American Institute of Mathematical Sciences

Advanced
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

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 & 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, (1992).   Google Scholar

[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.   Google Scholar

[3]

F. Brauer and P. van den Driessche, Models for transmission of disease with immigration of infectives,, Math. Biosci., 171 (2001), 143.  doi: 10.1016/S0025-5564(01)00057-8.  Google Scholar

[4]

A. Fall, A. Iggidr, G. Sallet and J. J. Tewa, Epidemiological models and Lyapunov functions,, Math. Model. Nat. Phenom., 2 (2007), 55.   Google Scholar

[5]

H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases,, Math. Biosci. Eng., 3 (2006), 513.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1090/S0002-9939-08-09341-6.  Google Scholar

[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.   Google Scholar

[9]

H. W. Hethcote, An immunization model for a heterogeneous population,, Theor. Popu. Biol., 14 (1978), 338.  doi: 10.1016/0040-5809(78)90011-4.  Google Scholar

[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.  doi: 10.1137/0152047.  Google Scholar

[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.  doi: 10.1016/S0025-5564(98)10057-3.  Google Scholar

[12]

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

[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.  doi: 10.1016/0025-5564(88)90031-4.  Google Scholar

[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.   Google Scholar

[15]

A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population,, Math. Biosci., 28 (1976), 221.  doi: 10.1016/0025-5564(76)90125-5.  Google Scholar

[16]

M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks,, J. Differential Equations, 248 (2010), 1.  doi: 10.1016/j.jde.2009.09.003.  Google Scholar

[17]

G. Li, W. Wang and Z. Jin, Global stability of an SEIR epidemicmodel with constant immigration,, Chaos, 30 (2006), 1012.   Google Scholar

[18]

A. L. Lloyd and R. M. May, Spatial heterogeneity in epidemic models,, J. Theor. Biol., 179 (1996), 1.  doi: 10.1006/jtbi.1996.0042.  Google Scholar

[19]

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

[20]

C. C. McCluskey and P. van den Driessche, Global analysis of two tuberculosis models,, J. Dyn. Differential Equations, 16 (2004), 139.  doi: 10.1023/B:JODY.0000041283.66784.3e.  Google Scholar

[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.  doi: 10.1016/j.aml.2009.11.014.  Google Scholar

[22]

H. R. Thieme, Local stability in epidemic models for heterogeneous populations,, Mathematics in Biology and Medicine, 57 (1985), 185.   Google Scholar

[23]

H. R. Thieme, "Mathematics in Population Biology,", Princeton Series in Theoretical and Computational Biology, (2003).   Google Scholar

show all references

References:
[1]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control,", Oxford University Press, (1992).   Google Scholar

[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.   Google Scholar

[3]

F. Brauer and P. van den Driessche, Models for transmission of disease with immigration of infectives,, Math. Biosci., 171 (2001), 143.  doi: 10.1016/S0025-5564(01)00057-8.  Google Scholar

[4]

A. Fall, A. Iggidr, G. Sallet and J. J. Tewa, Epidemiological models and Lyapunov functions,, Math. Model. Nat. Phenom., 2 (2007), 55.   Google Scholar

[5]

H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases,, Math. Biosci. Eng., 3 (2006), 513.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1090/S0002-9939-08-09341-6.  Google Scholar

[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.   Google Scholar

[9]

H. W. Hethcote, An immunization model for a heterogeneous population,, Theor. Popu. Biol., 14 (1978), 338.  doi: 10.1016/0040-5809(78)90011-4.  Google Scholar

[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.  doi: 10.1137/0152047.  Google Scholar

[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.  doi: 10.1016/S0025-5564(98)10057-3.  Google Scholar

[12]

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

[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.  doi: 10.1016/0025-5564(88)90031-4.  Google Scholar

[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.   Google Scholar

[15]

A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population,, Math. Biosci., 28 (1976), 221.  doi: 10.1016/0025-5564(76)90125-5.  Google Scholar

[16]

M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks,, J. Differential Equations, 248 (2010), 1.  doi: 10.1016/j.jde.2009.09.003.  Google Scholar

[17]

G. Li, W. Wang and Z. Jin, Global stability of an SEIR epidemicmodel with constant immigration,, Chaos, 30 (2006), 1012.   Google Scholar

[18]

A. L. Lloyd and R. M. May, Spatial heterogeneity in epidemic models,, J. Theor. Biol., 179 (1996), 1.  doi: 10.1006/jtbi.1996.0042.  Google Scholar

[19]

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

[20]

C. C. McCluskey and P. van den Driessche, Global analysis of two tuberculosis models,, J. Dyn. Differential Equations, 16 (2004), 139.  doi: 10.1023/B:JODY.0000041283.66784.3e.  Google Scholar

[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.  doi: 10.1016/j.aml.2009.11.014.  Google Scholar

[22]

H. R. Thieme, Local stability in epidemic models for heterogeneous populations,, Mathematics in Biology and Medicine, 57 (1985), 185.   Google Scholar

[23]

H. R. Thieme, "Mathematics in Population Biology,", Princeton Series in Theoretical and Computational Biology, (2003).   Google Scholar

[1]

Yicang Zhou, Zhien Ma. Global stability of a class of discrete age-structured SIS models with immigration. Mathematical Biosciences & Engineering, 2009, 6 (2) : 409-425. doi: 10.3934/mbe.2009.6.409
[2]

Andrey V. Melnik, Andrei Korobeinikov. Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility. Mathematical Biosciences & Engineering, 2013, 10 (2) : 369-378. doi: 10.3934/mbe.2013.10.369
[3]

Deqiong Ding, Wendi Qin, Xiaohua Ding. Lyapunov functions and global stability for a discretized multigroup SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1971-1981. doi: 10.3934/dcdsb.2015.20.1971
[4]

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
[5]

Qingming Gou, Wendi Wang. Global stability of two epidemic models. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 333-345. doi: 10.3934/dcdsb.2007.8.333
[6]

Attila Dénes, Gergely Röst. Global stability for SIR and SIRS models with nonlinear incidence and removal terms via Dulac functions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1101-1117. doi: 10.3934/dcdsb.2016.21.1101
[7]

Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101
[8]

Andrei Korobeinikov, Philip K. Maini. A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Mathematical Biosciences & Engineering, 2004, 1 (1) : 57-60. doi: 10.3934/mbe.2004.1.57
[9]

Yoichi Enatsu, Yukihiko Nakata, Yoshiaki Muroya. Global stability for a class of discrete SIR epidemic models. Mathematical Biosciences & Engineering, 2010, 7 (2) : 347-361. doi: 10.3934/mbe.2010.7.347
[10]

Jing-Jing Xiang, Juan Wang, Li-Ming Cai. Global stability of the dengue disease transmission models. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2217-2232. doi: 10.3934/dcdsb.2015.20.2217
[11]

Paul Georgescu, Hong Zhang, Daniel Maxin. The global stability of coexisting equilibria for three models of mutualism. Mathematical Biosciences & Engineering, 2016, 13 (1) : 101-118. doi: 10.3934/mbe.2016.13.101
[12]

Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631
[13]

C. Connell Mccluskey. Lyapunov functions for tuberculosis models with fast and slow progression. Mathematical Biosciences & Engineering, 2006, 3 (4) : 603-614. doi: 10.3934/mbe.2006.3.603
[14]

Sergio Grillo, Jerrold E. Marsden, Sujit Nair. Lyapunov constraints and global asymptotic stabilization. Journal of Geometric Mechanics, 2011, 3 (2) : 145-196. doi: 10.3934/jgm.2011.3.145
[15]

Burcu Adivar, Ebru Selin Selen. Compartmental disease transmission models for smallpox. Conference Publications, 2011, 2011 (Special) : 13-21. doi: 10.3934/proc.2011.2011.13
[16]

Volodymyr Pichkur. On practical stability of differential inclusions using Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1977-1986. doi: 10.3934/dcdsb.2017116
[17]

Hal L. Smith, Horst R. Thieme. Persistence and global stability for a class of discrete time structured population models. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4627-4646. doi: 10.3934/dcds.2013.33.4627
[18]

Cruz Vargas-De-León, Alberto d'Onofrio. Global stability of infectious disease models with contact rate as a function of prevalence index. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1019-1033. doi: 10.3934/mbe.2017053
[19]

Napoleon Bame, Samuel Bowong, Josepha Mbang, Gauthier Sallet, Jean-Jules Tewa. Global stability analysis for SEIS models with n latent classes. Mathematical Biosciences & Engineering, 2008, 5 (1) : 20-33. doi: 10.3934/mbe.2008.5.20
[20]

Eduardo Liz. Local stability implies global stability in some one-dimensional discrete single-species models. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 191-199. doi: 10.3934/dcdsb.2007.7.191

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences