Legionnaires’ disease is a very serious type of pneumonia (lung infection) caused by bacteria called Legionella. In this research work, a mathematical model for the transmission dynamics of Legionnaires’ disease is developed. Mathematical analysis is carried out to gain relevant insight on the basic features of the model. Some of the findings of this research indicate the existence of a legionnaire-free equilibrium, which was later shown to be globally stable, provided the control reproduction number is less than one. Further analysis showed legionnaires’ disease to be endemic among the human population, which we proved to be globally asymptotically stable whenever the effective basic reproduction number is greater than unity and unstable whenever the effective basic reproduction number is less than unity. Furthermore, administering effective treatments to humans exposed to Legionnaires’ disease should be prioritized, as shown in the simulation results.