  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.
