Asymptotic profile of endemic equilibrium to a diffusive epidemic model with saturated incidence rate

Yan’e Wang; Zhiguo Wang; Chengxia Lei; Yan’e Wang; Zhiguo Wang; Chengxia Lei

doi:10.3934/mbe.2019192

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 3885-3913. doi: 10.3934/mbe.2019192

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Asymptotic profile of endemic equilibrium to a diffusive epidemic model with saturated incidence rate

1.
Key Laboratory of Modern Teaching Technology, Ministry of Education, Xi’an, Shaanxi, 710062, China
2.
School of Computer Science, Shaanxi Normal University, Xi’an, Shaanxi, 710119, China
3.
School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi, 710119, China
4.
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu, 221116, China

Received: 24 December 2018 Accepted: 02 April 2019 Published: 30 April 2019

We study the existence and asymptotic profile of endemic equilibrium (EE) of a diffusive SIS epidemic model with saturated incidence rate. By introducing the basic reproduction number $\mathcal {R}_0$, the existence of EE is established when $\mathcal {R}_0>1$. The effects of diffusion rates and the saturated coefficient on asymptotic profile of EE are investigated. Our results indicate that when the diffusion rate of susceptible individuals is small and the total population $N$ is below a certain level, or the saturated coefficient is large, the infected population dies out, while the two populations persist if at least one of the diffusion rates of the susceptible and infected individuals is large. Finally, we illustrate the influences of the population diffusion and the saturation coefficient on this model numerically.
- SIS epidemic model,
- diffusion,
- saturated incidence rate,
- endemic equilibrium,
- asymptotic profile,
- extinction/persistence
Citation: Yan’e Wang , Zhiguo Wang, Chengxia Lei. Asymptotic profile of endemic equilibrium to a diffusive epidemic model with saturated incidence rate[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 3885-3913. doi: 10.3934/mbe.2019192

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Abstract

We study the existence and asymptotic profile of endemic equilibrium (EE) of a diffusive SIS epidemic model with saturated incidence rate. By introducing the basic reproduction number $\mathcal {R}_0$, the existence of EE is established when $\mathcal {R}_0>1$. The effects of diffusion rates and the saturated coefficient on asymptotic profile of EE are investigated. Our results indicate that when the diffusion rate of susceptible individuals is small and the total population $N$ is below a certain level, or the saturated coefficient is large, the infected population dies out, while the two populations persist if at least one of the diffusion rates of the susceptible and infected individuals is large. Finally, we illustrate the influences of the population diffusion and the saturation coefficient on this model numerically.

References

[1]	L. J. S. Allen, B. M. Bolker, Y. Lou, et al., A generalization of the Kermack–McKendrick deterministic epidemic model, Discrete Contin. Dyn. Syst., 21 (2008), 1–20.
[2]	F. Brauer and C. Castillo-Ch'avez, Mathematical models in population biology and epidemiology,Springer, 2001.
[3]	R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction–Diffusion Equations, John Wiley and Sons Ltd. Chichester, UK, 2003.
[4]	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.
[5]	R. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differ. Equations, 261 (2016), 3305–3343.
[6]	V. Capasso and G. Serio, A generalization of the Kermack–McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 41–61.
[7]	K. Deng and Y. Wu, Dynamics of an SIS epidemic reaction–diffusion model, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 929–946.
[8]	J.Ge, K.I.Kim, Z.Lin, etal., ASISreaction-diffusion-advectionmodelinalow-riskandhigh-risk domain, J. Differ. Equations, 259 (2015), 5486–5509.
[9]	J. Ge, L. Lin and L. Zhang, A diffusive SIS epidemic model incorporating the media coverage impact in the heterogeneous environment, Discrete Contin. Dyn. Syst. Ser B, 22 (2017), 2763–2776.
[10]	W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. Ser. A, 115 (1927), 700–721.
[11]	K. Kousuke, H. Matsuzawa and R. Peng, Concentration profile of endemic equilibrium of a reaction-diffusion-advection SIS epidemic model, Calc. Var. Part. D. E., 56 (2017), 112.
[12]	A. Lahrouz and L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Stat. Probabil. Lett., 83 (2013), 960–968.
[13]	C. Lei, Z. Lin and Q. Zhang, The spreading front of invasive species in favorable habitat or unfavorable habitat, J. Differ. Equations, 257 (2014), 145–166.
[14]	B. Li, H. Li and Y. Tong, Analysis on a diffusive SIS epidemic model with logistic source, Z. Angew. Math. Phys., 68 (2017), 96.
[15]	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.
[16]	H. Li, R. Peng and F.-B. Wang, On a diffusive susceptible-infected-susceptible 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.
[17]	C.-S. Lin, W.-W. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differ. Equations, 72 (1988), 1–27.
[18]	Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differ. Equations, 131 (1996), 79–131.
[19]	X. Meng, S. Zhao, T. Feng, et al., Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis, J. Math. Anal. Appl., 433 (2016), 227–242.
[20]	R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction–diffusion model. Part I, J. Differ. Equations, 247 (4) (2009), 1096–1119.
[21]	R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction–diffusion model, Nonlinear Anal., 71 (4) (2009), 239–247.
[22]	R. Peng and F. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction–diffusion model: effects of epidemic risk and population movement, Phys. D, 259 (2013), 8–25.
[23]	R. Peng and X.-Q. Zhao, A reaction–diffusion SIS epidemic model in a time–periodic environment, Nonlinearity, 25 (2012), 1451–1471.
[24]	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.
[25]	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.
[26]	R. Xu and Z. Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear Anal., 10 (2009), 3175–3189.
[27]	F. Zhang, Z. Jin and G. Sun, Bifurcation analysis of a delayed epidemic model, Appl. Math. Comput., 216 (2010), 753–767.

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