  Infectious disease epidemiologists routinely fit stochastic epidemic models
to time series data to elucidate infectious disease dynamics, evaluate
interventions, and forecast epidemic trajectories. To improve computational
tractability, many approximate stochastic models have been proposed. In this
paper, we focus on one class of such approximations -- time series
Susceptible-Infectious-Removed (TSIR) models. Infectious disease modeling often
starts with a homogeneous mixing assumption, postulating that the rate of
disease transmission is proportional to a product of the numbers of susceptible
and infectious individuals. One popular way to relax this assumption proceeds
by raising the number of susceptible and/or infectious individuals to some
positive powers. We show that when this technique is used within the TSIR
models they cannot be interpreted as approximate SIR models, which has
important implications for TSIR-based statistical inference. Our simulation
study shows that TSIR-based estimates of infection and mixing rates are
systematically biased in the presence of non-homogeneous mixing, suggesting
that caution is needed when interpreting TSIR model parameter estimates when
this class of models relaxes the homogeneous mixing assumption.
