Piecewise fractional derivatives and wavelets in epidemic modeling

In this paper, we propose a novel methodology for studying the dynamics of epidemic spread, focusing on the utilization of fundamental mathematical concepts related to piecewise differential and integral operators. These mathematical tools play a crucial role in the process of modeling epidemic phenomena, enabling us to investigate the behavior of infectious diseases within defined time intervals. Our primary objective is to enhance our understanding of epidemic dynamics and the underlying influencing factors. We introduce the Susceptible-Infectious-Recovered (SIR) model as the foundational framework, which is formulated as a system of differential equations. Our approach involves discretizing time and employing interpolation techniques for integrals, specifically utilizing the collocation method with Bernoulli wavelets. By incorporating piecewise derivatives, we are able to conduct comprehensive simulations and analyses of epidemic spread under various intervention strategies, including social distancing measures. The outcomes of our numerical simulations serve to validate the efficacy of our proposed methodology, offering valuable insights into the intricate dynamics of real -world epidemic scenarios. This contribution significantly advances the field of epidemic control optimization, providing an integrated framework that seamlessly integrates fractional calculus, piecewise differential derivatives, and the capabilities of wavelets. Our findings provide crucial guidance for policymakers and healthcare leaders, offering a deeper understanding of the effectiveness of different control strategies. By considering our innovative approach, we can better inform and shape epidemic control measures, ultimately enhancing public health and fortifying our defenses against infectious diseases.