Testing and isolation are key public health tools to reduce the spread of COVID-19 . However, imperfect diagnostic testing for COVID-19 poses many interesting statistical challenges. Symptomatic testing leads to selection bias in the reported number of cases with many infections remaining undetected and reported cases coming from a selected population with access to testing. In addition, RT-PCR tests, which are still the primary diagnostic test for COVID-19 are reported to have a high number of false negatives. These two issues make estimation of the true number of infections from reported case data to be challenging using a standard epidemiologic model like the Susceptible-Exposed-Infected-Recovered (SEIR model). In this paper, a new disease transmission model called the SEIR-fansy is proposed which takes these issues into account and offers more accurate estimation and predictions when these data issues are acute. Using this model, the spread of COVID-19 in India is modeled. It is estimated that, in India, for every reported case, there were approximately 11.1 and 19.2 infections that remained undetected in waves 1 and 2 of COVID-19 respectively. The corresponding underreporting factors for deaths are 3.6 and 4.6 respectively. These estimates of unreported infections and deaths help us to appreciate the real magnitude of the pandemic as well as design better intervention strategies. We make the software implementing this method available for broader use by the community. The mathematical model is relevant to any infectious disease with limited testing and imperfect tests.