Spread of infectious diseases: Effects of the treatment of population

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.