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

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 ith 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.