Compartmental models in epidemiology consist of a subdivision of individuals into disjoint subgroups and transition rules among these compartments. The Susceptible-Infected-Recovered (SIR) and Susceptible-ExposedInfected-Recovered (SEIR) models describe the spread of a disease characterized by permanent immunity, once the subject is recovered from the disease. The Susceptible-Infected-Susceptible (SIS) type models describe the spread of diseases characterized by a temporary immunity. All models possess equilibrium states characterized by the vanishing of the number of Infected individuals. In previous work, we modelled the sol-gel transition of chemical (irreversible) and physical (reversible) gels in terms of the SIR/SEIR and SIS models respectively. In all cases we have seen that the points ti where higher order derivatives of sigmoidal transition curves reach their absolute extreme values, have an accumulation point to. In the case of chemical gels, this point agrees qualitatively with the “gel point”. Based on these observations we defined the “critical point of a sigmoidal curve y(t)” as the limit of the sequence {ti}. We proved that, under certain general assumptions, the limit point t0 corresponds to the slope of the asymptotic phase angle of the Fourier transform of y’(t). This result is interpreted as the existence of a preferred origin of time that will make the Fourier transform of y’(t) asymptotically real. In the case of the SIR/SEIR models, the critical point is computed by symbolic and numeric computations. For the SIS model (with some modifications) the exact solution turns out to be the “generalized logistic growth curve” whose Fourier transform is computed analytically and the critical point is explicitly found.