In this work, we study the dynamics of monkeypox by developing and analyzing a deterministic mathematical model. The model consists of six ordinary differential equations describing the interaction between the human compartments and rodent compartment. We first establish the well-posedness of the mathematical model. The basic reproduction number is given as a function of the human infection and rodent infection reproduction numbers. Without control variables, we investigate the basic model’s local stability. The result shows that the disease-free equilibrium of the model is asymptotically stable when the reproduction number is less than unity. By examining an optimum control issue, we further examine the efficacy of the optimal control pair on monkeypox disease. We use Pontryagin’s maximal principle to determine the essential conditions for the existence of optimal control. We perform numerical simulation of the optimality system using the fourth-order Runge–Kutta forward–backward numerical scheme to show the effect of the optimal control on monkeypox spread. Prevention against rodent–human and human–human infection contact is essential to eradicate monkeypox disease.