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The impact of media awareness on a fractional-order SEIR epidemic model with optimal treatment and vaccination

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Abstract

During any pandemic, to decrease the spreading possibility of infectious diseases to people, boosting public awareness is crucial. Media helps to boost this awareness. This paper aims to check the impact of awareness of media regarding infectious diseases. For this reason, an SEIR-type epidemic model is proposed and examined in this research, along with the memory effect. Here, stability analysis and the existence and uniqueness of the non-negative and bounded solution of our model are explored. The basic reproduction number \(R_0\) is calculated using the next-generation matrix method. Depending on the value of \(R_0\), only one endemic equilibrium occurs and is stable for \(R_0>1\). In addition, it is found that the system encounters a transcritical bifurcation at \(R_0=1\). Here, the optimal treatment and vaccination are determined using the fractional-order optimal control. The findings are also displayed and corroborated by replicating the model with certain hypothetical parameter values, and it can be concluded that fractional order produces better outcomes when the sickness persists in the population. To determine how sensitive the parameters are to \(R_0\) and the state variables, we do local and global sensitivity analysis. Sensitivity analysis reveals that by reducing the rates of new recruitment and disease transmission, we might be able to lower the value of \(R_0\). We might be able to retain more susceptibles in the susceptible class by increasing media literacy as well.

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Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. W.O. Kermack, A.G. McKendrick, A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Ser. A Contain. Pap. Math. Phys. Char. 115(772), 700–721 (1927)

  2. R. Lu, B. Ramakrishnan, M.W. Falah, A.K. Farhan, N.M.G. Al-Saidi, V.T. Pham, Synchronization and different patterns in a network of diffusively coupled elegant Wang–Zhang–Bao circuits. Eur. Phys. J. Spec. Top. (2022). https://doi.org/10.1140/epjs/s11734-022-00690-8

    Article  Google Scholar 

  3. M. Jawad, Z. Khan, E. Bonyah, R. Jan, Analysis of hybrid nanofluid stagnation point flow over a stretching surface with melting heat transfer. Math. Probl. Eng. 2022, 9469164 (2022). https://doi.org/10.1155/2022/9469164

    Article  Google Scholar 

  4. N.T.J. Bailey, The Mathematical Theory of Infectious Disease and its Applications (Charles Griffin and Company, London, 1975)

    Google Scholar 

  5. R.H. Anderson, R.M. May, Infectious Disease of Humans (Oxford University Press, Oxford, 1991)

    Book  Google Scholar 

  6. M. Amaku, F.A.B. Coutinho, E. Massad, Why dengue and yellow fever coexist in some areas of the world and not in others? Biosystems 106(2–3), 111–120 (2011)

    Article  Google Scholar 

  7. S. Majee, S. Adak, S. Jana, M. Mandal, T.K. Kar, Complex dynamics of a fractional-order SIR system in the context of COVID-19. J. Appl. Math. Comput. 68, 4051–4074 (2022). https://doi.org/10.1007/s12190-021-01681-z

    Article  MathSciNet  MATH  Google Scholar 

  8. P.A. Naik, Global dynamics of a fractional order SIR epidemic model with memory. Int. J. Biomath. 13(8), 2050071 (2020). https://doi.org/10.1142/S1793524520500710

    Article  MathSciNet  MATH  Google Scholar 

  9. A.M. Yousef, S.M. Salman, Backward bifurcation in a fractional-order SIRS epidemic model with a non-linear incidence rate. Int. J. Nonlinear Sci. Numer. Simul. 7–8(17), 401–412 (2016)

    Article  MATH  Google Scholar 

  10. S. Majee, S. Jana, D.K. Das, T.K. Kar, Global dynamics of a fractional-order HFMD model incorporating optimal treatment and stochastic stability. Chaos Solitons Fractals 161, 112291 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Jan, S. Boulaaras, S. Alyobi, M. Jawad, Transmission dynamics of hand-foot-mouth disease with partial immunity through non-integer derivative. Int. J. Biomath. 16, 2250115 (2023). https://doi.org/10.1142/S1793524522501157

    Article  MathSciNet  MATH  Google Scholar 

  12. C. Ozcaglar, A. Shabbeer, S.L. Vandenberg, B. Yener, K.P. Bennett, Epidemiological models of Mycobacterium tuberculosis complex infections. Math. Biosci. 236(2), 77–96 (2012). https://doi.org/10.1016/j.mbs.2012.02.003

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Khatua, D.K. Das, T.K. Kar, Optimal control strategy for adherence to different treatment regimen in various stages of tuberculosis infection. Eur. Phys. J. Plus 136, 801 (2021)

    Article  Google Scholar 

  14. S. Majee, S. Jana, S. Barman, T.K. Kar, Transmission dynamics of monkeypox virus with treatment and vaccination controls: a fractional order mathematical approach. Phys. Scr. 98(2), 024002 (2023). https://doi.org/10.1088/1402-4896/acae64

    Article  ADS  Google Scholar 

  15. S. Majee, S. Jana, T.K. Kar, Dynamical analysis of monkeypox transmission incorporating optimal vaccination and treatment with cost-effectiveness. Chaos 33, 043103 (2023). https://doi.org/10.1063/5.0139157

    Article  ADS  MathSciNet  Google Scholar 

  16. S. Paul, A. Mahata, S. Mukherjee, B. Roy, M. Salimi, A. Ahmadian, Study of fractional order SEIR epidemic model and effect of vaccination on the spread of COVID-19. Int. J. Appl. Comput. Math. 8, 237 (2022). https://doi.org/10.1007/s40819-022-01411-4

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Adak, R. Majumder, S. Majee, S. Jana, T.K. Kar, An ANFIS model-based approach to investigate the effect of lockdown due to COVID-19 on public health. Eur. Phys. J. Spec. Top. 231, 3317–3327 (2022). https://doi.org/10.1140/epjs/s11734-022-00621-7

    Article  Google Scholar 

  18. T.K. Kar, S. Jana, A theoretical study on mathematical modelling of an infectious disease with application of optimal control. Biosystems 111(1), 37–50 (2013). https://doi.org/10.1016/j.biosystems.2012.10

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. H.W. Berhe, O.D. Makinde, D.M. Theuri, Co-dynamics of measles and dysentery diarrhea diseases with optimal control and cost-effectiveness analysis. Appl. Math. Comput. 347, 903–921 (2019)

    MathSciNet  MATH  Google Scholar 

  21. T.K. Kar, S.K. Nandi, S. Jana, M. Mandal, Stability and bifurcation analysis of an epidemic model with the effect of media. Chaos Solitons Fractals 120, 188–199 (2019). https://doi.org/10.1016/j.chaos.2019.01.025

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Y. Li, J. Cui, The impact of media coverage on the dynamics of infectious disease. Int. J. Biomath. 1(1), 65–74 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. R. Jan, S. Boulaaras, S.A.A. Shah, Fractional-calculus analysis of human immunodeficiency virus and CD4+ T-cells with control interventions. Commun. Theor. Phys. 74(10), 105001 (2022). https://doi.org/10.1088/1572-9494/ac7e2b

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. R. Jan, A. Alharbi, S. Boulaaras, S. Alyobi, Z. Khan, A robust study of the transmission dynamics of zoonotic infection through non-integer derivative. Demonstr. Math. 55(1), 922–938 (2022). https://doi.org/10.1515/dema-2022-0179

    Article  MathSciNet  MATH  Google Scholar 

  25. O. AbuArqub, Computational algorithm for solving singular Fredholm time-fractional partial integrodifferential equations with error estimates. J. Appl. Math. Comput. 59, 227–243 (2019). https://doi.org/10.1007/s12190-018-1176-x

    Article  MathSciNet  Google Scholar 

  26. S. Momani, O.A. Arqub, B. Maayah, Piecewise optimal fractional reproducing Kernel solution and convergence analysis for the Atangana–Baleanu–Caputo model of the Lienard’s equation. Fractals 28(08), 2040007 (2020)

    Article  ADS  MATH  Google Scholar 

  27. M. Borah, A. Gayan, J.S. Sharma, Y.Q. Chen, Z. Wei, V.T. Pham, Is fractional-order chaos theory the new tool to model chaotic pandemics as Covid-19? Nonlinear Dyn. 109, 1187–1215 (2022). https://doi.org/10.1007/s11071-021-07196-3

    Article  Google Scholar 

  28. R. Jan, Z. Shah, W. Deebani, E. Alzahrani, Analysis and dynamical behavior of a novel dengue model via fractional calculus. Int. J. Biomath. 15(06), 2250036 (2022). https://doi.org/10.1142/S179352452250036X

    Article  MathSciNet  MATH  Google Scholar 

  29. T.Q. Tang, Z. Shah, E. Bonyah, R. Jan, M. Shutaywi, N. Alreshidi, Modeling and analysis of breast cancer with adverse reactions of chemotherapy treatment through fractional derivative. Comput. Math. Methods Med. 2022, 5636844 (2022). https://doi.org/10.1155/2022/5636844

    Article  Google Scholar 

  30. M. Du, Z. Wang, H. Hu, Measuring memory with the order of fractional derivative. Sci. Rep. 3, 3431 (2013). https://doi.org/10.1038/srep03431

    Article  ADS  Google Scholar 

  31. H.A.A. El-Saka, The fractional-order SIS epidemic model with variable population size. J. Egypt. Math. Soc. 22(1), 50–54 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. F. Gao, X. Li, W. Li, X. Zhou, Stability analysis of a fractional-order novel hepatitis B virus model with immune delay based on Caputo–Fabrizio derivative. Chaos Solitons Fractals 142, 110436 (2021). https://doi.org/10.1016/j.chaos.2020.110436

    Article  MathSciNet  MATH  Google Scholar 

  33. I. Petras, Fractional-order Nonlinear Systems: Modeling Aanlysis and Simulation (Higher Education Press, Beijing, 2011)

    Book  MATH  Google Scholar 

  34. J.M. Tchuenche, N. Dube, C.P. Bhunu, R.J. Smith, C.T. Bauch, The impact of media coverage on the transmission dynamics of human influenza. BMC Public Health 11(S1), 1–14 (2011)

    Article  Google Scholar 

  35. A. Khatua, T.K. Kar, Impacts of media awareness on a stage structured epidemic model. Int. J. Appl. Comput. Math. 6(5), 1–22 (2020). https://doi.org/10.1007/s40819-020-00904-4

    Article  MathSciNet  MATH  Google Scholar 

  36. E. Ahmed, A.S. Elgazzar, On fractional order differential equations model for nonlocal epidemics. Physica A Stat. Mech. Appl. 379(2), 607–614 (2007)

    Article  ADS  Google Scholar 

  37. V.D. Djordjević, J. Jarić, B. Fabry et al., Fractional derivatives embody essential features of cell rheological behavior. Ann. Biomed. Eng. 31, 692–699 (2003)

    Article  Google Scholar 

  38. A.M.A. El-Sayed, S.M. Salman, N.A. Elabd, On a fractional-order delay Mackey–Glass equation. Adv. Differ. Equ. 137, 1–11 (2016). https://doi.org/10.1186/s13662-016-0863-x

    Article  MathSciNet  MATH  Google Scholar 

  39. S.M. Salman, A.M. Yousef, On a fractional-order model for HBV infection with cure of infected cells. J. Egypt. Math. Soc. 4(25), 445–451 (2017)

    MathSciNet  MATH  Google Scholar 

  40. H. Delavari, D. Baleanu, J. Sadati, Stability analysis of caputo fractional-order non-linear systems revisited. Nonlinear Dyn. 67, 2433–2439 (2012). https://doi.org/10.1007/s11071-011-0157-5

    Article  MATH  Google Scholar 

  41. O.A. Arqub, A. El-Ajou, Solution of the fractional epidemic model by homotopy analysis method. J. King Saud Univ. Sci. 25(1), 73–81 (2013)

    Article  Google Scholar 

  42. C. Huang, L. Cai, J. Cao, Linear control for synchronization of a fractional-order time-delayed chaotic financial system. Chaos Solitons Fractals 113, 326–332 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. J. Huo, H. Zhao, L. Zhu, The effect of vaccines on backward bifurcation in a fractional order HIV model. Nonlinear Anal. Real World Appl. 26, 289–305 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. R. Rakkiyappan, G. Velmurugan, J. Cao, Stability analysis of fractional order complex-valued neural networks with time delays. Chaos Solitons Fractals 78, 297–316 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Y. Li, Y. Chen, I. Podlubny, Stability of fractional-order non-linear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability. Comput. Math. Appl. 59, 1810–1821 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  46. R. Jan, S. Boulaaras, Analysis of fractional-order dynamics of dengue infection with non-linear incidence functions. Trans. Inst. Meas. Control 44(13), 2630–2641 (2022). https://doi.org/10.1177/01423312221085049

    Article  Google Scholar 

  47. P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  48. E. Ahmed, A.M.A. El-Sayed, H.A.A. El-Saka, On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems. Phys. Lett. A 358(1), 1–4 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. G. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, New York, 1983)

    Book  MATH  Google Scholar 

  50. L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko, The Mathematical Theory of Optimal Processes (Wiley, New York, 1962)

    MATH  Google Scholar 

  51. W.H. Fleming, R.W. Rishel, Deterministic and Stochastic Optimal Control (Springer, New York, 1975)

    Book  MATH  Google Scholar 

  52. D.L. Lukes, Differential Equations: Classical to Control, Mathematics in Science and Engineering, vol. 162 (Academic Press, New York, 1982)

    Google Scholar 

  53. R. Shi, T. Lu, Dynamic analysis and optimal control of a fractional order model for hand-foot-mouth Disease. J. Appl. Math. Comput. 64, 565–590 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  54. K. Diethelm, Efficient solution of multi-term fractional differential equations using \(P(EC)^mE\) methods. Computing 71(4), 305–319 (2003). https://doi.org/10.1007/s00607-003-0033-3

    Article  MathSciNet  MATH  Google Scholar 

  55. M. Mandal, S. Jana, S. Majee, A. Khatua, T.K. Kar, Forecasting the pandemic COVID-19 in India: a mathematical approach. J. Appl. Nonlinear Dyn. 11(3), 549–571 (2022). https://doi.org/10.5890/JAND.2022.09.004

    Article  MathSciNet  Google Scholar 

  56. J.K.K. Asamoah, E. Yankson, E. Okyere, G.Q. Sun, Z. Jin, R. Jan, Optimal control and cost-effectiveness analysis for dengue fever model with asymptomatic and partial immune individuals. Results Phys. 31, 104919 (2021). https://doi.org/10.1016/j.rinp.2021.104919

    Article  Google Scholar 

  57. A. Jan, R. Jan, H. Khan, M.S. Zobaer, R. Shah, Fractional-order dynamics of Rift Valley fever in ruminant host with vaccination. Commun. Math. Biol. Neurosci. 2020, 79 (2020). https://doi.org/10.28919/cmbn/5017

    Article  Google Scholar 

  58. J. Wu, R. Dhingra, M. Gambhir, J.V. Remais, Sensitivity analysis of infectious disease models: methods, advances and their application. J. R. Soc. Interface 10(86) (2013). https://doi.org/10.1098/rsif.2012.1018

  59. S. Marino, I.B. Hogue, C.J. Ray, D.E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology. J. Theor. Biol. 254(1), 178–196 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  60. F.K. Zadeh, J. Nossent, F. Sarrazin, F. Pianosi, A. van Griensven, T. Wagener, W. Bauwens, Comparison of variance-based and moment-independent global sensitivity analysis approaches by application to the SWAT model. Environ. Model. Softw. 91, 210–222 (2017)

    Article  Google Scholar 

  61. F.J. Massey Jr., The Kolmogorov–Smirnov test for goodness of fit. J. Am. Stat. Assoc. 46(253), 68–78 (1951)

    Article  MATH  Google Scholar 

  62. H. Abboubakar, R. Fandio, B.S. Sofack, H.P. Ekobena Fouda, Fractional dynamics of a measles epidemic model. Axioms 11(8), 363 (2022). https://doi.org/10.3390/axioms11080363

    Article  Google Scholar 

  63. S.M. Salman, Memory and media coverage effect on an HIV/AIDS epidemic model with treatment. J. Comput. Appl. Math. 385(15), 113203 (2021). https://doi.org/10.1016/j.cam.2020.113203

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Research work of Suvankar Majee is financially supported by Council of Scientific and Industrial Research (CSIR), India (File No. 08/003(0142)/2020-EMR-I, dated: 18th March 2020), the work of Snehasis Barman is financially supported by NATIONAL FELLOWSHIP FOR SCHEDULED CAST STUDENTS (UGC-NFSC, UGC-Ref.No.: 191620004584, Dated: 20/05/2020) and the work of Prof. Tapan Kumar Kar is financially supported by Science and Engineering Research Board (File No. MTR/2022/000734, dated: 19/12/2022), Department of Science and Technology, Government of India. The authors would like to thank the guest editors Salah Boulaaras, Rashid Jan and Viet-Thanh Pham of the journal and the anonymous reviewers for their valuable suggestions and comments toward significant improvement of the article.

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  1. Suvankar Majee, Snehasis Barman, Anupam Khatua, T. K. Kar, and Soovoojeet Jana have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal, 711103, India

    Suvankar Majee, Snehasis Barman & T. K. Kar

  2. Department of Applied Sciences and Humanities, National Institute of Advanced Manufacturing Technology, Ranchi, Jharkhand, 834003, India

    Anupam Khatua

  3. Department of Mathematics, Ramsaday College, Amta, Howrah, West Bengal, 711401, India

    Soovoojeet Jana

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Majee, S., Barman, S., Khatua, A. et al. The impact of media awareness on a fractional-order SEIR epidemic model with optimal treatment and vaccination. Eur. Phys. J. Spec. Top. 232, 2459–2483 (2023). https://doi.org/10.1140/epjs/s11734-023-00910-9

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