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/ωα
where ω is the angular frequency and α ∈ (0, 1) conforms to the absolute values of
a normalised zero mean Gaussian distribution. It is shown that α is a primary parameter
for evaluating the global status of the pandemic in the sense that the pandemic
will become extinguished as α → 0 for all countries. For this reason, and based on
the data currently available, a study is made of the variations in α 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.