doi: 10.3934/cpaa.2021179
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Dynamics of an SIRS epidemic model with cross-diffusion

1. 

School of Mathematical Sciences, Harbin Normal University, Harbin, Heilongjiang, 150025, China

2. 

School of Mathematical Sciences and Y.Y. Tseng Functional Analysis Research Center, Harbin Normal University, Harbin, Heilongjiang, 150025, China

* Corresponding author

Received  June 2021 Revised  September 2021 Early access October 2021

Fund Project: The authors are supported by National NSFC grant 11971135, NSFH grant LH2019A017, 2018-KYYWF-0999, XKYQ201403 and HSDSSCX2021-12

The dynamical behavior of an SIRS epidemic reaction-diffusion model with frequency-dependent mechanism in a spatially heterogeneous environment is studied, with a chemotaxis effect that susceptible individuals tend to move away from higher concentration of infected individuals. Regardless of the strength of the chemotactic coefficient and the spatial dimension $ n $, it is established the unique global classical solution which is uniformly-in-time bounded. The model still recognizes the threshold dynamics in terms of the basic reproduction number $ \mathcal{R}_{0} $ even in the case of chemotaxis effects: if $ \mathcal{R}_{0}<1 $, the unique disease free equilibrium is globally stable; if $ \mathcal{R}_{0}>1 $, the disease is uniformly persistent and there is at least one endemic equilibrium, which is globally stable in some special cases with weak chemotactic sensitivity. We also show the asymptotic profile of endemic equilibria (when exists) if the diffusion (migration) rate of the susceptible is small, which indicates that the disease always exists in the entire habitat in this case. Our results suggest that one cannot eradicate the SIRS disease model by only controlling the diffusion rate of susceptible individuals.

Citation: Yaru Hu, Jinfeng Wang. Dynamics of an SIRS epidemic model with cross-diffusion. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021179
References:
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[24]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differ. Equ., 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar

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J. F. WangS. N. Wu and J. P. Shi, Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 1273-1289.  doi: 10.3934/dcdsb.2020162.  Google Scholar

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show all references

References:
[1]

N. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differ. Equ., 33 (1979), 201-225.  doi: 10.1016/0022-0396(79)90088-3.  Google Scholar

[2]

L. J. S. AllenB. M. BolkerY. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.  Google Scholar

[3]

R. Anderson and R. May, Population biology of infectious diseases, Nature, 280 (1979), 361-367.   Google Scholar

[4]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar

[5]

X. R. Cao, Global bounded solutions of the higer-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891.  Google Scholar

[6]

R. H. CuiK. -Y Lam and Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J. Differ. Equ., 263 (2017), 2343-2373.  doi: 10.1016/j.jde.2017.03.045.  Google Scholar

[7]

M. De Jong, O. Diekmann and H. Heesterbeek, How Does Transmission of Infection Depend on Population Size? In Epidemic Models. Their Structure and Relation to Data, Cambridge University Press, New York, (1995), 84–89. Google Scholar

[8]

S. Y. Han, C. X. Lei and X. Y. Zhang, Qualitative analysis on a diffusive SIRS epidemic model with standard incidence infection mechanism, Z. Angew. Math. Phys., 71 (2020). doi: 10.1007/s00033-020-01418-1.  Google Scholar

[9]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[10]

W. J$\ddot{a}$ger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.   Google Scholar

[11]

H. Y. Jin and T. Xiang, Boundedness and exponential convergence in a chemotaxis model for tumor invasion, Nonlinearity, 29 (2016), 3579-3596.  doi: 10.1088/0951-7715/29/12/3579.  Google Scholar

[12]

W. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. II-The problem of endemicity, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 115 (1927), 700-721.  doi: 10.1098/rspa.1932.0171.  Google Scholar

[13]

K. KutoH. Matsuzawa and R. Peng, Concentration profile of endemic equilibrium of a reaction-diffusion-advection SIS epidemic model, Calc. Var. Partial Differ. Equations, 56 (4) (2017), 112.  doi: 10.1007/s00526-017-1207-8.  Google Scholar

[14]

B. Li and Q. Y. Bie, Long-time dynamics of an SIRS reaction-diffusion epidemic model, J. Math. Anal. Appl., 475 (2019), 1910-1926.  doi: 10.1016/j.jmaa.2019.03.062.  Google Scholar

[15]

H. C. LiR. Peng and T. Xiang, Dynamics and asymptotic profiles of endemic equilibrium for two frequency-dependent SIS epidemic models with cross-diffusion, European J. Appl. Math., 31 (2020), 26-56.  doi: 10.1017/S0956792518000463.  Google Scholar

[16]

G. 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.  Google Scholar

[17]

P. Magal and X. Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar

[18]

R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model, I. J. Differ. Equ., 247 (2009), 1096-1119.  doi: 10.1016/j.jde.2009.05.002.  Google Scholar

[19]

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.  Google Scholar

[20]

R. Peng and X. Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.   Google Scholar

[21]

Y. S. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443.  Google Scholar

[22]

C. Vargas-De-Le$\acute{o}$n, On the global stability of SIS, SIR, and SIRS epidemic models with standard incidence, Chaos Solitons Fractals, 44 (2011), 1106-1110.  doi: 10.1016/j.chaos.2011.09.002.  Google Scholar

[23]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[24]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differ. Equ., 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar

[25]

J. F. WangS. N. Wu and J. P. Shi, Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 1273-1289.  doi: 10.3934/dcdsb.2020162.  Google Scholar

[26]

Y. X. Wu and X. F. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J. Differ. Equ., 261 (2016), 4424-4447.  doi: 10.1016/j.jde.2016.06.028.  Google Scholar

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