A capture-recapture problem when information is obtained from two qualitatively different sources

We first consider a wildlife-related capture recapture problem with a known total number of fish. Our objective is to estimate the number of healthy fish, H. We first use a rod that only attracts healthy fish, which are tagged and returned to the water. Later, we use a net that scoops both sick and healthy fish. Three assumptions regarding the probability of being caught by the net, conditional on health status and being caught by the rod, lead to three different estimates of H. We give approximations to expected values of the three estimates and give a condition under which they bracket H. A potential application of these methods is to the follow-up of World Trade Center responders. Responders are disease-free when they arrive at the clean-up site and are asked to report for a visit after a fixed period of time, but some fail to do so. Some responders, whether they come to the scheduled return visit or not, spontaneously report a disease before the scheduled visit, but absence of disease is never reported in this manner. We use the methods developed to estimate the total number of subjects with disease by the time of the scheduled return visit.