Abstract
Fractional calculus has been widely used in mathematical modeling of evolutionary systems with memory effect on dynamics. The main interest of this work is to attest, through a statistical approach, how the hysteresis phenomenon, which describes a type of memory effect present in biological systems, can be treated by fractional calculus. We also analyse the contribution of the historical values of a function in the evaluation of fractional operators according to their order. To illustrate the efficiency of this non-integer order calculus, we consider the SIR (susceptible–infected–recovered) compartmental model which is widely used in epidemiology. We employ this compartmental model to study the dynamics of the spread of COVID-19 in some countries, one version with memory and one without memory.
Highlights
The fractional calculus was originated in 1695 as a generalization of the integer order calculus.This happened through a question asked byL’Hospital to Leibniz about the meaning of dn dxn, being n =In this way, it is possible to define integrals and derivatives with arbitrary orders.Since its inception, many contributions have been made by several researchers, including famous names such as Euler, Lagrange, Laplace, Fourier, Abel and Liouville (Camargo and Oliveira 2015; Srivastava and Saxena 2001).This calculus is one of the most effective current mathematical tools used to model realworld problems
To illustrate the potential of fractional differential equations in epidemiological processes with hysteresis, we investigate the evolution of COVID-19 in a population
We found out in the previous section that fractional calculus is capable of describing the hysteresis effect, which is more specific than the memory effect, since it involves the complete historical past to determine
Summary
The fractional calculus was originated in 1695 as a generalization of the integer order calculus.This happened through a question asked byL’Hospital to Leibniz about the meaning of dn dxn , being n =In this way, it is possible to define integrals and derivatives with arbitrary orders.Since its inception, many contributions have been made by several researchers, including famous names such as Euler, Lagrange, Laplace, Fourier, Abel and Liouville (Camargo and Oliveira 2015; Srivastava and Saxena 2001).This calculus is one of the most effective current mathematical tools used to model realworld problems. The fractional calculus was originated in 1695 as a generalization of the integer order calculus. L’Hospital to Leibniz about the meaning of dn dxn , being n = In this way, it is possible to define integrals and derivatives with arbitrary orders. Many contributions have been made by several researchers, including famous names such as Euler, Lagrange, Laplace, Fourier, Abel and Liouville (Camargo and Oliveira 2015; Srivastava and Saxena 2001). This calculus is one of the most effective current mathematical tools used to model realworld problems.
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