Scandinavian Journal of Statistics, Vol. 35: 650–663, 2008 doi: 10.1111/j.1467-9469.2008.00594.x  Board of the Foundation of the Scandinavian Journal of Statistics 2008. Published by Blackwell Publishing Ltd. Estimating the Basic Reproductive Number in the General Epidemic Model with an Unknown Initial Number of Susceptible Individuals ERIC H. Y. LAU Department of Community Medicine, The University of Hong Kong PAUL S. F. YIP Department of Social Work and Social Administration, The University of Hong Kong ABSTRACT. In any epidemic, there may exist an unidentified subpopulation which might be naturally immune or isolated and who will not be involved in the transmission of the disease. Estimation of key parameters, for example, the basic reproductive number, without accounting for this possibility would underestimate the severity of the epidemics. Here, we propose a procedure to estimate the basic reproductive number (R 0 ) in an epidemic model with an unknown initial number of susceptibles. The infection process is usually not completely observed, but is reconstructed by a kernel-smoothing method under a counting process framework. Simulation is used to evaluate the performance of the estimators for major epidemics. We illustrate the procedure using the Abakaliki smallpox data. Key words: basic reproductive number, counting process, epidemic model, kernel-smoothing method, martingale estimating equations, naturally immune, susceptible 1. Introduction In epidemiological studies, one of the important parameters is the basic reproductive num- ber R 0 (Riley et al., 2003; Longini et al., 2005), the expected number of secondary cases per primary case of infection in a completely susceptible population. When R 0 > 1, each genera- tion of infectious individuals is likely to infect the next generation with a larger number, until there is a significant depletion of the susceptible population. Interventions in the infection or the removal process, for example through a vaccination policy, will usually be implemented with the aim to reduce R 0 below unity. The initial number of susceptible individuals  is usually assumed known and available for statistical inference for R 0 (Becker, 1989; Becker & Hasofer, 1997; Daley & Gani, 1999; Hernandez-Suarez, 2002). However, there are some situations in which the number of sus- ceptibles might not be known; for example, HIV-positive cases are recorded, but the size of the group at risk is usually unknown, or a certain unknown proportion of the population are immune from the disease possibly due to prior vaccination. In these cases, the estimated basic reproductive number is biased downward if we regard the whole population as susceptible. Comparisons with previously estimated R 0 in other epidemic areas, with possibly different proportions of susceptibles, may lead to misleading perception in the severity of the disease. Only few attempts have been made to tackle this problem (Ferrari et al., 2005). This paper is motivated by the large discrepancies between different estimates of R 0 for the outbreak of smallpox in Abakaliki (Bailey, 1975; Eichner & Dietz, 2003). The estimated R 0 is around 1.1–1.4 when the whole population is assumed susceptible (Becker, 1989; Rida, 1991; Yip, 1989; O’Neill & Roberts, 1999). This is very different from the established values