Skip to main content
SpringerLink
Log in

Stochastic two-strain epidemic model with bilinear and non-monotonic incidence rates

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

This work is focused in the mathematical modeling and analysis of the diseases resulting from multiple strains. It is in this context that our aim is to formulate a stochastic model driven by white noise, where, the infection rate of the first and second strains are described by bilinear and non-monotone incidence functions, respectively. The paper begins by examining whether there is a unique global positive solution. After that, the paper moves to the investigation of the disease’s extinction and persistence in mean of the two-strain epidemic disease. Finally, diverse numerical simulations are achieved to validate the theoretical findings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Data Availability

No data associated in the manuscript.

References

  1. J. He, D. Gu, X. Wu, K. Reynolds, X. Duan, C. Yao, J. Wang, C.S. Chen, J. Chen, R.P. Wildman et al., Major causes of death among men and women in China. N. Engl. J. Med. 353(11), 1124–1134 (2005)

    Article  Google Scholar 

  2. N. Imran, M. Zeshan, Z. Pervaiz, Mental health considerations for children & adolescents in covid-19 pandemic. Pakistan J. Med. Sci. 36(COVID19–S4), S67 (2020)

    Google Scholar 

  3. J.B. Xavier, J.M. Monk, S. Poudel, C.J. Norsigian, A.V. Sastry, C. Liao, J. Bento, M.A. Suchard, M.L. Arrieta-Ortiz, E.J. Peterson et al., Mathematical models to study the biology of pathogens and the infectious diseases they cause. Iscience 25, 104079 (2022)

    Article  ADS  Google Scholar 

  4. W.O. Kermack, A.G. McKendrick, A contribution to the mathematical theory of epidemics. Proc. R. Soc. London Ser. A, Contain. Papers Math. Phys. Character 115(772), 700–721 (1927)

    ADS  MATH  Google Scholar 

  5. Y. Fang, Y. Nie, M. Penny, Transmission dynamics of the covid-19 outbreak and effectiveness of government interventions: a data-driven analysis. J. Med. Virol. 92(6), 645–659 (2020)

    Article  Google Scholar 

  6. N. Yoshida, Existence of exact solution of the susceptible-exposed-infectious-recovered (SEIR) epidemic model. J. Differ. Equ. 355, 103–143 (2023). https://doi.org/10.1016/j.jde.2023.01.017

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. S. Sharma, V. Volpert, M. Banerjee, Extended SEIQR type model for covid-19 epidemic and data analysis. medRxiv 2020–08 (2020)

  8. A. Adhikary, A. Pal, A six compartments with time-delay model SHIQRD for the COVID-19 pandemic in India: During lockdown and beyond. Alex. Eng. J. 61(2), 1403–1412 (2022)

    Article  Google Scholar 

  9. M. Kamat, V. Kurlawala, G. Ghosh, R. Vaishnav et al., Immune dynamics of SARS-CoV-2 virus evolution. Int. J. Mol. Immuno Oncol. 7(1), 3–15 (2022)

    Article  Google Scholar 

  10. H. Liu, P. Wei, J.W. Kappler, P. Marrack, G. Zhang, SARS-CoV-2 variants of concern and variants of interest receptor binding domain mutations and virus infectivity. Front. Immunol. 13, 50 (2022)

    Google Scholar 

  11. A. Crespo-Bellido, S. Duffy, The how of counter-defense: viral evolution to combat host immunity. Curr. Opin. Microbiol. 74, 102320 (2023). https://doi.org/10.1016/j.mib.2023.102320

    Article  Google Scholar 

  12. M.L. van de Weijer, R.D. Luteijn, E.J. Wiertz, Viral immune evasion: lessons in MHC class i antigen presentation. Seminars in immunology, Elsevier 27, 125–137 (2015)

  13. H.C. Li, S.Y. Lo, Hepatitis c virus: virology, diagnosis and treatment. World J. Hepatol. 7(10), 1377 (2015)

    Article  Google Scholar 

  14. T. Li, Y. Guo, Modeling and optimal control of mutated covid-19 (delta strain) with imperfect vaccination. Chaos, Solitons Fractals 156, 111825 (2022). https://doi.org/10.1016/j.chaos.2022.111825

    Article  MathSciNet  Google Scholar 

  15. J.M. Cuevas, R. Geller, R. Garijo, J. López-Aldeguer, R. Sanjuán, Extremely high mutation rate of HIV-1 in vivo. PLoS Biol. 13(9), e1002251 (2015)

    Article  Google Scholar 

  16. E.O. Alzahrani, W. Ahmad, M.A. Khan, S.J. Malebary, Optimal control strategies of ZIKA virus model with mutant. Commun. Nonlinear Sci. Numer. Simul. 93, 105532 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  17. J.C. Eckalbar, W.L. Eckalbar, Dynamics of an epidemic model with quadratic treatment. Nonlinear Anal. Real World Appl. 12(1), 320–332 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Sadki, S. Harroudi, K. Allali, Local and global stability of an HCV viral dynamics model with two routes of infection and adaptive immunity. Comput. Methods Biomech. Biomed. Eng. 1–28 (2023)

  19. M.L. Taylor, T.W. Carr, An SIR epidemic model with partial temporary immunity modeled with delay. J. Math. Biol. 59, 841–880 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Y. Muroya, Y. Enatsu, Y. Nakata, Global stability of a delayed sirs epidemic model with a non-monotonic incidence rate. J. Math. Anal. Appl. 377(1), 1–14 (2011). https://doi.org/10.1016/j.jmaa.2010.10.010

    Article  MathSciNet  MATH  Google Scholar 

  21. M.A. Kuddus, E.S. McBryde, A.I. Adekunle, M.T. Meehan, Analysis and simulation of a two-strain disease model with nonlinear incidence. Chaos, Solitons Fractals 155, 111637 (2022)

    Article  MathSciNet  Google Scholar 

  22. I.A. Baba, E. Hincal, Global stability analysis of two-strain epidemic model with bilinear and non-monotone incidence rates. Eur. Phys. J. Plus 132, 1–10 (2017)

    Article  Google Scholar 

  23. D. Bentaleb, S. Amine, Lyapunov function and global stability for a two-strain seir model with bilinear and non-monotone incidence. Int. J. Biomath. 12(02), 1950021 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. H.F. Huo, Z.P. Ma, Dynamics of a delayed epidemic model with non-monotonic incidence rate. Commun. Nonlinear Sci. Numer. Simul. 15(2), 459–468 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Q. Liu, D. Jiang, T. Hayat, A. Alsaedi, Dynamics of a stochastic multigroup SIQR epidemic model with standard incidence rates. J. Franklin Inst. 356(5), 2960–2993 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  27. M. Sadki, A. Ez-zetouni, K. Allali, Persistence and extinction for stochastic HBV epidemic model with treatment cure rate. Boletim da Sociedade Paranaense de Matemática Article in press (2022). https://doi.org/10.5269/bspm.64254

    Article  Google Scholar 

  28. Q. Yang, X. Mao, Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations. Nonlinear Anal. Real World Appl. 14(3), 1434–1456 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. H. Andersson, T. Britton, Stochastic Epidemic Models and their Statistical Analysis, vol. 151 (Springer Science & Business Media, Berlin, 2012)

    MATH  Google Scholar 

  30. C. Rajivganthi, F.A. Rihan, Global dynamics of a stochastic viral infection model with latently infected cells. Appl. Sci. 11(21), 10484 (2021)

    Article  Google Scholar 

  31. F.A. Rihan, C. Rajivganthi, Dynamics of tumor-immune system with random noise. Mathematics 9(21), 2707 (2021)

    Article  Google Scholar 

  32. B. Boukanjime, M. El Fatini, A. Laaribi, R. Taki, Analysis of a deterministic and a stochastic epidemic model with two distinct epidemics hypothesis. Phys. A 534, 122321 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Z. Chang, X. Meng, X. Lu, Analysis of a novel stochastic sirs epidemic model with two different saturated incidence rates. Phys. A 472, 103–116 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  34. E.M. Farah, S. Amine, S. Ahmad, K. Nonlaopon, K. Allali, Theoretical and numerical results of a stochastic model describing resistance and non-resistance strains of influenza. Eur. Phys. J. Plus 137(10), 1–15 (2022)

    Article  Google Scholar 

  35. A. Miao, X. Wang, T. Zhang, W. Wang, B. Sampath Aruna Pradeep, Dynamical analysis of a stochastic sis epidemic model with nonlinear incidence rate and double epidemic hypothesis. Adv. Difference Equ. 2017, 1–27 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  36. X. Wang, C. Huang, Y. Hao, Q. Shi, A stochastic mathematical model of two different infectious epidemic under vertical transmission. Math. Biosci. Eng. 19(3), 2179–2192 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  37. M.E. Craft, Infectious disease transmission and contact networks in wildlife and livestock. Philos. Trans. R. Soc. B: Biol. Sci. 370(1669), 20140107 (2015)

    Article  Google Scholar 

  38. M.B. Elowitz, A.J. Levine, E.D. Siggia, P.S. Swain, Stochastic gene expression in a single cell. Science 297(5584), 1183–1186 (2002)

    Article  ADS  Google Scholar 

  39. J.M. Raser, E.K. O’shea, Noise in gene expression: origins, consequences, and control. Science 309(5743), 2010–2013 (2005)

    Article  ADS  Google Scholar 

  40. Y. Zhao, D. Jiang, D. O’Regan, The extinction and persistence of the stochastic sis epidemic model with vaccination. Phys. A 392(20), 4916–4927 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  41. R.S. Liptser, A strong law of large numbers for local martingales. Stochastics 3(1–4), 217–228 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  42. I.A. Baba, H. Ahmad, M. Alsulami, K.M. Abualnaja, M. Altanji, A mathematical model to study resistance and non-resistance strains of influenza. Results Phys. 26, 104390 (2021)

    Article  Google Scholar 

  43. T. Baranovich, R. Saito, Y. Suzuki, H. Zaraket, C. Dapat, I. Caperig-Dapat, T. Oguma, I.I. Shabana, T. Saito, H. Suzuki et al., Emergence of h274y oseltamivir-resistant a (h1n1) influenza viruses in Japan during the 2008–2009 season. J. Clin. Virol. 47(1), 23–28 (2010)

    Article  Google Scholar 

  44. J. Carr, J. Ives, L. Kelly, R. Lambkin, J. Oxford, D. Mendel, L. Tai, N. Roberts, Influenza virus carrying neuraminidase with reduced sensitivity to oseltamivir carboxylate has altered properties in vitro and is compromised for infectivity and replicative ability in vivo. Antiviral Res. 54(2), 79–88 (2002)

    Article  Google Scholar 

  45. A.S. Monto, J.L. McKimm-Breschkin, C. Macken, A.W. Hampson, A. Hay, A. Klimov, M. Tashiro, R.G. Webster, M. Aymard, F.G. Hayden et al., Detection of influenza viruses resistant to neuraminidase inhibitors in global surveillance during the first 3 years of their use. Antimicrob. Agents Chemother. 50(7), 2395–2402 (2006)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Marya Sadki & Karam Allali

Authors
  1. Marya Sadki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Karam Allali
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marya Sadki.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sadki, M., Allali, K. Stochastic two-strain epidemic model with bilinear and non-monotonic incidence rates. Eur. Phys. J. Plus 138, 923 (2023). https://doi.org/10.1140/epjp/s13360-023-04563-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-023-04563-4

Search

Navigation