Mathematical analysis of fractional order co-abuse infection model using power law type kernel

In this study, we present a mathematical model designed to illustrate the simultaneous occurrence of smoking and heroin co-abuse infections.To explore non-negative solutions and identify a stable equilibrium point, as well as the fundamental reproductive number, we enhance the model by integrating Caputo fractional-order (FO) derivative operators.Employing functional analysis concepts, we derive several results pertaining to the existence of a unique solution.Additionally, we utilize the Ulam-Hyres (UH) notion to establish the stability of the model solutions.To offer further insights, we present numerical results for the fractional-order system using an Euler-type numerical technique.These results are visually represented in graphs, illustrating the diverse responses of the model under different parameter values.