Optimal curing policy for epidemic spreading over a community network with heterogeneous population

We investigate the influence of the contact network structure on the spread of epidemics over an heterogeneous population. In our model, the epidemic process spreads over a directed weighted graph. A continuous-time individualbased susceptible–infected–susceptible (SIS) is analyzed using a first-order mean-field approximation. First, we consider a network with general topology in order to investigate the epidemic threshold and the stability properties of the system. Then, we analyze the case of a community network relying on the graph-theoretical notion of equitable partition. We show that, in this case, the epidemic threshold can be computed using a lower-dimensional dynamical system. Moreover we prove that the positive steady-state of the original system, that appears above the threshold, can be computed using this lower-dimensional system. In the second part of the work, we leverage on our model to derive a cost-optimal curing policy, in order to prevent the disease from persisting indefinitely within the population. The solution of this optimization problem is obtained by formulating a convex minimization problem on a general but symmetric network structure. Finally, in the case of a two-level optimal curing problem an algorithm is designed with a polynomial time complexity in the network size.