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November  2019, 24(11): 5945-5957. doi: 10.3934/dcdsb.2019114

Asymptotic behavior of an SIR reaction-diffusion model with a linear source

Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504, USA

Received  October 2018 Revised  December 2018 Published  June 2019

In this paper, we consider an SIR reaction-diffusion model with a linear external source in spatially heterogeneous environment. We first study the global stability of the disease-free equilibrium in spatially heterogeneous environment and the global stability of the endemic equilibrium in spatially homogeneous environment. We then investigate the asymptotic profiles of the endemic equilibrium in spatially heterogeneous environment for small and large diffusion rates.

Citation: Keng Deng. Asymptotic behavior of an SIR reaction-diffusion model with a linear source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5945-5957. doi: 10.3934/dcdsb.2019114
References:
[1]

N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differential Equations, 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. Dynam. Systems, 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.  Google Scholar

[3]

F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-3516-1.  Google Scholar

[4]

H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.  Google Scholar

[5]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Wiley, Chichester, West Sussex, UK, 2003. doi: 10.1002/0470871296.  Google Scholar

[6]

S. Chinviriyasit and W. Chinviriyasit, Numerical modelling of an SIR epidemic model with diffusion, Appl. Math. Comput., 216 (2010), 395-409.  doi: 10.1016/j.amc.2010.01.028.  Google Scholar

[7]

K. Deng and Y. Wu, Dynamics of a susceptible-infected-susceptible epidemic reaction-diffusion model, Proc. Royal Soc. Edinburgh Sect. A, 146 (2016), 929-946.  doi: 10.1017/S0308210515000864.  Google Scholar

[8]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.  Google Scholar

[9]

W. E. Fitzgibbon and J. J. Morgan, A diffusive epidemic model on a bounded domain of arbitrary dimension, Differential Integral Equations, 1 (1988), 125-132.   Google Scholar

[10]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988.  Google Scholar

[11]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, 1981.  Google Scholar

[13]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics-Ⅰ, Proc. Roy. Soc. London Ser. A, 115 (1927), 700-721.   Google Scholar

[14]

T. Kuniya and J. Wang, Lyapunov functions and global stability for a spatially diffusive SIR epidemic model, Appl. Anal., 96 (2017), 1935-1960.  doi: 10.1080/00036811.2016.1199796.  Google Scholar

[15]

H. LiR. Peng and Z.-A. Wang, On a diffusive SIS 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.  doi: 10.1137/18M1167863.  Google Scholar

[16]

C.-S. LinW.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.  Google Scholar

[17]

Y. Lou and T. Nagylaki, Evolution of a semilinear parabolic system for migration and selection without dominance, J. Differential Equations, 225 (2006), 624-665.  doi: 10.1016/j.jde.2006.01.012.  Google Scholar

[18]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.  Google Scholar

[19] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.   Google Scholar
[20]

R. PengJ. Shi and M. 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

[21]

R. Peng and X. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.  doi: 10.1088/0951-7715/25/5/1451.  Google Scholar

[22]

G. F. Webb, A reaction-diffusion model for a deterministic diffusive epidemic, J. Math. Anal. Appl., 84 (1981), 150-161.  doi: 10.1016/0022-247X(81)90156-6.  Google Scholar

show all references

References:
[1]

N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differential Equations, 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. Dynam. Systems, 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.  Google Scholar

[3]

F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-3516-1.  Google Scholar

[4]

H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.  Google Scholar

[5]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Wiley, Chichester, West Sussex, UK, 2003. doi: 10.1002/0470871296.  Google Scholar

[6]

S. Chinviriyasit and W. Chinviriyasit, Numerical modelling of an SIR epidemic model with diffusion, Appl. Math. Comput., 216 (2010), 395-409.  doi: 10.1016/j.amc.2010.01.028.  Google Scholar

[7]

K. Deng and Y. Wu, Dynamics of a susceptible-infected-susceptible epidemic reaction-diffusion model, Proc. Royal Soc. Edinburgh Sect. A, 146 (2016), 929-946.  doi: 10.1017/S0308210515000864.  Google Scholar

[8]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.  Google Scholar

[9]

W. E. Fitzgibbon and J. J. Morgan, A diffusive epidemic model on a bounded domain of arbitrary dimension, Differential Integral Equations, 1 (1988), 125-132.   Google Scholar

[10]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988.  Google Scholar

[11]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, 1981.  Google Scholar

[13]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics-Ⅰ, Proc. Roy. Soc. London Ser. A, 115 (1927), 700-721.   Google Scholar

[14]

T. Kuniya and J. Wang, Lyapunov functions and global stability for a spatially diffusive SIR epidemic model, Appl. Anal., 96 (2017), 1935-1960.  doi: 10.1080/00036811.2016.1199796.  Google Scholar

[15]

H. LiR. Peng and Z.-A. Wang, On a diffusive SIS 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.  doi: 10.1137/18M1167863.  Google Scholar

[16]

C.-S. LinW.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.  Google Scholar

[17]

Y. Lou and T. Nagylaki, Evolution of a semilinear parabolic system for migration and selection without dominance, J. Differential Equations, 225 (2006), 624-665.  doi: 10.1016/j.jde.2006.01.012.  Google Scholar

[18]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.  Google Scholar

[19] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.   Google Scholar
[20]

R. PengJ. Shi and M. 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

[21]

R. Peng and X. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.  doi: 10.1088/0951-7715/25/5/1451.  Google Scholar

[22]

G. F. Webb, A reaction-diffusion model for a deterministic diffusive epidemic, J. Math. Anal. Appl., 84 (1981), 150-161.  doi: 10.1016/0022-247X(81)90156-6.  Google Scholar

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