Patrick
De Leenheer
Oregon State University
Title: The mathematics behind the basic
reproduction number R0
Abstract: We review some mathematical results
that are part of the folklore of the basic reproduction number, a
concept that is prevalent in epidemiology and population biology. The
basic reproduction number is commonly used in applications because it
is often easier to calculate than the spectral radius of the
nonnegative matrix to which it is associated. Moreover, its value
helps to establish the stability or instability of the linear recursion
defined by the matrix, because, as the saying goes, "The spectral
radius of a nonnegative matrix, and its associated basic reproduction
number, lie on the same side of 1". Consequently, controlling an
infectious disease amounts to making the basic reproduction number less
than 1. Perhaps not as wellknown is that these results had already
been obtained by Vandergraft in 1968 and are applicable to the more
general class of linear maps that preserve a cone in R^n and not just
to linear maps described by a nonnegative matrix. Vandergraft's work
was carried out decades before the notion of the basic reproduction
number became popular in mathematical biology, yet interestingly,
Vandergraft attributes the ideas to even earlier work in numerical
analysis by Varga in 1963. We strengthen one of Vandergraft's results,
albeit very slightly, using an idea of Li and Schneider that was
proposed for linear maps which preserve the nonnegative orthant cone.
Looming in the background, and grounding all the proofs of these
results, is the celebrated PerronFrobenius Theorem for linear maps
that preserve a cone, which is presented in a concise, yet
comprehensive way in a relatively recent book by Lemmens and
Nussbaum.

