Application of Tikhonov's theory to an age-structured singularly perturbed SIS model with a general treatment function

We consider a simple SIS model with a general nonlinear treatment function. We combine it with a demographic model, keeping in mind the difference in time scale. A slow-fast system emerges. Our purpose is to show how Tikhonov's Theorem for singularly perturbed systems can be applied to reduce the whole model to planar systems that approximate its slow phases. Based on assumptions made about the treatment function, we determine graphically the number and the attractivity of slow surfaces on which reduced problems are defined. Approximation results are given on bounded, arbitrarily large and unbounded time intervals. An example is examined for a function that combines a natural recovery rate and a homographic treatment function that takes into account the limitations of hospital capacities. Numerical simulations are interpreted in light of theoretical results, followed by a discussion of treatment effectiveness and healthcare capacity.