Skip to main content Accessibility help
×
Home

Dynamics and asymptotic profiles of endemic equilibrium for two frequency-dependent SIS epidemic models with cross-diffusion

  • HUICONG LI (a1), RUI PENG (a2) and TIAN XIANG (a3)

Abstract

This paper is concerned with two frequency-dependent susceptible–infected–susceptible epidemic reaction–diffusion models in heterogeneous environment, with a cross-diffusion term modelling the effect that susceptible individuals tend to move away from higher concentration of infected individuals. It is first shown that the corresponding Neumann initial-boundary value problem in an n-dimensional bounded smooth domain possesses a unique global classical solution which is uniformly in-time bounded regardless of the strength of the cross-diffusion and the spatial dimension n. It is further shown that, even in the presence of cross-diffusion, the models still admit threshold-type dynamics in terms of the basic reproduction number $\mathcal {R}_0$ – i.e. the unique disease-free equilibrium is globally stable if $\mathcal {R}_0\lt1$ , while if $\mathcal {R}_0\gt1$ , the disease is uniformly persistent and there is an endemic equilibrium (EE), which is globally stable in some special cases with weak chemotactic sensitivity. Our results on the asymptotic profiles of EE illustrate that restricting the motility of susceptible population may eliminate the infectious disease entirely for the first model with constant total population but fails for the second model with varying total population. In particular, this implies that such cross-diffusion does not contribute to the elimination of the infectious disease modelled by the second one.

Copyright

References

Hide All
[1]Alikakos, N. (1979) An application of the invariance principle to reaction-diffusion equations. J. Differ. Equations 33, 201225.
[2]Allen, L. J. S., Bolker, B. M., Lou, Y. & Nevai, A. L. (2007) Asymptotic profiles of the steady states for an SIS epidemic disease patch model. SIAM J. Appl. Math. 67, 12831309.
[3]Allen, L. J. S., Bolker, B. M., Lou, Y. & Nevai, A. L. (2008) Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete Contin. Dyn. Syst. 21, 120.
[4]Bellomo, N., Bellouquid, A., Tao, Y. & Winkler, M. (2015) Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues. Math. Models Methods Appl. Sci. 25, 16631763.
[5]Brown, K. J., Dunne, P. C. & Gardner, R. A. (1981) A semilinear parabolic system arising in the theory of superconductivity. J. Differ. Equations 40, 232252.
[6]Cao, X. (2015) Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces. Discrete Contin. Dyn. Syst. 35, 18911904.
[7]Cieślak, T., Laurencot, Ph. & Morales-Rodrigo, C. (2008) Global existence and convergence to steady states in a chemorepulsion system, equations, in parabolic and Navier–Stokes equations. Banach Center Publ. Polish Acad. Sci. Inst. Math. 81, 105117.
[8]Cui, J., Tao, X. & Zhu, H. (2008) An SIS infection model incorporating media coverage. Rocky Mount. J. Math. 38, 13231334.
[9]Cui, R., Lam, K.-Y. & Lou, Y. (2017) Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments. J. Differ. Equations 263, 23432373.
[10]Cui, R. & Lou, Y. (2016) A spatial SIS model in advective heterogeneous environments. J. Diff. Equations 261, 33053343.
[11]Deng, K. & Wu, Y. (2016) Dynamics of a susceptible-infected-susceptible epidemic reaction-diffusion model. Proc. Roy. Soc. Edinb. Sect. A 146, 929946.
[12]Ding, W., Huang, W. & Kansakar, S. (2013) Traveling wave solutions for a diffusive SIS epidemic model. Discrete Contin. Dyn. Syst. Ser. B 18, 12911304.
[13]Du, Y., Peng, R. & Wang, M. (2009) Effect of a protection zone in the diffusive Leslie predator-prey model. J. Differ. Equations 246, 39323956.
[14]Friedman, A. (1964) Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, NJ, xiv+347 pp.
[15]Gao, D. & Ruan, S. (2011) An SIS patch model with variable transmission coefficients. Math. Biosci. 232, 110115.
[16]Ge, J., Kim, K. I., Lin, Z. & Zhu, H. (2015) A SIS reaction-diffusion-advection model in a low-risk and high-risk domain. J. Differ. Equations 259, 54865509.
[17]Horstmann, D. & Winkler, M. (2005) Boundedness vs. blow-up in a chemotaxis system. J. Differ. Equations 215, 52107.
[18]Huang, W., Han, M. & Liu, K. (2010) Dynamics of an SIS reaction-diffusion epidemic model for disease transmission. Math. Biosci. Eng. 7, 5166.
[19]Hutson, V., Lou, Y. & Mischaikow, K. (2005) Convergence in competition models with small diffusion coefficients. J. Differ. Equations 211, 135161.
[20]Jäger, W. & Luckhaus, S. (1992) On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Amer. Math. Soc. 329, 819824.
[21]Jin, H.-Y. & Xiang, T. (2016) Boundedness and exponential convergence in a chemotaxis model for tumor invasion. Nonlinearity 29, 35793596.
[22]Kuto, K., Matsuzawa, H. & Peng, R. (2017) Concentration profile of endemic equilibrium of a reaction-diffusion-advection SIS epidemic model. Calc. Var. Partial Differ. Equations 56(4), Art. 112, 128.
[23]Ladyzhenskaya, O., Solonnikov, V. & Uralceva, N. (1968) Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI.
[24]Li, H., Peng, R. & Wang, F.-B. (2017) Varying total population enhances disease persistence: qualitative analysis on a diffusive SIS epidemic model. J. Differ. Equations 262, 885913.
[25]Li, T., Pan, R. & Zhao, K. (2012) Global dynamics of a hyperbolic-parabolic model arising from chemotaxis. SIAM J. Appl. Math. 72, 417443.
[26]Lou, Y. & Ni, W.-M. (1996) Diffusion, self-diffusion and cross-diffusion. J. Differ. Equations 131, 79131.
[27]Magal, P. & Zhao, X.-Q. (2005) Global attractors and steady states for uniformly persistent dynamical systems. SIAM. J. Math. Anal. 37, 251275.
[28]Peng, R. (2009) Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. I. J. Differ. Equations 247, 10961119.
[29]Peng, R. & Liu, S. (2009) Global stability of the steady states of an SIS epidemic reaction-diffusion model. Nonlinear Anal. 71, 239247.
[30]Peng, R. & Yi, F. (2013) Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: effects of epidemic risk and population movement. Phys. D 259, 825.
[31]Peng, R. & Zhao, X.-Q. (2012) A reaction-diffusion SIS epidemic model in a time-periodic environment. Nonlinearity 25, 14511471.
[32]Porzio, M. M. & Vespri, V. (1993) Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations. J. Differ. Equations 103, 146178.
[33]Tao, Y. (2013) Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity. Discrete Contin. Dyn. Syst. Ser. B 18, 27052722.
[34]Tao, Y. & Wang, Z.-A. (2013) Competing effects of attraction vs. repulsion in chemotaxis. Math. Models Methods Appl. Sci. 23, 136.
[35]Tao, Y. & Winkler, M. (2012) Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant. J. Differ. Equations 252, 25202543.
[36]Winkler, M. (2010) Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model. J. Differ. Equations 248, 28892905.
[37]Winkler, M. (2010) Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source. Comm. Partial Differ. Equations 35, 15161537.
[38]Winkler, M. (2011) Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction. J. Math. Anal. Appl. 384, 261272.
[39]Winkler, M. (2013) Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system. J. Math. Pures Appl. 100, 748767.
[40]Winkler, M. (2014) Stabilization in a two-dimensional chemotaxis-Navier-Stokes system. Arch. Ration. Mech. Anal. 211, 455487.
[41]Wu, Y. & Zou, X. (2016) Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism. J. Differ. Equations 261, 44244447.
[42]Zhao, X. (2017) Dynamical Systems in Population Biology, second edition, Springer-Verlag, New York.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed