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May  2016, 21(3): 977-996. doi: 10.3934/dcdsb.2016.21.977

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

 1 School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049

Received  June 2014 Revised  May 2015 Published  January 2016

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.
Citation: Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with generalized nonlinear incidence and vaccination age. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 977-996. doi: 10.3934/dcdsb.2016.21.977
##### References:
 [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. doi: 10.1016/j.nonrwa.2010.12.030. [2] 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. doi: 10.1126/science.8073289. [3] Y. Xiao and S. Tang, Dynamics of infection with nonlinear incidence in a simple vaccination model,, Nonlinear Anal.: Real World Appl., 11 (2010), 4154. doi: 10.1016/j.nonrwa.2010.05.002. [4] 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. doi: 10.1016/j.amc.2009.04.005. [5] 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. doi: 10.1016/j.nonrwa.2010.12.015. [6] 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. doi: 10.1016/j.apm.2011.07.044. [7] F. Hoppensteadt, An age-dependent epidemic model,, J. Franklin Inst., 297 (1974), 325. doi: 10.1016/0016-0032(74)90037-4. [8] F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics and Epiemics,, SIAM Publications, (1975). doi: 10.1137/1.9781611970487. [9] 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. doi: 10.1016/j.mbs.2005.01.004. [10] 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. doi: 10.1016/j.apm.2009.06.002. [11] 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. doi: 10.1016/j.amc.2013.10.073. [12] G. Webb, Theory of Nonlinear Age-Dependent Population Dynamics,, Marcel Dekker. New York, (1985). [13] P. Magal, Compact attractors for time periodic age-structured population models,, Electron. J. Diff. Equs., 65 (2001), 1. [14] 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. [15] 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. doi: 10.1080/00036810903208122. [16] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence,, Graduate Studies in Mathematics, (2011). doi: 10.1090/gsm/118. [17] 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. doi: 10.1137/S0036139995289842. [18] 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. doi: 10.1016/j.amc.2011.05.056. [19] 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. doi: 10.1016/j.amc.2011.10.015. [20] T. Kuniya, Global stability of a multi-group SVIR epidemic model,, Nonlinear Anal.: Real World Appl., 14 (2013), 1135. doi: 10.1016/j.nonrwa.2012.09.004. [21] 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. doi: 10.1016/j.nonrwa.2011.11.016. [22] 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. [23] 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. doi: 10.1090/S0002-9939-08-09341-6. [24] M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks,, J. Differential Equations, 248 (2010), 1. doi: 10.1016/j.jde.2009.09.003. [25] 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. doi: 10.3934/dcdsb.2012.17.2413. [26] A. Berman and R. J. Plemmons, Nonnegative Matrices in Mathematical Science,, Academic Press, (1979). [27] J. P. LaSalle, The Stability of Dynamical Systems,, SIAM, (1976). [28] 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. [29] 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. [30] N. P. Bhatia and G. P. Szego, Dynamical Systems: Stability Theory and Applications,, Lecture Notes in Mathematics, 35 (1967). [31] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511530043.

show all references

##### References:
 [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. doi: 10.1016/j.nonrwa.2010.12.030. [2] 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. doi: 10.1126/science.8073289. [3] Y. Xiao and S. Tang, Dynamics of infection with nonlinear incidence in a simple vaccination model,, Nonlinear Anal.: Real World Appl., 11 (2010), 4154. doi: 10.1016/j.nonrwa.2010.05.002. [4] 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. doi: 10.1016/j.amc.2009.04.005. [5] 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. doi: 10.1016/j.nonrwa.2010.12.015. [6] 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. doi: 10.1016/j.apm.2011.07.044. [7] F. Hoppensteadt, An age-dependent epidemic model,, J. Franklin Inst., 297 (1974), 325. doi: 10.1016/0016-0032(74)90037-4. [8] F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics and Epiemics,, SIAM Publications, (1975). doi: 10.1137/1.9781611970487. [9] 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. doi: 10.1016/j.mbs.2005.01.004. [10] 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. doi: 10.1016/j.apm.2009.06.002. [11] 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. doi: 10.1016/j.amc.2013.10.073. [12] G. Webb, Theory of Nonlinear Age-Dependent Population Dynamics,, Marcel Dekker. New York, (1985). [13] P. Magal, Compact attractors for time periodic age-structured population models,, Electron. J. Diff. Equs., 65 (2001), 1. [14] 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. [15] 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. doi: 10.1080/00036810903208122. [16] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence,, Graduate Studies in Mathematics, (2011). doi: 10.1090/gsm/118. [17] 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. doi: 10.1137/S0036139995289842. [18] 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. doi: 10.1016/j.amc.2011.05.056. [19] 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. doi: 10.1016/j.amc.2011.10.015. [20] T. Kuniya, Global stability of a multi-group SVIR epidemic model,, Nonlinear Anal.: Real World Appl., 14 (2013), 1135. doi: 10.1016/j.nonrwa.2012.09.004. [21] 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. doi: 10.1016/j.nonrwa.2011.11.016. [22] 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. [23] 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. doi: 10.1090/S0002-9939-08-09341-6. [24] M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks,, J. Differential Equations, 248 (2010), 1. doi: 10.1016/j.jde.2009.09.003. [25] 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. doi: 10.3934/dcdsb.2012.17.2413. [26] A. Berman and R. J. Plemmons, Nonnegative Matrices in Mathematical Science,, Academic Press, (1979). [27] J. P. LaSalle, The Stability of Dynamical Systems,, SIAM, (1976). [28] 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. [29] 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. [30] N. P. Bhatia and G. P. Szego, Dynamical Systems: Stability Theory and Applications,, Lecture Notes in Mathematics, 35 (1967). [31] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511530043.
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