Near real-time monitoring of outbreak transmission dynamics and evaluation of public health interventions are critical for interrupting the spread of the novel coronavirus (SARS-CoV-2) and mitigating morbidity and mortality caused by coronavirus disease (COVID-19). Formulating a regional mechanistic model of SARS-CoV-2 transmission dynamics and frequently estimating parameters of this model using streaming surveillance data offers one way to accomplish data-driven decision making. For example, to detect an increase in new SARS-CoV-2 infections due to relaxation of previously implemented mitigation measures one can monitor estimates of the basic and effective reproductive numbers. However, parameter estimation can be imprecise, and sometimes even impossible, because surveillance data are noisy and not informative about all aspects of the mechanistic model, even for reasonably parsimonious epidemic models. To overcome this obstacle, at least partially, we propose a Bayesian modeling framework that integrates multiple surveillance data streams. Our model uses both COVID-19 incidence and mortality time series to estimate our model parameters. Importantly, our data generating model for incidence data takes into account changes in the total number of tests performed. We apply our Bayesian data integration method to COVID-19 surveillance data collected in Orange County, California. Our results suggest that California Department of Public Health stay-at-home order, issued on March 19, 2020, lowered the SARS-CoV-2 effective reproductive number $R_e$ in Orange County below 1.0, which means that the order was successful in suppressing SARS-CoV-2 infections. However, subsequent "re-opening" steps took place when thousands of infectious individuals remained in Orange County, so $R_e$ increased to approximately 1.0 by mid-June and above 1.0 by mid-July.