## Journal of Applied Probability

### Respondent-driven sampling and an unusual epidemic

#### Abstract

Respondent-driven sampling (RDS) is frequently used when sampling from hidden populations. In RDS, sampled individuals pass on participation coupons to at most $c$ of their acquaintances in the community ($c=3$ being a common choice). If these individuals choose to participate, they in turn pass coupons on to their acquaintances, and so on. The process of recruiting is shown to behave like a new Reed–Frost-type network epidemic, in which becoming infected' corresponds to study participation. We calculate $R_0$, the probability of a major outbreak', and the relative size of a major outbreak for $c\lt\infty$ in the limit of infinite population size and compare to the standard Reed–Frost epidemic. Our results indicate that $c$ should often be chosen larger than in current practice.

#### Article information

Source
J. Appl. Probab., Volume 53, Number 2 (2016), 518-530.

Dates
First available in Project Euclid: 17 June 2016

https://projecteuclid.org/euclid.jap/1466172871

Mathematical Reviews number (MathSciNet)
MR3514295

Zentralblatt MATH identifier
1342.92256

Subjects
Primary: 92D30: Epidemiology
Secondary: 91D30: Social networks

#### Citation

Malmros, J.; Liljeros, F.; Britton, T. Respondent-driven sampling and an unusual epidemic. J. Appl. Probab. 53 (2016), no. 2, 518--530. https://projecteuclid.org/euclid.jap/1466172871

#### References

• Andersson, H. and Britton, T. (2000). Stochastic Epidemic Models and Their Statistical Analysis (Lecture Notes Statist. 151). Springer, New York.
• Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
• Ball, F. and Lyne, O. D. (2001). Stochastic multitype SIR epidemics among a population partitioned into households. Adv. Appl. Prob. 33, 99–123.
• Ball, F. and Neal, P. (2002). A general model for stochastic SIR epidemics with two levels of mixing. Math. Biosci. 180, 73–102.
• Ball, F. and Sirl, D. (2013). Acquaintance vaccination in an epidemic on a random graph with specified degree distribution. J. Appl. Prob. 50, 1147–1168. (Correction: 52 (2015), 908.)
• Ball, F., Sirl, D. and Trapman, P. (2009). Threshold behaviour and final outcome of an epidemic on a random network with household structure. Adv. Appl. Prob. 41, 765–796.
• Ball, F. G., Sirl, D. J. and Trapman, P. (2014). Epidemics on random intersection graphs. Ann. Appl. Prob. 24, 1081–1128.
• Barbour, A. D. and Reinert, G. (2013). Approximating the epidemic curve. Electron. J. Prob. 18, 30 pp.
• Bengtsson, L. et al. (2012). Implementation of web-based respondent-driven sampling among men who have sex with men in Vietnam. PLoS ONE 7, e49417.
• Britton, T., Deijfen, M. and Martin-Löf, A. (2006). Generating simple random graphs with prescribed degree distribution. J. Statist. Phys. 124, 1377–1397.
• Britton, T., Janson, S. and Martin-Löf, A. (2007). Graphs with specified degree distributions, simple epidemics, and local vaccination strategies. Adv. Appl. Prob. 39, 922–948.
• Csardi, G. and Nepusz, T. (2006). The igraph software package for complex network research. InterJournal Complex Systems 1695.
• Gile, K. J. (2011). Improved inference for respondent-driven sampling data with application to HIV prevalence estimation. J. Amer. Statist. Assoc. 106, 135–146.
• Gile, K. J. and Handcock, M. S. (2015). Network model-assisted inference from respondent-driven sampling data. J. R. Statist. Soc. A 178, 619–639.
• Heckathorn, D. D. (1997). Respondent-driven sampling: a new approach to the study of hidden populations. Social Problems 44, 174–199.
• Heckathorn, D. D. (2002). Respondent-driven sampling II: deriving valid population estimates from chain-referral samples of hidden populations. Social Problems 49, 11–34.
• Lu, X., Malmros, J., Liljeros, F. and Britton, T. (2013). Respondent-driven sampling on directed networks. Electron. J. Statist. 7, 292–322.
• Malekinejad, M. et al. (2008). Using respondent-driven sampling methodology for HIV biological and behavioral surveillance in international settings: a systematic review. AIDS Behavior 12, 105–130.
• Martin-Löf, A. (1986). Symmetric sampling procedures, general epidemic processes and their threshold limit theorems. J. Appl. Prob. 23, 265–282.
• Molloy, M. and Reed, B. (1995). A critical point for random graphs with a given degree sequence. Random Structures Algorithms 6, 161–179.
• Molloy, M. and Reed, B. (1998). The size of the giant component of a random graph with a given degree sequence. Comb. Prob. Comput. 7, 295–305.
• Newman, M. E. J. (2002). Spread of epidemic disease on networks. Phys. Rev. E (3) 66, 016128.
• Salganik, M. J. and Heckathorn, D. D. (2004). Sampling and estimation in hidden populations using respondent-driven sampling. Sociological Methodol. 34, 193–239.
• Van der Hofstad, R. (2014). Random graphs and complex networks. Vol. I. Available at http://www.win.tue.nl/\raise.17ex$\scriptstyle\sim$hofstad/NotesRGCN.html.
• Volz, E. and Heckathorn, D. D. (2008). Probability based estimation theory for respondent driven sampling. J. Official Statist. 24, 79–97.
• Wejnert, C. (2009). An empirical test of respondent-driven sampling: point estimates, variance, degree measures, and out-of-equilibrium data. Sociological Methodol. 39, 73–116.
• Wejnert, C. and Heckathorn, D. D. (2008). Web-based network sampling: Efficiency and efficacy of respondent-driven sampling for online research. Sociological Meth. Res. 37, 105–134.