Annealed mean-field epidemiological model on scale-free networks with a mitigating factor

An annealed version of the quenched mean-field model for epidemic spread is introduced and investigated analytically and assisted by numerical calculations. The interaction between individuals follows a prescription that is used to generate a scale-free network, and we have adjusted the number of connections to produce a sparse network. Specifically, the model's behavior near the infection threshold is examined, as well as the behavior of the stationary prevalence and the probability that a connection between individuals encounters an infected one. We found that these functions display a monotonically increasing dependence on the infection rate. Subsequently, a modification that mimics the mitigation in the probability of encountering an infected individual is introduced, following an old idea rooted in the Malthus-Verhulst model. We found that this modification drastically changes the probability that a connection meets an infected individual. However, despite this change, it does not alter the monotonically increasing behavior of the stationary prevalence.