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

Qualitative analysis of solutions for a degenerate PDE model of epidemic dynamics