\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Asymptotic behaviour of the nonautonomous SIR equations with diffusion

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 35B40 37B55; Secondary: 37N25 92D30.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

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

    [2]

    J. Arino, R. Jordan and P. van den Driessche, Quarantine in a multi-species epidemic model with spatial dynamics, Math. Biosci., 206 (2007), 46-60.doi: 10.1016/j.mbs.2005.09.002.

    [3]

    J. Arino, Diseases in metapopulations, in "Modeling and Dynamics of Infectious Diseases," Ser. Contemp. Appl. Math. CAM, Vol. 11, 64-122, Higher Education Press and World Scientific, Beijing Singapore, (2009).

    [4]

    F. Brauer, P. van den Driessche and Jianhong Wu (editors), "Mathematical Epidemiology," Springer Lecture Notes in Mathematics, Vol. 1945, Springer-Verlag, Heidelberg, 2008.doi: 10.1007/978-3-540-78911-6.

    [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., Springer, Berlin, (1989).doi: 10.1007/978-3-642-93454-4_14.

    [6]

    T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis, 64 (2006), 484-498.doi: 10.1016/j.na.2005.03.111.

    [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), 263-268.doi: 10.1016/j.crma.2005.12.015.

    [8]

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

    [9]

    C. Castillo-Chavez and H. R. Thieme, On the role of variable infectivity in the dynamics of the human immunodeficiency virus epidemic, in "Mathematical and Statistical Approaches to AIDS Epidemiology," Vol. 83, 157-176, Lecture Notes in Biomath., Springer, Berlin, (1989).doi: 10.1007/978-3-642-93454-4_7.

    [10]

    I. Chueshov, "Monotone Random Systems: Theory and Applications," Lecture Notes in Mathematics, Vol. 1779, Springer-Verlag, Berlin, 2002.

    [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), 363-378.doi: 10.1080/17513750701605465.

    [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), 92-101.

    [13]

    M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49.doi: 10.1016/0025-5564(77)90062-1.

    [14]

    D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, Vol. 840, Springer, Berlin, 1981.

    [15]

    G. Herzog and R. Redheffer, Nonautonomous SEIRS and Thron models for epidemiology and cell biology, Nonlinear Anal. RWA., 5 (2004), 33-44.doi: 10.1016/S1468-1218(02)00075-5.

    [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), 585-593.doi: 10.1007/BF00146381.

    [17]

    M. J. Keeling, P. Rohani and B. T. Grenfell, Seasonally forced disease dynamics explored as switching between attractors, Physica D, 148 (2001), 317-335.doi: 10.1016/S0167-2789(00)00187-1.

    [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), 700-721.doi: 10.1098/rspa.1927.0118.

    [19]

    P. E. Kloeden and V. Kozyakin, The dynamics of epidemiological systems with nonautonomous and random coefficients, Mathematics in Engineering, Science and Aerospace, 2 (2011), 105-118.

    [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), 163-181.doi: 10.1098/rspa.2006.1753.

    [21]

    P. E. Kloeden and M. Rasmussen, "Nonautonomous Dynamical Systems," Amer. Math. Soc., Providence, 2011.

    [22]

    F. Li and N. K. Yip, Long time behavior of some epidemic models, Discrete and Continuous Dynamical Systems, Series B, 16 (2011), 867-881.doi: 10.3934/dcdsb.2011.16.867.

    [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), 191-213.doi: 10.1016/S0025-5564(99)00030-9.

    [24]

    R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.doi: 10.2307/2001590.

    [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), 1-35.doi: 10.1515/crll.1991.413.1.

    [26]

    R. Peng and X. Q. 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.

    [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), 289-337.doi: 10.1016/j.jde.2007.03.028.

    [28]

    H. L. Smith and H. R. Thieme, "Dynamical Systems and Population Persistence," Graduate Studies in Mathematics, 118. American Mathematical Society, Providence, RI, 2011.

    [29]

    L. Stone, R. Olinky and A. Huppert, Seasonal dynamics of recurrent epidemics, Nature, 446 (2007), 533-536.doi: 10.1038/nature05638.

    [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), 237-268.doi: 10.1016/0022-0396(88)90107-6.

    [31]

    H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066.

    [32]

    H. R. Thieme, Uniform weak implies uniform strong persistence for non-autonomous semiflows, Proc. Am. Math. Soc., 127 (1999), 2395-2403.

    [33]

    H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosci., 166 (2000), 173-201.doi: 10.1016/S0025-5564(00)00018-3.

    [34]

    H. R. Thieme, "Mathematics in Population Biology," Princeton University Press, Princeton, 2003.

    [35]

    Y. Wang, C. K. Zhong and S. Zhou, Pullback attractors of nonautonomous dynamical systems, Discrete and Continuous Dynamical Systems, 16 (2006), 587-614.doi: 10.3934/dcds.2006.16.587.

    [36]

    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.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(133) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return