This paper examined the features of an infection therapy for fractional-order quarry-hunter systems in order to control sickness. It focused especially on how illnesses and several populations combine to affect how well harvesting policies work. We created a new dynamic model full of such ideas by examining systems with fractional-order non-integer systems and introducing fractional-order systems that can remember in order to comprehend that specific system. These thresholds are essential for directing management strategies, according to research on the presence, uniqueness, and stability of solutions to these models. Additionally, we presented particular MATLAB-based numerical methods for fractional-order model. Through a series of numerical application experiments, we validated the method's efficacy and its value in guiding strategy modifications regarding harvesting rates in the face of epidemic infections. This demonstrates the necessity of using a fractional approach in ecosystem research in order to improve the methods used for resource management. This paper primarily focused on the unique insight brought into the quarry-hunter models with infectious diseases by the fractional-order dynamics in ecology. The results are meaningful especially since they can be utilized to come up with effective measures to control diseases and even promote the sustainability of ecological systems.

This work presents a numerical approach for handling a fractional Lienard equation (FLE) arising in an oscillating circuit. The scheme is based on the Vieta Lucas operational matrix of the fractional Liouville-Caputo derivative and the collocation method. This methodology involves a systematic approach wherein the operational matrix aids in expressing the fractional problem in terms of non-linear algebraic equations. The proposed numerical approach utilizing the operational matrix method offers a vital solution framework for efficiently tackling the fractional Lienard equation, addressing a key challenge in mathematical modeling. To analyze the fractional order system, we derive an approximate solution for the FLE. The solutions are explained graphically and in tabular form.