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August  2021, 26(8): 4479-4491. doi: 10.3934/dcdsb.2020296

Lyapunov functions for disease models with immigration of infected hosts

Connell McCluskey

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

Received  June 2020 Revised  July 2020 Published  August 2021 Early access  October 2020

Fund Project: The author is supported by an NSERC Discovery Grant

Recent work has produced examples where models of the spread of infectious disease with immigration of infected hosts are shown to be globally asymptotically stable through the use of Lyapunov functions. In each case, the Lyapunov function was similar to a Lyapunov function that worked for the corresponding model without immigration of infected hosts.

We distill the calculations from the individual examples into a general result, finding algebraic conditions under which the Lyapunov function for a model without immigration of infected hosts extends to be a valid Lyapunov function for the corresponding system with immigration of infected hosts.

Finally, the method is applied to a multi-group $ SIR $ model.

Keywords: Lyapunov functions, global stability, immigration, infectious disease, epidemiology.
Mathematics Subject Classification: Primary: 34D23; Secondary: 92D30.
Citation: Connell McCluskey. Lyapunov functions for disease models with immigration of infected hosts. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4479-4491. doi: 10.3934/dcdsb.2020296
References:
[1]

R. M. Almarashi and C. C. McCluskey, The effect of immigration of infectives on disease-free equilibria, J. Math. Biol., 79 (2019), 1015-1028.  doi: 10.1007/s00285-019-01387-8.  Google Scholar

[2]

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), 815-821.  doi: 10.1038/nm0895-815.  Google Scholar

[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.  Google Scholar

[4]

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.  doi: 10.1080/17513750802120877.  Google Scholar

[5]

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.  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), 259-284.   Google Scholar

[7]

J. K. Hale, Ordinary Differential Equations, Pure and Applied Mathematics, 21, John Wiley & Sons, New York-London-Sydney, 1969.  Google Scholar

[8]

S. Henshaw and C. C. McCluskey, Global stability of a vaccination model with immigration, Electron J. Differential Equations, 2015 (2015), 1-10.   Google Scholar

[9]

A. Iggidr, J. Mbang, G. Sallet and J.-J. Tewa, Multi-compartment models, Discrete Contin. Dyn. Syst., Proceedings of the 6th AIMS International Conference, 2007,506–519. doi: 10.3934/proc.2007.2007.506.  Google Scholar

[10]

A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math. Med. Biol., 21 (2004), 75-83.  doi: 10.1093/imammb/21.2.75.  Google Scholar

[11]

J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1976.  Google Scholar

[12]

C. C. McCluskey, Lyapunov functions for tuberculosis models with fast and slow progression, Math. Biosci. Eng., 3 (2006), 603-614.  doi: 10.3934/mbe.2006.3.603.  Google Scholar

[13]

C. C. McCluskey, Global stability for an SEI model of infectious disease with age structure and immigration of infecteds, Math. Biosci. Eng., 13 (2016), 381-400.  doi: 10.3934/mbe.2015008.  Google Scholar

[14]

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.  Google Scholar

[15]

R. Zhang, D. Li and S. Liu, Global analysis of an age-structured SEIR model with immigration of population and nonlinear incidence rate, J. Appl. Anal. Comput., 9 (2019), 1470-1492.  doi: 10.11948/2156-907X.20180281.  Google Scholar

show all references

References:
[1]

R. M. Almarashi and C. C. McCluskey, The effect of immigration of infectives on disease-free equilibria, J. Math. Biol., 79 (2019), 1015-1028.  doi: 10.1007/s00285-019-01387-8.  Google Scholar

[2]

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), 815-821.  doi: 10.1038/nm0895-815.  Google Scholar

[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.  Google Scholar

[4]

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.  doi: 10.1080/17513750802120877.  Google Scholar

[5]

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.  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), 259-284.   Google Scholar

[7]

J. K. Hale, Ordinary Differential Equations, Pure and Applied Mathematics, 21, John Wiley & Sons, New York-London-Sydney, 1969.  Google Scholar

[8]

S. Henshaw and C. C. McCluskey, Global stability of a vaccination model with immigration, Electron J. Differential Equations, 2015 (2015), 1-10.   Google Scholar

[9]

A. Iggidr, J. Mbang, G. Sallet and J.-J. Tewa, Multi-compartment models, Discrete Contin. Dyn. Syst., Proceedings of the 6th AIMS International Conference, 2007,506–519. doi: 10.3934/proc.2007.2007.506.  Google Scholar

[10]

A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math. Med. Biol., 21 (2004), 75-83.  doi: 10.1093/imammb/21.2.75.  Google Scholar

[11]

J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1976.  Google Scholar

[12]

C. C. McCluskey, Lyapunov functions for tuberculosis models with fast and slow progression, Math. Biosci. Eng., 3 (2006), 603-614.  doi: 10.3934/mbe.2006.3.603.  Google Scholar

[13]

C. C. McCluskey, Global stability for an SEI model of infectious disease with age structure and immigration of infecteds, Math. Biosci. Eng., 13 (2016), 381-400.  doi: 10.3934/mbe.2015008.  Google Scholar

[14]

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.  Google Scholar

[15]

R. Zhang, D. Li and S. Liu, Global analysis of an age-structured SEIR model with immigration of population and nonlinear incidence rate, J. Appl. Anal. Comput., 9 (2019), 1470-1492.  doi: 10.11948/2156-907X.20180281.  Google Scholar

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