Dynamical diversity induced by individual responsive immunization

A voluntary vaccination allows for a healthy individual to choose vaccination according to the individual’s local information. Hence, vaccination has the potential to provide a complex negative feedback (non-infection decreases propensity for vaccination, hence increasing infection and vice versa). In this paper, we investigate a kind of SIS epidemic model with a deterministic and voluntary vaccination scheme in scale-free networks. We first study a threshold model with no historical information. By using the comparative method we confirm that under some conditions there exist two critical values of infection rates to determine three kinds of epidemic dynamical behaviors: the epidemic spread, the asymptotical decay and the exponential decay. Furthermore, a mean-field approximation model can predict the maximal infection level but cannot predict the existence of two critical infection rates. In numerical simulations, we observe a maximum in epidemic duration as a function of the model parameter. A similar phenomenon has been found in the model with historical information. Finally, we study a degree-weighted model with a nonnegative exponent α$\alpha$ where α=0$\alpha =0$ corresponds to the threshold model. We find that at the steady state the infection density increases with α$\alpha$, while the variation of the vaccination fraction is less straightforward.