257, 1372-1402 (2014) · Zbl 1320.35060 |
| [43] | Sun, X.; Cui, R., Analysis on a diffusive SIS epidemic model with saturated incidence rate and linear source in a heterogeneous environment, J. Math. Anal. Appl., 490, Article 124212 pp. (2020) · Zbl 1444.92124 |
| [44] | Tong, Y.; Lei, C., An SIS epidemic reaction-diffusion model with spontaneous infection in a spatially heterogeneous environment, Nonlinear Anal., Real World Appl., 41, 443-460 (2018) · Zbl 1380.92086 |
| [45] | Webb, G. F., A reaction-diffusion model for a deterministic diffusive epidemic, J. Math. Anal. Appl., 84, 150-161 (1981) · Zbl 0484.92019 |
| [46] | Wen, X.; Ji, J.; Li, B., Asymptotic profiles of the endemic steady state to a diffusive SIS epidemic model with mass action infection mechanism, J. Math. Anal. Appl., 458, 715-729 (2018) · Zbl 1379.92068 |
| [47] | Wu, Y.; Zou, X., Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J. Differ. Equ., 261, 4424-4447 (2016) · Zbl 1346.35199 |
| [48] | Wu, Y.; Zou, X., Dynamics and profiles of a diffusive host-pathogen system with distinct dispersal rates, J. Differ. Equ., 264, 4989-5024 (2018) · Zbl 1387.35362 |
| [49] | Xu, W.-B.; Li, W.-T.; Ruan, S., Spatial propagation in an epidemic model with nonlocal diffusion: the influences of initial data and dispersal, Sci. China Math., 63, 2177-2206 (2020) · Zbl 1464.35384 |
| [50] | 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 |
| [51] | Yang, F.-Y.; Li, W.-T.; Ruan, S., Dynamics of a nonlocal dispersal SIS epidemic model with Neumann boundary conditions, J. Differ. Equ., 267, 2011-2051 (2019) · Zbl 1416.35285 |
| [52] | Zhang, J.; Cui, R., Asymptotic behavior of an SIS reaction-diffusion-advection model with saturation and spontaneous infection mechanism, Z. Angew. Math. Phys., 71, 150 (2020) · Zbl 1447.35070 |
| [53] | Zhang, J.; Cui, R., Asymptotic profiles of the endemic steady state of a diffusive SIS epidemic system with saturated incidence rate and spontaneous infection, Math. Methods Appl. Sci., 44, 517-532 (2021) · Zbl 1475.35368 |
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.
