  The importance of a strict quarantine has been widely debated during the
COVID-19 epidemic even from the purely epidemiological point of view. One
argument against strict lockdown measures is that once the strict quarantine is
lifted, the epidemic comes back, and so the cumulative number of infected
individuals during the entire epidemic will stay the same. We consider an SIR
model on a network and follow the disease dynamics, modeling the phases of
quarantine by changing the node degree distribution. We show that the system
reaches different steady states based on the history: the outcome of the
epidemic is path-dependent despite the same final node degree distribution. The
results indicate that two-phase route to the final node degree distribution (a
strict phase followed by a soft phase) are always better than one phase (the
same soft one) unless all the individuals have the same number of connections
at the end (the same degree); in the latter case, the overall number of
infected is indeed history-independent. The modeling also suggests that the
optimal procedure of lifting the quarantine consists of releasing nodes in the
order of their degree - highest first.
