A Solution for the Principle Integral of the Kermack McKenrick Epidemiological Model

Abstract

The Kermack–McKendrick integral is accurately computed here without resorting to numerical techniques. This integral appears in the process of solving the so-called SIR epidemiological model. To our knowledge, no closed form solution for this integral has been tabulated. We derive here a closed form solution for the Kermack McKendrick integral valid over the range of the variables useful for epidemiological modeling. The final result contains three closed-form terms one of which contains an infinite series of indexed integrals that can be written to arbitrary order in terms of hypergeometric functions. Using a finite number of terms in the infinite series, this approximation is compared with the numerical results for replacement numbers 1.5 through 5.5, which are typical of the values governing epidemics. Over this range, the analytical forms for S(t), I(t) and R(t) are nearly identical with those obtained by numerical evaluation of the Kermack–McKendrick integral. As the number of terms used from the infinite series is increased, the analytical result appears to converge with the numerical solution. For large values of the replacement number, we show that the Kermack–McKendrick integral can be approximated by only one term.