Mathematical modeling and optimal control analysis of Monkeypox virus in contaminated environment

Monkeypox is a rare zoonotic disease similar to smallpox though less severe. The 2022 Monkeypox disease presented a great threat to global health as outbreaks spread mostly to countries in which Monkeypox was non-endemic. This study presents a new mathematical model with the effect of environmental and direct transmission of the Monkeypox virus in human and rodent populations. Monkeypox-Free and Monkeypox-Endemic equilibrium points are established. The basic reproduction number \(\mathbb {R}_{0}\) ${\mathbb{R}}_0$ , is derived and used to predict the future of the disease. This led to the findings: Monkeypox-Free equilibrium is globally asymptotically stable whenever \(\mathbb {R}_0 \le 1\) ${\mathbb{R}}_0\le 1$ . Infected rodents’ only endemic equilibrium point is globally asymptotically stable whenever \(\mathbb {R}_{0n}^r >1\) ${\mathbb{R}}_{0n}^r>1$ and \(\mathbb {R}_{0h} <1\) ${\mathbb{R}}_{0h}<1$ . The infected humans only endemic equilibrium point is globally asymptotically stable whenever \(\mathbb {R}_{0n}^r <1\) ${\mathbb{R}}_{0n}^r<1$ and \(\mathbb {R}_{0h} >1\) ${\mathbb{R}}_{0h}>1$ , as confirmed by Lyapunov’s method and LaSalle’s invariant principle. Additionally, five time-dependent control variables to reduce the disease are adopted. The existence of optimal control is established and using Pontryagin’s maximum principle, the optimality conditions are derived. Runge-Kutta fourth-order schemes forward and backward in time are used to obtain the solutions of the state and co-state systems respectively. Numerical simulations showing different combinations of control variables in reducing Monkeypox infection are presented. The findings of this study give useful measures for health practitioners and policymakers to implement effective and optimal control ways to curtail the Monkeypox outbreak.