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. 2024 Oct 12;14(1):23871.
doi: 10.1038/s41598-024-74767-w.

Time delayed fractional diabetes mellitus model and consistent numerical algorithm

Mudassar Rafique  1 , Muhammad Aziz Ur Rehamn  1 , Muhammad Rafiq  2   3 , Zafar Iqbal  4 , Nauman Ahmed  5   6 , Hadil Alhazmi  7 , Shafiullah Niazai  8 , Ilyas Khan  9
Affiliations

Time delayed fractional diabetes mellitus model and consistent numerical algorithm

Mudassar Rafique et al. Sci Rep. .

Abstract

The diabetes mellitus model (DMM) is explored in this study. Many health issues are caused by this disease. For this reason, the integer order DMM is converted into the time delayed fractional order model by fitting the fractional order Caputo differential operator and delay factor in the model. It is proved that the generalized model has the advantage of a unique solution for every time t. Moreover, every solution of the system is positive and bounded. Two equilibrium states of the fractional model are worked out i.e. disease free equilibrium state and the endemic equilibrium state. The risk factor indicator, R0 is computed for the system. The stability analysis is carried out for the underlying system at both the equilibrium states. The key role of R0 is investigated for the disease dynamics and stability of the system. The hybridized finite difference numerical method is formulated for obtaining the numerical solutions of the delayed fractional DMM. The physical features of the numerical method are examined. Simulated graphs are presented to assess the biological behavior of the numerical method. Lastly, the outcomes of the study are furnished in the conclusion section.

Keywords: Fractional time delayed differential equation; GL non-standard finite difference schemes; Global stability.; LaSalle principal; Volterra type Lyapunov function.

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Conflict of interest statement

The authors declare no competing interests.

Figures

Fig. 1
Fig. 1
Simulated graphs at endemic equilibrium point.
Fig. 2
Fig. 2
Visualization of the disease-free equilibrium point.
Fig. 3
Fig. 3
Effect of delay parameter formula image on exposed (pre-diabetic).
See this image and copyright information in PMC

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