American Institute of Mathematical Sciences

Advanced
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

C. Connell McCluskey 1,

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.
Keywords: Lyapunov functional, tuberculosis., immigration, epidemiology, age-structure, Global stability.
Mathematics Subject Classification: Primary: 34K20, 92D3.
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.

show all references

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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