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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

Andrey V. Melnik 1, and  Andrei Korobeinikov 2,

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.
Keywords: age structured model, Lyapunov function, age structure, global stability, Infectious disease, endemic equilibrium state, compartment model, direct Lyapunov method, distributed population model..
Mathematics Subject Classification: Primary: 92D30; Secondary: 35F31, 34D2.
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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

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.  Google Scholar

[22]

A. M. Lyapunov, "The General Problem of the Stability of Motion," Taylor & Francis, Ltd., London, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

H. R. Thieme, "Mathematics in Population Biology," Princeton University Press, Princeton, 2003.  Google Scholar

[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.  Google Scholar

[33]

V. Volterra, "Leçons Sur la Théorie Mathématique de la Lutte Pour la Vie," Gauthier-Villars, Paris, 1931.  Google Scholar

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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

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.  Google Scholar

[22]

A. M. Lyapunov, "The General Problem of the Stability of Motion," Taylor & Francis, Ltd., London, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

H. R. Thieme, "Mathematics in Population Biology," Princeton University Press, Princeton, 2003.  Google Scholar

[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.  Google Scholar

[33]

V. Volterra, "Leçons Sur la Théorie Mathématique de la Lutte Pour la Vie," Gauthier-Villars, Paris, 1931.  Google Scholar

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