A path-specific approach to SEIR modeling

Despite being developed in the late 1920s, compartmental epidemic modeling is still a rich and fruitful area of research. The original compartmental epidemic models were SIR (Susceptible, Infectious, Removed) models, which assume permanent immunity after recovery. SIR models, along with the more recent SEIR (Susceptible, Exposed, Infectious, Removed) models are still the gold standard in modeling pathogens that confer permanent immunity. This dissertation expands the SEIR structure to include a new class of spatial SEIR models. The exponential assumption of these models states that the latent and infectious times of the pathogen are exponentially distributed. Work that relaxes this assumption and still allows for mixing to occur at the population level is limited, thereby making strong assumptions about these times. We relax this assumption in a flexible way, by considering a hybrid approach that contains characteristics of both population level and individual level approaches. Next, we expand the Conditional Autoregressive (CAR) class of spatial models. This is to account for the Mumps data set we have procured, which contains mismatched lattice structures that cannot be handled by traditional CAR models. The use of CAR models is desirable here, as these models are known to produce spatial smoothing on lattices, and are a natural way to draw strength spatially in estimating spatial effects. Finally, we develop a pair of spatial SEIR models utilizing our CAR structure. The first utilizes the exponential assumption, which is very robust. The second develops a highly flexible spatial SEIR model by embedding the CAR structure into the SEIR structure. This allows for a realistic analysis of epidemic data occurring on a lattice.