Advanced Search
Article Contents
Article Contents

Global stability of a multi-group model with generalized nonlinear incidence and vaccination age

Abstract Related Papers Cited by
  • A multi-group epidemic model with general nonlinear incidence and vaccination age structure has been formulated and studied. Mathematical analysis shows that the global stability of disease-free equilibrium and endemic equilibrium of the model are determined by the basic reproduction number $\mathcal{R}_0$: the disease-free equilibrium is globally asymptotically stable if $\mathcal{R}_0<1$, the endemic equilibrium is globally asymptotically stable if $\mathcal{R}_0>1$. The Lyapunov functionals for the global dynamics of the multi-group model are constructed by applying the theory of non-negative matrices and a novel grouping technique in estimating the derivative.
    Mathematics Subject Classification: Primary: 92D25, 92D30; Secondary: 35B35, 37B25.


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

    J. Q. Li, Y. L. Yang and Y. C. Zhou, Global stability of an epidemic model with latent stage and vaccination, Nonlinear Anal.: Real World Appl., 12 (2011), 2163-2173.doi: 10.1016/j.nonrwa.2010.12.030.


    S. M. Blower and A. R. McLean, Prophylactic vaccines, risk behavior change, and the probability of eradicating HIV in San Francisco, Science, 265 (1994), 1451-1454.doi: 10.1126/science.8073289.


    Y. Xiao and S. Tang, Dynamics of infection with nonlinear incidence in a simple vaccination model, Nonlinear Anal.: Real World Appl., 11 (2010), 4154-4163.doi: 10.1016/j.nonrwa.2010.05.002.


    X. Y. Song, Y. Jiang and H. M. Wei, Analysis of a saturation incidence SVEIRS epidemic model with pulse and two time delays, Appl. Math. Comput., 214 (2009), 381-390.doi: 10.1016/j.amc.2009.04.005.


    D. Q. Ding and X. H. Ding, Global stability of multi-group vaccination epidemic models with delays, Nonlinear Anal.: Real World Appl., 12 (2011), 1991-1997.doi: 10.1016/j.nonrwa.2010.12.015.


    G. P. Sahu and J. Dhar, Analysis of an SVEIS epidemic model with partial temporary immunity and saturation incidence rate, Appl. Math. Model., 36 (2012), 908-923.doi: 10.1016/j.apm.2011.07.044.


    F. Hoppensteadt, An age-dependent epidemic model, J. Franklin Inst., 297 (1974), 325-333.doi: 10.1016/0016-0032(74)90037-4.


    F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics and Epiemics, SIAM Publications, Philadelphia, 1975.doi: 10.1137/1.9781611970487.


    M. Iannelli, M. Martcheva and X. Z. Li, Strain replacement in an epidemic model with super-infection and perfect vaccination, Math. Biosci., 195 (2005), 23-46.doi: 10.1016/j.mbs.2005.01.004.


    X. Z. Li, J. Wang and M. Ghosh, Stability and bifurcation of an SIVS epidemic model with treatment and age of vaccination, Appl. Math. Model., 34 (2010), 437-450.doi: 10.1016/j.apm.2009.06.002.


    X. C. Duan, S. L. Yuan and X. Z. Li, Global stability of an SVIR model with age of vaccination, Appl. Math. Comput., 226 (2014), 528-540.doi: 10.1016/j.amc.2013.10.073.


    G. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker. New York, 1985.


    P. Magal, Compact attractors for time periodic age-structured population models, Electron. J. Diff. Equs., 65 (2001), 1-35.


    Z. Liu, P. Magal and S. Ruan, Center-unstable manfold theorem for non-densely defined Cauchy problem, and the stability of bifurcation periodic orbits by Hopf bifurcation, Canadian Applied Mathematics Quarterly, 20 (2012), 135-178.


    P. Magal, C. C. McCluske and G. F. Webb, Liapunov functional and global asymptotic stability for an infection-age model, Applicable Analysis, 89 (2010), 1109-1140.doi: 10.1080/00036810903208122.


    H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, 118. American Mathematical Society, Providence, RI, 2011.doi: 10.1090/gsm/118.


    G. S. Wolkowicz, H. Xia and S. Ruan, Competition in the chemostat: A distributed delay model and its global asymptotic behavior, SIAM J. Appl. Math., 57 (1997), 1281-1310.doi: 10.1137/S0036139995289842.


    R. Y. Sun and J. P. Shi, Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates, Appl. Math. Comput., 218 (2011), 280-286.doi: 10.1016/j.amc.2011.05.056.


    H. Chen and J. T. Sun, Global stability of delay multigroup epidemic models with group mixing and nonlinear incidence rates, Appl. Math. Comput., 218 (2011), 4391-4400.doi: 10.1016/j.amc.2011.10.015.


    T. Kuniya, Global stability of a multi-group SVIR epidemic model, Nonlinear Anal.: Real World Appl., 14 (2013), 1135-1143.doi: 10.1016/j.nonrwa.2012.09.004.


    H. Y. Shu, D. J. Fan and J. J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Anal.: Real World Appl., 13 (2012), 1581-1592.doi: 10.1016/j.nonrwa.2011.11.016.


    H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canad. Appl. Math. Quart., 14 (2006), 259-284.


    H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802.doi: 10.1090/S0002-9939-08-09341-6.


    M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 1-20.doi: 10.1016/j.jde.2009.09.003.


    H. Guo and M. Y. Li, Impacts of migration and immigration on disease transmission dynamics in heterogeneous populations, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2413-2430.doi: 10.3934/dcdsb.2012.17.2413.


    A. Berman and R. J. Plemmons, Nonnegative Matrices in Mathematical Science, Academic Press, New York, 1979.


    J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, 1976.


    H. I. Freedman, M. X. Tang and S. G. Ruan, Uniform persistence and flows near a closed positively invariant set, J. Dyn. Differ. Equat., 6 (1994), 583-600.


    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.


    N. P. Bhatia and G. P. Szego, Dynamical Systems: Stability Theory and Applications, Lecture Notes in Mathematics, vol. 35, Springer, Berlin, 1967.


    H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995.doi: 10.1017/CBO9780511530043.

  • 加载中

Article Metrics

HTML views() PDF downloads(272) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint