Epidemic spreading phenomena on a scale-free network with time-varying transmission rate due to social responses

In this paper, we study and simulate the effect of individual social responses, as a collective factor, on the epidemic spreading processes. We formally define the problem based on the traditional SI and SIR compartmental models considering the time-varying infection probability dependent on the social responses. In this study, models of generic and special case scenarios are developed. While in the generic case the effective parameter of behavioral response is demonstrated as one collective factor, in the special case the behavioral response is assumed as the combination of two collective factors: social cost and transfer rate of social awareness. With social cost, we refer to the costs incurred by a certain population to prevent or mitigate an epidemic. With transfer rate of social awareness, we describe the averaged rate of received information and knowledge regarding a disease that individuals hold and make use to avoid negative consequences. We show that, while in both SI and SIR models the density of infected agents grows exponentially during the initial time steps, the inclusion of our models of social responses, either generic or special one, leads to mitigation of the spreading. As a result of both generic and special cases, the density of infected agents in the stationary state and the maximum number of infected agents decrease according to power-law functions for different values of collective factors. In the special case results, we also witnessed significant changes in the slope of decreasing trends of stationary density of states happening for a critical value of transfer rate of social awareness, approximately at about the inverse of the time interval of transmission rate update. With this result, we point out that increasing the transfer rate of social awareness to about this critical point outperforms any slight increase in social cost in reducing the number of infected agents.