review · Alexandria Engineering Journal
Mathematical biology increasingly uses fractional operators to model how infectious diseases spread through populations. This research investigates the use of innovative fractional operators to represent infectious disease dynamics. Diverse numerical methods and parameter adjustments for time fractional components were used to establish the mechanical properties of the model. Computational simulations were conducted across a range of fractional and fractal orders to confirm the theoretical findings. The existence of solutions for the epidemic model was established using the Banach contraction principle, and the reproductive number was determined under specific state conditions. Ultimately, the framework uses numerical simulations to evaluate disease transmission within a community, offering insights to support future disease analysis, decision-making, and public health control strategies.
Understanding how infectious diseases spread is essential for protecting communities during outbreaks. Applying fractional calculus to epidemiological modelling allows researchers to capture complex disease dynamics more accurately. This mathematical approach provides foundational tools that can inform public health decision-making and support the development of effective disease control strategies.
This work represents early-stage theoretical and computational research. It could eventually inform the algorithms behind epidemic simulation software used by public health agencies, academic researchers, and healthcare planners to assess transmission risk. However, the abstract describes mathematical proofs and simulations, indicating that practical software products or market applications remain distant.
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Both mathematics and science exhibit a reciprocally advantageous and close relationship. The undeniable significance of mathematics in diverse scientific fields, including electrical engineering, physics, biology, and medicine, is beyond dispute. Due to the multitude of applications of mathematical biology in the present era, scholars have demonstrated a marked intrigue in this specific realm. In the past few years, there has been a discernible rise in the application of different fractional operators as a method of expressing the given problem. The utilization of this approach has gained significant acceptance as a customary technique for examining the propagation of epidemics. The current investigation explores innovative fractional operators in the framework of a model for an infectious disease. The mechanical characteristics of the model with fractional orders are established by employing diverse numerical methods and adjusting parameters related to time fractional components. To substantiate the theoretical discoveries, computational simulations are implemented for the suggested approach over a variety of fractional orders, displaying the outcomes of both fractional and fractal orders. We employed a highly efficient numerical methodology to obtain solutions for both the model and simulations. Additionally, we provide the criteria for the existence of a solution to the proposed epidemic model, and we determine the reproductive number under specific state conditions of the analyzed dynamic system. A fractional-order model of infectious disease is proposed to be analyzed through simulations, aiming to ascertain the possible efficacy of disease transmission within the community. These aspects are tackled through the application of the Banach contraction principle. Moreover, we have proposed future research directions incorporating the utilization of a novel hybrid fractional operator. Fractional calculus emerges as a significant area of investigation within applied mathematics, offering valuable tools for problem-solving across diverse fields such as medicine, science, and engineering. Recent scholars have emphasized the importance of mathematical techniques, especially those related to fractional and integer calculus modeling, as extremely advantageous when examining the dynamics of different disease models. The utilization of the fractional operator allows for the investigation of the dynamic impact of diseases on society, providing valuable insights for analysis, decision-making, and disease control.
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DOI: 10.1016/j.aej.2024.03.059
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