Because of its important role in health policy-shaping, population health monitoring (PHM) is considered a fundamental block for public health services. However, traditional public health data collection approaches, such as clinic-visit-based data integration or health surveys, could be very costly and time-consuming. To address this challenge, this article proposes a cost-effective approach called Compressive Population Health (CPH), where a subset of a given area is selected in terms of regions within the area for data collection in the traditional way, while leveraging inherent spatial correlations of neighboring regions to perform data inference for the rest of the area. By alternating selected regions longitudinally, this approach can validate and correct previously assessed spatial correlations. To verify whether the idea of CPH is feasible, we conduct an in-depth study based on spatiotemporal morbidity rates of chronic diseases in more than 500 regions around London for over 10 years. We introduce our CPH approach and present three extensive analytical studies. The first confirms that significant spatiotemporal correlations do exist. In the second study, by deploying multiple state-of-the-art data recovery algorithms, we verify that these spatiotemporal correlations can be leveraged to do data inference accurately using only a small number of samples. Finally, we compare different methods for region selection for traditional data collection and show how such methods can further reduce the overall cost while maintaining high PHM quality.