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

Lyapunov functions for disease models with immigration of infected hosts

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.

Citation: Connell McCluskey. Lyapunov functions for disease models with immigration of infected hosts. Discrete and 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.

[2]

S. M. BlowerA. R. McLeanT. C. PorcoP. M. SmallP. C. HopwellM. A. Sanchez and A. R. Moss, The intrinsic transmission dynamics of tuberculosis epidemics, Nature Medicine, 1 (1995), 815-821.  doi: 10.1038/nm0895-815.

[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]

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.

[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.

[6]

H. GuoM. 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]

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

[8]

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

[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.

[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.

[11]

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

[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.

[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.

[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.

[15]

R. ZhangD. 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.

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.

[2]

S. M. BlowerA. R. McLeanT. C. PorcoP. M. SmallP. C. HopwellM. A. Sanchez and A. R. Moss, The intrinsic transmission dynamics of tuberculosis epidemics, Nature Medicine, 1 (1995), 815-821.  doi: 10.1038/nm0895-815.

[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]

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.

[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.

[6]

H. GuoM. 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]

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

[8]

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

[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.

[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.

[11]

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

[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.

[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.

[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.

[15]

R. ZhangD. 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.

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