Updating the Inverse of a Matrix When Removing the $i$th Row and Column with an Application to Disease Modeling

TL;DR

This paper derives a new analytical formula for the inverse of a matrix after removing a specific row and column, and applies it to compute the reproductive ratio in a disease spread model involving relapsing hosts.

Contribution

It introduces a novel limit-based expression for matrix inversion after row and column removal and demonstrates its application in epidemiological modeling.

Findings

Derived an explicit limit-based formula for matrix inversion after removing a row and column.

Applied the formula to calculate the reproductive ratio in a relapsing disease model.

Provided analytical insights into disease dynamics involving multiple host species.

Abstract

The Sherman-Woodbury-Morrison (SWM) formula gives an explicit formula for the inverse perturbation of a matrix in terms of the inverse of the original matrix and the perturbation. This formula is useful for numerical applications. We have produced similar results, giving an expression for the inverse of a matrix when the $i$th row and column are removed. However, our expression involves taking a limit, which inhibits use in similar applications as the SWM formula. However, using our expression to find an analytical result on the spectral radius of a special product of two matrices leads to an application. In particular, we find a way to compute the fundamental reproductive ratio of a relapsing disease being spread by a vector among two species of host that undergo a different number of relapses.