An Immunization Strategy for Hidden Populations

Hidden populations, such as injecting drug users (IDUs), sex workers (SWs) and men who have sex with men (MSM), are considered at high risk of contracting and transmitting infectious diseases such as AIDS, gonorrhea, syphilis etc. However, public health interventions to such groups are prohibited due to strong privacy concerns and lack of global information, which is a necessity for traditional strategies such as targeted immunization and acquaintance immunization. In this study, we introduce an innovative intervention strategy to be used in combination with a sampling approach that is widely used for hidden populations, Respondent-driven Sampling (RDS). The RDS strategy is implemented in two steps: First, RDS is used to estimate the average degree (personal network size) and degree distribution of the target population with sample data. Second, a cut-off threshold is calculated and used to screen the respondents to be immunized. Simulations on model networks and real-world networks reveal that the efficiency of the RDS strategy is close to that of the targeted strategy. As the new strategy can be implemented with the RDS sampling process, it provides a cost-efficient and feasible approach for disease intervention and control for hidden populations.


Ⅰ The speed of immunizing top nodes
We calculated the top ranked nodes as a fraction of nodes considered in each immunization strategy. Fig. S1 shows that the RDS strategy can immunize the high degree individuals (the top 10% individuals in the ranking of degree) almost as quickly as targeted immunization, and is much more quickly than acquaintance immunization and random immunization.
In RDS, the inclusion probability of a node is proportional to its degree in RDS; therefore high degree nodes can be sampled more quickly in RDS than in the random selection; Meanwhile, the cut-off threshold which is obtained based on the immunization threshold of targeted immunization can partition the sampled nodes into two parts; in this way, the nodes with higher degree in the sample can be targeted almost as quickly as targeted immunization.

Ⅱ immunization in Watts-Strogatz (WS) network and Erdős-Ré nyi (ER) network
In the main article, we implemented the RDS strategy in the Barabasi-Albert (BA) network due to the fact that the most social networks in real world are scale-free networks, i.e., heterogeneous networks. And compared to the homogeneous network in which each node has approximately the same degree, the heterogeneous network is prone to the spreading and the persistence of infections because of its diverging connectivity fluctuations [1] [2] .
Beside the Barabasi-Albert (BA) network used in the main article, we have implemented the RDS strategy in another two classical network models, i.e., the Watts-Strogatz (WS) network and Erdős-Ré nyi (ER) network. The WS network is constructed in consistence with the study in [3]: The starting point is a ring with 10000 nodes, in which each node is symmetrically connected with its 10 nearest neighbors; Then, for every node each edge connected to a clockwise neighbor is kept as originating from the original node and rewired to a randomly chosen target node with probability 0.5. The ER network is generated by the algorithm in [4]: it starts with 10000 nodes and each pair of nodes is connected randomly with probability 0.001. Fig. S2 shows the simulation results that the reduced prevalence ρ f /ρ 0 varies with the number of the immunized nodes. The efficiency of RDS strategy is also following that obtained with the targeted strategy and better than that obtained with the acquaintance strategy and random strategy. However, the difference of these efficiencies is not very remarkable; we can see the immunization thresholds of the four immunization strategies are almost equivalent. The results are consistent with the conclusion in Pastor's research [3] which is verified that in case of immunizing on the homogeneous networks (e.g., the Watts-Strogatz (WS) network) the immunization threshold is almost equivalent in the random and targeted immunization.

Ⅲ Immunization with multiple seeds and coupons
Besides implementing RDS strategy with 1 seed and 1 coupon in main article, we have explored the effect of increasing seeds and coupons. Fig. S3 and Fig. S4 show that the efficiency of the RDS strategy varies from the different number of seeds and coupons. We can see that the efficiency of RDS strategy is not affected by the seed number or coupon number. That's because the number of seeds or coupons does not change the inclusion probability of the individuals [5][6][7] .   Fig. S5. We can see that the two general methods can reduce the length of the referral effectively. However, in practice we usually have only a small number of seeds in the implementation due to the fact that the seeds typically are the current or former members of the targeted population (e.g., the current IDUs). Meanwhile, the rejection rate of distributing coupon is usually high in practice, which makes long referral chains rarely appear.