Fractional calculus: numerical methods and SIR models

Fractional calculus is ”the theory of integrals and derivatives of arbitrary order, which unify and generalize the notions of integer-order differentiation and n-fold integration”. The idea of generalizing differential operators to a non-integer order, in particular to the order 1/2, first appears in the correspondence of Leibniz with L’Hopital (1695), Johann Bernoulli (1695), and John Wallis (1697) as a mere question or maybe even play of thoughts. In the following three hundred years a lot of mathematicians contributed to the fractional calculus: Laplace (1812), Lacroix (1812), Fourier (1822), Abel (1823-1826), Liouville (1832-1837), Riemann (1847), Grunwald (1867-1872), Letnikov (1868-1872), Sonin (1869), Laurent (1884), Heaviside (1892-1912), Weyl (1917), Davis (1936), Erde`lyi (1939-1965), Gelfand and Shilov (1959-1964), Dzherbashian (1966), Caputo (1969), and many others. Yet, it is only after the First Conference on Fractional Calculus and its applications that the fractional calculus becomes one of the most intensively developing areas of mathematical analysis. Recently, many mathematicians and applied researchers have tried to model real processes using the fractional calculus. This is because of the fact that the realistic modeling of a physical phenomenon does not depend only on the instant time, but also on the history of the previous time which can be successfully achieved by using fractional calculus. In other words, the nature of the definition of the fractional derivatives have provided an excellent instrument for the modeling of memory and hereditary properties of various materials and processes