Abstract
<title>Abstract</title> <p> In group theory, symmetry belongs to a single model or to a whole family of models, and most time, conflating the two invalidates any reduction built on it. This paper pin points the difference clearly for a three-compartment susceptible–infected–AIDS model of HIV transmission as the case may be. The nonnegative octant is revealed to be forward invariant, every solution is bounded, with disease-free and endemic equilibria are separated by the basic reproduction number. Putting to test Lie point symmetry directly through the first prolongation, we establish that time translation is the only elementary symmetry admitted when the parameters are held fixed, and that the population scalings commonly assumed for compartmental models violate the determining equations outright. Those scalings are rehabilitated only as equivalence transformations, once the recruitment and transmission parameters are rescaled in step with the populations. Together with time dilation this yields a solvable three-parameter equivalence group whose invariants collapse five parameters to three and return a compact dimensionless model. An independent numerical experiment confirms the equivalence group to eleven significant figures, and the reduced flow reproduces the transcritical threshold, the endemic equilibrium, and the parameter sensitivities of the full system exactly. The analysis absolutely gives a reproducible template for distinguishing genuine symmetries from modelfamily equivalences in richer HIV models, including those with treatment, age structure, then spatial spread. <bold>Mathematics Subject Classification</bold> 92D30 · 34C14 · 22E70 · 34A34 </p>