In this article, we propose a new monotonically decreasing count model which includes Gauss hypergeometric function and is suitable for Zero vertex, over dispersed dataset. Though the proposed model includes Gauss hypergeometric function, nevertheless it possesses simple and closed expressions for various distributional characteristics. An application to count regression modeling using a well-known dataset from the National Medical Expenditure Survey is discussed by considering the length of stay in hospitalization as a dependent variable and following the proposed count model. We compare our result with the classical negative binomial regression model and uniform-Poisson regression model.