Public Health Tool Predicts Effects of a Pandemic and Mitigation Efforts
“If the reverse is true, the disease will go extinct,” he says. “Although Kermack and McKendrick gave a proof of this assertion, they were unable to provide an exact, closed form, analytic solution of the basic differential equation that governed the time development of the various populations.”
No one since has been able to find an exact solution to that equation that doesn’t require several pages to write down.
“My coauthor Doug Barlow and I fared no differently from our predecessors in the search for an exact solution,” says Dr. Baird. “We did, however, find the next best thing, which was an approximate analytic solution that can be written down on one line.”
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.
“During the onset, the number of infected individuals steadily increases, and as a result the rate of increase in the number of recovered individuals accelerates,” Dr. Baird says. “To slow down the spread, mitigation effects need to focus their attention on a parameter known as R-zero.”
R-zero is the rate of spread divided by the rate of recovery.
“If the value of R-zero is greater than unity, the disease is spreading,” says Dr. Baird. “If the value of R-zero is less than unity, the disease is going extinct.”
To reach the extinction phase, public health officials want to decrease the rate of spread while increasing the rate of recovery. “The period of subsidence of the disease begins after the number of infected individuals has reached its maximum,” Dr. Baird says.
During subsidence, the number of infected individuals begins to decrease, while the number of recovered individuals is still increasing but at a rate which is decelerating.
“Among our results is also a formula, which when combined with population data collected during onset, can be used to predict the time when the number of infected individuals needing hospitalization is expected to reach its maximum,” Dr. Baird says.
“To get an idea of the accuracy of our formula, we compared its predictions with that of a numerical solution of the Kermack-McKendrick equation that we generated using MathCad computer software,” he says. “In all cases tried, the difference between our analytic solution and the computer solution was never greater than 2%.
Dr. Baird’s curiosity was sparked in 2020 by news reports describing the rapid increase in numbers of people infected by COVID-19.
“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.”
