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Dynamics of an epidemic model with advection and free boundaries

1 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, 730000, P.R. China
2 Department of Mathematics, Harbin Engineering University, Harbin, 150001, P.R. China

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

This paper deals with the propagation dynamics of an epidemic model, which is modeled by a partially degenerate reaction-diffusion-advection system with free boundaries and sigmoidal function. We focus on the effect of small advection on the propagation dynamics of the epidemic disease. At first, the global existence and uniqueness of solution are obtained. And then, the spreading-vanishing dichotomy and the criteria for spreading and vanishing are given. Our results imply that the small advection make the disease spread more difficult.
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1. V. Capasso and S. L. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Rev. d'Epidemiol. Santé Publique, 27 (1979), 32–121.

2. H. H. Wilson, Ordinary Differential Equations, Addison-Wesley Publ. Comp., London, 1971.

3. V. Capasso and R. E. Wilson, Analysis of a reaction-diffusion system modeling man-environment-man epidemics, SIAM J. Appl. Math., 57 (1997), 327–346.

4. D. Xu and X. Q. Zhao, Erratum to: "Bistable waves in an epidemic model", J. Dyn. Differ. Eq., 17 (2005), 219–247.

5. Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377–405.

6. Y. Du, Z. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089–2142.

7. Y. Du and Z. Lin, The diffusive competition model with a free boundary: invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3105–3132.

8. Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673–2724.

9. J. Ge, K. I. Kim, Z. Lin, et al., A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differ. Eq., 259 (2015), 5486–5509.

10. H. Gu, B. Lou and M. Zhou, Long time behavior of solutions of Fisher-KPP equation with advec-tion and free boundaries, J. Funct. Anal., 269 (2015), 1714–1768.

11. J. Guo and C. Wu, On a free boundary problem for a two-species weak competition system, J. Dynam. Differ. Eq., 24 (2012), 873–895.

12. K. I. Kim, Z. Lin and Q. Zhang, An SIR epidemic model with free boundary, Nonlinear Anal. Real. World Appl., 14 (2013), 1992–2001.

13. J. Wang and L. Zhang, Invasion by an inferior or superior competitor: a diffusive competition model with a free boundary in a heterogeneous environment, J. Math. Anal. Appl., 423 (2015), 377–398.

14. M. Wang, On some free boundary problems of the prey-predator model, J. Differ. Eq., 256 (2014), 3365–3394.

15. M. Wang and J. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dyn. Differ. Eq., 26 (2014), 655–672.

16. M. Wang and J. Zhao, A free boundary problem for the predator-prey model with double free boundaries, J. Dyn. Differ. Eq., 29 (2017), 957–979.

17. M. Wang and Y. Zhang, Dynamics for a diffusive prey-predator model with different free bound- aries, J. Differ. Eq., 264 (2018), 3527–3558.

18. W. T. Li, M. Zhao and J. Wang, Spreading fronts in a partially degenerate integro-differential reaction-diffusion system, Z. Angew. Math. Phys., 68 (2017), Art. 109, 28 pp.

19. A. K. Tarboush, Z. Lin and M. Zhang, Spreading and vanishing in a West Nile virus model with expanding fronts, Sci. China Math., 60 (2017), 841–860.

20. J. Wang and J. F. Cao, The spreading frontiers in partially degenerate reaction-diffusion systems, Nonlinear Anal., 122 (2015), 215–238.

21. X. Bao, W. Shen and Z. Shen, Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems, Commun. Pure Appl. Anal., 18 (2019), 361–396.

22. B. S. Han and Y. Yang, An integro-PDE model with variable motility, Nonlinear Anal. Real. World Appl., 45 (2019), 186–199.

23. I. Ahn, S. Beak and Z. Lin, The spreading fronts of an infective environment in a man-environment-man epidemic model, Appl. Math. Model., 40 (2016), 7082–7101.

24. M. Zhao, W. T. Li and W. Ni, Spreading speed of a degenerate and cooperative epidemic model with free boundaries, Discrete Contin. Dyn. Syst. Ser. B, in press (2019).

25. N. A. Maidana and H. Yang, Spatial spreading of West Nile Virus described by traveling waves, J. Theoret. Biol., 258 (2009), 403–417.

26. H. Gu, Z. Lin and B. Lou, Long time behavior of solutions of a diffusion-advection logistic model with free boundaries, Appl. Math. Lett., 37 (2014), 49–53.

27. H. Gu, Z. Lin and B. Lou, Different asymptotic spreading speeds induced by advection in a diffusion problem with free boundaries, Proc. Amer. Math. Soc., 143 (2015), 1109–1117.

28. 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.

29. H. Gu and B. Lou, Spreading in advective environment modeled by a reaction diffusion equation with free boundaries, J. Differ. Eq., 260 (2016), 3991–4015.

30. Y. Kaneko and H. Matsuzawa, Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations, J. Math. Anal. Appl., 428 (2015), 43–76.

31. H. Monobe and C. H. Wu, On a free boundary problem for a reaction-diffusion-advection logistic model in heterogeneous environment, J. Differ. Eq., 261 (2016), 6144–6177.

32. N. Sun, B. Lou and M. Zhou, Fisher-KPP equation with free boundaries and time-periodic advec-tions, Calc. Var. Partial Differ. Eq., 56 (2017), 61–96.

33. L. Wei, G. Zhang and M. Zhou, Long time behavior for solutions of the diffusive logistic equation with advection and free boundary, Calc. Var. Partial Differ. Eq., 55 (2016), 95–128.

34. Y. Zhao and M. Wang, A reaction-diffusion-advection equation with mixed and free boundary conditions, J. Dynam. Differ. Eq., 30 (2018), 743–777.

35. Q. Chen, F. Li and F. Wang, A reaction-diffusion-advection competition model with two free boundaries in heterogeneous time-periodic environment, IMA J. Appl. Math., 82 (2017), 445–470.

36. M. Li and Z. Lin, The spreading fronts in a mutualistic model with advection, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2089–2105.

37. C. Tian and S. Ruan, A free boundary problem for Aedes aegypti mosquito invasion, Appl. Math. Model., 46 (2017), 203–217.

38. M. Zhang, J. Ge and Z. Lin, The invasive dynamics of Aedes aegypti mosquito in a heterogenous environment (in Chinese), Sci. Sin. Math., 48 (2018), 999–1018.

39. L. Zhou, S. Zhang and Z. Liu, A free boundary problem of a predator-prey model with advection in heterogeneous environment, Appl. Math. Comput., 289 (2016), 22–36.

40. 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.

41. G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583–603.

42. Z. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883–1892.

43. M. Wang, H. Huang and S. Liu, A logistic SI epidemic model with degenerate diffusion and free boundary, preprint, (2019).

44. J. F. Cao, Y. Du, F. Li, et al., The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries, J. Funct. Anal., (2019), https://doi.org/10.1016/j.jfa.2019.02.013.

45. M. Wang, Existence and uniqueness of solutions of free boundary problems in heterogeneous environments, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 415–421.

46. M. Wang, A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment, J. Funct. Anal., 270 (2016), 483–508.

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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