Mobile health apps are revolutionizing the healthcare ecosystem by improving communication, efficiency, and quality of service. In low- and middle-income countries, they also play a unique role as a source of information about health outcomes and behaviors of patients and healthcare workers, while providing a suitable channel to deliver both personalized and collective policy interventions. We propose a framework to study user engagement with mobile health, focusing on healthcare workers and digital health apps designed to support them in resource-poor settings. The behavioral logs produced by these apps can be transformed into daily time series characterizing each user's activity. We use probabilistic and survival analysis to build multiple personalized measures of meaningful engagement, which could serve to tailor content and digital interventions suiting each health worker's specific needs. Special attention is given to the problem of detecting churn, understood as a marker of complete disengagement. We discuss the application of our methods to the Indian and Ethiopian users of the Safe Delivery App, a capacity-building tool for skilled birth attendants. This work represents an important step towards a full characterization of user engagement in mobile health applications, which can significantly enhance the abilities of health workers and, ultimately, save lives.