Nonlinear effects of resource allocation delay on epidemic spreading in complex networks.

The impact of resource allocation on the dynamics of epidemic spreading is an important topic. In real-life scenarios, individuals usually prioritize their own safety, and this self-protection consciousness will lead to delays in resource allocation. However, there is a lack of systematic research on the impact of resource allocation delay on epidemic spreading. To this end, a coupled model for resource allocation and epidemic spreading is proposed, which considers both the allocation decisions and delay behavior of individuals with limited resources. Through theoretical analysis, the influence mechanism of resource allocation delay on epidemic spreading is deduced, and the relationship among epidemic threshold, delay time, and the fraction of cautious individuals is obtained, and finally, the stability of the solution under different conditions is proven. Furthermore, the dynamic characteristics of epidemic spreading under the influence of the two factors are systematically studied by combining numerical simulation and theoretical analysis. The results show that the impact of delay behavior exhibits nonlinear characteristics, namely, appropriate delay can enhance control effectiveness, while excessive delay results in insufficient resource allocation and consequently increases infection risk. Particularly, an optimal delay that maximizes the epidemic threshold is identified. In addition, an increase in the proportion of cautious individuals can significantly increase the epidemic threshold, but an excessively high proportion can severely constrain resource allocation, which reduces the control effectiveness. The results of this study provide scientific evidence for developing more effective epidemic control strategies, particularly in optimizing resource allocation and improving control outcomes.