<p>Swine flu is a viral communicable disease that spreads from animals to humans and from humans to humans through air droplets caused by coughing or sneezing. This type of infectious viral disease is usually analyzed through integer-order mathematical modeling. For better understanding of transmission dynamics, recently the fractal fractional order derivative modeling has is a topic of interest for the interdisciplinary research field due to its long-term predictions. In this paper, the swine flu disease is modeled via a fractal fractional operator with a Mittag-Leffler type kernel. In this work, the swine flu model is modified and analyzed by using the harmonic mean of the state variables <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40808_2025_2461_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="7" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {I}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">I</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40808_2025_2461_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {S}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">S</mi> </math></EquationSource> </InlineEquation>. The numerical simulation with the Adams-Bashforth method for this model is established to achieve the accurate findings. This approach enhances stability, robustness, and precision in disease dynamics. Firstly, the positive bounded solutions and invariant area are determined by providing the equilibrium point and basic reproduction number <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40808_2025_2461_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$R_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>, respectively. The fixed theorem is applied to determine the existence and uniqueness of the solution and Hyer-Ulam stability of the model. Our results highlight the potential of fractional derivatives in the analysis of viral diseases for future strategies and control policies. In our model, interactions among susceptible, exposed, symptomatic, asymptomatic, quarantined, and recovered individuals drive the dynamics of swine flu. Over a 10-day period, the susceptible class drops from 70 to 48, indicating a decrease in the number of people at risk for infection. It appears that the number of people exposed to the virus reaches a peak before falling as the exposed class peaks and then falls to 35. The effectiveness of both the natural recovery process and quarantine measures is demonstrated by the notable increase in the recovered class from 10 to 50 over the same time period. These patterns fit the predicted behavior of a fractal-fractional model, which effectively depicts the intricate dynamics of the spread of swine flu.</p>

Fractional modeling of a communicable through air droplets based disease in Mittag–Leffler kernel sense and its numerical simulation

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发表日期：

2025年6月16日

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Razia Begum,
Sajjad Ali,
Kamal Shah,
Thabet Abdeljawad

摘要

Swine flu is a viral communicable disease that spreads from animals to humans and from humans to humans through air droplets caused by coughing or sneezing. This type of infectious viral disease is usually analyzed through integer-order mathematical modeling. For better understanding of transmission dynamics, recently the fractal fractional order derivative modeling has is a topic of interest for the interdisciplinary research field due to its long-term predictions. In this paper, the swine flu disease is modeled via a fractal fractional operator with a Mittag-Leffler type kernel. In this work, the swine flu model is modified and analyzed by using the harmonic mean of the state variables $\mathbb {I}$ $I$ and $\mathbb {S}$ $S$ . The numerical simulation with the Adams-Bashforth method for this model is established to achieve the accurate findings. This approach enhances stability, robustness, and precision in disease dynamics. Firstly, the positive bounded solutions and invariant area are determined by providing the equilibrium point and basic reproduction number $R_0$ $R_{0}$ , respectively. The fixed theorem is applied to determine the existence and uniqueness of the solution and Hyer-Ulam stability of the model. Our results highlight the potential of fractional derivatives in the analysis of viral diseases for future strategies and control policies. In our model, interactions among susceptible, exposed, symptomatic, asymptomatic, quarantined, and recovered individuals drive the dynamics of swine flu. Over a 10-day period, the susceptible class drops from 70 to 48, indicating a decrease in the number of people at risk for infection. It appears that the number of people exposed to the virus reaches a peak before falling as the exposed class peaks and then falls to 35. The effectiveness of both the natural recovery process and quarantine measures is demonstrated by the notable increase in the recovered class from 10 to 50 over the same time period. These patterns fit the predicted behavior of a fractal-fractional model, which effectively depicts the intricate dynamics of the spread of swine flu.