Unexpectedly Intriguing!
11 November 2022

Early during the coronavirus pandemic, researchers in many fields got a crash course in epidemiology. More specifically, they got a crash course in how to apply the math behind the SIR model, which describes how fast a contagious condition might spread through a population before becoming endemic.

The SIR model divides a population into three categories, the Susceptible, the Infectious, and the Recovered (or Removed). Once basic data about the rates of infection and recovery are determined, the model can simulate how many people will fall within each of these categories at different points of time. Here's a primer introducing that basic math, in which we featured the following 22-minute video from Numberphile's Brady Haran and Ben Sparks on how to build your own SIR model from scratch using the online GeoGebra application:

Although each of the individual equations for each component of the SIR model involves relatively simple relationships, their interactions lead to much more complex math. Math that cannot be done simply by plugging numbers into an algebraic formula. Instead, the SIR model's math requires serious computing power to apply numerical methods, running thousands or millions of calculations to progressively reach reasonably accurate, but still not exact solutions.

That's why the press release for a paper recently published in the International Journal of Chemical Kinetics caught our attention. Its authors recognized part of the SIR model's math developed by epidemiologial pioneers W. O. Kermack and A. G. McKendrick is identical to the math used to describe the progress of an autocatalytic reaction in chemistry. Here's the Kermack-McKendrick integral, which has no direct solution:

In this equation, So and Io represent initial values for the number of Susceptible and Infectious portions of the popuation, R represents the Recovered (or Removed) portion of the population as a function of time (t), while the Greek letter lambda (λ) represents the ratio of the rate of spread among susceptible population to the rate of recovery. The letter e is Euler's constant.

That was math for which chemists James Baird, Douglas Barlow and Buddhi Pantha had developed the next best thing to a direct solution. They derived an approximate algebraic formula for quickly solving the Kermack-McKendrick integral with a small margin of error. Here's their simplified formulation:

Better still, they identified where their simplified formulation will work best:

In this report, a description is given of an accurate approximation to the Kermack-McKendrick integral which in turn can be used to determine values for R(t), I(t) and S(t) in the SIR epidemic model. The result is shown to be effective for situations where 1.5 ≤ Ro ≤ 10 with no need to numerically compute an integral.

The press release better describes how their formulation meshes with the chemistry of autocatalytic reactions:

Dr. Baird presented the model in May at the Southeastern Theoretical Chemistry Association meeting in Atlanta.

"The World Health Organization could program our equation into a hand-held computer," Dr. Baird says. "Our formula is able to predict the time required for the number of infected individuals to achieve its maximum. In the chemical analog, this is known as the induction time."

The formula is capable of predicting the number of hospitalizations, death rates, community exposure rates and related variables. It also calculates the populations of susceptible, infectious and recovered individuals, and predicts a clean separation between the period of onset of the disease and the period of subsidence....

"The rate of infection initially accelerates until it reaches a point where the infection rate is balanced by the recovery rate of infected individuals, at which point the number of infected people peaks and then starts to decay," he says.

That mechanism reminded him of the mechanism that governs an autocatalytic reaction.

"I subsequently learned that the mathematical description of the spread of infectious diseases was first described by Kermack and McKendrick," Dr. Baird says.

"When I read their paper, I realized that their mechanism was exactly the same as that of an autocatalytic reaction, where a catalyst molecule combines with a reactant molecule to produce two catalyst molecules," he says. "The rate of production of catalyst molecules accelerates until it is balanced by the rate of decay of the catalyst to form the product."

And that's how the algebraic formula that can quickly approximate the solution to the Kermack-McKendrick integral for the epidemiological SIR model with minimal computing power came to be published in a chemistry journal.

### References

James K. Baird et al, Analytic solution to the rate law for a fundamental autocatalytic reaction mechanism operating in the "efficient" regime, International Journal of Chemical Kinetics (2022). DOI: 10.1002/kin.21598.

James K. Baird, Douglas A. Barlow, Buddhi Pantha. A Solution for the Principle Integral of the Kermack-McKenrick Epidemiological Model. [Preprint (PDF)]. DOI: 10.31224/2264. April 2022. [This second paper is an ungated preprint that focuses on the epidemiological application of the authors' approximation of the Kermack-McKendrick integral.]

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