At the time of this post, we are over two years into the COVID-19 pandemic. Since the virus’s worldwide spread of early 2020, we’ve seen various experts and agencies project the duration and severity of the disease. The public was confused throughout the pandemic as it saw both underestimates of the spread of the disease and overestimates of it. Frustrating for all was how we experience several waves of the disease as incidents increased to a peak and then suddenly decrease only to be followed by another wave in a few months.
This chart is the cases in the United States (1/31/2022) as tracked by the Johns Hopkins University Coronavirus Resource Center: New COVID-19 Cases Worldwide - Johns Hopkins Coronavirus Resource Center (jhu.edu)
The last two peaks of the chart reflect the “delta” and “omicron” variants of COVID-19.
To understand the rise and fall of the disease, it helps to begin with a very simple model. Let’s assume there is a very large population of susceptible victims. To begin, we’ll assume that the population is infinite (not realistic, but just a way to begin modeling the spread of the disease). Next there will be one initial patient who has the disease and introduces it to the general population. If over the course of that initial patient’s experience with the disease, the patient infects two other people, then the basic reproduction number, R0, of the disease is 2. In this very simplified example, the first patient infects two others who go on to infect four more and the disease spreads exponentially.
This first model isn’t realistic because while the initial population may be very large, it is not infinite. However, the pattern seen early in the disease does match with the relatively low number of infections followed by a very steep increase in the number of cases.
The SIR mathematical model was introduced in the 1920s to better describe and predict how a disease spreads through a population. The acronym SIR stands for Susceptible, Infected, and Recovered (sometimes R also stands for Removed in the cases where a subject dies). At each point in time every member of the population can be divided into one of the three groups, S, I, and R.
The next iteration of developing the mathematical model, is the assumption that members of the recovered group acquire immunity. In that case, the portion of the population in the susceptible group (S) decreases as the disease spreads.
As the size the susceptible population decreases, the effective rate at which each infected person spreads the disease decreases. This continues to a point where initially the disease was spreading exponentially, the rate of spread decreases and eventually the number of current cases (the Infection group) drops.
A link to the spreadsheet to produce the above graphic is
given here – one can change the initial parameters, such as the R0 and
population size: SIR Model
The graph shown above resembles the various peaks we’ve seen in the COVID-19 spread. To explain the recurrence of peaks, if members of the “recovered” group, R, lose their immunity over time or if the virus mutates so former members of the R group have less immunity, then the population of the susceptible group increases after initially decreasing, therefore allowing the caseloads to increase again in successive waves. This is what we experience with the annual flu. Within a given season, if one catches the flu, the person generally has immunity but not immunity for the next subsequent seasons as the flu mutates.
Suggested References:
Waves of disease: What Makes a “Wave” of Disease?
My post was a simplified SIR model. Those wishes to explore the differential equation SIR model are referred to: Extending the basic SIR Model in R | by Najma Ashraf | Towards Data Science or The SIR Model for Spread of Disease - The Differential Equation Model | Mathematical Association of America (maa.org)
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