Respondent-driven detection is a chain recruitment method used to sample contact persons of infected persons in order to enhance case finding. It starts with initial individuals, so-called seeds, who are invited for participation. Afterwards, seeds receive a fixed number of coupons to invite individuals with whom they had contact during a specific time period. Recruitees are then asked to do the same, resulting in successive waves of contact persons who are connected in one recruitment tree. However, often the majority of participants fail to invite others, or invitees do not accept an invitation, and recruitment stops after several waves. A mathematical model can help to analyse how various factors influence peer recruitment and to understand under which circumstances sustainable recruitment is possible. We implemented a stochastic simulation model, where parameters were suggested by empirical data from an online survey, to determine the thresholds for obtaining large recruitment trees and the number of waves needed to reach a steady state in the sample composition for individual characteristics. We also examined the relationship between mean and variance of the number of invitations sent out by participants and the probability of obtaining a large recruitment tree. Our main finding is that a situation where participants send out any number of coupons between one and the maximum number is more effective in reaching large recruitment trees, compared to a situation where the majority of participants does not send out any invitations and a smaller group sends out the maximum number of invitations. The presented model is a helpful tool that can assist public health professionals in preparing research and contact tracing using online respondent-driven detection. In particular, it can provide information on the required minimum number of successfully sent invitations to reach large recruitment trees, a certain sample composition or certain number of waves.