Convergence Speed and Robustness Analysis of Epidemic Spreading Processes With Time-Delay

The networked Susceptible-Infected-Susceptible (SIS) model is a well-studied model for epidemic spreading processes in complex networks. A critical problem in the networked SIS model with time delay is to simultaneously optimize the network's convergence rate to the healthy state and its robustness to time delay while there is a constraint on the total curing resources that can be utilized throughout the network. Here, this problem has been derived as a multi-objective optimization problem with additional constraints guaranteeing a minimum convergence rate and a maximum tolerable time delay. It is shown that the problem's Pareto frontier consists of three segments, where for complete bipartite network and uniform infection rates their details have been derived analytically. By incorporating both objective functions into one, a corresponding convex optimization problem is developed, which is solved over tree reducible bipartite networks, using semidefinite programming (SDP). Furthermore, a convex combination approximation has been provided for the second segment of the Pareto frontier. The trade-off between the convergence rate and robustness to time-delay has been demonstrated via numerical simulations, where, for scale-free networks, the impact of network's size on this trade-off has been investigated numerically. Additionally, it is shown how the convergence rate is dictated by the eigenvalue of the
$M$
-matrix of the network, and the first segment of the Pareto frontier provides the most diverse results.