SIR model with local and global infective contacts: A deterministic approach and applications

An epidemic model with births and deaths is considered on a two-dimensional L×L lattice. Each individual can have global infective contacts according to the standard susceptible–infected–recovered (SIR) model rules or local infective contacts with their nearest neighbors. We propose a deterministic approach to this model and, for the parameters corresponding to pertussis and rubella in the prevaccine era, verify that there is a close agreement with the stochastic simulations when epidemic spread or endemic stationarity is considered. We also find that our approach captures the characteristic features of the dynamic behavior of the system after a sudden decrease in global contacts that may arise as a consequence of health care measures. By using the deterministic approach, we are able to characterize the exponential growth of the epidemic behavior and analyze the stability of the system at the stationary values. Since the deterministic approximation captures the essential features of the disease transmission dynamics of the stochastic model, it provides a useful tool for performing systematic studies as a function of the model parameters. We give an example of this potentiality by analyzing the likelihood of the endemic state to become extinct when the weight of the global contacts is drastically reduced.