  In a metapopulation network, infectious diseases spread widely because of the
travel of individuals. In the present study, we consider a modified
metapopulation Susceptible-Infected-Removed (SIR) model with a latent period,
which we call the SHIR model. In the SHIR model, an infectious period is
divided into two stages. In the first stage, which corresponds to the latent
period, infectious individuals can travel. However, in the second stage, the
same individuals cannot travel since they are seriously ill. Final size
distributions of the metapopulation SIR and SHIR models are simulated with two
different methods and compared. In Monte Carlo simulations, in which the
population is treated as an integer, the distributions show similar behavior.
However, in reaction-diffusion systems, in which the population is treated as a
real number, the final size distribution of the SHIR model has a discontinuous
jump, and that of the SIR model shows a continuous transition. The
discontinuous jump is found to be an artifact that occurs owing to an
inappropriate termination condition.
