Influence of Social Contacts on Endemic Dynamics in the Extended SEIS Model

In the framework of mean-field approximation, the influence of social contacts on the spread of an epidemic in a population of constant size is discussed. The key feature of the proposed model is that it includes two infection–transmission mechanisms depending on the physical nature of the contact between people. We separate the transfer mechanism related directly to the movement of people (the so-called transport processes) from the one occurring when the relative velocity between individuals is negligible (the so-called social contacts). Based on the developed physicochemical analogy, this approach allows us to derive, in a unified manner, expressions for the rate constants of infection–transmission of different nature. The resulting transmission rate constants are used to modify the SEIS model to examine the influence of social activity on the formation of an endemic equilibrium in the population under consideration. The frequency of social contacts is estimated using Dunbar’s approach and a direct statistical calculation based on the binomial distribution. These relations are then used to discuss the formation of quasi-stationary states, which can be interpreted as endemic equilibria. A qualitative analysis of the resulting dynamical regimes is carried out. The necessary conditions for the existence of this equilibrium, depending on both social and medical–biological factors, are also derived. The analytical results are illustrated by numerical simulations. The present results should be interpreted as a necessary step to establish a link between purely transport and social mechanisms of epidemic development.