Abstract
In this paper, we are concerned with an SIS epidemic reaction–diffusion model with logistic source in spatially heterogeneous environment. We first discuss some basic properties of the parabolic system, including the uniform upper bound of solutions and global stability of the endemic equilibrium when spatial environment is homogeneous. Our primary focus is to determine the asymptotic profile of endemic equilibria (when exist) if the diffusion (migration) rate of the susceptible or infected population is small or large. Combined with the results of Li et al. (J Differ Equ 262:885–913, 2017) where the case of linear source is studied, our analysis suggests that varying total population enhances persistence of infectious disease.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alikakos, N.: \(L^p\) bounds of solutions of reaction–diffusion equation. Commun. Partial Differ. Equ. 4, 827–868 (1979)
Allen, L.J.S., Bolker, B.M., Lou, Y., Nevai, A.L.: Asymptotic profiles of the steady states for an SIS epidemic disease patch model. SIAM J. Appl. Math. 67, 1283–1309 (2007)
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)
Brown, K.J., Dunne, P.C., Gardner, R.A.: A semilinear parabolic system arising in the theory of superconductivity. J. Differ. Equ. 40, 232–252 (1981)
Cui, J., Tao, X., Zhu, H.: An SIS infection model incorporating media coverage. Rocky Mount. J. Math. 38, 1323–1334 (2008)
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)
Cui, R., Lou, Y.: A spatial SIS model in advective heterogeneous environments. J. Differ. Equ. 261, 3305–3343 (2016)
Deng, K., Wu, Y.: Dynamics of a susceptible-infected-susceptible epidemic reaction–diffusion model. Proc. R. Soc. Edinb. Sect. A 146, 929–946 (2016)
Ding, W., Huang, W., Kansakar, S.: Traveling wave solutions for a diffusive SIS epidemic model. Discrete Contin. Dyn. Syst. Ser. B 18, 1291–1304 (2013)
Du, Y., Peng, R., Wang, M.: Effect of a protection zone in the diffusive Leslie predator-prey model. J. Differ. Equ. 246, 3932–3956 (2009)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equation of Second Order. Springer, New York (2001)
Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer, New York (1981)
Huang, W., Han, M., Liu, K.: Dynamics of an SIS reaction–diffusion epidemic model for disease transmission. Math. Biosci. Eng. 7, 51–66 (2010)
Li, H., Peng, R., Wang, F.-B.: Varying total population enhances disease persistence: qualitative analysis on a diffusive SIS epidemic model. J. Differ. Equ. 262, 885–913 (2017)
Lieberman, G.M.: Bounds for the steady-state Sel’kov model for arbitrary \(p\) in any number of dimensions. SIAM J. Math. Anal. 36, 1400–1406 (2005)
Lou, Y., Ni, W.-M.: Diffusion, self-diffusion and cross-diffusion. J. Differ. Equ. 131, 79–131 (1996)
Nirenberg, L.: Topic in Nonlinear Functional Analysis. American Mathematical Society, Providence, RI (1974)
Peng, R.: Asymptotic profiles of the positive steady state for an SIS epidemic reaction–diffusion model. Part I. J. Differ. Equ. 247, 1096–1119 (2009)
Peng, R., Liu, S.: Global stability of the steady states of an SIS epidemic reaction–diffusion model. Nonlinear Anal. 71, 239–247 (2009)
Peng, R., Shi, J., Wang, M.: On stationary patterns of a reaction–diffusion model with autocatalysis and saturation law. Nonlinearity 21, 1471–1488 (2008)
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)
Peng, R., Zhao, X.-Q.: A reaction–diffusion SIS epidemic model in a time-periodic environment. Nonlinearity 25, 1451–1471 (2012)
Wang, B.-G., Li, W.-T., Wang, Z.-C.: A reaction–diffusion SIS epidemic model in an almost-periodic environment. Z. Angew. Math. Phys. 66, 3085–3108 (2015)
Wang, W., Zhao, X.-Q.: Basic reproduction numbers for reaction–diffusion epidemic models. SIAM J. Appl. Dyn. Syst. 11, 1652–1673 (2012)
Wu, Y.X., Zou, X.F.: Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism. J. Differ. Equ. 261, 4424–4447 (2016)
Ye, Q.X., Li, Z.Y., Wang, M.X., Wu, Y.P.: Introduction to Reaction–Diffusion Equations, 2nd edn. Science Press, Beijing (2011). (in Chinese)
Zhao, X.: Dynamical Systems in Population Biology, 2nd edn. Springer, New York (2017)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, B., Li, H. & Tong, Y. Analysis on a diffusive SIS epidemic model with logistic source. Z. Angew. Math. Phys. 68, 96 (2017). https://doi.org/10.1007/s00033-017-0845-1
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00033-017-0845-1