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

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.

A fractional approach to study the pure-temporal Epidemic Type Aftershock Sequence (ETAS) process for earthquakes modeling

“…Some recent papers have been devoted to the applications of fractional differential equations in modelling the temporal decay of aftershocks, we refer for example to [6,7]. De facto, fractional derivatives seem to universally appear in mathematical models of epidemic processes (e.g., see [8,9,10,11]), thus playing an important role in handling diffusion and memory mechanisms. In particular, the Caputo fractional derivative is a very useful tool to describe natural processes with memory and an underlying power-law behavior, as it is defined by the convolution between a power law kernel and the ordinary derivative of a function.…”

A fractional approach to study the pure-temporal Epidemic Type Aftershock Sequence (ETAS) process for earthquakes modeling

“…Some recent papers have been devoted to the applications of fractional differential equations in modelling the temporal decay of aftershocks, we refer for example to [24,41]. De facto, fractional derivatives seem to universally appear in mathematical models of epidemic processes (e.g., see [1,3,4,33]), thus playing an important role in handling diffusion and memory mechanisms. In particular, the Caputo fractional derivative is a very useful tool to describe natural processes with memory and an underlying power-law behavior, as it is defined by the convolution between a power law kernel and the ordinary derivative of a function.…”

A SEIR Epidemic Model of Whooping Cough‐Like Infections and Its Dynamically Consistent Approximation

“…e unreported cases and their modeling regarding COVID-19 are studied in reference [21]. Wellknown models studied with stochastic techniques are presented in references [22][23][24][25]. e main focus of this research is to study the efficiencies of the implicit numerical integration scheme for the whooping cough disease.…”