2012, 9(4): 819-841. doi: 10.3934/mbe.2012.9.819

Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes

1. 

Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada

Received  November 2011 Revised  June 2012 Published  October 2012

We study a model of disease transmission with continuous age-structure for latently infected individuals and for infectious individuals. The model is very appropriate for tuberculosis. Key theorems, including asymptotic smoothness and uniform persistence, are proven by reformulating the system as a system of Volterra integral equations. The basic reproduction number $\mathcal{R}_{0}$ is calculated. For $\mathcal{R}_{0}<1$, the disease-free equilibrium is globally asymptotically stable. For $\mathcal{R}_{0}>1$, a Lyapunov functional is used to show that the endemic equilibrium is globally stable amongst solutions for which the disease is present. Finally, some special cases are considered.
Citation: C. Connell McCluskey. Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes. Mathematical Biosciences & Engineering, 2012, 9 (4) : 819-841. doi: 10.3934/mbe.2012.9.819
References:
[1]

S. M. Blower, A. R. McLean, T. C. Porco, P. M. Small, P. C. Hopwell, M. A. Sanchez and A. R. Moss, The intrinsic transmission dynamics of tuberculosis epidemics,, Nature Medicine, 1 (1995), 815.

[2]

E. M. C. D'Agata, P. Magal, D. Olivier, S. Ruan and G. F. Webb, Modeling antibiotic resistance in hospitals: The impact of minimizing treatment duration,, J. Theoret. Biol., 249 (2007), 487. doi: 10.1016/j.jtbi.2007.08.011.

[3]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $\mathcal R_0$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365.

[4]

Z. Feng and H. Thieme, Endemic models with arbitrarily distributed periods of infection I: fundamental properties of the model,, SIAM J. Appl. Math., 61 (2000), 803. doi: 10.1137/S0036139998347834.

[5]

H. Guo and M. Y. Li, Global dynamics of a staged progression model with amelioration for infectious diseases,, J. of Biol. Dyn., 2 (2008), 154.

[6]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", Amer. Math. Soc., (1988).

[7]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599. doi: 10.1137/S0036144500371907.

[8]

F. Hoppensteadt, An age dependent epidemic problem,, J. Franklin Inst., 297 (1974), 325. doi: 10.1016/0016-0032(74)90037-4.

[9]

J. Hyman, J. Li and E. Stanley, The differential infectivity and staged progression models for the transmission of HIV,, Math. Biosci., 155 (1999), 77. doi: 10.1016/S0025-5564(98)10057-3.

[10]

A. Iggidr, J. Mbang, G. Sallet and J.-J. Tewa, Multi-compartment models,, Discrete Contin. Dyn. Syst. Ser. Supplement, (2007), 506.

[11]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics,, Proc. R. Soc. London, 115 (1927), 700.

[12]

A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models,, Math. Med. and Biol., 21 (2004), 75. doi: 10.1093/imammb/21.2.75.

[13]

M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology,, Math. Biosci., 125 (1995), 155. doi: 10.1016/0025-5564(95)92756-5.

[14]

X. Lin, H. W. Hethcote and P. van den Driessche, An epidemiological model for HIV/AIDS with proportional recruitment,, Math. Biosci., 118 (1993), 181. doi: 10.1016/0025-5564(93)90051-B.

[15]

P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model,, Applicable Analysis, 89 (2010), 1109. doi: 10.1080/00036810903208122.

[16]

C. C. McCluskey, A model of HIV/AIDS with staged progression and amelioration,, Math. Biosci., 181 (2003), 1. doi: 10.1016/S0025-5564(02)00149-9.

[17]

C. C. McCluskey, Lyapunov functions for tuberculosis models with fast and slow progression,, Math. Biosci. and Eng., 3 (2006), 603.

[18]

C. C. McCluskey, Global stability for a class of mass action systems allowing for latency in tuberculosis,, J. Math. Anal. Appl., 338 (2008), 518. doi: 10.1016/j.jmaa.2007.05.012.

[19]

C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. and Eng., 6 (2009), 603.

[20]

C. C. McCluskey, Complete global stability for an SIR epidemic model with delay - distributed or discrete,, Nonlinear Anal. RWA, 11 (2010), 55. doi: 10.1016/j.nonrwa.2008.10.014.

[21]

C. C. McCluskey, Global stability for an SIR epidemic model with delay and general nonlinear incidence,, Math. Biosci. and Eng., 7 (2010), 837.

[22]

C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence,, Nonlinear Anal. RWA, 11 (2010), 3106. doi: 10.1016/j.nonrwa.2009.11.005.

[23]

G. Röst, SEI model with varying transmission and mortality rates,, in, (2011), 489.

[24]

G. Röst and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. and Eng., 5 (2008), 389.

[25]

H. L. Smith and H. R. Thieme, "Dynamical Systems and Population Persistence,", Amer. Math. Soc., (2011).

[26]

H. R. Thieme and C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?,, SIAM J. Appl. Math., 53 (1993), 1447. doi: 10.1137/0153068.

[27]

G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics,", Marcel Dekker, (1985).

show all references

References:
[1]

S. M. Blower, A. R. McLean, T. C. Porco, P. M. Small, P. C. Hopwell, M. A. Sanchez and A. R. Moss, The intrinsic transmission dynamics of tuberculosis epidemics,, Nature Medicine, 1 (1995), 815.

[2]

E. M. C. D'Agata, P. Magal, D. Olivier, S. Ruan and G. F. Webb, Modeling antibiotic resistance in hospitals: The impact of minimizing treatment duration,, J. Theoret. Biol., 249 (2007), 487. doi: 10.1016/j.jtbi.2007.08.011.

[3]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $\mathcal R_0$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365.

[4]

Z. Feng and H. Thieme, Endemic models with arbitrarily distributed periods of infection I: fundamental properties of the model,, SIAM J. Appl. Math., 61 (2000), 803. doi: 10.1137/S0036139998347834.

[5]

H. Guo and M. Y. Li, Global dynamics of a staged progression model with amelioration for infectious diseases,, J. of Biol. Dyn., 2 (2008), 154.

[6]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", Amer. Math. Soc., (1988).

[7]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599. doi: 10.1137/S0036144500371907.

[8]

F. Hoppensteadt, An age dependent epidemic problem,, J. Franklin Inst., 297 (1974), 325. doi: 10.1016/0016-0032(74)90037-4.

[9]

J. Hyman, J. Li and E. Stanley, The differential infectivity and staged progression models for the transmission of HIV,, Math. Biosci., 155 (1999), 77. doi: 10.1016/S0025-5564(98)10057-3.

[10]

A. Iggidr, J. Mbang, G. Sallet and J.-J. Tewa, Multi-compartment models,, Discrete Contin. Dyn. Syst. Ser. Supplement, (2007), 506.

[11]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics,, Proc. R. Soc. London, 115 (1927), 700.

[12]

A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models,, Math. Med. and Biol., 21 (2004), 75. doi: 10.1093/imammb/21.2.75.

[13]

M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology,, Math. Biosci., 125 (1995), 155. doi: 10.1016/0025-5564(95)92756-5.

[14]

X. Lin, H. W. Hethcote and P. van den Driessche, An epidemiological model for HIV/AIDS with proportional recruitment,, Math. Biosci., 118 (1993), 181. doi: 10.1016/0025-5564(93)90051-B.

[15]

P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model,, Applicable Analysis, 89 (2010), 1109. doi: 10.1080/00036810903208122.

[16]

C. C. McCluskey, A model of HIV/AIDS with staged progression and amelioration,, Math. Biosci., 181 (2003), 1. doi: 10.1016/S0025-5564(02)00149-9.

[17]

C. C. McCluskey, Lyapunov functions for tuberculosis models with fast and slow progression,, Math. Biosci. and Eng., 3 (2006), 603.

[18]

C. C. McCluskey, Global stability for a class of mass action systems allowing for latency in tuberculosis,, J. Math. Anal. Appl., 338 (2008), 518. doi: 10.1016/j.jmaa.2007.05.012.

[19]

C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. and Eng., 6 (2009), 603.

[20]

C. C. McCluskey, Complete global stability for an SIR epidemic model with delay - distributed or discrete,, Nonlinear Anal. RWA, 11 (2010), 55. doi: 10.1016/j.nonrwa.2008.10.014.

[21]

C. C. McCluskey, Global stability for an SIR epidemic model with delay and general nonlinear incidence,, Math. Biosci. and Eng., 7 (2010), 837.

[22]

C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence,, Nonlinear Anal. RWA, 11 (2010), 3106. doi: 10.1016/j.nonrwa.2009.11.005.

[23]

G. Röst, SEI model with varying transmission and mortality rates,, in, (2011), 489.

[24]

G. Röst and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. and Eng., 5 (2008), 389.

[25]

H. L. Smith and H. R. Thieme, "Dynamical Systems and Population Persistence,", Amer. Math. Soc., (2011).

[26]

H. R. Thieme and C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?,, SIAM J. Appl. Math., 53 (1993), 1447. doi: 10.1137/0153068.

[27]

G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics,", Marcel Dekker, (1985).

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