Dynamics and asymptotic profiles of steady states of an SIRS epidemic model in spatially heterogenous environment

Baoxiang Zhang; Yongli Cai; Bingxian Wang; Weiming Wang

doi:10.3934/mbe.2020047

Mathematical Biosciences and Engineering, 2020, 17(1): 893-909. doi: 10.3934/mbe.2020047. Research article Special Issues

Export file:

Format

RIS(for EndNote,Reference Manager,ProCite)
BibTex
Text

Content

Citation Only
Citation and Abstract

Dynamics and asymptotic profiles of steady states of an SIRS epidemic model in spatially heterogenous environment

Baoxiang Zhang, Yongli Cai, Bingxian Wang, Weiming Wang

School of Mathematics and Statistics, Huaiyin Normal University, Huaian, 223300, China

Received: , Accepted: , Published:

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

Abstract Full Text(HTML) Figure/Table Related pages

This paper performs qualitative analysis on a reaction–diffusion SIRS epidemic system with ratio–dependent incidence rate in spatially heterogeneous environment. The threshold dynamics in the term of the basic reproduction number R₀ is established. And the asymptotic profile of endemic equilibrium is determined if the diffusion rate of the susceptible individuals is small. The results show that restricting the movement of susceptible individuals can effectively control the number of infectious individuals.

Figure/Table

Supplementary

Article Metrics

Keywords: ratio–dependent incidence rate; spatial heterogeneity; basic reproduction number; threshold dynamics; asymptotic profile

Citation: Baoxiang Zhang, Yongli Cai, Bingxian Wang, Weiming Wang. Dynamics and asymptotic profiles of steady states of an SIRS epidemic model in spatially heterogenous environment. Mathematical Biosciences and Engineering, 2020, 17(1): 893-909. doi: 10.3934/mbe.2020047

References:

1. R. May and R. Anderson, Spatial heterogeneity and the design of immunization programs, Math. Biosci., 72 (1984), 83-111.
2. H. Hethcote and J. W. Van Ark, Epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation, and immunization programs, Math. Biosci., 84 (1987), 85-118.
3. V. Capasso, Mathematical structures of epidemic systems, volume 97. Springer, 1993.
4. M. E. Alexander and S. M. Moghadas. Periodicity in an epidemic model with a generalized nonlinear incidence, Math. Biosci., 189 (2004), 75-96.
5. W. D. Wang, Epidemic models with nonlinear infection forces, Math. Biosci. Eng., 3 (2006), 267-279.
6. D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429.
7. Y. Cai, Y. Kang and W. M. Wang, A stochastic SIRS epidemic model with nonlinear incidence rate, Appl. Math. Comp., 305 (2017), 221-240.
8. W. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of sirs epidemiological models, J. Math. Biol., 23 (1986), 187-204.
9. W. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380.
10. H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.
11. B. Fred and C.-C. Carlos, Mathematical models in population biology and epidemiology (Second Edition). Springer, 2012.
12. Y. Cai, Y. Kang, M. Banerjee, et al., A stochastic SIRS epidemic model with infectious force under intervention strategies, J. Differ. Equations, 259 (2015), 7463-7502.
13. S. Yuan and B. Li, Global dynamics of an epidemic model with a ratio-dependent nonlinear incidence rate, Discrete Dyn. Nat. Soc., 2009 (2009), 609306.
14. C. Neuhauser, Mathematical Challenges in Spatial Ecology, Notices AMS, 48 (2001), 1304-1314.
15. S. Ruan, Spatial-Temporal Dynamics in Nonlocal Epidemiological Models, In: Takeuchi Y., Iwasa Y., Sato K. (eds) Mathematics for Life Science and Medicine. Springer, Berlin, Heidelberg, 2007.
16. W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A reaction-diffusion system modeling direct and indirect transmission of diseases, Discrete Cont. Dyn.-B, 4 (2004), 893-910.
17. Z. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, P. Roy. Soc. A-Math. Phy., 466 (2010), 237-261.
18. Y. Cai and W. M. Wang, Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment, Discrete Cont. Dyn.-B, 20 (2015), 989-1013.
19. Y. Cai and W. M. Wang, Fish-hook bifurcation branch in a spatial heterogeneous epidemic model with cross-diffusion, Nonl. Anal. Real World Appl., 30 (2016), 99-125.
20. W. M. Wang, X. Gao, Y. Cai, et al., Turing patterns in a diffusive epidemic model with saturated infection force, J. Franklin Inst., 355 (2018), 7226-7245.
21. Y. Cai, Y. Kang, M. Banerjee, et al., Complex dynamics of a host-parasite model with both horizontal and vertical transmissions in a spatial heterogeneous environment, Nonl. Anal. Real World Appl., 40 (2018), 444-465.
22. P. Magal, G. Webb and Y. Wu, On a vector-host epidemic model with spatial structure, Nonlinearity, 31 (2018), 5589-5614.
23. Y. Cai, Z. Ding, B. Yang, et al., Transmission dynamics of Zika virus with spatial structure-A case study in Rio de Janeiro, Brazil, Phys. A, 514 (2019), 729-740.
24. J. Ge, K. Kim, Z. Lin, et al., An SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differ. Equations, 259 (2015), 5486-5509.
25. Y. Cai, X. Lian, Z. Peng, et al., Spatiotemporal transmission dynamics for influenza disease in a heterogenous environment, Nonl. Anal. Real, 46 (2019), 178-194.
26. E. E. Holmes, M. A. Lewis, J. E. Banks, et al., Partial differential equations in ecology: Spatial interactions and population dynamics, Ecology, 75 (1994), 17-29.
27. T. Caraco, M. Duryea, G. Gardner, et al., Host spatial heterogeneity and extinction of an SIS epidemic, J. Theor. Biol., 192 (1998), 351-361.
28. A. L. LLoyd and R. M. May, Spatial heterogeneity in epidemic models, J. Theor. Biol., 179 (1996), 1-11.
29. J. Dushoff and S. Levin, The effects of population heterogeneity on disease invasion, Math. Biosci., 128 (1995), 25-40.
30. S. Merler and M. Ajelli, The role of population heterogeneity and human mobility in the spread of pandemic influenza, Proc. R. Soc. B, 277 (2010), 557-565.
31. B. T. Grenfell, O. N. Bjornstad and J. Kappey, Travelling waves and spatial hierarchies in measles epidemics, Nature, 414 (2001), 716-723.
32. M. J. Keeling, M. E. Woolhouse; D. J. Shaw, et al., Dynamics of the 2001 UK foot and mouth epidemic: stochastic dispersal in a heterogeneous landscape, Science, 294 (2001), 813-817.
33. V. Colizza, A. Barrat, M. Barthelemy, et al., The role of the airline transportation network in the prediction and predictability of global epidemics, Proc. Natl. Acad. Sci. USA, 103 (2006), 2015-2020.
34. L. Hufnagel, D. Brockmann and T. Geisel, Forecast and control of epidemics in a globalized world, Proc. Natl. Acad. Sci. USA, 101 (2004), 15124-15129.
35. D. Henry and D. B. Henry, Geometric theory of semilinear parabolic equations, Springer-Verlag Berlin, 1981.
36. M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Prentice-Hall, New Jersey, 1967.
37. K. Yamazaki and X. Wang, Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model, Discrete Cont. Dyn-B, 21 (2017), 1297-1316.
38. 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.
39. N. K. Vaidya, F.-B. Wang and X. Zou, Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment, Discrete Cont. Dyn.-B, 17 (2013), 2829-2848.
40. O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R₀ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.
41. P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.
42. L. J. S. Allen, B. M. Bolker, Y. Lou, et al., Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Cont. Dyn-A, 21 (2008), 1-20.
43. R. Peng and X. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.
44. W. D. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Sys., 11 (2012), 1652-1673.
45. Y. Lou and T. Nagylaki, Evolution of a semilinear parabolic system for migration and selection without dominance, J. Differ. Equations, 225 (2006), 624-665.
46. X. Zhao, J. Borwein and P. Borwein, Dynamical systems in population biology, volume 16. Springer, 2003.
47. P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.
48. M. A. Pino, A priori estimates and applications to existence-nonexistence for a semilinear elliptic system, Indiana U. Math. J., 43 (1994), 77-129.

show all references

Reader Comments

your name: * your email: *

Download full text in PDF

Associated material

Metrics