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Global dynamics of a time-delayed two-strain epidemic model with general incidence rates and vaccination

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Abstract

This paper investigates a two strains time-delayed epidemic model with general incidence rates and vaccination. Five compartments will be included to the studied model that are the susceptible, the first strain infected subpopulation, the second strain infected ones, the vaccinated and the recovered individuals. Two time delays will be taken into account in our model in order to illustrate the incubation time infection period for each infected strain. In addition, a part of susceptible population is assumed to be vaccinated, but the vaccine is supposed to be efficient against only the first strain. Hence, the vaccinated individuals can be infected only by the second strain only. Since our problem deals with a population dynamics issue, we will demonstrate first that the solution is positive and bounded. In addition to this, the existence result will ensure the wellposedness of our suggested model. The basic reproduction number is given by utilizing the next generation matrix method and five steady states are determined. The global stability of each equilibrium is fulfilled using some suitable Lyapunov functionals. It was demonstrated that the equilibria global stability depends mainly on the first and second strain reproduction numbers. The numerical simulation will end the work by illustrating the equilibria stabilities and highlighting the effect of vaccination. Numerical tests were performed by using come classical biological incidences rates, such as bilinear one, Crowley–Martin, Beddington–DeAngelis and non-monotonic incidence functions. Besides illustrating the equilibria stability, it was shown that the vaccination strategy plays an essential role in controlling the disease spread. The vaccination, even its efficiency against only one strain infected individuals, have demonstrated its importance in eradicating the disease leading to an extinction of the infection. Acting on only one strain infected individuals, by administrating good vaccines, may give good results in terms of stopping the disease spread.

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Data Availability Statement

The author declares no data is used in this research.

References

  • Ahumada M, Ledesma-Araujo A, Gordillo L, Marín JF (2023) Mutation and SARS-CoV-2 strain competition under vaccination in a modified SIR model. Chaos Solitons Fractals 166:112964

    Article  CAS  Google Scholar 

  • Avila-Ponce de León U, Avila-Vales E, Huang KL (2022) Modeling COVID-19 dynamic using a two-strain model with vaccination. Chaos Solitons Fractals 157:111927

    Article  Google Scholar 

  • Baba IA, Hincal E (2017) Global stability analysis of two-strain epidemic model with bilinear and non-monotone incidence rates. Eur Phys J Plus 132:208

    Article  Google Scholar 

  • Baba IA, Hincal E (2018) A model for influenza with vaccination and awareness. Chaos Solitons Fractals 106:49–55

    Article  Google Scholar 

  • Baba IA, Hincal E, Alsaadi SHK (2018a) Global stability analysis of a two-strain epidemic model with awareness. Adv Differ Equ Control Process 19(2):83–100

    Google Scholar 

  • Baba IA, Kaymakamzade B, Hincal E (2018b) Two-strain epidemic model with two vaccinations. Chaos Solitons Fractals 106:342–348

    Article  Google Scholar 

  • Beddington JR (1975) Mutual interference between parasites or predators and its effect on searching efficiency. J. Animal Ecol. 44:331–341

    Article  Google Scholar 

  • Bentaleb D, Amine S (2019) Lyapunov function and global stability for a two-strain SEIR model with bilinear and non-monotone. Int J Biomath 12(2):1950021

    Article  Google Scholar 

  • Brenchley JM, Price DA, Schacker TW, Asher TE, Silvestri G, Rao S et al (2006) Microbial translocation is a cause of systemic immune activation in chronic HIV infection. Nat Med 12:1365–1371

    Article  CAS  Google Scholar 

  • Capasso V, Serio G (1978) A generalization of the Kermack-Mckendrick deterministic epidemic model. Math Biosci 42:43–61

    Article  Google Scholar 

  • Chen D, Xu Z (2016) Global dynamics of a delayed diffusive two-strain disease model. Differ Equ Appl 8(1):99–122

    Google Scholar 

  • Crowley PH, Martin EK (1989) Functional responses and interference within and between year classes of a dragonfly population. J N Am Benth Soc 8:211–221

    Article  Google Scholar 

  • DeAngelis DL, Goldstein RA, O’Neill RV (1975) A model for tropic interaction. Ecology 56:881–892

    Article  Google Scholar 

  • Farah EM, Amine S, Allali K (2021) Dynamics of a time-delayed two-strain epidemic model with general incidence rates. Chaos Solitons Fractals 153:111527

    Article  Google Scholar 

  • Gobalenya AE, Baker SC, Baric RS, de Groot RJ, Drosten C, Gulyaeva AA et al (2020) The species severe acute respiratory syndrome-related coronavirus: classifying 2019- nCoV and naming it SARS-CoV-2. Nat Microbiol 5:536–544

    Article  Google Scholar 

  • Golub JE, Bur S, Cronin W, Gange S, Baruch N, Comstock G, Chaisson RE (2006) Delayed tuberculosis diagnosis and tuberculosis transmission. Int J Tuber 10:24–30

    CAS  Google Scholar 

  • Gubler DJ (1998) Dengue and dengue hemorrhagic fever. Clin Microbiol Rev 11:480–96

    Article  CAS  Google Scholar 

  • Hale J, Verduyn Lunel SM (1993) Introduction to functional differential equations. Springer, New York

    Book  Google Scholar 

  • Ji C, Jiang D (2014) Threshold behaviour of a stochastic SIR model. Appl Math Model 38:5067–5079

    Article  Google Scholar 

  • Kaymakamzade B, Hincal E (2018) Two-strain epidemic model with two vaccinations and two time delayed. Qual Quant 52:695–709

    Article  Google Scholar 

  • Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. Proc R Soc Lond Ser A Contain Paper Math Phys Charact 115(772):700–721

    Google Scholar 

  • Khyar O, Allali K (2020) Global dynamics of a multi-strain SEIR epidemic model with general incidence rates: application to COVID-19 pandemic. Nonlinear Dyn 102(1):489–509

    Article  Google Scholar 

  • Kirby T (2021) New variant of SARS-CoV-2 in UK causes surge of COVID-19. Lancet Respir Med 9(2):e20–e21

    Article  CAS  Google Scholar 

  • Layne SP, Monto SP, Taubenberger JK (2020) Pandemic influenza: an inconvenient mutation. Science N.Y. 323:1560–1561

    Article  Google Scholar 

  • Liu X, Yang L (2012) Stability analysis of an SEIQV epidemic model with saturated incidence rate. Nonlinear Anal Real World Appl 13:2671–2679

    Article  Google Scholar 

  • Meskaf A, Khyar O, Danane J, Allali K (2020) Global stability analysis of a two-strain epidemic model with non-monotone incidence rates. Chaos Solitons Fractals 133:109647

    Article  Google Scholar 

  • Munster VJ, de Wit E, van den Brand JMA, Herfst S, Schrauwen EJA, Bestebroer TM, van de Vijver D, Boucher CA, Koopmans M, Rimmelzwaan GF et al (2009) Pathogenesis and transmission ofswine-origin a (h1n1) influenza virus, ferrets. Science. 325:481–483

    Article  CAS  Google Scholar 

  • Norder H, Couroucé AM, Magnius LO (1994) Complete genomes, phylogenetic relatedness, and structural proteins of six strains of the hepatitis B virus, four of which represent two new genotypes. Virology 198(2):489–503

    Article  CAS  Google Scholar 

  • Ruan S, Wang W (2003) Dynamical behavior of an epidemic model with a nonlinear incidence rate. J Differ Equ 188:135–163

    Article  Google Scholar 

  • Smith HL (1995) Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. In: Math. Surreys Monographs., vol. 41, AMS, Providence. RI

  • Tchoumi SY, Rwezaura H, Tchuenche JM (2022) Dynamic of a two-strain COVID-19 model with vaccination. Results Phys 39:105777

    Article  CAS  Google Scholar 

  • Van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci, pp 29–48

  • Van Oosterhout C, Hall N, Ly H, Tyler KM (2021) COVID-19 evolution during the pandemic-Implications of new SARS-CoV-2 variants on disease control and public health policies

  • Wang JJ, Zhang JZ, Jin Z (2010) Analysis of an SIR model with bilinear incidence rate. Nonlinear Anal Real World Appl 11:2390–2402

    Article  Google Scholar 

  • Xu Z, Qu L, Huang Y (2016) Global dynamics of a two-strain flu model with delay. Math Comput Simul 124:44–59

    Article  Google Scholar 

  • Zhao Y, Jiang D (2014) The threshold of a stochastic SIRS epidemic model with saturated incidence. Appl Math Lett 34:90–93

    Article  Google Scholar 

  • Zhou X, Cui J (2011) Global stability of the viral dynamics with Crowley-Martin functional response. Bull Korean Math Soc 48:555–574

    Article  Google Scholar 

Download references

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  1. Laboratory of Mathematics, Computer Science and Applications, Faculty of Sciences and Technologies, Hassan II University of Casablanca, PO Box 146, 20650, Mohammedia, Morocco

    Karam Allali

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  1. Karam Allali
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Allali, K. Global dynamics of a time-delayed two-strain epidemic model with general incidence rates and vaccination. Model. Earth Syst. Environ. 10, 4295–4315 (2024). https://doi.org/10.1007/s40808-024-02011-5

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