SIR model on a dynamical network and the endemic state of an infectious disease

• The quasi-stationary state of an SIR model on a dynamical network is characterized.
• Local contacts decrease the effective transmission rate (ETR) of the disease.
• An explicit relation between ETR and correlation among S–I individuals is obtained.
• Tails of histograms of infected individuals (outbreaks) behave as qq-exponentials.
• Network effects can be partially absorbed by rescaling of an SIR-model parameter.

In this work we performed a numerical study of an epidemic model that mimics the endemic state of whooping cough in the pre-vaccine era. We considered a stochastic SIR model on dynamical networks that involve local and global contacts among individuals and analysed the influence of the network properties on the characterization of the quasi-stationary state. We computed probability density functions (PDF) for infected fraction of individuals and found that they are well fitted by gamma functions, excepted the tails of the distributions that are q-exponentials. We also computed the fluctuation power spectra of infective time series for different networks. We found that network effects can be partially absorbed by rescaling the rate of infective contacts of the model. An explicit relation between the effective transmission rate of the disease and the correlation of susceptible individuals with their infective nearest neighbours was obtained. This relation quantifies the known screening of infective individuals observed in these networks. We finally discuss the goodness and limitations of the SIR model with homogeneous mixing and parameters taken from epidemiological data to describe the dynamic behaviour observed in the networks studied.