Fractional-Order Susceptible-Infected Model: Definition and Applications to the Study of COVID-19 Main Protease

Abstract

We propose a model for the transmission of perturbations across the amino acids of a protein represented as an interaction network. The dynamics consists of a Susceptible-Infected (SI) model based on the Caputo fractional-order derivative. We find an upper bound to the analytical solution of this model which represents the worse-case scenario on the propagation of perturbations across a protein residue network. This upper bound is expressed in terms of Mittag-Leffler functions of the adjacency matrix of the network of inter-amino acids interactions. We then apply this model to the analysis of the propagation of perturbations produced by inhibitors of the main protease of SARS CoV-2. We find that the perturbations produced by strong inhibitors of the protease are propagated far away from the binding site, confirming the long-range nature of intra-protein communication. On the contrary, the weakest inhibitors only transmit their perturbations across a close environment around the binding site. These findings may help to the design of drug candidates against this new coronavirus.

COVID-19 and SARS-CoV-2. Modeling the present, looking at the future

“…Attention to potential combinations between these new approaches and those existing for drug design and repurposing is also important. For instance, the investigation of ligand–protein interactions from a network perspective seems to reveal aspects not revealed by other approaches (see [108] , [112] ).…”

“…In order to explain the mechanism by which these perturbations are transmitted across the structure of the main protease, Abadias et al. [112] developed a fractional Susceptible–Infected (SI) model based on the assumption that there are similarities between epidemic spreading and a diffusive process on a protein residue network to prove the capability of propagating information in complex 3D protein structures [113] . The new fractional SI model on a network was defined as: with initial condition , and where is the infection rate, is the adjacency matrix, is an all-ones column vector, and is the Caputo fractional derivative defined as where , and where is the smallest integer greater than or equal to .…”

A fractional-order SIRD model with time-dependent memory indexes for encompassing the multi-fractional characteristics of the COVID-19

“…In both cases, they are ”full-memory indexes” because their contributions are since the initial conditions. In literature, recent works have proposed fractional-order mathematical models for studying the COVID-19 [19] , [20] , [21] , [22] , [23] , [24] , [25] , [26] , [27] . Based on the fractional-order derivative with singular kernel we found that Rezapour et al.…”

“…proposed a fractional-order SEIRD model for the spread of COVID-19 and compared it with the real data and integer-order cases [25] ; Abadias et al. analyzed the transmission of perturbations across the amino acids of a protein represented as an interaction network applying the Caputo derivative in a SI model [26] ; and Lu et al. presented a fractional SEIHDR model based on the coupling effect of inter-city networks to cope with the mortality rates (exposure, infection and hospitalization) and the infectivity of individuals during the incubation period [27] .…”

Data Analytics and Sir Modeling of Covid-19 in Bulgaria

“…Since the beginning of the COVID-19 epidemic, there has been various mathematical and statistical modelling that have predicted the global and national epidemic with varying degrees of accuracy and reliability (see [1], [2], [4], [10], [21], [25], [27], [28], [29], [32], [35], [40]). The accuracy of prediction and its uncertainty depend on the assumptions, availability and quality of data (see [31]).…”