A Mathematical Model For the Spread of a Virus

TL;DR

This paper introduces a mathematical RGB-based model for virus spread in isolated populations, mirroring the classical SIR model, and derives explicit formulas for key infection dynamics, including effects of vaccination.

Contribution

The paper presents a novel RGB color-coded mathematical model for virus transmission that parallels the SIR model and provides explicit formulas for infection progression and vaccination effects.

Findings

The model captures three infection phases: initial, transition ramp, and plateau.

Key temporal milestones depend on population size and contagious lifetime ratio.

Explicit formulas describe the timing and magnitude of infection peaks.

Abstract

This paper describes a mathematical model for the spread of a virus through an isolated population of a given size. The model uses three, color-coded components, called molecules (red for infected and still contagious; green for infected, but no longer contagious; and blue for uninfected). In retrospect, the model turns out to be a digital analogue for the well-known SIR model of Kermac and McKendrick (1927). In our RGB model, the number of accumulated infections goes through three phases, beginning at a very low level, then changing to a transition ramp of rapid growth, and ending in a plateau of final values. Consequently, the differential change or growth rate begins at 0, rises to a peak corresponding to the maximum slope of the transition ramp, and then falls back to 0. The properties of these time variations, including the slope, duration, and height of the transition ramp, and the width and height of the infection rate, depend on a single parameter - the time that a red molecule is contagious divided by the average time between collisions of the molecules. Various temporal milestones, including the starting time of the transition ramp, the time that the accumulating number of infections obtains its maximum slope, and the location of the peak of the infection rate depend on the size of the population in addition to the contagious lifetime ratio. Explicit formulas for these quantities are derived and summarized. Finally, Appendix E has been added to describe the effect of vaccinations.