In this paper, with the assumptions that an infectious disease has a fixed
latent period in a population and the latent individuals of the population may
disperse, we reformulate an SIR model for the population living in two patches
(cities, towns, or countries etc.), which is a generalization of the classic
Kermack-McKendrick SIR model. The model is given by a system of delay
differential equations with a fixed delay accounting for the latency and
non-local terms caused by the mobility of the individuals during the latent
period. We analytically show that the model preserves some properties that the
classic Kermack-McKendrick SIR model possesses: the disease always dies out,
leaving a certain portion of the susceptible population untouched (called
final sizes). Although we can not determine the two final sizes, we are able to
show that the ratio of the final sizes in the two patches is totally determined
by the ratio of the dispersion rates of the susceptible individuals between the
two patches. We also explore numerically the patterns by which the disease dies
out, and find that the new model may have very rich patterns for the disease
to die out. In particular, it allows multiple outbreaks of the disease before it
goes to extinction, strongly contrasting to the classic Kermack-McKendrick SIR
model.