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Analysis on a diffusive SIS epidemic system with linear source and frequency-dependent incidence function in a heterogeneous environment

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu Province, China

Special Issues: Spatial dynamics for epidemic models with dispersal of organisms and heterogenity of environment

In this paper, we consider a diffusive SIS epidemic reaction-diffusion model with linear source in a heterogeneous environment in which the frequency-dependent incidence function is SI/(c + S + I) with c a positive constant. We first derive the uniform bounds of solutions, and the uniform persistence property if the basic reproduction number $\mathcal{R}_{0}>1$. Then, in some cases we prove that the global attractivity of the disease-free equilibrium and the endemic equilibrium. Lastly, we investigate the asymptotic profile of the endemic equilibrium (when it exists) as the diffusion rate of the susceptible or infected population is small. Compared to the previous results [1, 2] in the case of c=0, some new dynamical behaviors appear in the model studied here; in particular, $\mathcal{R}_{0}$ is a decreasing function in c∈[0, ∞) and the disease dies out once c is properly large. In addition, our results indicate that the linear source term can enhance the disease persistence.
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Keywords SIS model with linear source; frequency-dependent incidence function; basic reproduction number; disease-free equilibrium and endemic equilibrium; global attractivity; uniform persistence; asymptotic profile

Citation: Jinzhe Suo, Bo Li. Analysis on a diffusive SIS epidemic system with linear source and frequency-dependent incidence function in a heterogeneous environment. Mathematical Biosciences and Engineering, 2020, 17(1): 418-441. doi: 10.3934/mbe.2020023


  • 1. L. J. S. Allen, B. M. Bolker, Y. Lou, et al., Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Dis. Contin. Dyn. Syst. A, 21 (2008), 1-20.
  • 2. R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. Part I, J. Differ. Equations, 247 (2009), 1096-1119.
  • 3. R. M. Anderson and R. M. May, Populaition biology of infectious diseases, Nature 280 (1979), 361-367.
  • 4. R. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differ. Equations, 261 (2016), 3305-3343.
  • 5. R. Cui, K.-Y. Lam and Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J. Differ. Equations, 263 (2017), 2343-2373.
  • 6. Z. Du and R. Peng, A priori L-estimates for solutions of a class of reaction-diffusion systems, J. Math. Biol., 72 (2016), 1429-1439.
  • 7. H. W. Hethcote, Epidemiology models with variable population size, Mathematical understanding of infectious disease dynamics, 63-89, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 16, World Sci. Publ., Hackensack, NJ, 2009.
  • 8. W. O. Kermack and A. G. McKendrick, Contribution to the mathematical theory of epidemics-I, Proc. Roy. Soc. London Ser. A, 115 (1927), 700-721.
  • 9. M. Martcheva, An introduction to mathmatical epidemiology, Springer,New York, (2015).
  • 10. H. W Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.
  • 11. B. Li, H. Li and Y. Tong, Analysis on a diffusive SIS epidemic model with logistic source, Z. Angew. Math. Phys., 68 (2017), Art. 96, 25pp.
  • 12. H. Li, R. Peng and F. B. Wang, Varying total population enhances disease persistence: qualitative analysis on a diffusive SIS epidemic model, J. Differ. Equations, 262 (2017), 885-913.
  • 13. M. E. Alexander and S. M. Moghadas, Bifurcation Analysis of an SIRS Epidemic Model with Generalized Incidence, SIAM J. Appl. Math., 65 (2001), 1794-1816.
  • 14. R. M. Anderson and R. M. May, Regulation and stability of host-parasite interactions.I. Regulatory processes, J. Anim. Ecol., 47 (1978), 219-247.
  • 15. O. Diekmann and M. Kretzschmar, Patterns in the effects of infectious diseases on population growth, J. Math. Biol., 29 (1991), 539-570.
  • 16. J. A. P. Heesterbeck and J. A. J. Metz, The saturating contact rate in marriage and epidemic models, J. Math. Biol., 31 (1993), 529-539.
  • 17. M. G. Roberts, The dynamics of bovine tuberculosis in possum populations and its eradication or control by culling or vaccination, J. Anim. Ecol., 65 (1996), 451-464.
  • 18. Y. Cai, K. Wang and W. Wang, Global transmission dynamics of a Zika virus model, Appl. Math. Lett., 92 (2019), 190-195.
  • 19. L. Chen and J. Sun, Optimal vaccination and treatment of an epidemic network model, Physics Lett. A, 378 (2014), 3028-3036.
  • 20. L. Chen and J. Sun, Global stability and optimal control of an SIRS epidemic model on heterogeneous networks, Physica A, 410 (2014), 196-204.
  • 21. K. Deng and Y. Wu, Dynamics of a susceptible-infected-susceptible epidemic reaction-diffusion model, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 929-946.
  • 22. X. Gao, Y. Cai, F. Rao, et al., Positive steady states in an epidemic model with nonlinear incidence rate, Comput. Math. Appl., 75 (2018), 424-443.    
  • 23. J. Ge, C. Lei and Z. Lin, Reproduction numbers and the expanding fronts for a diffusion-advection SIS model in heterogeneous time-periodic environment, Nonlinear Anal. Real World Appl., 33 (2017), 100-120.
  • 24. K. Kuto, H. Matsuzawa and R. Peng, Concentration profile of endemic equilibrium of a reactiondiffusion-advection SIS epidemic model, Calc. Var. Partial Dif., 56 (2017), Art. 112, 28 pp.
  • 25. C. Lei, F. Li and J. Liu, Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4499- 4517.
  • 26. B. Li and Q. Bie, Long-time dynamics of an SIRS reaction-diffusion epidemic model, J. Math. Anal. Appl., 475 (2019), 1910-1926.
  • 27. H. Li, R. Peng and Z. Wang, On a diffusive SIS 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.
  • 28. H. Li, R. Peng and T. Xiang, Dynamics and asymptotic profiles of endemic equilibrium for two frequency-dependent SIS epidemic models with cross-diffusion, Eur. J. Appl. Math., 2019, https://doi.org/10.1017/S0956792518000463, in press.
  • 29. Z. Lin, Y. Zhao and P. Zhou, The infected frontier in an SEIR epidemic model with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2355-2376.
  • 30. R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Anal., 71 (2009), 239-247.
  • 31. L. Pu and Z. Lin, A diffusive SIS epidemic model in a heterogeneous and periodically evolving environment, 16 (2019), 3094-3110.
  • 32. X. Wen, J. Ji and B. Li, Asymptotic profiles of the endemic equilibrium to a diffusive SIS epidemic model with mass action infection mechanism, J. Math. Anal. Appl., 458 (2018), 715-729.
  • 33. Y. Wu and X. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J. Differ. Equations, 261 (2016), 4424-4447.
  • 34. M. Zhu, X. Guo and Z. Lin, The risk index for an SIR epidemic model and spatial spreading of the infectious disease, Math. Biosci. Eng., 14 (2017), 1565-1583.
  • 35. R. Peng and X. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.
  • 36. P. Magal and X.-Q Zhao, Global attractive and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275.
  • 37. X.-Q. Zhao, Dynamical Systems in Populaition Biology, Springer-Verlag, New York,(2003)
  • 38. M. Wang, Nonlinear Partial Differential Equations of Parabolic Type, Science Press, Beijing, 1993(in chinese).
  • 39. K. J. Brown, P. C. Dunne and R. A. Gardner, A semilinear parabolic system arising in the theory of superconductivity, J. Differ. Equations, 40 (1981), 232-252.
  • 40. 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.
  • 41. R. Peng, J. Shi and M. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity, 21 (2008), 1471-1488.
  • 42. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, Springer, (2001).
  • 43. Y. Du, R. Peng and M. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Differ. Equations, 246 (2009), 3932-3956.
  • 44. Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differ. Equations, 131 (1996), 79-131.
  • 45. W.-M. Ni and I. Takagi, On the Neumann problem for some semilinear elliptic equations and eystems of activator-inhibitor type, Trans. Amer. Math. Soc., 297 (1986), 351-368.
  • 46. H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in L1, J. Math. Soc. Japan, 25 (1973), 565-590.
  • 47. R. Peng and F. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reactiondiffusion model: Effects of epidemic risk and population movement, Phys. D, 259 (2013), 8-25.


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