A Novel Technique to Control the Accuracy of a Nonlinear Fractional Order Model of COVID-19: Application of the CESTAC Method and the CADNA Library

Abstract

In this paper, a nonlinear fractional order model of COVID-19 is approximated. For this aim, at first we apply the Caputo–Fabrizio fractional derivative to model the usual form of the phenomenon. In order to show the existence of a solution, the Banach fixed point theorem and the Picard–Lindelof approach are used. Additionally, the stability analysis is discussed using the fixed point theorem. The model is approximated based on Indian data and using the homotopy analysis transform method (HATM), which is among the most famous, flexible and applicable semi-analytical methods. After that, the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library, which are based on discrete stochastic arithmetic (DSA), are applied to validate the numerical results of the HATM. Additionally, the stopping condition in the numerical algorithm is based on two successive approximations and the main theorem of the CESTAC method can aid us analytically to apply the new terminations criterion instead of the usual absolute error that we use in the floating-point arithmetic (FPA). Finding the optimal approximations and the optimal iteration of the HATM to solve the nonlinear fractional order model of COVID-19 are the main novelties of this study.

Extension of SEIR Compartmental Models for Constructive Lyapunov Control of COVID-19 and Analysis in Terms of Practical Stability

“…Some authors [24,25] propose to keep the model as simple as possible because of the thin data availability in early periods, with the purpose of focusing on specific local peculiarities. Simultaneously, some authors [8,[26][27][28][29][30] use fractional-order models and/or controllers instead of classical integer-order calculus to model the dynamics of the compartments, at the cost of increased complexity and required computational resources, because of the infinite memory problem and other aspects. Nevertheless, to control the system using a fractional model is a new trend which can be generally justified by the chaotic nature of the considered phenomena, as already tested for other nonlinear oscillating systems.…”

Mathematical Analysis of a Fractional COVID-19 Model Applied to Wuhan, Spain and Portugal

“…This effect is fully captured by flexibility of the order of differentiation for fractional derivatives and can be seen as a hereditary property or a arXiv:2106.15407v1 [math.DS] 27 Jun 2021 variety on strains and genomes of viruses (as conjectured in [23]), which is useful for epidemic models. In [25], a model for controlling the Coronavirus pandemic 2019 in India with Caputo-Fabrizio fractional derivatives and the homotopy analysis transform method is investigated. Often, fractional-order models give rise to theoretical models that allow a significant improvement in the fitting of real data, when compared with analogous classical models [26].…”

Multi-Model Selection and Analysis for COVID-19

“…Based on the compartment hypothesis of the population, most models include compartments of susceptible (S), exposed (E), infected (I), recovered (R), death (D), quarantined (Q), asymptomatic (A), insusceptible (P), and hospitalized (H) individuals, such as SIR [8], SEIR [9], SEIR-Q [10], SEIR-QD [11], and SEIR-AHQ [12] models. In addition, many sophisticated models also have been proposed [13][14][15][16].…”