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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  November 2019 Early access  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 and 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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[8]

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

[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. 

[10]

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

[11]

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

[12]

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

[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. 

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[8]

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

[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. 

[10]

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

[11]

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

[12]

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

[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. 

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

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