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January  2014, 13(1): 157-173. doi: 10.3934/cpaa.2014.13.157

Asymptotic behaviour of the nonautonomous SIR equations with diffusion

1. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080-Sevilla

2. 

FB Mathematik, Johann Wolfgang Goethe Universität, Postfach 11 19 32, D-60054 Frankfurt a.M.

Received  August 2012 Revised  April 2013 Published  July 2013

The existence and uniqueness of positive solutions of a nonautonomous system of SIR equations with diffusion are established as well as the continuous dependence of such solutions on initial data. The proofs are facilitated by the fact that the nonlinear coefficients satisfy a global Lipschitz property due to their special structure. An explicit disease-free nonautonomous equilibrium solution is determined and its stability investigated. Uniform weak disease persistence is also shown. The main aim of the paper is to establish the existence of a nonautonomous pullback attractor is established for the nonautonomous process generated by the equations on the positive cone of an appropriate function space. For this an energy method is used to determine a pullback absorbing set and then the flattening property is verified, thus giving the required asymptotic compactness of the process.
Citation: María Anguiano, P.E. Kloeden. Asymptotic behaviour of the nonautonomous SIR equations with diffusion. Communications on Pure & Applied Analysis, 2014, 13 (1) : 157-173. doi: 10.3934/cpaa.2014.13.157
References:
[1]

R. M. Anderson and R. M. May, "Infectious Disease of Humans, Dynamics and Control,", Oxford University Press, (1992).   Google Scholar

[2]

J. Arino, R. Jordan and P. van den Driessche, Quarantine in a multi-species epidemic model with spatial dynamics,, Math. Biosci., 206 (2007), 46.  doi: 10.1016/j.mbs.2005.09.002.  Google Scholar

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T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for non-autonomous 2D Navier-Stokes equations in unbounded domains,, C.R. Math. Acad. Sci. Paris, 342 (2006), 263.  doi: 10.1016/j.crma.2005.12.015.  Google Scholar

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A. N. Carvalho, J. A. Langa and J. C. Robinson, "Attractors for Infinite-Dimensional Non-Autonomous Systems,", Springer-Verlag, (2013).  doi: 10.1007/978-1-4614-4581-4.  Google Scholar

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I. Chueshov, "Monotone Random Systems: Theory and Applications,", Lecture Notes in Mathematics, (1779).   Google Scholar

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T. Dhirasakdanon, H. R. Thieme and P. van den Driessche, A sharp threshold for disease persistence in host metapopulations,, J. Biol. Dyn., 1 (2007), 363.  doi: 10.1080/17513750701605465.  Google Scholar

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M. A. Efendiev and H. J. Eberl, On positivity of solutions of semi-linear convection-diffusion-reaction systems, with applications in ecology and environmental engineering,, RIMS Kyoto Kokyuroko, 1542 (2007), 92.   Google Scholar

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D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, (1981).   Google Scholar

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G. Herzog and R. Redheffer, Nonautonomous SEIRS and Thron models for epidemiology and cell biology,, Nonlinear Anal. RWA., 5 (2004), 33.  doi: 10.1016/S1468-1218(02)00075-5.  Google Scholar

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M. Iannelli, R. Loro, F. A. Milner, A. Pugliese and G. Rabbiolo, An AIDS model with distributed incubation and variable infectiousness: applications to IV drug users in Latium, Italy,, Eur. J. Epidemiol., 8 (1992), 585.  doi: 10.1007/BF00146381.  Google Scholar

[17]

M. J. Keeling, P. Rohani and B. T. Grenfell, Seasonally forced disease dynamics explored as switching between attractors,, Physica D, 148 (2001), 317.  doi: 10.1016/S0167-2789(00)00187-1.  Google Scholar

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W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics (part I),, Proc. R. Soc. Lond. Ser. A, 115 (1927), 700.  doi: 10.1098/rspa.1927.0118.  Google Scholar

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P. E. Kloeden and V. Kozyakin, The dynamics of epidemiological systems with nonautonomous and random coefficients,, Mathematics in Engineering, 2 (2011), 105.   Google Scholar

[20]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors,, Proceedings of the Royal Society of London. Series A., 463 (2007), 163.  doi: 10.1098/rspa.2006.1753.  Google Scholar

[21]

P. E. Kloeden and M. Rasmussen, "Nonautonomous Dynamical Systems,", Amer. Math. Soc., (2011).   Google Scholar

[22]

F. Li and N. K. Yip, Long time behavior of some epidemic models,, Discrete and Continuous Dynamical Systems, 16 (2011), 867.  doi: 10.3934/dcdsb.2011.16.867.  Google Scholar

[23]

M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size,, Math. Biosci., 160 (1999), 191.  doi: 10.1016/S0025-5564(99)00030-9.  Google Scholar

[24]

R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems,, Trans. Amer. Math. Soc., 321 (1990), 1.  doi: 10.2307/2001590.  Google Scholar

[25]

R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence,, J. Reine Angew. Math., 413 (1991), 1.  doi: 10.1515/crll.1991.413.1.  Google Scholar

[26]

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

[27]

J. C. Robinson, A. Rodríguez-Bernal and A. Vidal-López, Pullback attractors and extremal complete trajectories for non-autonomous reaction-diffusion problems,, J. Differential Equations, 238 (2007), 289.  doi: 10.1016/j.jde.2007.03.028.  Google Scholar

[28]

H. L. Smith and H. R. Thieme, "Dynamical Systems and Population Persistence,", Graduate Studies in Mathematics, (2011).   Google Scholar

[29]

L. Stone, R. Olinky and A. Huppert, Seasonal dynamics of recurrent epidemics,, Nature, 446 (2007), 533.  doi: 10.1038/nature05638.  Google Scholar

[30]

H. R. Thieme, Asymptotic proportionality (weak ergodicity) and conditional asymptotic equality of solutions to time-heterogeneous sublinear difference and differential equations,, J. Diff. Equations, 73 (1988), 237.  doi: 10.1016/0022-0396(88)90107-6.  Google Scholar

[31]

H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators,, Differential Integral Equations, 3 (1990), 1035.   Google Scholar

[32]

H. R. Thieme, Uniform weak implies uniform strong persistence for non-autonomous semiflows,, Proc. Am. Math. Soc., 127 (1999), 2395.   Google Scholar

[33]

H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology,, Math. Biosci., 166 (2000), 173.  doi: 10.1016/S0025-5564(00)00018-3.  Google Scholar

[34]

H. R. Thieme, "Mathematics in Population Biology,", Princeton University Press, (2003).   Google Scholar

[35]

Y. Wang, C. K. Zhong and S. Zhou, Pullback attractors of nonautonomous dynamical systems,, Discrete and Continuous Dynamical Systems, 16 (2006), 587.  doi: 10.3934/dcds.2006.16.587.  Google Scholar

[36]

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

show all references

References:
[1]

R. M. Anderson and R. M. May, "Infectious Disease of Humans, Dynamics and Control,", Oxford University Press, (1992).   Google Scholar

[2]

J. Arino, R. Jordan and P. van den Driessche, Quarantine in a multi-species epidemic model with spatial dynamics,, Math. Biosci., 206 (2007), 46.  doi: 10.1016/j.mbs.2005.09.002.  Google Scholar

[3]

J. Arino, Diseases in metapopulations,, in, (2009), 64.   Google Scholar

[4]

F. Brauer, P. van den Driessche and Jianhong Wu (editors), "Mathematical Epidemiology,", Springer Lecture Notes in Mathematics, (1945).  doi: 10.1007/978-3-540-78911-6.  Google Scholar

[5]

S. Busenberg and C. Castillo-Chavez, Interaction, pair formation and force of infection terms in sexually transmitted diseases, in "Mathematical and Statistical Approaches to AIDS Epidemiology," Vol. 83, 289锟?300,, Lecture Notes in Biomath., (1989).  doi: 10.1007/978-3-642-93454-4_14.  Google Scholar

[6]

T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems,, Nonlinear Analysis, 64 (2006), 484.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[7]

T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for non-autonomous 2D Navier-Stokes equations in unbounded domains,, C.R. Math. Acad. Sci. Paris, 342 (2006), 263.  doi: 10.1016/j.crma.2005.12.015.  Google Scholar

[8]

A. N. Carvalho, J. A. Langa and J. C. Robinson, "Attractors for Infinite-Dimensional Non-Autonomous Systems,", Springer-Verlag, (2013).  doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[9]

C. Castillo-Chavez and H. R. Thieme, On the role of variable infectivity in the dynamics of the human immunodeficiency virus epidemic,, in, (1989), 157.  doi: 10.1007/978-3-642-93454-4_7.  Google Scholar

[10]

I. Chueshov, "Monotone Random Systems: Theory and Applications,", Lecture Notes in Mathematics, (1779).   Google Scholar

[11]

T. Dhirasakdanon, H. R. Thieme and P. van den Driessche, A sharp threshold for disease persistence in host metapopulations,, J. Biol. Dyn., 1 (2007), 363.  doi: 10.1080/17513750701605465.  Google Scholar

[12]

M. A. Efendiev and H. J. Eberl, On positivity of solutions of semi-linear convection-diffusion-reaction systems, with applications in ecology and environmental engineering,, RIMS Kyoto Kokyuroko, 1542 (2007), 92.   Google Scholar

[13]

M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations,, Math. Biosci., 33 (1977), 35.  doi: 10.1016/0025-5564(77)90062-1.  Google Scholar

[14]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, (1981).   Google Scholar

[15]

G. Herzog and R. Redheffer, Nonautonomous SEIRS and Thron models for epidemiology and cell biology,, Nonlinear Anal. RWA., 5 (2004), 33.  doi: 10.1016/S1468-1218(02)00075-5.  Google Scholar

[16]

M. Iannelli, R. Loro, F. A. Milner, A. Pugliese and G. Rabbiolo, An AIDS model with distributed incubation and variable infectiousness: applications to IV drug users in Latium, Italy,, Eur. J. Epidemiol., 8 (1992), 585.  doi: 10.1007/BF00146381.  Google Scholar

[17]

M. J. Keeling, P. Rohani and B. T. Grenfell, Seasonally forced disease dynamics explored as switching between attractors,, Physica D, 148 (2001), 317.  doi: 10.1016/S0167-2789(00)00187-1.  Google Scholar

[18]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics (part I),, Proc. R. Soc. Lond. Ser. A, 115 (1927), 700.  doi: 10.1098/rspa.1927.0118.  Google Scholar

[19]

P. E. Kloeden and V. Kozyakin, The dynamics of epidemiological systems with nonautonomous and random coefficients,, Mathematics in Engineering, 2 (2011), 105.   Google Scholar

[20]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors,, Proceedings of the Royal Society of London. Series A., 463 (2007), 163.  doi: 10.1098/rspa.2006.1753.  Google Scholar

[21]

P. E. Kloeden and M. Rasmussen, "Nonautonomous Dynamical Systems,", Amer. Math. Soc., (2011).   Google Scholar

[22]

F. Li and N. K. Yip, Long time behavior of some epidemic models,, Discrete and Continuous Dynamical Systems, 16 (2011), 867.  doi: 10.3934/dcdsb.2011.16.867.  Google Scholar

[23]

M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size,, Math. Biosci., 160 (1999), 191.  doi: 10.1016/S0025-5564(99)00030-9.  Google Scholar

[24]

R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems,, Trans. Amer. Math. Soc., 321 (1990), 1.  doi: 10.2307/2001590.  Google Scholar

[25]

R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence,, J. Reine Angew. Math., 413 (1991), 1.  doi: 10.1515/crll.1991.413.1.  Google Scholar

[26]

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

[27]

J. C. Robinson, A. Rodríguez-Bernal and A. Vidal-López, Pullback attractors and extremal complete trajectories for non-autonomous reaction-diffusion problems,, J. Differential Equations, 238 (2007), 289.  doi: 10.1016/j.jde.2007.03.028.  Google Scholar

[28]

H. L. Smith and H. R. Thieme, "Dynamical Systems and Population Persistence,", Graduate Studies in Mathematics, (2011).   Google Scholar

[29]

L. Stone, R. Olinky and A. Huppert, Seasonal dynamics of recurrent epidemics,, Nature, 446 (2007), 533.  doi: 10.1038/nature05638.  Google Scholar

[30]

H. R. Thieme, Asymptotic proportionality (weak ergodicity) and conditional asymptotic equality of solutions to time-heterogeneous sublinear difference and differential equations,, J. Diff. Equations, 73 (1988), 237.  doi: 10.1016/0022-0396(88)90107-6.  Google Scholar

[31]

H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators,, Differential Integral Equations, 3 (1990), 1035.   Google Scholar

[32]

H. R. Thieme, Uniform weak implies uniform strong persistence for non-autonomous semiflows,, Proc. Am. Math. Soc., 127 (1999), 2395.   Google Scholar

[33]

H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology,, Math. Biosci., 166 (2000), 173.  doi: 10.1016/S0025-5564(00)00018-3.  Google Scholar

[34]

H. R. Thieme, "Mathematics in Population Biology,", Princeton University Press, (2003).   Google Scholar

[35]

Y. Wang, C. K. Zhong and S. Zhou, Pullback attractors of nonautonomous dynamical systems,, Discrete and Continuous Dynamical Systems, 16 (2006), 587.  doi: 10.3934/dcds.2006.16.587.  Google Scholar

[36]

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

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