Recent literature on economic growth points towards the possibility of the existence of multiple equilibria, under certain conditions. In this paper we use a growth model with a public health infrastructure to analyze, in detail, its local dynamics. The discussion of the existence of equilibria in the model reveals that some candidates for equilibria turn out to be non-relevant and are therefore ruled out. Indeed, multiple equilibria may only arise under restrictive parameter values, meaning that the so called "growth miracles" in some literature may be less likely to occur. Numerical computations illustrate that each relevant equilibrium exhibits local saddle-path stability.