Abstract
Fractional-order epidemic models have become widely studied in the literature. Here, we consider the generalization of a simple SIR 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 SIR 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.
Highlights
Starting from the seminal paper by Kermack and McKendrick [1], epidemic models have been a fruitful field of applications of dynamical systems
Epidemic models with information variables have been considered, in the context of behavioral epidemiology
This paper focuses on introducing some general non-local SIR model by such kind of naive substitution in terms of Caputo-type non-local derivative induced by Bernstein functions
Summary
Starting from the seminal paper by Kermack and McKendrick [1], epidemic models have been a fruitful field of applications of dynamical systems. Concerning population models, one obtains analogous fractional-order stochastic models (or more general non-local models) by using time-changed continuous-time Markov chains, as one can see, for instance, in [22,23,24,25]. Let us anticipate an important feature one has to take in consideration when working with non-local SIR models (both deterministic and stochastic) Such kind of models do not adapt well to epidemics for which the number of infective is approximatively diffusive or super-diffusive in time (i.e., if the number of infective exhibits a variance of the form Ctγ with γ ≥ 1, see, e.g., [29]), limiting the field of applications. There is evidence that the Coronavirus infectious disease 2019 (COVID-2019) admits a super-diffusive behavior in some contexts (see, e.g., [30,31]), these models should not be applied to describe such infectious disease in the super-spreading environment
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