Impacts of migration and immigration on disease transmission dynamics in heterogeneous populations

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October 2012, 17(7): 2413-2430. doi: 10.3934/dcdsb.2012.17.2413

Impacts of migration and immigration on disease transmission dynamics in heterogeneous populations

Hongbin Guo ^1, and Michael Yi Li ^2,

1.	Centre for Disease Modeling, Department of Mathematics and Statistics, York University, 4700 Keele Street Toronto, ON, M3J 1P3
2.	Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Received June 2011 Revised March 2012 Published July 2012

Abstract
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Population migration and immigration have greatly increased the spread and transmission of many infectious diseases at a regional, national and global scale. To investigate quantitatively and qualitatively the impact of migration and immigration on the transmission dynamics of infectious diseases, especially in heterogeneous host populations, we incorporate immigration/migration terms into all sub-population compartments, susceptible and infected, of two types of well-known heterogeneous epidemic models: multi-stage models and multi-group models for HIV/AIDS and other STDs. We show that, when migration or immigration into infected sub-population is present, the disease always becomes endemic in the population and tends to a unique asymptotically stable endemic equilibrium $P^*.$ The global stability of $P^*$ is established under general and biological meaningful conditions, and the proof utilizes a global Lyapunov function and the graph-theoretic techniques developed in Guo et al. (2008).

Keywords: heterogeneity, immigration, Compartmental models, global Lyapunov functions., global stability.

Mathematics Subject Classification: Primary: 92D30; Secondary: 34D2.

Citation: Hongbin Guo, Michael Yi Li. Impacts of migration and immigration on disease transmission dynamics in heterogeneous populations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2413-2430. doi: 10.3934/dcdsb.2012.17.2413

References:

[1]	R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, Oxford, 1992.
[2]	N. Bame, S. Bowong, J. Mbang, G. Sallet and J. Tewa, Global stability analysis for SEIS models with n latent classes, Math. Biosci. Eng., 5 (2008), 20-33.
[3]	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.
[4]	A. Fall, A. Iggidr, G. Sallet and J. J. Tewa, Epidemiological models and Lyapunov functions, Math. Model. Nat. Phenom., 2 (2007), 55-73.
[5]	H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng., 3 (2006), 513-525.
[6]	H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q., 14 (2006), 259-284.
[7]	H. Guo, M. Y. Li and Z. Shuai, A graph-theoretical approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802. doi: 10.1090/S0002-9939-08-09341-6.
[8]	H. Guo and M. Y. Li, Global dynamics of a staged progression model with amelioration for infectious diseases, J. Biol. Dyn., 2 (2008), 154-168.
[9]	H. W. Hethcote, An immunization model for a heterogeneous population, Theor. Popu. Biol., 14 (1978), 338-349. doi: 10.1016/0040-5809(78)90011-4.
[10]	W. Huang, K. L. Cooke and C. Castillo-Chavez, Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. Math., 52 (1992), 835-854. doi: 10.1137/0152047.
[11]	J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci., 155 (1999), 77-109. doi: 10.1016/S0025-5564(98)10057-3.
[12]	A. Iggidr, J. Mbang, G. Sallet and J.-J. Tewa, Multi-compartment models, Discrete Contin. Dyn. Syst., 2007, Dynamical Systems and Differential Equations, Proceedings of the 6$^th$ AIMS International Conference, suppl., 506-519.
[13]	J. A. Jacquez, C. P. Simon, J. Koopman, L. Sattenspiel and T. Perry, Modeling and analysis of HIV transmission: The effect of contact patterns, Math. Biosci., 92 (1988), 119-199. doi: 10.1016/0025-5564(88)90031-4.
[14]	A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models withnonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60.
[15]	A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5.
[16]	M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 1-20. doi: 10.1016/j.jde.2009.09.003.
[17]	G. Li, W. Wang and Z. Jin, Global stability of an SEIR epidemicmodel with constant immigration, Chaos, Solitons & Fractals, 30 (2006), 1012-1019.
[18]	A. L. Lloyd and R. M. May, Spatial heterogeneity in epidemic models, J. Theor. Biol., 179 (1996), 1-11. doi: 10.1006/jtbi.1996.0042.
[19]	C. C. McCluskey, A model of HIV/AIDS with staged progression and amelioration, Math. Biosci., 181 (2003), 1-16. doi: 10.1016/S0025-5564(02)00149-9.
[20]	C. C. McCluskey and P. van den Driessche, Global analysis of two tuberculosis models, J. Dyn. Differential Equations, 16 (2004), 139-166. doi: 10.1023/B:JODY.0000041283.66784.3e.
[21]	S. M. O'Regan, T. C. Kelly, A. Korobeinikov, M. J. A. O'Callaghan and A. V. Pokrovskii, Lyapunov functions for SIR and SIRS epidemic models, Appl. Math. Lett., 23 (2010), 446-448. doi: 10.1016/j.aml.2009.11.014.
[22]	H. R. Thieme, Local stability in epidemic models for heterogeneous populations, "Mathematics in Biology and Medicine" (Bari, 1983) (eds. V. Capasso, E. Grosso and S. L. Paveri-Fontana), Lecture Notes in Biomath., 57, Springer, Berlin, (1985), 185-189.
[23]	H. R. Thieme, "Mathematics in Population Biology," Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003.

show all references

References:

[1]	R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, Oxford, 1992.
[2]	N. Bame, S. Bowong, J. Mbang, G. Sallet and J. Tewa, Global stability analysis for SEIS models with n latent classes, Math. Biosci. Eng., 5 (2008), 20-33.
[3]	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.
[4]	A. Fall, A. Iggidr, G. Sallet and J. J. Tewa, Epidemiological models and Lyapunov functions, Math. Model. Nat. Phenom., 2 (2007), 55-73.
[5]	H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng., 3 (2006), 513-525.
[6]	H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q., 14 (2006), 259-284.
[7]	H. Guo, M. Y. Li and Z. Shuai, A graph-theoretical approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802. doi: 10.1090/S0002-9939-08-09341-6.
[8]	H. Guo and M. Y. Li, Global dynamics of a staged progression model with amelioration for infectious diseases, J. Biol. Dyn., 2 (2008), 154-168.
[9]	H. W. Hethcote, An immunization model for a heterogeneous population, Theor. Popu. Biol., 14 (1978), 338-349. doi: 10.1016/0040-5809(78)90011-4.
[10]	W. Huang, K. L. Cooke and C. Castillo-Chavez, Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. Math., 52 (1992), 835-854. doi: 10.1137/0152047.
[11]	J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci., 155 (1999), 77-109. doi: 10.1016/S0025-5564(98)10057-3.
[12]	A. Iggidr, J. Mbang, G. Sallet and J.-J. Tewa, Multi-compartment models, Discrete Contin. Dyn. Syst., 2007, Dynamical Systems and Differential Equations, Proceedings of the 6$^th$ AIMS International Conference, suppl., 506-519.
[13]	J. A. Jacquez, C. P. Simon, J. Koopman, L. Sattenspiel and T. Perry, Modeling and analysis of HIV transmission: The effect of contact patterns, Math. Biosci., 92 (1988), 119-199. doi: 10.1016/0025-5564(88)90031-4.
[14]	A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models withnonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60.
[15]	A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5.
[16]	M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 1-20. doi: 10.1016/j.jde.2009.09.003.
[17]	G. Li, W. Wang and Z. Jin, Global stability of an SEIR epidemicmodel with constant immigration, Chaos, Solitons & Fractals, 30 (2006), 1012-1019.
[18]	A. L. Lloyd and R. M. May, Spatial heterogeneity in epidemic models, J. Theor. Biol., 179 (1996), 1-11. doi: 10.1006/jtbi.1996.0042.
[19]	C. C. McCluskey, A model of HIV/AIDS with staged progression and amelioration, Math. Biosci., 181 (2003), 1-16. doi: 10.1016/S0025-5564(02)00149-9.
[20]	C. C. McCluskey and P. van den Driessche, Global analysis of two tuberculosis models, J. Dyn. Differential Equations, 16 (2004), 139-166. doi: 10.1023/B:JODY.0000041283.66784.3e.
[21]	S. M. O'Regan, T. C. Kelly, A. Korobeinikov, M. J. A. O'Callaghan and A. V. Pokrovskii, Lyapunov functions for SIR and SIRS epidemic models, Appl. Math. Lett., 23 (2010), 446-448. doi: 10.1016/j.aml.2009.11.014.
[22]	H. R. Thieme, Local stability in epidemic models for heterogeneous populations, "Mathematics in Biology and Medicine" (Bari, 1983) (eds. V. Capasso, E. Grosso and S. L. Paveri-Fontana), Lecture Notes in Biomath., 57, Springer, Berlin, (1985), 185-189.
[23]	H. R. Thieme, "Mathematics in Population Biology," Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003.

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