# American Institute of Mathematical Sciences

• Previous Article
Analysis of a free boundary problem for avascular tumor growth with a periodic supply of nutrients
• DCDS-B Home
• This Issue
• Next Article
Oscillations of many interfaces in the near-shadow regime of two-component reaction-diffusion systems
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 and 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), 2163-2173. 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), 1451-1454. 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), 4154-4163. 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), 381-390. 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), 1991-1997. 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), 908-923. doi: 10.1016/j.apm.2011.07.044. [7] F. Hoppensteadt, An age-dependent epidemic model, J. Franklin Inst., 297 (1974), 325-333. doi: 10.1016/0016-0032(74)90037-4. [8] F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics and Epiemics, SIAM Publications, Philadelphia, 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), 23-46. 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), 437-450. 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), 528-540. 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), 1-35. [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), 135-178. [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), 1109-1140. doi: 10.1080/00036810903208122. [16] 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. [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), 1281-1310. 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), 280-286. 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), 4391-4400. 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), 1135-1143. 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), 1581-1592. 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), 259-284. [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), 2793-2802. 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), 1-20. 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), 2413-2430. doi: 10.3934/dcdsb.2012.17.2413. [26] A. Berman and R. J. Plemmons, Nonnegative Matrices in Mathematical Science, Academic Press, New York, 1979. [27] J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, 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), 583-600. [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), 191-213. 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, vol. 35, Springer, Berlin, 1967. [31] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 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), 2163-2173. 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), 1451-1454. 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), 4154-4163. 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), 381-390. 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), 1991-1997. 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), 908-923. doi: 10.1016/j.apm.2011.07.044. [7] F. Hoppensteadt, An age-dependent epidemic model, J. Franklin Inst., 297 (1974), 325-333. doi: 10.1016/0016-0032(74)90037-4. [8] F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics and Epiemics, SIAM Publications, Philadelphia, 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), 23-46. 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), 437-450. 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), 528-540. 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), 1-35. [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), 135-178. [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), 1109-1140. doi: 10.1080/00036810903208122. [16] 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. [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), 1281-1310. 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), 280-286. 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), 4391-4400. 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), 1135-1143. 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), 1581-1592. 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), 259-284. [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), 2793-2802. 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), 1-20. 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), 2413-2430. doi: 10.3934/dcdsb.2012.17.2413. [26] A. Berman and R. J. Plemmons, Nonnegative Matrices in Mathematical Science, Academic Press, New York, 1979. [27] J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, 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), 583-600. [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), 191-213. 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, vol. 35, Springer, Berlin, 1967. [31] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.
 [1] Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with vaccination age, distributed delay and random perturbation. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1083-1106. doi: 10.3934/mbe.2015.12.1083 [2] Yoshiaki Muroya, Toshikazu Kuniya, Yoichi Enatsu. Global stability of a delayed multi-group SIRS epidemic model with nonlinear incidence rates and relapse of infection. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3057-3091. doi: 10.3934/dcdsb.2015.20.3057 [3] Toshikazu Kuniya, Jinliang Wang, Hisashi Inaba. A multi-group SIR epidemic model with age structure. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3515-3550. doi: 10.3934/dcdsb.2016109 [4] Jinliang Wang, Xianning Liu, Toshikazu Kuniya, Jingmei Pang. Global stability for multi-group SIR and SEIR epidemic models with age-dependent susceptibility. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2795-2812. doi: 10.3934/dcdsb.2017151 [5] Gunduz Caginalp, Mark DeSantis. Multi-group asset flow equations and stability. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 109-150. doi: 10.3934/dcdsb.2011.16.109 [6] Chunmei Zhang, Wenxue Li, Ke Wang. Graph-theoretic approach to stability of multi-group models with dispersal. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 259-280. doi: 10.3934/dcdsb.2015.20.259 [7] Jinliang Wang, Hongying Shu. Global analysis on a class of multi-group SEIR model with latency and relapse. Mathematical Biosciences & Engineering, 2016, 13 (1) : 209-225. doi: 10.3934/mbe.2016.13.209 [8] Rui Wang, Xiaoyue Li, Denis S. Mukama. On stochastic multi-group Lotka-Volterra ecosystems with regime switching. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3499-3528. doi: 10.3934/dcdsb.2017177 [9] Toshikazu Kuniya, Yoshiaki Muroya. Global stability of a multi-group SIS epidemic model for population migration. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1105-1118. doi: 10.3934/dcdsb.2014.19.1105 [10] Jing Hui, Lansun Chen. Impulsive vaccination of sir epidemic models with nonlinear incidence rates. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 595-605. doi: 10.3934/dcdsb.2004.4.595 [11] Yoshiaki Muroya. A Lotka-Volterra system with patch structure (related to a multi-group SI epidemic model). Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 999-1008. doi: 10.3934/dcdss.2015.8.999 [12] Lili Liu, Xianning Liu, Jinliang Wang. Threshold dynamics of a delayed multi-group heroin epidemic model in heterogeneous populations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2615-2630. doi: 10.3934/dcdsb.2016064 [13] Xiaomei Feng, Zhidong Teng, Fengqin Zhang. Global dynamics of a general class of multi-group epidemic models with latency and relapse. Mathematical Biosciences & Engineering, 2015, 12 (1) : 99-115. doi: 10.3934/mbe.2015.12.99 [14] Andrei Korobeinikov, Philip K. Maini. A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Mathematical Biosciences & Engineering, 2004, 1 (1) : 57-60. doi: 10.3934/mbe.2004.1.57 [15] Xin Zhao, Tao Feng, Liang Wang, Zhipeng Qiu. Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6131-6154. doi: 10.3934/dcdsb.2021010 [16] Yoji Otani, Tsuyoshi Kajiwara, Toru Sasaki. Lyapunov functionals for multistrain models with infinite delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 507-536. doi: 10.3934/dcdsb.2017025 [17] Jinling Zhou, Yu Yang, Cheng-Hsiung Hsu. Traveling waves for a nonlocal dispersal vaccination model with general incidence. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1469-1495. doi: 10.3934/dcdsb.2019236 [18] Yu Yang, Jinling Zhou, Cheng-Hsiung Hsu. Critical traveling wave solutions for a vaccination model with general incidence. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1209-1225. doi: 10.3934/dcdsb.2021087 [19] Yoji Otani, Tsuyoshi Kajiwara, Toru Sasaki. Lyapunov functionals for virus-immune models with infinite delay. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3093-3114. doi: 10.3934/dcdsb.2015.20.3093 [20] Xichao Duan, Sanling Yuan, Kaifa Wang. Dynamics of a diffusive age-structured HBV model with saturating incidence. Mathematical Biosciences & Engineering, 2016, 13 (5) : 935-968. doi: 10.3934/mbe.2016024

2020 Impact Factor: 1.327