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A fractional SEIQR model on diphtheria disease

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Abstract

The purpose of this paper is to study a fractional mathematical model on diphtheria disease, considering the parameters of natural immunity, treatment, and vaccination. This model includes five compartments, namely, the susceptible, exposed, infected, quarantined, and recovered sub population. All compartments involve the memory effect and long-rate interactions that is modeled by a Caputo fractional derivative. This paper starts with the study of some analytical results. We first present the preliminary concepts of fractional calculus. The well-posedness of our fractional model is proved based on the boundedness, non-negativity, existence, and uniqueness. The existence of local, and global stability are also studied based on the Magniton’s theorem and the appropriate Lyapunov function. We further apply the predictor–corrector scheme to establish the numerical simulations. The validation of our fractional model is based on the real data by using the least square technique. We also present the comparison results of root mean square error for varying fractional order \(\alpha =1\), \(\alpha =0.95\), \(\alpha =0.9\), \(\alpha =0.85\), and \(\alpha =0.8\).

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Data availability statement

This study uses secondary data from the Ministry of Health of the Republic of Indonesia, which is published online. The reference used as secondary data information for the purposes of this study refers to Sariadji (2017).

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Acknowledgements

The authors received financial support for the research under contract number 121/UN3.1.17/PT/2022 by the Faculty of Advanced Technology and Multidiscipline, Universitas Airlangga, Indonesia. The authors would also like to thank the reviewers for their valuable comments and suggestions which helped to improve the paper.

Funding

Mohammad Ghani received financial support for the research under contract number 121/UN3.1.17/PT/2022 by the Faculty of Advanced Technology and Multidiscipline, Universitas Airlangga, Indonesia.

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Authors and Affiliations

  1. Faculty of Advanced Technology and Multidiscipline, Universitas Airlangga, Surabaya, 60115, Indonesia

    Mohammad Ghani, Ika Qutsiati Utami, Fadillah Willis Triyayuda & Mutiara Afifah

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  1. Mohammad Ghani
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  2. Ika Qutsiati Utami
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  3. Fadillah Willis Triyayuda
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  4. Mutiara Afifah
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Contributions

MG: formal analysis, investigation, methodology, software, writing—original draft, writing—review and editing. IQU: formal analysis, investigation, methodology, software, writing—original draft, writing—review and editing. AWT: conceptualization, formal analysis, investigation, writing—original draft, writing—review and editing. MA: conceptualization, formal analysis, investigation, writing—original draft, writing—review and editing.

Corresponding author

Correspondence to Mohammad Ghani.

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None of the authors of this paper has a financial or personal relationship with other people or organizations that could inappropriately influence or bias the content of the paper. It is to specifically state that “No Competing interests are at stake and there is No Conflict of Interest” with other people or organizations that could inappropriately influence or bias the content of the paper.

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Ghani, M., Utami, I.Q., Triyayuda, F.W. et al. A fractional SEIQR model on diphtheria disease. Model. Earth Syst. Environ. 9, 2199–2219 (2023). https://doi.org/10.1007/s40808-022-01615-z

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