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2013, 10(2): 369-378. doi: 10.3934/mbe.2013.10.369

## Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility

 1 OCCAM, Mathematical Institute, 24 - 29 St Giles', Oxford, OX1 3LB, United Kingdom 2 Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra, Barcelona, Spain

Received  January 2012 Revised  September 2012 Published  January 2013

We consider global asymptotic properties for the SIR and SEIR age structured models for infectious diseases where the susceptibility depends on the age. Using the direct Lyapunov method with Volterra type Lyapunov functions, we establish conditions for the global stability of a unique endemic steady state and the infection-free steady state.
Citation: 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
##### References:
 [1] P. Adda, J. L. Dimi, A. Iggidr, J. C. Kamgang, G. Sallet and J. J. Tewa, General models of host-parasite systems. Global analysis, Disc. Cont. Dyn. Syst. Ser. B, 8 (2007), 1-17. [2] A. Fall, A. Iggidr, G. Sallet and J. J. Tewa, Epidemiological models and Lyapunov functions, Math. Model. Nat. Phenom., 2 (2007), 62-83. doi: 10.1051/mmnp:2008011. [3] P. Georgescu and Y.-H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal, SIAM J. Appl. Math., 2 (2006), 337-353. doi: 10.1137/060654876. [4] H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng., 3 (2006), 513-525. doi: 10.3934/mbe.2006.3.513. [5] H. Guo and M. Y. Li, Global dynamics of a staged-progression model with amelioration for infectious diseases, J. Biol. Dynamics, 2 (2008), 154-168. doi: 10.1080/17513750802120877. [6] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [7] G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 22 (2009), 1690-1693. doi: 10.1016/j.aml.2009.06.004. [8] G. Huang, W. Ma and Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 24 (2011), 1199-1203. doi: 10.1016/j.aml.2011.02.007. [9] G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 129-139. doi: 10.1007/s00285-010-0368-2. [10] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differentical equations models of viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821. [11] G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207. doi: 10.1007/s11538-009-9487-6. [12] A. Iggidr, J. Mbang and G. Sallet, Stability analysis of within-host parasite models with delays, Math. Biosci., 209 (2007), 51-75. doi: 10.1016/j.mbs.2007.01.008. [13] 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. [14] A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. Math. Biol., 71 (2009), 75-83. doi: 10.1007/s11538-007-9196-y. [15] A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886. doi: 10.1007/s11538-007-9196-y. [16] A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose dependent parasite reproduction and virulence, and nonlinear incidence rate, Math. Med. Biol., 26 (2009), 225-239. doi: 10.1093/imammb/dqp009. [17] A. Korobeinikov, Stability of ecosystem: Global properties of a general prey-predator model, Math. Med. Biol., 26 (2009), 309-321. doi: 10.1093/imammb/dqp009. [18] A. Korobeinikov, Global properties of a general predator-prey model with non-symmetric attack and consumption rate, Discrete Cont. Dyn. Syst. Ser. B, 14 (2010), 1095-1103. doi: 10.3934/dcdsb.2010.14.1095. [19] J. P. LaSalle, "The Stability of Dynamical Systems," SIAM, Philadelphia, 1976. [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. [21] S. Liu and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy, Math. Biosci. Eng., 7 (2010), 675-685. doi: 10.3934/mbe.2010.7.675. [22] A. M. Lyapunov, "The General Problem of the Stability of Motion," Taylor & Francis, Ltd., London, 1992. [23] P. Magal, C. C. McCluskey and G. F. Webb, Liapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140. doi: 10.1080/00036810903208122. [24] V. G. Matsenko and V. N. Rubanovskii, The Lyapunov direct method for analyzing the dynamics of the age structure of biological populations, USSR Comput Maths Math. Phys., 23 (1983), 45-49. [25] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610. doi: 10.3934/mbe.2009.6.603. [26] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay - distributed or discrete, Nonlinear Anal. Real World Appl., 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014. [27] C. C. McCluskey, Delay versus age-of-infection-global stability, Appl. Math. Comput., 217 (2010), 3046-3049. doi: 10.1016/j.amc.2010.08.037. [28] C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. Real World Appl., 11 (2010), 3106-3109. doi: 10.1016/j.nonrwa.2009.11.005. [29] A. V. Melnik and A. Korobeinikov, Global asymptotic properties of staged models with multiple progression pathways for infectious diseases, Math. Biosci. Eng., 8 (2011), 1019-1034. doi: 10.3934/mbe.2011.8.1019. [30] H. R. Thieme, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators, J. Differ. Equations, 250 (2011), 3772-3801. doi: 10.1016/j.jde.2011.01.007. [31] H. R. Thieme, "Mathematics in Population Biology," Princeton University Press, Princeton, 2003. [32] H. R. Thieme, Disease extinction and disease persistence in age structured epidemic models, Nonlinear Anal., 47 (2001), 6181-6194. doi: 10.1016/S0362-546X(01)00677-0. [33] V. Volterra, "Leçons Sur la Théorie Mathématique de la Lutte Pour la Vie," Gauthier-Villars, Paris, 1931.

show all references

##### References:
 [1] P. Adda, J. L. Dimi, A. Iggidr, J. C. Kamgang, G. Sallet and J. J. Tewa, General models of host-parasite systems. Global analysis, Disc. Cont. Dyn. Syst. Ser. B, 8 (2007), 1-17. [2] A. Fall, A. Iggidr, G. Sallet and J. J. Tewa, Epidemiological models and Lyapunov functions, Math. Model. Nat. Phenom., 2 (2007), 62-83. doi: 10.1051/mmnp:2008011. [3] P. Georgescu and Y.-H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal, SIAM J. Appl. Math., 2 (2006), 337-353. doi: 10.1137/060654876. [4] H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng., 3 (2006), 513-525. doi: 10.3934/mbe.2006.3.513. [5] H. Guo and M. Y. Li, Global dynamics of a staged-progression model with amelioration for infectious diseases, J. Biol. Dynamics, 2 (2008), 154-168. doi: 10.1080/17513750802120877. [6] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [7] G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 22 (2009), 1690-1693. doi: 10.1016/j.aml.2009.06.004. [8] G. Huang, W. Ma and Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 24 (2011), 1199-1203. doi: 10.1016/j.aml.2011.02.007. [9] G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 129-139. doi: 10.1007/s00285-010-0368-2. [10] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differentical equations models of viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821. [11] G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207. doi: 10.1007/s11538-009-9487-6. [12] A. Iggidr, J. Mbang and G. Sallet, Stability analysis of within-host parasite models with delays, Math. Biosci., 209 (2007), 51-75. doi: 10.1016/j.mbs.2007.01.008. [13] 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. [14] A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. Math. Biol., 71 (2009), 75-83. doi: 10.1007/s11538-007-9196-y. [15] A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886. doi: 10.1007/s11538-007-9196-y. [16] A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose dependent parasite reproduction and virulence, and nonlinear incidence rate, Math. Med. Biol., 26 (2009), 225-239. doi: 10.1093/imammb/dqp009. [17] A. Korobeinikov, Stability of ecosystem: Global properties of a general prey-predator model, Math. Med. Biol., 26 (2009), 309-321. doi: 10.1093/imammb/dqp009. [18] A. Korobeinikov, Global properties of a general predator-prey model with non-symmetric attack and consumption rate, Discrete Cont. Dyn. Syst. Ser. B, 14 (2010), 1095-1103. doi: 10.3934/dcdsb.2010.14.1095. [19] J. P. LaSalle, "The Stability of Dynamical Systems," SIAM, Philadelphia, 1976. [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. [21] S. Liu and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy, Math. Biosci. Eng., 7 (2010), 675-685. doi: 10.3934/mbe.2010.7.675. [22] A. M. Lyapunov, "The General Problem of the Stability of Motion," Taylor & Francis, Ltd., London, 1992. [23] P. Magal, C. C. McCluskey and G. F. Webb, Liapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140. doi: 10.1080/00036810903208122. [24] V. G. Matsenko and V. N. Rubanovskii, The Lyapunov direct method for analyzing the dynamics of the age structure of biological populations, USSR Comput Maths Math. Phys., 23 (1983), 45-49. [25] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610. doi: 10.3934/mbe.2009.6.603. [26] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay - distributed or discrete, Nonlinear Anal. Real World Appl., 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014. [27] C. C. McCluskey, Delay versus age-of-infection-global stability, Appl. Math. Comput., 217 (2010), 3046-3049. doi: 10.1016/j.amc.2010.08.037. [28] C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. Real World Appl., 11 (2010), 3106-3109. doi: 10.1016/j.nonrwa.2009.11.005. [29] A. V. Melnik and A. Korobeinikov, Global asymptotic properties of staged models with multiple progression pathways for infectious diseases, Math. Biosci. Eng., 8 (2011), 1019-1034. doi: 10.3934/mbe.2011.8.1019. [30] H. R. Thieme, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators, J. Differ. Equations, 250 (2011), 3772-3801. doi: 10.1016/j.jde.2011.01.007. [31] H. R. Thieme, "Mathematics in Population Biology," Princeton University Press, Princeton, 2003. [32] H. R. Thieme, Disease extinction and disease persistence in age structured epidemic models, Nonlinear Anal., 47 (2001), 6181-6194. doi: 10.1016/S0362-546X(01)00677-0. [33] V. Volterra, "Leçons Sur la Théorie Mathématique de la Lutte Pour la Vie," Gauthier-Villars, Paris, 1931.
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