# American Institute of Mathematical Sciences

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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", Amer. Math. Soc., (1988). Google Scholar [7] H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599. doi: 10.1137/S0036144500371907. Google Scholar [8] F. Hoppensteadt, An age dependent epidemic problem,, J. Franklin Inst., 297 (1974), 325. doi: 10.1016/0016-0032(74)90037-4. Google Scholar [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. Google Scholar [10] A. Iggidr, J. Mbang, G. Sallet and J.-J. Tewa, Multi-compartment models,, Discrete Contin. Dyn. Syst. Ser. Supplement, (2007), 506. Google Scholar [11] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics,, Proc. R. Soc. London, 115 (1927), 700. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [17] C. C. McCluskey, Lyapunov functions for tuberculosis models with fast and slow progression,, Math. Biosci. and Eng., 3 (2006), 603. Google Scholar [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. Google Scholar [19] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. and Eng., 6 (2009), 603. Google Scholar [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. Google Scholar [21] C. C. McCluskey, Global stability for an SIR epidemic model with delay and general nonlinear incidence,, Math. Biosci. and Eng., 7 (2010), 837. Google Scholar [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. Google Scholar [23] G. Röst, SEI model with varying transmission and mortality rates,, in, (2011), 489. Google Scholar [24] G. Röst and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. and Eng., 5 (2008), 389. Google Scholar [25] H. L. Smith and H. R. Thieme, "Dynamical Systems and Population Persistence,", Amer. Math. Soc., (2011). Google Scholar [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. Google Scholar [27] G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics,", Marcel Dekker, (1985). Google Scholar

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##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", Amer. Math. Soc., (1988). Google Scholar [7] H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599. doi: 10.1137/S0036144500371907. Google Scholar [8] F. Hoppensteadt, An age dependent epidemic problem,, J. Franklin Inst., 297 (1974), 325. doi: 10.1016/0016-0032(74)90037-4. Google Scholar [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. Google Scholar [10] A. Iggidr, J. Mbang, G. Sallet and J.-J. Tewa, Multi-compartment models,, Discrete Contin. Dyn. Syst. Ser. Supplement, (2007), 506. Google Scholar [11] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics,, Proc. R. Soc. London, 115 (1927), 700. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [17] C. C. McCluskey, Lyapunov functions for tuberculosis models with fast and slow progression,, Math. Biosci. and Eng., 3 (2006), 603. Google Scholar [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. Google Scholar [19] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. and Eng., 6 (2009), 603. Google Scholar [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. Google Scholar [21] C. C. McCluskey, Global stability for an SIR epidemic model with delay and general nonlinear incidence,, Math. Biosci. and Eng., 7 (2010), 837. Google Scholar [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. Google Scholar [23] G. Röst, SEI model with varying transmission and mortality rates,, in, (2011), 489. Google Scholar [24] G. Röst and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. and Eng., 5 (2008), 389. Google Scholar [25] H. L. Smith and H. R. Thieme, "Dynamical Systems and Population Persistence,", Amer. Math. Soc., (2011). Google Scholar [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. Google Scholar [27] G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics,", Marcel Dekker, (1985). Google Scholar
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