The outbreak of COVID-19 disrupts the life of many people in the world. The state of Arizona in the U.S. emerges as one of the country's newest COVID-19 hot spots. Accurate forecasting for COVID-19 cases will help governments to implement necessary measures and convince more people to take personal precautions to combat the virus. It is difficult to accurately predict the COVID- 19 cases due to many human factors involved. This paper aims to provide a forecasting model for COVID-19 cases with the help of human activity data from the Google Community Mobility Reports. To achieve this goal, a specific partial differential equation (PDE) is developed and validated with the COVID-19 data from the New York Times at the county level in the state of Arizona in the U.S. The proposed model describes the combined effects of transboundary spread among county clusters in Arizona and human activities on the transmission of COVID-19. The results show that the prediction accuracy of this model is well acceptable (above 94%). Furthermore, we study the effectiveness of personal precautions such as wearing face masks and practicing social distancing on COVID-19 cases at the local level. The localized analytical results can be used to help to slow the spread of COVID- 19 in Arizona. To the best of our knowledge, this work is the first attempt to apply PDE models on COVID-19 prediction with the Google Community Mobility Reports.