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Abstract
Boolean networks are used for modeling networks whose node activity can be described as a binary value, such as on-off, active-nonactive, or infected-susceptible. We study the dynamics of an epidemic model of a so-called Susceptible-Infected- Susceptible (SIS) network in which nodes represent people that are initially susceptible to catching a virus, then they may become infected upon interaction with other people and thus become capable of spreading the virus, and return to being susceptible once the illness ends. We generate a mathematical Boolean model for the fraction of infected people at any given time and show it is a good fit for the network we model. The model allows us to assess the impact of the variation of the network parameters: the infection time that ranges from two to fourteen days, as well as a fixed or variable infection rate. We use our model to observe the dynamics of the epidemic. We utilize pattern formation plots and density of ones graphs to determine the behavior of the epidemic for varying parameters. We use mean-field approximation to generate a more useful approximation to the model and we assess the goodness of fit of the approximation. We provide some statistics that examine the differences between the model and the mean-field approximation for varying individual susceptibility and length of infection. Finally, we use bifurcation diagrams to study the long term behavior of infection. The pattern formation plots and density of ones plots indicate that the period of infection or infection rate may have a significant impact on the survival of the epidemic, while the number of infected people on the first day has little impact on the long term survival of the epidemic. The bifurcation diagrams show that there is some level of stability of the epidemic and that no specific attractors can be observed in the model. Health practices driving low levels of individual susceptibility or infection time may have significant impact on the dynamics of the network.
