Event Detail

Event Type: 
Department Colloquium
Monday, November 23, 2020 - 16:00 to 16:50

Speaker Info

Local Speaker: 

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 non-negative 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 non-negative 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 well-known, 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 non-negative 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 performed 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 non-negative orthant cone. Looming in the background, and grounding all the proofs of these results, is the celebrated Perron-Frobenius 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.