Minimum-Effort Control of a Networked SIS Model

A stability problem becomes a resource-allocation problem

In the susceptible–infected–susceptible model, a node can recover and later become infected again. Infection travels through an undirected, connected network with transmission matrix B; the diagonal matrix Δ contains existing curing rates. Additional nonnegative rates d change the balance between infection and recovery. Stabilizing the healthy state becomes the matrix condition Δ − B + diag(d) ⪰ 0.

The objective minimizes a weighted sum of these additions, allowing different intervention costs and a fixed set of nodes where rates cannot change. Feasibility therefore comes before optimization: the principal submatrix of Δ − B on the uncontrollable nodes must be positive definite. If that condition fails, increasing resources only at the permitted nodes cannot stabilize this model.

min Σᵢ cᵢdᵢ subject to Δ − B + diag(d) ⪰ 0, d ≥ 0

How the reductions identify where control is needed

A change of variables absorbs the cost weights, and Schur complements eliminate constrained portions of the matrix while preserving the optimization problem. Further diagonal-dominance checks identify nodes requiring no additional curing rate in the reduced problem. Repeating this process decreases the matrix dimension until the remaining allocation can be recovered explicitly.

The manuscript proves finite termination and optimality for its undirected model and illustrates the algorithm with an example. The structural conclusions refer to the transformed, reduced matrices; they are more specific than simply allocating resources to highly connected nodes. Directed networks and more general spreading dynamics remain extensions.