On the Construction of Some Deterministic and Stochastic Non-Local SIR Models

• Main Author: Giacomo Ascione
• Format: Article
• Language: English
• Published: MDPI AG 2020-11-01
• Series: Mathematics
• Subjects: epidemic model; Bernstein function; semi-Markov process; subordinator; fluid limit
• Online Access: https://www.mdpi.com/2227-7390/8/12/2103

Fractional-order epidemic models have become widely studied in the literature. Here, we consider the generalization of a simple <inline-formula><math display="inline"><semantics><mrow><mi>S</mi><mi>I</mi><mi>R</mi></mrow></semantics></math></inline-formula> model in the context of generalized fractional calculus and we study the main features of such model. Moreover, we construct semi-Markov stochastic epidemic models by using time changed continuous time Markov chains, where the parent process is the stochastic analog of a simple <inline-formula><math display="inline"><semantics><mrow><mi>S</mi><mi>I</mi><mi>R</mi></mrow></semantics></math></inline-formula> epidemic. In particular, we show that, differently from what happens in the classic case, the deterministic model does not coincide with the large population limit of the stochastic one. This loss of fluid limit is then stressed in terms of numerical examples.