  Structure-preserving discretizations of the SIR model are presented by
focusing on the hodograph transformation and the conditions for integrability
for their discrete SIR models are given. For those integrable discrete SIR
models, we derive their exact solutions as well as conserved quantities. If we
choose the parameter appropriately for one of our proposed discrete SIR models,
it conserves the conserved quantities of the SIR model. We also investigate an
ultradiscretizable discrete SIR model.
