This work comprehensively analyzed the monkeypox virus utilizing a deterministic mathematical model within a constant proportional-Caputo derivative framework. The suggested model considered the interplay of human and rodent populations by incorporating certain realistic vaccination parameters. Our study was a testament to the thoroughness of this work. We explored the uniqueness result using Banach's contraction principle. The solution's positivity and boundedness were studied in detail, as were the basic reproduction number and the stability analysis of the system's equilibrium conditions. We performed a variety of Ulam's stability analyses to guarantee the solution existed. Additionally, we implemented a decomposition formula to obtain the numerical scheme. This numerical approach allowed for numerical simulation as a graphical representation for certain real data sets and different parameter values in order to understand the model's dynamic behavior.