Optimal Allocation of Resources for Suppressing Epidemic Spreading on Networks

Efficient allocation of limited medical resources is crucial for controlling epidemic spreading on networks. Based on the susceptible-infected-susceptible model, we solve an optimization problem as how best to allocate the limited resources so as to minimize the prevalence, providing that the curing rate of each node is positively correlated to its medical resource. By quenched mean-field theory and heterogeneous mean-field (HMF) theory, we prove that epidemic outbreak will be suppressed to the greatest extent if the curing rate of each node is directly proportional to its degree, under which the effective infection rate $\lambda$ has a maximal threshold $\lambda_c^{opt}=1/\left\langle k \right\rangle$ where $\left\langle k \right\rangle$ is average degree of the underlying network. For weak infection region ($\lambda\gtrsim\lambda_c^{opt}$), we combine a perturbation theory with Lagrange multiplier method (LMM) to derive the analytical expression of optimal allocation of the curing rates and the corresponding minimized prevalence. For general infection region ($\lambda>\lambda_c^{opt}$), the high-dimensional optimization problem is converted into numerically solving low-dimensional nonlinear equations by the HMF theory and LMM. Counterintuitively, in the strong infection region the low-degree nodes should be allocated more medical resources than the high-degree nodes to minimize the prevalence. Finally, we use simulated annealing to validate the theoretical results.