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

TL;DR

This paper analyzes a degenerate PDE epidemic model, establishing conditions for solutions' existence, regularity, and long-term behavior, supported by numerical simulations.

Contribution

It provides new theoretical insights into the existence, regularity, and asymptotic behavior of solutions for a degenerate SIS-PDE epidemic model.

Findings

Existence of classical and weak solutions under certain conditions

Solutions vanish at the origin, aligning with conservation properties

Long-term behavior includes convergence to Dirac delta functions

Abstract

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.