On the Evolution Equation for Modelling the Covid-19 Pandemic

Abstract

The paper introduces and discusses the evolution equation, and, based exclusively on this equation, considers random walk models for the time series available on the daily confirmed Covid-19 cases for different countries. It is shown that a conventional random walk model is not consistent with the current global pandemic time series data, which exhibits non-ergodic properties. A self-affine random walk field model is investigated, derived from the evolutionary equation for a specified memory function which provides the non-ergodic fields evident in the available Covid-19 data. This is based on using a spectral scaling relationship of the type $1/\omega ^{\alpha }$ where $\omega$ is the angular frequency and $\alpha \in (0,1)$ conforms to the absolute values of a normalised zero mean Gaussian distribution. It is shown that $\alpha$ is a primary parameter for evaluating the global status of the pandemic in the sense that the pandemic will become extinguished as $\alpha \rightarrow 0$ for all countries. For this reason, and based on the data currently available, a study is made of the variations in $\alpha$ for 100 randomly selected countries. Finally, in the context of the Bio-dynamic Hypothesis, a parametric model is considered for simulating the three-dimensional structure of a spike protein which may be of value in the development of a vaccine.