A novel predictive modeling framework for the spread of infectious diseases using high-dimensional partial differential equations is developed and implemented. A scalar function representing the infected population is defined on a high-dimensional space and its evolution over all the directions is described by a population balance equation (PBE). New infections are introduced among the susceptible population from a non-quarantined infected population based on their interaction, adherence to distancing norms, hygiene levels and any other societal interventions. Moreover, recovery, death, immunity and all aforementioned parameters are modeled on the high-dimensional space. To epitomize the capabilities and features of the above framework, prognostic estimates of Covid-19 spread using a six-dimensional (time, 2D space, infection severity, duration of infection, and population age) PBE is presented. Further, scenario analysis for different policy interventions and population behavior is presented, throwing more insights into the spatio-temporal spread of infections across duration of disease, infection severity and age of the population. These insights could be used for science-informed policy planning.