  Compartmental models are widely used in mathematical epidemiology to describe
dynamics of infection disease. A new SIS-PDE model, recently derived by Chalub
and Souza, is based on a diffusion-drift approximation of probability density
in a well-known discrete - time Markov chain SIS-DTMC model. This new SIS-PDE
model is conservative due to degeneracy of the diffusion term at the origin. We
analyze a class of degenerate PDE models and obtain sufficient conditions for
existence of classical solutions with certain regularity properties. Also, we
show that under some additional assumptions about coefficients and initial data
classical solutions vanish at the origin at any finite time. Vanishing at the
origin of solutions is consistent with the conservation property of the model.
The main results of this article are: sufficient conditions for existence of
weak solutions, analysis of their asymptotic behavior at the origin, and the
proof of existence of weak solutions convergent to Dirac delta function.
Moreover, we study long-time behavior of solutions and confirm our analysis by
numerical computations.
