Controllability of large-scale network systems has considerably received attention in recent years. We introduce a qualitative study on the controllability using graph theory and then a quantitative study using optimization.

The dynamics of disease transmission models is often characterized by the basic reproduction number ℛ₀, the average number of new cases generated by a typical infected individual introduced into a completely susceptible population. Typically, the phenomenon forward bifurcation is observed, where the disease-free equilibrium (DFE) loses its stability and a stable endemic equilibrium (EE) appears as ℛ₀ increases through one. In this paper, we discuss the effect of the treatment capacity (such as limited beds in hospitals, or an insufficient supply of medicine) on the disease spread in a simple disease transmission model with two types of infective individuals, which are divided into severe ones who need treatment and the others. It is shown that a backward bifurcation occurs, where a stable EE co-exists with a stable DFE when ℛ₀<1, if the treatment capacity is relatively small. This epidemiological implication is that, when there is not enough capacity for treatment, the requirement ℛ₀<1 while necessary, is not sufficient for effective disease control. And also, even though ℛ₀<1, disease outbreak can happen to a high endemic level. In this case, decreasing ℛ₀ to its former level will not necessarily make the disease disappear.