  The contact structure of a population plays an important role in transmission
of infection. Many ``structured models'' capture aspects of the contact
structure through an underlying network or a mixing matrix. An important
observation in such models, is that once a fraction $1-1/\mathcal{R}_0$ has
been infected, the residual susceptible population can no longer sustain an
epidemic. A recent observation of some structured models is that this threshold
can be crossed with a smaller fraction of infected individuals, because the
disease acts like a targeted vaccine, preferentially immunizing higher-risk
individuals who play a greater role in transmission. Therefore, a limited
``first wave'' may leave behind a residual population that cannot support a
second wave once interventions are lifted. In this paper, we systematically
analyse a number of mean-field models for networks and other structured
populations to address issues relevant to the Covid-19 pandemic. In particular,
we consider herd-immunity under several scenarios. We confirm that, in networks
with high degree heterogeneity, the first wave confers herd-immunity with
significantly fewer infections than equivalent models with lower degree
heterogeneity. However, if modelling the intervention as a change in the
contact network, then this effect might become more subtle. Indeed, modifying
the structure can shield highly connected nodes from becoming infected during
the first wave and make the second wave more substantial. We confirm this
finding by using an age-structured compartmental model parameterised with real
data and comparing lockdown periods implemented either as a global scaling of
the mixing matrix or age-specific structural changes. We find that results
regarding herd immunity levels are strongly dependent on the model, the
duration of lockdown and how lockdown is implemented.

Populations and Evolution;Physics and Society