\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Analysis of a diffusive SIS epidemic model with spontaneous infection and a linear source in spatially heterogeneous environment

Abstract / Introduction Full Text(HTML) Related Papers Cited by
  • In this paper, we investigate a diffusive SIS epidemic model with spontaneous infection and a linear source in spatially heterogeneous environment. We first prove that the solution of the model is bounded when the susceptible and infected individuals have same or distinct dispersal rates. The global stability of the constant endemic equilibrium is proved by constructing suitable Lyapunov functionals when all parameters are positive constants. We employ the topological degree argument to show the existence of positive steady state. Most importantly, we have also investigated the asymptotic profiles of the positive steady state as the dispersal rate of susceptible or infected individuals tends to zero or infinity. Our result reveals that a linear source and spontaneous infection can significantly enhance disease persistence no matter what dispersal rate of the susceptible or infected population is small or large, which leads to the situation that when total population number allows to vary, disease becomes more difficult to control.

    Mathematics Subject Classification: 35K57, 37N25, 35B40.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] L. J. S. AllenB. M. BolkerY. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Disc. Cont. Dyn. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.
    [2] R. M. Anderson and R. M. May, Population biology of infectious diseases: Part Ⅰ, Nature, 280 (1979), 361-367.  doi: 10.1038/280361a0.
    [3] H. Brézis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.
    [4] R. H. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differential Equations, 261 (2016), 3305-3343.  doi: 10.1016/j.jde.2016.05.025.
    [5] Z. J. Du and R. Peng, A priori $L^\infty$ estimates for solutions of a class of reaction-diffusion systems, J. Math. Biol., 72 (2016), 1429-1439.  doi: 10.1007/s00285-015-0914-z.
    [6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001.
    [7] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2006), 599-653.  doi: 10.1137/S0036144500371907.
    [8] A. L. Hill, D. G. Rand, M. A. Nowak and N. A. Christakis, Infectious disease modeling of social contagion in networks, PLoS Comput. Biol., 6 (2010), e1000968, 15 pp. doi: 10.1371/journal.pcbi.1000968.
    [9] L. Dung, Dissipativity and global attractors for a class of quasilinear parabolic systems, Commun. Partial Differ. Equ., 22 (1997), 413-433.  doi: 10.1080/03605309708821269.
    [10] H. C. LiR. Peng and F.-B. Wang, Varying total population enhances disease persistence: Qualitative analysis on a diffusive SIS epidemic model, J. Differential Equations, 262 (2017), 885-913.  doi: 10.1016/j.jde.2016.09.044.
    [11] B. Li, H. C. Li and Y. C. Tong, Analysis on a diffusive SIS epidemic model with logistic source, Z. Angew. Math. Phys., 68 (2017), Art. 96, 25 pp. doi: 10.1007/s00033-017-0845-1.
    [12] G. M. Lieberman, Bounds for the steady-state Sel'kov model for arbitrary $p$ in any number of dimensions, SIAM J. Math. Anal., 36 (2005), 1400-1406.  doi: 10.1137/S003614100343651X.
    [13] Y. J. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Bio., 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8.
    [14] L. Nirenberg, Topic in Nonlinear Functional Analysis, Courant Lecture Notes in Mathematics, 6. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/cln/006.
    [15] R. Peng and X.-Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.  doi: 10.1088/0951-7715/25/5/1451.
    [16] R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. Part Ⅰ, J. Differential Equations, 247 (2009), 1096-1119.  doi: 10.1016/j.jde.2009.05.002.
    [17] R. Peng and F. Q. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: Effects of epidemic risk and population movement, Physica D, 259 (2013), 8-25.  doi: 10.1016/j.physd.2013.05.006.
    [18] R. PengJ. P. Shi and M. X. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity, 21 (2008), 1471-1488.  doi: 10.1088/0951-7715/21/7/006.
    [19] Y. C. Tong and C. X. Lei, An SIS epidemic reaction-diffusion model with spontaneous infection in a spatially heterogeneous environment, Nonlinear Analysis: RWA, 41 (2018), 443-460.  doi: 10.1016/j.nonrwa.2017.11.002.
    [20] H. F. Weinberger, Invariant sets for weakly coupled parabolic and elliptic systems, Rend. Mat., 8 (1975), 295-310. 
    [21] Y. X. Wu and X. F. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J. Differential Equations, 261 (2016), 4424-4447.  doi: 10.1016/j.jde.2016.06.028.
  • 加载中
SHARE

Article Metrics

HTML views(1770) PDF downloads(521) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return