American Institute of Mathematical Sciences

Advanced
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

[1]

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
[2]

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
[3]

Hiroshi Ito. Input-to-state stability and Lyapunov functions with explicit domains for SIR model of infectious diseases. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 5171-5196. doi: 10.3934/dcdsb.2020338
[4]

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
[5]

Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631
[6]

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
[7]

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
[8]

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
[9]

Tsuyoshi Kajiwara, Toru Sasaki, Yasuhiro Takeuchi. Construction of Lyapunov functions for some models of infectious diseases in vivo: From simple models to complex models. Mathematical Biosciences & Engineering, 2015, 12 (1) : 117-133. doi: 10.3934/mbe.2015.12.117
[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]

Roberto A. Saenz, Herbert W. Hethcote. Competing species models with an infectious disease. Mathematical Biosciences & Engineering, 2006, 3 (1) : 219-235. doi: 10.3934/mbe.2006.3.219
[12]

S.M. Moghadas. Modelling the effect of imperfect vaccines on disease epidemiology. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 999-1012. doi: 10.3934/dcdsb.2004.4.999
[13]

Gang Huang, Edoardo Beretta, Yasuhiro Takeuchi. Global stability for epidemic model with constant latency and infectious periods. Mathematical Biosciences & Engineering, 2012, 9 (2) : 297-312. doi: 10.3934/mbe.2012.9.297
[14]

Andreas Widder. On the usefulness of set-membership estimation in the epidemiology of infectious diseases. Mathematical Biosciences & Engineering, 2018, 15 (1) : 141-152. doi: 10.3934/mbe.2018006
[15]

Michael Schönlein. Asymptotic stability and smooth Lyapunov functions for a class of abstract dynamical systems. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4053-4069. doi: 10.3934/dcds.2017172
[16]

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
[17]

Shangbing Ai. Global stability of equilibria in a tick-borne disease model. Mathematical Biosciences & Engineering, 2007, 4 (4) : 567-572. doi: 10.3934/mbe.2007.4.567
[18]

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
[19]

C. Connell McCluskey. Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes. Mathematical Biosciences & Engineering, 2012, 9 (4) : 819-841. doi: 10.3934/mbe.2012.9.819
[20]

Yoshiaki Muroya, Yoichi Enatsu, Huaixing Li. A note on the global stability of an SEIR epidemic model with constant latency time and infectious period. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 173-183. doi: 10.3934/dcdsb.2013.18.173

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (185)
  • HTML views (275)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences