Estimating Micro-Behavior's Impact on Transmission

Unlike with macro-distancing behaviour and contact rates, there is no simple mathematical framework linking change in precautionary micro-behaviours to changes in non-household transmission probabilities. We must therefore estimate the effect of precautionary micro-behaviour on transmission via case data. We implicitly assume that any reduction in local-to-local transmission potential that is not explained by changes to the numbers of non-household contacts, the duration of household contacts, or improved disease surveillance is explained by the effect of precautionary micro-behaviour on non-household transmission probabilities.
Whilst it is not necessary to use ancillary data to estimate the effect that precautionary micro-behaviour has at its peak, we use behavioural survey data to estimate the temporal trend in precautionary micro-behaviour, in order to estimate to what extent adoption of that behaviour has waned and how that has affected transmission potential.
We therefore model ${\gamma }_t$ (a time-varying index of change relative to baseline of transmission probability per non-household contact, see Equation (12)), as a function of the proportion of the population adhering to precautionary micro-behaviours. We consider adherence to the ‘1.5 m rule’ as indicative of this broader suite of behaviours due to the availability of data on this behaviour in a series of weekly behavioural surveys beginning prior to the last distancing restriction being implemented Department of the Prime Minister and Cabinet, 2020 . We consider the number $m_{i,t}^{+}$ of respondents in region $i$ on survey wave commencing at time $t$ replying that they ‘always’ keep 1.5 m distance from non-household members, as a binomial sample with sample size $m_{i,t}$ . We use a generalised additive model to estimate $c_i(t)$ , the proportion of the population in region $i$ responding that they always comply as a the intervention stage, smoothed over time. Intervention stages are defined as periods of a continuous state of stay-at-home order, and this state thus switches each time a stay-at-home order is started, ended, or significantly changed. This state switching allows the model to react to sudden changes in compliance behaviour when orders are made or rescinded. We assume that the temporal pattern in the initial rate of adoption of the behaviour is the same as for macro-distancing behaviours — the adoption curve estimated from the mobility model. In other words, we assume that all macro-distancing and precautionary micro-behaviours were adopted simultaneously around the time the first population-wide restrictions were put in place in March and April 2020. However we do not assume that these behaviours peaked at the same time or subsequently followed the same temporal trend. The model for the proportion complying with this behaviour is therefore: $m_{i,t}^{+}=\mathrm{Binomial}(m_{i,t},c_i(t))$ $logit(c_i(t))={\zeta }_{i,j}+s(t)$
where ${\zeta }_{i,j}$ is intervention state $j$ in region $i$ , and $s$ is a smoothing function over time $t$ .
Given $c_i(t)$ , we model ${\gamma }_i(t)$ as a function of the degree of precautionary micro-behaviour relative to the peak: ${\gamma }_i(t)=1-\beta (c_i(t)/{\kappa }_i)$
where ${\kappa }_i$ is the peak of compliance, or maximum of $c_i(t)$ , and $\beta$ is inferred from case data in the main $R_{\mathrm{eff}}$ model.