The discrete fractional order difference applied to an epidemic model with indirect transmission

Abstract

[EN] A discrete fractional order model is proposed to analyze the behaviour of an epidemic process with indirect transmission. This model is based on a discrete version of the GrunwaldLetnikov fractional derivative operator. Some properties of this operator are shown and used to derive a truncated version of the operator, which is used to propose a model with short-term memory. Based on the biological meaning of the problem, some bounds have been obtained to assure the nonnegativity of the model solution. The (alpha, k ) -Basic Reproduction Number has been introduced and used to analyze the stability of the solution around its equilibrium points. Moreover, the influence of the fractional order, alpha, and the memory steps, k , on the behaviour of the solution has been analyzed. Finally, the results obtained have been clarified by means of numerical simulations of a model for the evolution of an infection by Salmonella in a hens farm. (c) 2021 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )