Qualitative analysis of generalized multistage epidemic model with immigration

Miller Cerón Gómez; Felipe Alves Rubio; Eduardo Ibarguen Mondragón; Miller Cerón Gómez; Felipe Alves Rubio; Eduardo Ibarguen Mondragón

doi:10.3934/mbe.2023702

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 9: 15765-15780. doi: 10.3934/mbe.2023702

Previous Article Next Article

Research article Special Issues

Qualitative analysis of generalized multistage epidemic model with immigration

1.
Department of Mathematics and Statistics, University of Nariño, Pasto, Nariño, Colombia
2.
Barcelona Institute for Global Health, Barcelona, Spain

Received: 25 April 2023 Revised: 05 July 2023 Accepted: 12 July 2023 Published: 31 July 2023

A model with multiple disease stages is discussed; its main feature is that it considers a general incidence rate, functions for death and immigration rates in all populations. We show via a suitable Lyapunov function that the unique endemic equilibrium is globally asymptotically stable. We conclude that, in order to obtain the existence and global stability of the equilibrium point of general models, conditions must be imposed on the functions present in the model. In addition, the model has no basic reproduction number due to the constant flow of infected people, which makes its eradication impossible; therefore, there is no equilibrium point free of infection.
- immigration,
- infectious latent state,
- nonlinear incidence rate,
- global stability,
- multistage model
Citation: Miller Cerón Gómez, Felipe Alves Rubio, Eduardo Ibarguen Mondragón. Qualitative analysis of generalized multistage epidemic model with immigration[J]. Mathematical Biosciences and Engineering, 2023, 20(9): 15765-15780. doi: 10.3934/mbe.2023702

Related Papers:

Abstract

A model with multiple disease stages is discussed; its main feature is that it considers a general incidence rate, functions for death and immigration rates in all populations. We show via a suitable Lyapunov function that the unique endemic equilibrium is globally asymptotically stable. We conclude that, in order to obtain the existence and global stability of the equilibrium point of general models, conditions must be imposed on the functions present in the model. In addition, the model has no basic reproduction number due to the constant flow of infected people, which makes its eradication impossible; therefore, there is no equilibrium point free of infection.

References

[1]	D. L. Blazes, S. F. Dowell, The role of disease surveillance in precision public health, in Genomic and Precision Medicine, Elsevier, (2019), 257–265. https://doi.org/10.1016/B978-0-12-801496-7.00015-0
[2]	H. Esmail, C. Barry, D. Young, R. Wilkinson, The ongoing challenge of latent tuberculosis, Philosoph. Transact. Royal Soc. B Biol. Sci., 369 (2014), 20130437. https://doi.org/10.1098/rstb.2013.0437 doi: 10.1098/rstb.2013.0437
[3]	F. Castelli, G. Sulis, Migration and infectious diseases, Clin. Microbiol. Infect., 23 (2017), 283–289. https://doi.org/10.1016/j.cmi.2017.03.012 doi: 10.1016/j.cmi.2017.03.012
[4]	I. Mokrousov, Major impact of massive migration on spread of Mycobacterium tuberculosis strains, Human Migrat. Biocultur. Perspect., 255 (2021). https://doi.org/10.1093/oso/9780190945961.003.0020 doi: 10.1093/oso/9780190945961.003.0020
[5]	M. Eckhardt, J. F. Hultquist, R. M. Kaake, R. Hüttenhain, N. J. Krogan, A systems approach to infectious disease, Nat. Rev. Genet., 21 (2020), 339–354. https://doi.org/10.1038/s41576-020-0212-5 doi: 10.1038/s41576-020-0212-5
[6]	D. M. Tobin, Modelling infectious disease to support human health, 2022. https://doi.org/10.1242/dmm.049824
[7]	A. L. Jenner, R. A. Aogo, C. L. Davis, A. M. Smith, M. Craig, Leveraging computational modeling to understand infectious diseases, Current Pathobiol. Rep., 8 (2020), 149–161. https://doi.org/10.1007/s40139-020-00213-x doi: 10.1007/s40139-020-00213-x
[8]	C. C. McCluskey, A model of HIV/AIDS with staged progression and amelioration, Math. Biosci., 181 (2003), 1–16. https://doi.org/10.1016/S0025-5564(02)00149-9 doi: 10.1016/S0025-5564(02)00149-9
[9]	H. Guo, M. Y. Li, Global dynamics of a staged-progression model with amelioration for infectious diseases, J. Biol. Dynam., 2 (2008), 154–168. https://doi.org/10.1080/17513750802120877 doi: 10.1080/17513750802120877
[10]	H. Guo, M. Y. Li, Z. Shuai, Global dynamics of a general class of multistage models for infectious diseases, SIAM J. Appl. Math., 72 (2012), 261–279. https://doi.org/10.1137/110827028 doi: 10.1137/110827028
[11]	Y. Wang, J. Cao, Global stability of general cholera models with nonlinear incidence and removal rates, J. Franklin Institute, 352 (2015), 2464–2485. https://doi.org/10.1016/j.jfranklin.2015.03.030 doi: 10.1016/j.jfranklin.2015.03.030
[12]	Y. Wang, S. Zhai, M. Du, P. Zhao, Dynamic behaviour of multi-stage epidemic model with imperfect vaccine, IAENG Int. J. Appl. Math., 52 (2022), 1–9.
[13]	M. C. Gómez, E. I. Mondragon, F. A. Rubio, Mathematical model of a SCIR epidemic system with migration and nonlinear incidence function, J. Math. Computer Sci., 31 (2023), 345–352. https://doi.org/10.22436/jmcs.031.04.01 doi: 10.22436/jmcs.031.04.01
[14]	S. Henshaw, C. C. McCluskey, Global stability of a vaccination model with immigration, Electr. J. Differ. Equat., 92 (2015), 1–10.
[15]	Z. A. Khan, A. L. Alaoui, A. Zeb, M. Tilioua, S. Djilali, Global dynamics of a SEI epidemic model with immigration and generalized nonlinear incidence functional, Results Phys., 27 (2021), 104477. https://doi.org/10.1016/j.rinp.2021.104477 doi: 10.1016/j.rinp.2021.104477
[16]	R. P. Sigdel, C. C. McCluskey, Global stability for an SEI model of infectious disease with immigration, Appl. Math. Comput., 243 (2014), 684–689. https://doi.org/10.1016/j.amc.2014.06.020 doi: 10.1016/j.amc.2014.06.020
[17]	P. J. Witbooi, An SEIR model with infected immigrants and recovered emigrants, Adv. Diffe. Equat., 2021 (2021), 1–15. https://doi.org/10.1186/s13662-020-03162-2 doi: 10.1186/s13662-020-03162-2
[18]	J. Zhang, J. Li, Z. Ma, Global dynamics of an SEIR epidemic model with immigration of different compartments, Acta Math. Sci., 26 (2006), 551–567. https://doi.org/10.1016/S0252-9602(06)60081-7 doi: 10.1016/S0252-9602(06)60081-7
[19]	H. Guo, M. Y. Li, Impacts of migration and immigration on disease transmission dynamics in heterogenous populations, Discrete Continuous Dynam. Syst. B, 17 (2012), 2413–2430. https://doi.org/10.3934/dcdsb.2012.17.2413 doi: 10.3934/dcdsb.2012.17.2413
[20]	V. Capasso, G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43–61. https://doi.org/10.1016/0025-5564(78)90006-8 doi: 10.1016/0025-5564(78)90006-8
[21]	A. Korobeinikov, M. Philip, Non-linear incidence and stability of infectious disease models, Math. Med. Biol. J. IMA, 22 (2005), 113–128. https://doi.org/10.1093/imammb/dqi001 doi: 10.1093/imammb/dqi001
[22]	World Health Organization: Health data overview for the Federative Republic of Brazil, 2023. Available from: https://data.who.int/countries/076
[23]	National Health Service: Overview HIV and AIDS, 2017. Available from: https://www.nhs.uk/conditions/hiv-and-aids/
[24]	National Institutes of Health: The Stages of HIV Infection, 2023. Available from: https://hivinfo.nih.gov/understanding-hiv/fact-sheets/stages-hiv-infection#:~:text=The%5C%20three%5C%20stages%5C%20of%5C%20HIV,one%5C%20stage%5C%20to%5C%20the%5C%20next
[25]	National Institutes of Health: What Are HIV and AIDS?, 2023. Available from: https://www.hiv.gov/hiv-basics/overview/about-hiv-and-aids/what-are-hiv-and-aids/
[26]	F. Brauer, P. van den Driessche, Models for transmission of disease with immigration of infectives, Math. Biosci., 171 (2001), 143–154. https://doi.org/10.1016/S0025-5564(01)00057-8 doi: 10.1016/S0025-5564(01)00057-8

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)