The space-time spread of infectious disease in geographical units depends on both transmission between neighboring units and the underlying distribution of susceptibility, which can be spatially dependent. Spatially correlated underlying risk may arise from known factors, such as population density, or unknown (or unmeasured) factors such as commuter flows, environmental conditions, or health disparities. We propose a general infectious disease model that accounts for both space-time transmission and the underlying susceptibility, and model the unknown, spatially correlated susceptibility as a Gaussian Process. We show that the unknown susceptibility surface can be estimated from observed, geo-located time series of infection events and use PICAR to achieve dimension reduction which improves computational stability and speed. We illustrate this method using incidents of measles outbreaks in England and Wales in the pre-vaccine era and show that the resulting susceptibility surface is strongly correlated with population size, consistent with prior analyses. We further apply this method to foot-and-mouth disease outbreaks in Turkey to illustrate spatial hot-spots of susceptibility.