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In this study, we conducted a thorough and in-depth investigation into the existence and stability of the boundary equilibria of a deterministic mathematical model for mpox transmission to gain a deeper understanding of the disease dynamics. The investigation begins with the identification and analysis of the mpox boundary equilibria, which include the rodent-only boundary equilibrium, the human-only boundary equilibrium, and the co-existence of the rodent and human boundary equilibria. Using centre manifold analysis, it was demonstrated that both the rodent equilibrium and the human equilibrium exhibit backward bifurcation when the basic reproduction number of the mpox transmission model is less than unity. This backward bifurcation phenomenon complicates the control and eradication of mpox in both rodent and human populations, even when the basic reproduction number is below one. To evaluate the robustness of the model results, uncertainty and sensitivity analyses were performed using Latin Hypercube Sampling and Partial Rank Correlation Coefficient methods. The results indicate that the basic reproduction number is most sensitive to the human-to-human and rodent-to-rodent transmission rates, as well as the proportion of quarantined humans who progress to active infection, while increases in the treatment rate of infected humans and the rodent death rate significantly reduce the transmission potential of mpox. Additionally, to underscore the importance of community education, public awareness, and enlightenment campaigns in curbing the spread of mpox, we introduced two time-dependent control measures to the mpox model, namely, precautionary actions taken by susceptible individuals to hinder the spread of Mpox (these actions include regular hand washing, wearing hand gloves when handling rodents, and avoiding direct contact with the open sores of infected humans), and the use of disinfectants (in washing of clothes and cleaning of surfaces) and proper sanitation practices to increase the decay rate of the mpox virus in the environment. Our numerical simulations indicate that each control measure is effective in reducing the spread of mpox. However, the combined implementation of both control measures proves to be particularly effective in significantly reducing the prevalence of the disease.