Fractional SIS Epidemic Models

Abstract

In this paper, we consider the fractional SIS (susceptible-infectious-susceptible) epidemic model (α-SIS model) in the case of constant population size. We provide a representation of the explicit solution to the fractional model and we illustrate the results by numerical schemes. A comparison with the limit case when the fractional order α converges to 1 (the SIS model) is also given. We analyze the effects of the fractional derivatives by comparing the SIS and the α-SIS models.

Power series solution of the fractional logistic equation

“…We obtain a power series representation of the solution of the fractional logistic equation (2.1), giving a new approach as compared with the relevant results of D'Ovidio, and Loreti in [3] and a different proof that the radius of convergence is indeed positive (see Theorem 2.1 below) providing the existence of solution for the fractional logistic equation. We also point out the recent estimates of Balzotti, D'Ovidio, and Loreti, see equation ( 25) and the proof of Theorems 1 and 2 in [9].…”

Solutions of Bernoulli equations in the fractional setting

“…As it is well known, also the solution of the fractional logistic equation -corresponding to p = 1 and a 0 = a 1 = −1 in (1)-was an open problem and in [4] the first and the third author were able to solve the fractional logistic equation by series representation, giving a detailed formula involving Euler numbers for u 0 = 1/2. This approach was then applied to SIS epidemic models in [2] and also further investigated in [1]. The present study extends the result in [4] to general initial data and to Bernoulli equations of general degree p + 1: we present a recursive formula for the coefficients of the solutions and explicit closed formulas for the first terms.…”

Effects of fractional derivatives in epidemic models

“…For instance, in Figure 1, N (t) first decreases and then increases when α 1 > α 2 , while it first increases and then decreases when α 1 < α 2 (a symmetrical behavior emerges in Figure 2). In case α 1 = α 2 then the sum is constant and we recover the theory developed in [BDL20,HOEK18]. A rigorous study of the symmetries emerging in the simulations and, in particular the intersections of all the functions N (t) at the same time, is still under investigation.…”