# American Institute of Mathematical Sciences

2016, 13(2): 381-400. doi: 10.3934/mbe.2015008

## Global stability for an $SEI$ model of infectious disease with age structure and immigration of infecteds

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

Received  May 2015 Revised  September 2015 Published  December 2015

We study a model of disease transmission with continuous age-structure for latently infected individuals and for infectious individuals and with immigration of new individuals into the susceptible, latent and infectious classes. The model is very appropriate for tuberculosis. A Lyapunov functional is used to show that the unique endemic equilibrium is globally stable for all parameter values.
Citation: C. Connell McCluskey. Global stability for an $SEI$ model of infectious disease with age structure and immigration of infecteds. Mathematical Biosciences & Engineering, 2016, 13 (2) : 381-400. doi: 10.3934/mbe.2015008
##### References:
 [1] F. Brauer and P. van den Driessche, Models for transmission of disease with immigration of infectives, Math. Biosci., 171 (2001), 143-154. doi: 10.1016/S0025-5564(01)00057-8. [2] R. D. Demasse and A. Ducrot, An age-structured within-host model for multistrain malaria infections, SIAM J. Appl. Math., 73 (2013), 572-593. doi: 10.1137/120890351. [3] 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), 803-833. doi: 10.1137/S0036139998347834. [4] H. Guo and M. Y. Li, Impacts of migration and immigration on disease transmission dynamics in heterogeneous populations, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2413-2430. doi: 10.3934/dcdsb.2012.17.2413. [5] H. Guo and J. Wu, Persistent high incidence of tuberculosis among immigrants in a low-incidence country: impact of immigrants with early or late latency. Math. Biosci. Eng., 8 (2011), 695-709. doi: 10.3934/mbe.2011.8.695. [6] S. Henshaw and C. C. McCluskey, Global stability of a vaccination model with immigration, Elect. J. Diff. Eqns., 2015 (2015), 1-10. [7] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. London, Ser. A, 115 (1927), 700-721. [8] A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60. doi: 10.3934/mbe.2004.1.57. [9] P. Magal and C. C. McCluskey, Two-group infection age model including an application to nosocomial infection, SIAM J. Appl. Math., 73 (2013), 1058-1095. doi: 10.1137/120882056. [10] P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Applicable Analysis, 89 (2010), 1109-1140. doi: 10.1080/00036810903208122. [11] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay - distributed or discrete, Nonlinear Anal. RWA, 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014. [12] C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Math. Biosci. Eng., 9 (2012), 819-841. doi: 10.3934/mbe.2012.9.819. [13] C. C. McCluskey and P. van den Driessche, Global analysis of two tuberculosis models, J. Dynam. Differential Equations, 16 (2004), 139-166. doi: 10.1023/B:JODY.0000041283.66784.3e. [14] G. Röst and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. and Eng., 5 (2008), 389-402. doi: 10.3934/mbe.2008.5.389. [15] R. P. Sigdel and C. C. McCluskey, Disease dynamics for the hometown of migrant workers, Math. Biosci. Eng., 11 (2014), 1175-1180. doi: 10.3934/mbe.2014.11.1175. [16] R. P. Sigdel and C. C. McCluskey, Global stability for an SEI model of infectious disease with immigration, Appl. Math. Comput., 243 (2014), 684-689. doi: 10.1016/j.amc.2014.06.020. [17] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Amer. Math. Soc., Providence, 2011. [18] 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), 1447-1479. doi: 10.1137/0153068. [19] Lin Wang and Xiao Wang, Influence of temporary migration on the transmission of infectious diseases in a migrants' home village, J. Theoret. Biol., 300 (2012), 100-109. doi: 10.1016/j.jtbi.2012.01.004. [20] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.

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##### References:
 [1] F. Brauer and P. van den Driessche, Models for transmission of disease with immigration of infectives, Math. Biosci., 171 (2001), 143-154. doi: 10.1016/S0025-5564(01)00057-8. [2] R. D. Demasse and A. Ducrot, An age-structured within-host model for multistrain malaria infections, SIAM J. Appl. Math., 73 (2013), 572-593. doi: 10.1137/120890351. [3] 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), 803-833. doi: 10.1137/S0036139998347834. [4] H. Guo and M. Y. Li, Impacts of migration and immigration on disease transmission dynamics in heterogeneous populations, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2413-2430. doi: 10.3934/dcdsb.2012.17.2413. [5] H. Guo and J. Wu, Persistent high incidence of tuberculosis among immigrants in a low-incidence country: impact of immigrants with early or late latency. Math. Biosci. Eng., 8 (2011), 695-709. doi: 10.3934/mbe.2011.8.695. [6] S. Henshaw and C. C. McCluskey, Global stability of a vaccination model with immigration, Elect. J. Diff. Eqns., 2015 (2015), 1-10. [7] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. London, Ser. A, 115 (1927), 700-721. [8] A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60. doi: 10.3934/mbe.2004.1.57. [9] P. Magal and C. C. McCluskey, Two-group infection age model including an application to nosocomial infection, SIAM J. Appl. Math., 73 (2013), 1058-1095. doi: 10.1137/120882056. [10] P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Applicable Analysis, 89 (2010), 1109-1140. doi: 10.1080/00036810903208122. [11] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay - distributed or discrete, Nonlinear Anal. RWA, 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014. [12] C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Math. Biosci. Eng., 9 (2012), 819-841. doi: 10.3934/mbe.2012.9.819. [13] C. C. McCluskey and P. van den Driessche, Global analysis of two tuberculosis models, J. Dynam. Differential Equations, 16 (2004), 139-166. doi: 10.1023/B:JODY.0000041283.66784.3e. [14] G. Röst and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. and Eng., 5 (2008), 389-402. doi: 10.3934/mbe.2008.5.389. [15] R. P. Sigdel and C. C. McCluskey, Disease dynamics for the hometown of migrant workers, Math. Biosci. Eng., 11 (2014), 1175-1180. doi: 10.3934/mbe.2014.11.1175. [16] R. P. Sigdel and C. C. McCluskey, Global stability for an SEI model of infectious disease with immigration, Appl. Math. Comput., 243 (2014), 684-689. doi: 10.1016/j.amc.2014.06.020. [17] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Amer. Math. Soc., Providence, 2011. [18] 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), 1447-1479. doi: 10.1137/0153068. [19] Lin Wang and Xiao Wang, Influence of temporary migration on the transmission of infectious diseases in a migrants' home village, J. Theoret. Biol., 300 (2012), 100-109. doi: 10.1016/j.jtbi.2012.01.004. [20] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.
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