# American Institute of Mathematical Sciences

• Previous Article
Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony with volume filling
• DCDS-B Home
• This Issue
• Next Article
Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems
September  2017, 22(7): 2795-2812. doi: 10.3934/dcdsb.2017151

## Global stability for multi-group SIR and SEIR epidemic models with age-dependent susceptibility

 1 School of Mathematics and Statistics, Southwest University, Chongqing 400715, China 2 China, School of Mathematical Science, Heilongjiang University, Harbin 150080, China 3 Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe-shi, Hyogo 657-0067, Japan

corresponding author: liuxn@swu.edu.cn (X. Liu)

Received  September 2013 Revised  March 2017 Published  May 2017

In this paper, we investigate the global asymptotic stability of multi-group SIR and SEIR age-structured models. These models allow the infectiousness and the death rate of susceptible individuals to vary and depend on the susceptibility, with which we can consider the heterogeneity of population. We establish global dynamics and demonstrate that the heterogeneity does not alter the dynamical structure of the basic SIR and SEIR with age-dependent susceptibility. Our results also demonstrate that, for age structured multi-group models considered, the graph-theoretic approach can be successfully applied by choosing an appropriate weighted matrix as well.

Citation: Jinliang Wang, Xianning Liu, Toshikazu Kuniya, Jingmei Pang. Global stability for multi-group SIR and SEIR epidemic models with age-dependent susceptibility. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2795-2812. doi: 10.3934/dcdsb.2017151
##### References:
 [1] V. Andreasen, Instability in an SIR-model with age-dependent susceptibility, in: O. Arino, D. Axelrod, M. Kimmel, M. Langlais (Eds.), Mathematical Population Dynamics: Analysis of Heterogeneity, in: Theory of Epidemics, Wuerz Publ., Winnipeg, (1995), 3-14. Google Scholar [2] H. Andersson and T. Britton, Heterogeneity in epidemic models and its effect on the spread of infection, J. App. Prob., 35 (1998), 651-661. doi: 10.1017/S0021900200016302. Google Scholar [3] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control Oxford University Press, Oxford, 1991.Google Scholar [4] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences Academic Press, New York, 1979. Google Scholar [5] S. Busenberg and K. Cooke, Vertically Transmitted Diseases: Models and Dynamics Springer, Berlin, 1993. doi: 10.1007/978-3-642-75301-5. Google Scholar [6] S. N. Busenberg, M. Iannelli and H. R. Thieme, Global behavior of an age-structured epidemic model, SIAM J. Math. Anal., 22 (1991), 1065-1080. doi: 10.1137/0522069. Google Scholar [7] Y. Cha, M. Iannelli and F. A. Milner, Stability change of an epidemic model, Dynam. Systems Appl., 9 (2000), 361-376. Google Scholar [8] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation John Wiley and Sons, Chichester, 2000. Google Scholar [9] 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. Google Scholar [10] 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. Google Scholar [11] T. J. Hagenaars, C. A. Donnelly and N. M. Ferguson, Spatial heterogeneity and the persistence of infectious diseases, J. Theor. Biol., 229 (2004), 349-359. doi: 10.1016/j.jtbi.2004.04.002. Google Scholar [12] J. K. Hale, Asymptotic Behavior of Dissipative Systems in: Math. Surv. Monogr. , vol. 25, Am. Math. Soc. , Providence, RI, 1988. Google Scholar [13] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. Google Scholar [14] J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci., 155 (1999), 77-109. doi: 10.1016/S0025-5564(98)10057-3. Google Scholar [15] H. Inaba, Endemic threshold results in an age-duration-structured population model for HIV infection, Math. Biosci., 201 (2006), 15-47. doi: 10.1016/j.mbs.2005.12.017. Google Scholar [16] H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411-434. doi: 10.1007/BF00178326. Google Scholar [17] A. Korobeinikov, P. K. Maini and W. J. Walker, Estimation of effective vaccination rate: Pertussis in New Zealand as a case study, J. Theor. Biol., 224 (2003), 269-275. doi: 10.1016/S0022-5193(03)00163-2. Google Scholar [18] T. Kuniya, Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model, Nonlinear Anal. RWA, 12 (2011), 2640-2655. doi: 10.1016/j.nonrwa.2011.03.011. Google Scholar [19] A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5. Google Scholar [20] M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 38-47. doi: 10.1016/j.jmaa.2009.09.017. Google Scholar [21] 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. Google Scholar [22] A. L. Lloyd and R. M. May, Spatial heterogeneity in epidemic models, J. Theor. Biol., 179 (1996), 1-11. doi: 10.1006/jtbi.1996.0042. Google Scholar [23] P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140. doi: 10.1080/00036810903208122. Google Scholar [24] P. Magal, Compact attractors for time periodic age-structured population models, Electron. J. Differential Equations, 65 (2001), 1-35. Google Scholar [25] P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models Memoirs of the American Mathematical Society, 202 (2009), vi+71 pp. doi: 10.1090/S0065-9266-09-00568-7. Google Scholar [26] P. Magal and H. R. Thieme, Eventual compactness for semiflows generated by nonlinear age-structured models, Commun. Pure Appl. Anal., 3 (2004), 695-727. doi: 10.3934/cpaa.2004.3.695. Google Scholar [27] A. G. McKendrick, Applications of mathematics to medical problems, Proc. Edinb. Math. Soc., 44 (1925), 98-130. doi: 10.1017/S0013091500034428. Google Scholar [28] C. C. McCluskey, Delay versus age-of-infection-Global stability, Applied Math. Comput., 217 (2010), 3046-3049. doi: 10.1016/j.amc.2010.08.037. Google Scholar [29] A. V. Melnik and A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility, Math. Biosci. Eng., 10 (2013), 369-378. doi: 10.3934/mbe.2013.10.369. Google Scholar [30] D. Schenzle, An age-structured model of pre-and post-vaccination measles transmission, IMA J. Math. Appl. Med. Biol., 1 (1984), 169-191. doi: 10.1093/imammb/1.2.169. Google Scholar [31] H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Anal. RWA, 13 (2012), 1581-1592. doi: 10.1016/j.nonrwa.2011.11.016. Google Scholar [32] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence Amer. Math. Soc. , Providence, 2011. Google Scholar [33] R. Sun, Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence, Comput. Math. Appl., 60 (2010), 2286-2291. doi: 10.1016/j.camwa.2010.08.020. Google Scholar [34] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066. Google Scholar [35] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [36] J. Wang, J. Zu, X. Liu, G. Huang and J. Zhang, Global dynamics of a multi-group epidemic model with general relapse distribution and nonlinear incidence rate, J. Biol. Syst., 20 (2012), 235-258. doi: 10.1142/S021833901250009X. Google Scholar [37] J. Wang, X. Liu, J. Pang and D. Hou, Global dynamics of a multi-group epidemic model with general exposed distribution and relapse, Osaka J. Math., 52 (2015), 117-138. Google Scholar [38] Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear Anal. RWA, 11 (2010), 995-1004. doi: 10.1016/j.nonrwa.2009.01.040. Google Scholar [39] G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics Marcel Dekker, New York, 1985. Google Scholar

show all references

##### References:
 [1] V. Andreasen, Instability in an SIR-model with age-dependent susceptibility, in: O. Arino, D. Axelrod, M. Kimmel, M. Langlais (Eds.), Mathematical Population Dynamics: Analysis of Heterogeneity, in: Theory of Epidemics, Wuerz Publ., Winnipeg, (1995), 3-14. Google Scholar [2] H. Andersson and T. Britton, Heterogeneity in epidemic models and its effect on the spread of infection, J. App. Prob., 35 (1998), 651-661. doi: 10.1017/S0021900200016302. Google Scholar [3] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control Oxford University Press, Oxford, 1991.Google Scholar [4] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences Academic Press, New York, 1979. Google Scholar [5] S. Busenberg and K. Cooke, Vertically Transmitted Diseases: Models and Dynamics Springer, Berlin, 1993. doi: 10.1007/978-3-642-75301-5. Google Scholar [6] S. N. Busenberg, M. Iannelli and H. R. Thieme, Global behavior of an age-structured epidemic model, SIAM J. Math. Anal., 22 (1991), 1065-1080. doi: 10.1137/0522069. Google Scholar [7] Y. Cha, M. Iannelli and F. A. Milner, Stability change of an epidemic model, Dynam. Systems Appl., 9 (2000), 361-376. Google Scholar [8] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation John Wiley and Sons, Chichester, 2000. Google Scholar [9] 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. Google Scholar [10] 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. Google Scholar [11] T. J. Hagenaars, C. A. Donnelly and N. M. Ferguson, Spatial heterogeneity and the persistence of infectious diseases, J. Theor. Biol., 229 (2004), 349-359. doi: 10.1016/j.jtbi.2004.04.002. Google Scholar [12] J. K. Hale, Asymptotic Behavior of Dissipative Systems in: Math. Surv. Monogr. , vol. 25, Am. Math. Soc. , Providence, RI, 1988. Google Scholar [13] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. Google Scholar [14] J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci., 155 (1999), 77-109. doi: 10.1016/S0025-5564(98)10057-3. Google Scholar [15] H. Inaba, Endemic threshold results in an age-duration-structured population model for HIV infection, Math. Biosci., 201 (2006), 15-47. doi: 10.1016/j.mbs.2005.12.017. Google Scholar [16] H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411-434. doi: 10.1007/BF00178326. Google Scholar [17] A. Korobeinikov, P. K. Maini and W. J. Walker, Estimation of effective vaccination rate: Pertussis in New Zealand as a case study, J. Theor. Biol., 224 (2003), 269-275. doi: 10.1016/S0022-5193(03)00163-2. Google Scholar [18] T. Kuniya, Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model, Nonlinear Anal. RWA, 12 (2011), 2640-2655. doi: 10.1016/j.nonrwa.2011.03.011. Google Scholar [19] A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5. Google Scholar [20] M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 38-47. doi: 10.1016/j.jmaa.2009.09.017. Google Scholar [21] 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. Google Scholar [22] A. L. Lloyd and R. M. May, Spatial heterogeneity in epidemic models, J. Theor. Biol., 179 (1996), 1-11. doi: 10.1006/jtbi.1996.0042. Google Scholar [23] P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140. doi: 10.1080/00036810903208122. Google Scholar [24] P. Magal, Compact attractors for time periodic age-structured population models, Electron. J. Differential Equations, 65 (2001), 1-35. Google Scholar [25] P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models Memoirs of the American Mathematical Society, 202 (2009), vi+71 pp. doi: 10.1090/S0065-9266-09-00568-7. Google Scholar [26] P. Magal and H. R. Thieme, Eventual compactness for semiflows generated by nonlinear age-structured models, Commun. Pure Appl. Anal., 3 (2004), 695-727. doi: 10.3934/cpaa.2004.3.695. Google Scholar [27] A. G. McKendrick, Applications of mathematics to medical problems, Proc. Edinb. Math. Soc., 44 (1925), 98-130. doi: 10.1017/S0013091500034428. Google Scholar [28] C. C. McCluskey, Delay versus age-of-infection-Global stability, Applied Math. Comput., 217 (2010), 3046-3049. doi: 10.1016/j.amc.2010.08.037. Google Scholar [29] A. V. Melnik and A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility, Math. Biosci. Eng., 10 (2013), 369-378. doi: 10.3934/mbe.2013.10.369. Google Scholar [30] D. Schenzle, An age-structured model of pre-and post-vaccination measles transmission, IMA J. Math. Appl. Med. Biol., 1 (1984), 169-191. doi: 10.1093/imammb/1.2.169. Google Scholar [31] H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Anal. RWA, 13 (2012), 1581-1592. doi: 10.1016/j.nonrwa.2011.11.016. Google Scholar [32] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence Amer. Math. Soc. , Providence, 2011. Google Scholar [33] R. Sun, Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence, Comput. Math. Appl., 60 (2010), 2286-2291. doi: 10.1016/j.camwa.2010.08.020. Google Scholar [34] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066. Google Scholar [35] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [36] J. Wang, J. Zu, X. Liu, G. Huang and J. Zhang, Global dynamics of a multi-group epidemic model with general relapse distribution and nonlinear incidence rate, J. Biol. Syst., 20 (2012), 235-258. doi: 10.1142/S021833901250009X. Google Scholar [37] J. Wang, X. Liu, J. Pang and D. Hou, Global dynamics of a multi-group epidemic model with general exposed distribution and relapse, Osaka J. Math., 52 (2015), 117-138. Google Scholar [38] Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear Anal. RWA, 11 (2010), 995-1004. doi: 10.1016/j.nonrwa.2009.01.040. Google Scholar [39] G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics Marcel Dekker, New York, 1985. Google Scholar
 [1] Jianxin Yang, Zhipeng Qiu, Xue-Zhi Li. Global stability of an age-structured cholera model. Mathematical Biosciences & Engineering, 2014, 11 (3) : 641-665. doi: 10.3934/mbe.2014.11.641 [2] Geni Gupur, Xue-Zhi Li. Global stability of an age-structured SIRS epidemic model with vaccination. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 643-652. doi: 10.3934/dcdsb.2004.4.643 [3] Mohammed Nor Frioui, Tarik Mohammed Touaoula, Bedreddine Ainseba. Global dynamics of an age-structured model with relapse. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019226 [4] Yicang Zhou, Zhien Ma. Global stability of a class of discrete age-structured SIS models with immigration. Mathematical Biosciences & Engineering, 2009, 6 (2) : 409-425. doi: 10.3934/mbe.2009.6.409 [5] Yu Yang, Shigui Ruan, Dongmei Xiao. Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function. Mathematical Biosciences & Engineering, 2015, 12 (4) : 859-877. doi: 10.3934/mbe.2015.12.859 [6] Cameron J. Browne, Sergei S. Pilyugin. Global analysis of age-structured within-host virus model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 1999-2017. doi: 10.3934/dcdsb.2013.18.1999 [7] Toshikazu Kuniya, Mimmo Iannelli. $R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission. Mathematical Biosciences & Engineering, 2014, 11 (4) : 929-945. doi: 10.3934/mbe.2014.11.929 [8] Jinliang Wang, Jiying Lang, Yuming Chen. Global dynamics of an age-structured HIV infection model incorporating latency and cell-to-cell transmission. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3721-3747. doi: 10.3934/dcdsb.2017186 [9] Shaoli Wang, Jianhong Wu, Libin Rong. A note on the global properties of an age-structured viral dynamic model with multiple target cell populations. Mathematical Biosciences & Engineering, 2017, 14 (3) : 805-820. doi: 10.3934/mbe.2017044 [10] Shengqin Xu, Chuncheng Wang, Dejun Fan. Stability and bifurcation in an age-structured model with stocking rate and time delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2535-2549. doi: 10.3934/dcdsb.2018264 [11] Andrey V. Melnik, Andrei Korobeinikov. Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility. Mathematical Biosciences & Engineering, 2013, 10 (2) : 369-378. doi: 10.3934/mbe.2013.10.369 [12] Ryszard Rudnicki, Radosław Wieczorek. On a nonlinear age-structured model of semelparous species. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2641-2656. doi: 10.3934/dcdsb.2014.19.2641 [13] Ovide Arino, Manuel Delgado, Mónica Molina-Becerra. Asymptotic behavior of disease-free equilibriums of an age-structured predator-prey model with disease in the prey. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 501-515. doi: 10.3934/dcdsb.2004.4.501 [14] Yuming Chen, Junyuan Yang, Fengqin Zhang. The global stability of an SIRS model with infection age. Mathematical Biosciences & Engineering, 2014, 11 (3) : 449-469. doi: 10.3934/mbe.2014.11.449 [15] Guangrui Li, Ming Mei, Yau Shu Wong. Nonlinear stability of traveling wavefronts in an age-structured reaction-diffusion population model. Mathematical Biosciences & Engineering, 2008, 5 (1) : 85-100. doi: 10.3934/mbe.2008.5.85 [16] Georgi Kapitanov, Christina Alvey, Katia Vogt-Geisse, Zhilan Feng. An age-structured model for the coupled dynamics of HIV and HSV-2. Mathematical Biosciences & Engineering, 2015, 12 (4) : 803-840. doi: 10.3934/mbe.2015.12.803 [17] 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 [18] Diène Ngom, A. Iggidir, Aboudramane Guiro, Abderrahim Ouahbi. An observer for a nonlinear age-structured model of a harvested fish population. Mathematical Biosciences & Engineering, 2008, 5 (2) : 337-354. doi: 10.3934/mbe.2008.5.337 [19] Georgi Kapitanov. A double age-structured model of the co-infection of tuberculosis and HIV. Mathematical Biosciences & Engineering, 2015, 12 (1) : 23-40. doi: 10.3934/mbe.2015.12.23 [20] Hisashi Inaba. Mathematical analysis of an age-structured SIR epidemic model with vertical transmission. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 69-96. doi: 10.3934/dcdsb.2006.6.69

2018 Impact Factor: 1.008

## Metrics

• PDF downloads (30)
• HTML views (24)
• Cited by (1)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]