Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order

Abstract

In this paper we construct nonstandard finite difference (NSFD) schemes to obtain numerical solutions of the susceptible–infected (SI) and susceptible–infected–recovered (SIR) fractional-order epidemic models. In order to deal with fractional derivatives we apply the Caputo operator and use the Grünwald–Letnikov method to approximate the fractional derivatives in the numerical simulations. According to the principles of dynamic consistency we construct NSFD schemes to numerically integrate the fractional-order epidemic models. These NSFD schemes preserve the positivity that other classical methods do not guarantee. Additionally, the NSFD schemes hold other conservation properties of the solution corresponding to the continuous epidemic model. We run some numerical comparisons with classical methods to test the behavior of the NSFD schemes using the short memory principle. We conclude that the NSFD schemes, which are explicit and computationally inexpensive, are reliable methods to obtain realistic positive numerical solutions of the SI and SIR fractional-order epidemic models.