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Asymptotic profiles of a nonlocal dispersal SIS epidemic model with saturated incidence

Part of: Parabolic equations and systems Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  14 May 2024

Yan-Xia Feng ,
Wan-Tong Li  and
Fei-Ying Yang
Yan-Xia Feng
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, P. R. China (fengyx2021@lzu.edu.cn; wtli@lzu.edu.cn; yangfy@lzu.edu.cn)
Wan-Tong Li
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, P. R. China (fengyx2021@lzu.edu.cn; wtli@lzu.edu.cn; yangfy@lzu.edu.cn)
Fei-Ying Yang
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, P. R. China (fengyx2021@lzu.edu.cn; wtli@lzu.edu.cn; yangfy@lzu.edu.cn)
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Abstract

Infection mechanism plays a significant role in epidemic models. To investigate the influence of saturation effect, a nonlocal (convolution) dispersal susceptible-infected-susceptible epidemic model with saturated incidence is considered. We first study the impact of dispersal rates and total population size on the basic reproduction number. Yang, Li and Ruan (J. Differ. Equ. 267 (2019) 2011–2051) obtained the limit of basic reproduction number as the dispersal rate tends to zero or infinity under the condition that a corresponding weighted eigenvalue problem has a unique positive principal eigenvalue. We remove this additional condition by a different method, which enables us to reduce the problem on the limiting profile of the basic reproduction number into that of the spectral bound of the corresponding operator. Then we establish the existence and uniqueness of endemic steady states by a equivalent equation and finally investigate the asymptotic profiles of the endemic steady states for small and large diffusion rates to provide reference for disease prevention and control, in which the lack of regularity of the endemic steady state and Harnack inequality makes the limit function of the sequence of the endemic steady state hard to get. Finally, we find whether lowing the movements of susceptible individuals can eradicate the disease or not depends on not only the sign of the difference between the transmission rate and the recovery rate but also the total population size, which is different from that of the model with standard or bilinear incidence.

Keywords

Nonlocal dispersalsusceptible-infected-susceptible (SIS) epidemic modelbasic reproduction numberendemic steady stateasymptotic profiles

MSC classification

Primary: 35K57: Reaction-diffusion equations
Secondary: 35R20: Partial operator-differential equations (i.e., PDE on finite-dimensional spaces for abstract space valued functions) 92D25: Population dynamics (general)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 33
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Altenberg, L.. Resolvent positive linear operators exhibit the reduction phenomenon. Proc. Natl. Acad. Sci. USA 109 (2012), 3705–3710.CrossRefGoogle ScholarPubMed
Allen, L. J. S., Bolker, B. M., Lou, Y. and Nevai, A. L.. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete Contin. Dyn. Syst. 21 (2008), 1–20.CrossRefGoogle Scholar
Andreu-Vaillo, F., Mazón, J. M., Rossi, J. D. and Toledo-Melero, J.. Nonlocal diffusion problems, mathematical surveys and monographs (AMS, Providence, Rhode Island, 2010).CrossRefGoogle Scholar
Anderson, R. M. and May, R. M.. Population biology of infectious diseases. Nature 280 (1979), 361–367.CrossRefGoogle ScholarPubMed
Berestycki, H., Coville, J. and Vo, H.-H.. On the definition and the properties of the principal eigenvalue of some nonlocal operators. J. Funct. Anal. 271 (2016), 2701–2751.CrossRefGoogle Scholar
Brockmann, D., Hufnagel, L. and Geisel, T.. The scaling laws of human travel. Nature 439 (2006), 462–465.CrossRefGoogle ScholarPubMed
Capasso, V. and Serio, G.. A generalization of the Kermack–McKendrick deterministic epidemic model. Math. Biosci. 42 (1978), 43–61.CrossRefGoogle Scholar
Castellano, K. and Salako, R. B.. On the effect of lowering population's movement to control the spread of an infectious disease. J. Differ. Equ. 316 (2022), 1–27.CrossRefGoogle Scholar
Coville, J.. On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators. J. Differ. Equ. 249 (2010), 2921–2953.CrossRefGoogle Scholar
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 (2021), 2997–3022.Google Scholar
de Jong, M. C. M., Diekmann, O. and Heesterbeek, H.. How does transmission of infection depend on population size?, in Epidemic Models: Their Structure and Relation to Data (Cambridge University Press, New York, 1995), pp. 84–94.Google Scholar
Deng, K. and Wu, Y.. Dynamics of a susceptible-infected-susceptible epidemic reaction-diffusion model. Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), 929–946.CrossRefGoogle Scholar
Deng, K. and Wu, Y.. Corrigendum: dynamics of a susceptible-infected-susceptible epidemic reaction-diffusion model. Proc. Roy. Soc. Edinburgh Sect. A 153 (2023), 718–720.CrossRefGoogle Scholar
Diekmann, O. and Kretzschmar, M.. Patterns in the effects of infectious diseases on population growth. J. Math. Biol. 29 (1991), 539–570.CrossRefGoogle ScholarPubMed
Feng, Y.-X., Li, W.-T., Ruan, S. and Yang, F.-Y.. Dynamics and asymptotic profiles of a nonlocal dispersal SIS epidemic model with bilinear incidence and Neumann boundary conditions. J. Differ. Equ. 335 (2022), 294–346.CrossRefGoogle Scholar
Fife, P.. Some nonclassical trends in parabolic and parabolic-like evolutions, in: Trends in Nonlinear Analysis (Springer, Berlin, 2003), pp. 153–191.CrossRefGoogle Scholar
Fitzgibbon, W. E. and Langlais, M.. Simple models for the transmission of microparasites between host populations living on non coincident spatial domain, in ‘Structured Population Models in Biology and Epidemiology’, P. Magal and S. Ruan (eds.), Lecture Notes in Math. Vol. 1936 (Springer-Verlag, Berlin), pp. 115–164.Google Scholar
Gao, D., Lei, C., Peng, R. and Zhang, B.. A diffusive SIS epidemic model with saturated incidence function in a heterogeneous environment. Nonlinearity 37 (2024), 025002.CrossRefGoogle Scholar
Guo, Y., Wang, J. and Ji, D.. Asymptotic profiles of a diffusive SIS epidemic model with vector-mediated infection and logistic source. Z. Angew. Math. Phys. 73 (2022), 255.CrossRefGoogle Scholar
Hallatscheka, O. and Fisher, D. S.. Acceleration of evolutionary spread by long-range dispersal. Proc. Natl. Acad. Sci. USA 111 (2014), E4911–E4919.Google Scholar
Heesterbeek, J. A. P. and Metz, J. A. J.. The saturating contact rate in marriage- and epidemic models. J. Math. Biol. 31 (1993), 529–539.CrossRefGoogle ScholarPubMed
Huo, X. and Cui, R.. A reaction-diffusion SIS epidemic model with saturated incidence rate and logistic source. Appl. Anal. 101 (2022), 4492–4511.CrossRefGoogle Scholar
Kendall, D. G.. Deterministic and stochastic epidemics in closed populations, in ‘Contributions to Biology and Problems of Health’, J. Neymann (eds.), Proc. 3rd Berkeley Symp. Math. Stat. and Prob. Vol. 4 (Berkeley, University of California Press, 1956), pp. 149–165.CrossRefGoogle Scholar
Kendall, D. G.. Mathematical models of the spread of infection, in ‘Mathematics and Computer Science in Biology and Medicine’ (London, Medical Research Council, 1965), pp. 213–225.Google Scholar
Kermack, W. O. and McKendrick, A. G.. A contribution to the mathematical theory of epidemics. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 115 (1927), 700–721.Google Scholar
Kuniya, T. and Wang, J.. Global dynamics of an SIR epidemic model with nonlocal diffusion. Nonlinear Anal. Real World Appl. 43 (2018), 262–282.CrossRefGoogle Scholar
Li, B., Li, H. and Tong, Y.. Analysis on a diffusive SIS epidemic model with logistic source. Z. Angew. Math. Phys. 68 (2017), 96.CrossRefGoogle Scholar
Li, H., Peng, R. and Wang, F.-B.. Varying total population enhances disease persistence: qualitative analysis on a diffusive SIS epidemic model. J. Differ. Equ. 262 (2017), 885–913.CrossRefGoogle Scholar
Li, H., Peng, R. and Wang, Z.-A.. 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 (2018), 2129–2153.CrossRefGoogle Scholar
Liu, W. M., Levin, S. A. and Iwas, Y.. Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol. 23 (1986), 187–204.CrossRefGoogle ScholarPubMed
Murray, J. D.. Mathematical Biology, II, Spatial Models and Biomedical Applications, 3rd ed., Interdisciplinary Applied Mathematics Vol. 18 (Springer-Verlag, New York, 2003).CrossRefGoogle Scholar
Peng, R.. Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model, I. J. Differ. Equ. 247 (2009), 1096–1119.CrossRefGoogle Scholar
Peng, R. and Liu, S.. Global stability of the steady states of an SIS epidemic reaction-diffusion model. Nonlinear Anal. 71 (2009), 239–247.CrossRefGoogle Scholar
Peng, R., Wang, Z.-A., Zhang, G. and Zhou, M.. Novel spatial profiles of some diffusive SIS epidemic models. J. Math. Biol. 87 (2023), 81.CrossRefGoogle ScholarPubMed
Peng, R. and 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 (2013), 8–25.CrossRefGoogle Scholar
Rass, L. and Radcliffe, J.. Spatial Deterministic Epidemics, Math. Surveys Monogr. Vol. 102 (Amer. Math. Soc., Providence, PI, 2003).CrossRefGoogle Scholar
Ruan, S.. Spatial-temporal dynamics in nonlocal epidemiological models, in: Takeuchi Y, Sato K. and Iwasa Y. (eds.), ‘Mathematics for Life Science and Medicine’ (Springer-Verlag, Berlin, 2007), pp. 99–122.Google Scholar
Ruan, S. and Wu, J.. Modeling spatial spread of communicable diseases involving animal hosts, in ‘Spatial Ecology’, S. Cantrell, C. Cosner and S. Ruan (eds.) (Chapman & Hall/CRC, Boca Raton, FL, 2009), pp. 293–316.CrossRefGoogle Scholar
Shen, W. and Xie, X.. On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications. Discrete Contin. Dyn. Syst. 35 (2015), 1665–1696.CrossRefGoogle Scholar
Sun, X. and 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 (2020), 124212.CrossRefGoogle Scholar
Suo, J. and Li, B.. Analysis on a diffusive SIS epidemic system with linear source and frequency-dependent incidence function in a heterogeneous environment. Math. Biosci. Eng. 17 (2020), 418–441.CrossRefGoogle Scholar
Wen, X., Ji, J. and 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 (2018), 715–729.CrossRefGoogle Scholar
Wu, Y. and Zou, X.. Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism. J. Differ. Equ. 261 (2016), 4424–4447.CrossRefGoogle Scholar
Yang, F.-Y. and Li, W.-T.. Dynamics of a nonlocal dispersal SIS epidemic model. Commun. Pure Appl. Anal. 16 (2017), 781–797.CrossRefGoogle Scholar
Yang, F.-Y., Li, W.-T. and Ruan, S.. Dynamics of a nonlocal dispersal SIS epidemic model with Neumann boundary conditions. J. Differ. Equ. 267 (2019), 2011–2051.CrossRefGoogle Scholar
Zhang, L. and Zhao, X.-Q.. Asymptotic behavior of the basic reproduction ratio for periodic reaction-diffusion systems. SIAM J. Math. Anal. 53 (2021), 6873–6909.CrossRefGoogle Scholar