A constant proportional caputo operator for modeling childhood disease epidemics

We propose a constant proportional-Caputo (CPC) operator for modeling childhood disease epidemics by treating positivity, boundedness, well-posedness, and biological viability and checking the reproductive and strength numbers for the biological system. For local and global stability analysis, we used the Volterra-type Lyapunov function with the first and second derivatives, providing symmetric results for fractional-order derivatives. We used different techniques to invert the proportional-Caputo (PC) and constant proportional-Caputo operators to evaluate the fractional integral operator. We also used our suggested model's fractional differential equations to derive the eigenfunctions of the CPC operator. Modern mathematical control with the fractional operator has many applications, including the complex and crucial study of systems with symmetry. Symmetry analysis is a powerful tool that enables the user to construct numerical solutions to a given fractional differential equation fairly systematically. Therefore, we construct the comparative results with the Caputo derivative and constant proportional-Caputo operator to understand the dynamic behavior of the disease. We also present an additional analysis of the constant proportional-Caputo and Hilfer generalized proportional operators. Using the Laplace Adomian Decomposition Method, we numerically simulate a system of fractional differential equations through tables and graphs.