False negative rates of severe acute respiratory coronavirus 2 diagnostic tests, together with selection bias due to prioritized testing can result in inaccurate modeling of COVID-19 transmission dynamics based on reported "case" counts. We propose an extension of the widely used Susceptible-Exposed-Infected-Removed (SEIR) model that accounts for misclassification error and selection bias, and derive an analytic expression for the basic reproduction number R0 as a function of false negative rates of the diagnostic tests and selection probabilities for getting tested. Analyzing data from the first two waves of the pandemic in India, we show that correcting for misclassification and selection leads to more accurate prediction in a test sample. We provide estimates of undetected infections and deaths between April 1, 2020 and August 31, 2021. At the end of the first wave in India, the estimated under-reporting factor for cases was at 11.1 (95% CI: 10.7,11.5) and for deaths at 3.58 (95% CI: 3.5,3.66) as of February 1, 2021, while they change to 19.2 (95% CI: 17.9, 19.9) and 4.55 (95% CI: 4.32, 4.68) as of July 1, 2021. Equivalently, 9.0% (95% CI: 8.7%, 9.3%) and 5.2% (95% CI: 5.0%, 5.6%) of total estimated infections were reported on these two dates, while 27.9% (95% CI: 27.3%, 28.6%) and 22% (95% CI: 21.4%, 23.1%) of estimated total deaths were reported. Extensive simulation studies demonstrate the effect of misclassification and selection on estimation of R0 and prediction of future infections. A R-package SEIRfansy is developed for broader dissemination.