Solution of deterministic–stochastic epidemic models by dynamical Monte Carlo method

This work is concerned with dynamical Monte Carlo (MC) method and its application to models originally formulated in a continuous-deterministic approach. Specifically, a susceptible–infected–removed–susceptible (SIRS) model is used in order to analyze aspects of the dynamical MC algorithm and achieve its applications in epidemic contexts. We first examine two known approaches to the dynamical interpretation of the MC method and follow with the application of one of them in the SIRS model. The working method chosen is based on the Poisson process where hierarchy of events, properly calculated waiting time between events, and independence of the events simulated, are the basic requirements. To verify the consistence of the method, some preliminary MC results are compared against exact steady-state solutions and other general numerical results (provided by Runge–Kutta method): good agreement is found. Finally, a space-dependent extension of the SIRS model is introduced and treated by MC. The results are interpreted under and in accordance with aspects of the herd-immunity concept.