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

The impact of network properties and mixing on control measures and disease-induced herd immunity in epidemic models: a mean-field model perspective