Reproduction number (R), defined as the average number of people that will be infected by an individual who has the infection, plays a central role in predicting the evolution of an infectious disease outbreak. However, the R most certainly varies by location and by time due to multiple factors, including time-varying social distancing, varying public policies, difference in population density, and even changing of weather. In order to study the dynamics of disease transmission, we modeled the instantaneous reproduction number Rt, where t>=0, which was allowed to change over time. We proposed an online algorithm to iteratively estimate the Rt using an observation-driven Poisson regression model with an autocorrelated latent process, and to study the impact of covariates on the variation of Rt. Specifically, we assumed that, conditioning on existing infection and assumptions on the distribution of how new infection events were distributed as a function of time, the observed daily new cases followed a Poisson distribution with mean depending on Rt. The associations between Rt and potential covariates, e.g., social distancing order, masking mandate, and temperature, were assessed using a semi-parametric latent regression with an embedded autoregressive term to model the evolution of Rt over time. Our estimators allow a close monitor of Rt and dynamic update of knowledge whenever new data were available.