  Motivated by the swift global transmission of infectious diseases, we present
a comprehensive framework for network-based epidemic control. Our aim is to
curb epidemics using two different approaches. In the first approach, we
introduce an optimization strategy that optimally reduces travel rates. We
analyze the convergence of this strategy and show that it hinges on the network
structure to minimize infection spread. In the second approach, we expand the
classic SIR model by incorporating and optimizing quarantined states to
strategically contain the epidemic. We show that this problem reduces to the
problem of matrix balancing. We establish a link between optimization
constraints and the epidemic's reproduction number, highlighting the
relationship between network structure and disease dynamics. We demonstrate
that applying augmented primal-dual gradient dynamics to the optimal quarantine
problem ensures exponential convergence to the KKT point. We conclude by
validating our approaches using simulation studies that leverage public data
from counties in the state of Massachusetts.
