Eigenvalue analysis of SARS-CoV-2 viral load data: illustration for eight COVID-19 patients

Abstract

Eigenvalue analysis is an important tool in economics and nonlinear physics to analyze industrial processes and instability phenomena, respectively. A model-based eigenvalue analysis of viral load data from eight symptomatic COVID-19 patients was conducted. The eigenvalues and eigenvectors of the instabilities were determined that give rise to COVID-19. For all eight patients, it was found that the virus dynamics followed the unstable eigenvectors until the viral load reached the respective peak values. At the peak virus values, the virus dynamics branched off from the directions specified by the eigenvectors. The temporal course of the unstable eigenvalues was determined as well. For all patients, it was found that the eigenvalues switched from positive to negative values just when the virus load reached peak values. These findings suggest that the fixed, instability-related eigenvalues and eigenvectors determine initial stages of SARS-CoV-2 infections during which virus load increases. In contrast, the time-dependent eigenvalues show a sign-switching phenomenon that indicates when the virus dynamics switches from the growth stage (increasing virus load) to the decay stage (decreasing virus load). The virus dynamics model was a standard three-variable virus dynamics model frequently used in the literature.

Unstable eigenvectors and reduced amplitude spaces specifying limit cycles of coupled oscillators with simultaneously diagonalizable matrices: with applications from electric circuits to gene regulation

“…Unstable eigenvectors and their amplitudes have been proven to be key analysis tools for a variety of studies on nonlinear systems ranging from studies in physics, chemistry, and biology [1,2,4,10,11,54,[57][58][59] to studies of human reactions [13] and research on COVID-19 [18][19][20]. In the current study, a new application of the concept of unstable eigenvectors and amplitudes was presented for coupled oscillators with simultaneously diagonalizable matrices.…”

“…(see also Eqs. (19) and (20). Since M −1 (by assumption) exists, the eigenvectors are linearly independent and the state vector as function of time can be expressed in the eigenvector basis like…”

“…This fundamental framework for the description of systems close to instabilities has been introduced several decades ago [1,[5][6][7][8]. Since then, it has produced a steady stream of new applications in contemporary research [9][10][11][12][13][14][15][16][17] (see in particular recent applications in COVID-19 research [18][19][20]), which illustrates the important task to make the eigenvector/eigenfunction framework available in research fields that have not yet taken fully advantage of this approach. On the other hand, coupled oscillator systems have been comprehensively studied [21,22], in particular, to identify their characteristic a e-mail: till.frank@uconn.edu (corresponding author) attractors.…”

Design and Development of an Optimal Control Model in System Dynamics through State-Space Representation

“…These vectors are used to analyze the behavior and interactions of different system variables. In short, eigenvectors and eigenvalues are important concepts in system dynamics, as they are used to analyze and understand the behavior of systems dynamics [27,28].…”