Susceptible-infectious-recovered models revisited: from the individual level to the population level

The classical susceptible-infectious-recovered (SIR) model, originated from the seminal papers of Ross [51] and Ross and Hudson [52,53] in 1916-1917 and the fundamental contributions of Kermack and McKendrick [36-38] in 1927-1932, describes the transmission of infectious diseases between susceptible and infective individuals and provides the basic framework for almost all later epidemic models, including stochastic epidemic models using Monte Carlo simulations or individual-based models (IBM). In this paper, by defining the rules of contacts between susceptible and infective individuals, the rules of transmission of diseases through these contacts, and the time of transmission during contacts, we provide detailed comparisons between the classical deterministic SIR model and the IBM stochastic simulations of the model. More specifically, for the purpose of numerical and stochastic simulations we distinguish two types of transmission processes: that initiated by susceptible individuals and that driven by infective individuals. Our analysis and simulations demonstrate that in both cases the IBM converges to the classical SIR model only in some particular situations. In general, the classical and individual-based SIR models are significantly different. Our study reveals that the timing of transmission in a contact at the individual level plays a crucial role in determining the transmission dynamics of an infectious disease at the population level.