January  2022, 21(1): 315-336. doi: 10.3934/cpaa.2021179

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 Published  January 2022 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 and Applied Analysis, 2022, 21 (1) : 315-336. doi: 10.3934/cpaa.2021179
References:
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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.

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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.

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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

[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.

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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.

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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.

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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.

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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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

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