Prioritizing high-contact occupations raises effectiveness of vaccination campaigns – Supplementary information –

A twenty-year-old idea from network science is that vaccination campaigns would be more effective if high-contact individuals were preferentially targeted. Implementation is impeded by the ethical and practical problem of differentiating vaccine access based on a personal characteristic that is hard-to-measure and private. Here, we propose the use of occupational category as a proxy for connectedness in a contact network. Using survey data on occupation-specific contact frequencies, we calibrate a model of disease propagation in populations undergoing varying vaccination campaigns. We find that vaccination campaigns that prioritize high-contact occupational groups achieve similar infection levels with half the number of vaccines, while also reducing and delaying peaks. The paper thus identifies a concrete, operational strategy for dramatically improving vaccination efficiency in ongoing pandemics.


Supplementary Results
reports detailed descriptive statistics on the properties of input networks (section I.), network position of initially infected nodes/index cases (section II.), course of epidemics split by normal vs. lockdown networks (section III.), and course of epidemics split by counter measures (section IV.).
Tables S2 -S5 report how well size and degree of occupational groups in the generated networks match the empirical data. Table S6 reports the average final sizes of epidemics across conditions (columns) and control variables (rows). These data are the basis for the differences of final size between the baseline and test conditions presented in Table 2 of the main text. Table S7 reports the average peak sizes of epidemics across conditions (columns) and control variables (rows). Table S8 reports the average peak sizes of epidemics of the baseline condition (column 2) and differences by test condition in percent points (columns 3 -5), similar to differences of final size reported in Table 2 of the main text. Table S9 reports the average duration of epidemics across conditions (columns) and control variables (rows). Table S10 reports the average duration of epidemics of the baseline condition (column 2) and differences by test condition in percent points (columns 3 -5), similar to differences of final size reported in Table 2 of the main text.          with x i denoting the actual proportion of closed triads i belongs to in i's ego-network, α the preferred proportion of closed triads of agent i, and t i the number of ties agent i possesses: Social maintenance costs are assumed to be quadratic in the number of ties t i to model increasing marginal costs of additional ties: Network generation for reported contact numbers prior to lockdown was realized using the NIDM network formation procedure. That is, time is modeled as discrete time steps.
Within each time step, we simulate the formation and dissolution of ties: • Create an empty set of unprocessed agents A 1 and repeat until all agents have been · Belonging to the same occupational group (with probability ω), or · a randomly selected agent (with probability 1 − ω). * If j is directly tied to i: · Dissolve tie ij, if i's utility excluding ij exceeds the utility including ij. * else: · Form tie ij, if both i's and j's utilities including ij exceed the utilities excluding ij. * Add j to A if j is directly tied to i.
Generation of lockdown networks was realized using a network pruning algorithm: • For all networks N : -Repeat until N is "in lockdown" (av. degree of all occupational groups ≤ reported av. lockdown degree + 3%): * For all edges e connecting nodes n 1 and n 2 in N (in random order): · If n 1 and n 2 are not yet "in lockdown" (degree > reported av. lockdown degree of occupational group + 3%): · Remove e with a probability depending on the reported percentage decrease in degree between prior and during lockdown (t − ):

Disease Spread
Disease states (susceptible, infected, recovered, or vaccinated) are defined for each node in the network: The probability for a node i to get infected per time step depends on the probability to get infected per single contact (γ) and the number of infected neighbors: As before, time is modeled as discrete time steps. Within each time step, we simulate disease transmission events: • Repeat until all nodes have been processed: -Randomly select an unprocessed node i: -If i is infected, compute whether node recovers: passed time steps since infection ≥ τ .
-If i is susceptible, compute whether i gets infected from infected neighbors (see Equation 7).

Parameters and Submodels
Table S11 presents all parameters including the range of possible settings and the initial settings used for network generation and the simulation of epidemics. We systematically varied the settings for α (to control clustering) and ω (to control occupational group homophily) and selected for each combination the 10 best fits, resulting in 90 normal (prior to lockdown) networks. Afterwards, we used the pruning algorithm described earlier to generate 90 lockdown networks.
Network size was selected to ensure large enough networks to make meaningful inferences, while limiting computational demands to allow a large number of network variations and simulated epidemics. Other fixed parameters were selected because of one of two reasons. First, they are backed by empirical data. Recent US labor market numbers were taken to set occupational group size [3]. Clustering in contact networks was found to be around 0.46 with clustering getting lower for older persons [4]. We, therefore, varied α between 0.3, 0.4, and 0.5. Average degree per contact in our model can be realized in the model by keeping the b 1 and c 1 constant, and setting c 2 to define the optimum number of ties: b 1 −c 1 2·ttarget . This is done for each node individually, dependent on reported contact numbers per occupational group. Other fixed parameters were selected because pilot runs showed that they produce informative and interesting variations of epidemics regarding final size, epidemic peak size, and epidemic peak time. Table S11: Parameters, ranges, and initial settings.

Parameter
Range Initial setting