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Analysis and Optimal Control of a Multistrain SEIR Epidemic Model with Saturated Incidence Rate and Treatment

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Abstract

In this paper, we study the dynamic of a multi-strain SEIR model with both saturated incidence and treatment functions. Two basic reproduction numbers are extracted from the epidemic model, noted \(R_{0,1}\) and \(R_{0,2}\). Using the Lyapunov method, we investigate the global stability of the disease free equilibrium and prove that it is globally asymptotically stable when \(R_{0,1}\) and \(R_{0,2}\) are less than one. Moreover, we formulate the optimal control problem, solve it, and perform some numerical simulations, to support the analytical results and test how well the proposed model may be applied in practice.

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References

  1. Arino, Julien, Cooke, K.L., Van Den Driessche, P., Velasco-Hernández, J.: An epidemiology model that includes a leaky vaccine with a general waning function. Discrete Contin. Dyn. Syst. Ser. B 4(2), 479–495 (2004)

    MathSciNet  MATH  Google Scholar 

  2. Bacaër, N.: Daniel bernoulli, d’alembert and the inoculation of smallpox (1760). A Short History of Mathematical Population Dynamics, pp. 21–30. Springer, Berlin (2011)

    Chapter  MATH  Google Scholar 

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

  4. Bhunu, C.P.: Mathematical analysis of a three-strain tuberculosis transmission model. Appl. Math. Modelling 35(9), 4647–4660 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bhunu, C.P., Garira, W.: A two strain tuberculosis transmission model with therapy and quarantine. Math. Modell. Anal. 14(3), 291–312 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brenchley, J.M., Price, D.A., Schacker, T.W., Asher, T.E., Silvestri, G., SR, Zachary Kazzaz, E Bornstein, O Lambotte, D Altmann, et al.: Microbial translocation is a cause of systemic immune activation in chronic hiv infection. Nat. Med. 12(12), 1365 (2006)

  7. Buonomo, B., Lacitignola, D.: On the backward bifurcation of a vaccination model with nonlinear incidence. Nonlinear Anal. 16(1), 30–46 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Capasso, V., Serio, G.: A generalization of the kermack-mckendrick deterministic epidemic model. Mathe. Biosci. 42(1–2), 43–61 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  9. Danane, J., Allali, K.: Optimal control of an hiv model with ctl cells and latently infected cells. Numer. Algebra Control Opt. 10(2), 207 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  10. Das, Anjana, Pal, M.: A mathematical study of an imprecise sir epidemic model with treatment control. J. Appl. Math. Comput. 56(1–2), 477–500 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Feng, Z., Thieme, H.R.: Recurrent outbreaks of childhood diseases revisited: the impact of isolation. Math. Biosci. 128(1–2), 93–130 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gao, D-p, Huang, N-j: Optimal control analysis of a tuberculosis model. Appl. Math. Modell. 58, 47–64 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Golub, J.E., Bur, S., Cronin, W.A., Gange, S., Baruch, N., Comstock, G.W., Chaisson, R.E.: Delayed tuberculosis diagnosis and tuberculosis transmission. Int. J. Tuberculosis Lung Dis. 10(1), 24–30 (2006)

    Google Scholar 

  14. D. J. Gubler. Epidemic dengue and dengue hemorrhagic fever: a global public health problem in the 21st century. In Emerging infections 1, pages 1–14. American Society of Microbiology, 1998

  15. Hethcote, H.W., Thieme, H.R.: Stability of the endemic equilibrium in epidemic models with subpopulations. Math. Biosci. 75(2), 205–277 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hethcote, H.W., Van Den Driessche, P.: Some epidemiological models with nonlinear incidence. J. Math. Biol. 29(3), 271–287 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhixing, H., Ma, W., Ruan, S.: Analysis of sir epidemic models with nonlinear incidence rate and treatment. Math. Biosci. 238(1), 12–20 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jana, Soovoojeet, Kumar Nandi, Swapan, Kar, T.K.: Complex dynamics of an sir epidemic model with saturated incidence rate and treatment. Acta. Biotheoretica 64(1), 65–84 (2016)

    Article  Google Scholar 

  19. Kar, T.K., Jana, S.: Application of three controls optimally in a vector-borne disease-a mathematical study. Commun. Nonlinear Sci. Numer. Simul. 18(10), 2868–2884 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kar, T.K., Jana, Soovoojeet: A theoretical study on mathematical modelling of an infectious disease with application of optimal control. Biosystems 111(1), 37–50 (2013)

    Article  Google Scholar 

  21. William O Kermack and Anderson G McKendrick. A contribution to the mathematical theory of epidemics. In Proceedings of the Royal Society of London A: mathematical, physical and engineering sciences, volume 115, pages 700–721. The Royal Society, 1927

  22. Khan, M.A., Khan, Y., Islam, S.: Complex dynamics of an seir epidemic model with saturated incidence rate and treatment. Phys. A 493, 210–227 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. S. Lenhart, J. T, Workman. Optimal control applied to biological models. Chapman and Hall/CRC, 2007

  24. Liu, W-m, Levin, S.A., Iwasa, Y.: Influence of nonlinear incidence rates upon the behavior of sirs epidemiological models. J. Math. Biol. 23(2), 187–204 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lukes, D.L.: Differential Equations: Classical to Controlled. Elsevier, Amsterdam (1982)

    MATH  Google Scholar 

  26. Daniel Makinde, O.: Adomian decomposition approach to a sir epidemic model with constant vaccination strategy. Appl. Math. Comput. 184(2), 842–848 (2007)

    MathSciNet  MATH  Google Scholar 

  27. Okosun, Kazeem O., Ouifki, Rachid, Marcus, Nizar: Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity. Biosystems 106(2–3), 136–145 (2011)

    Article  Google Scholar 

  28. LS Pontryagin, VG Boltyanskii, RV Gamkrelidze, and EF Mishchenko. The mathematical theory of optimal processes, ser. translated from the russian by kn trirogoff; edited by lw neustadt, 1962

  29. Qiu, Z., Feng, Z.: Transmission dynamics of an influenza model with vaccination and antiviral treatment. Bull. Math. Biol. 72(1), 1–33 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Rachah, Amira, Torres, Delfim F.M.: Mathematical modelling, simulation, and optimal control of the: ebola outbreak in west africa. Discrete Dyn. Nat. Soc. 2015, 2015 (2014)

    Google Scholar 

  31. H. S. Rodrigues, M. T. T. Monteiro, and D. F. M. Torres. Optimal control and numerical software: an overview. arXiv preprint arXiv:1401.7279, 2014

  32. Ruan, S., Wang, W.: Dynamical behavior of an epidemic model with a nonlinear incidence rate. J. Differ. Equ. 188(1), 135–163 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sweilam, N.H., AL-Mekhlafi, S.M.: Numerical study for multi-strain tuberculosis (tb) model of variable-order fractional derivatives. J. Adv. Res. 7(2), 271–283 (2016)

    Article  Google Scholar 

  34. Thomasey, D.H., Martcheva, M.: Serotype replacement of vertically transmitted diseases through perfect vaccination. J. Biol. Syst. 16(02), 255–277 (2008)

    Article  MATH  Google Scholar 

  35. Van den Driessche, Pauline, Watmough, James: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180(1–2), 29–48 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, J.-J., Zhang, J.-Z., Jin, Z.: Analysis of an sir model with bilinear incidence rate. Nonlinear Anal. 11(4), 2390–2402 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, Wendi: Backward bifurcation of an epidemic model with treatment. Math. Biosci. 201(1–2), 58–71 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. Lih-Ing, W., Feng, Z.: Homoclinic bifurcation in an siqr model for childhood diseases. J. Differ. Equ. 168(1), 150–167 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  39. Xiao, D., Ruan, S.: Global analysis of an epidemic model with nonmonotone incidence rate. Math. Biosci. 208(2), 419–429 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhang, Xu, Liu, Xianning: Backward bifurcation of an epidemic model with saturated treatment function. J. Math. Anal. Appl. 348(1), 433–443 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratory of Mathematics and Applications, Faculty of Sciences and Technologies, University Hassan II of Casablanca, Mohammedia, Po. Box 146, Morocco

    Dounia Bentaleb, Saida Amine & Karam Allali

  2. ENCG of Casablanca, University Hassan II, Casablanca, Morocco

    Sanaa Harroudi

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  1. Dounia Bentaleb
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  2. Sanaa Harroudi
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  3. Saida Amine
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  4. Karam Allali
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Correspondence to Dounia Bentaleb.

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Appendix. Numerical Algorithm

Appendix. Numerical Algorithm

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Bentaleb, D., Harroudi, S., Amine, S. et al. Analysis and Optimal Control of a Multistrain SEIR Epidemic Model with Saturated Incidence Rate and Treatment. Differ Equ Dyn Syst 31, 907–923 (2023). https://doi.org/10.1007/s12591-020-00544-6

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