Biracks and biquandles, which are useful for studying the knot theory, are special families of solutions of the set-theoretic Yang-Baxter equation. A homology theory for the set-theoretic Yang-Baxter equation was developed by Carter, Elhamdadi, and Saito in order to construct knot invariants. In this paper, we construct a normalized (co)homology theory of a set-theoretic solution of the Yang-Baxter equation. We obtain some concrete examples of non-trivial $n$-cocycles for Alexander biquandles. For a biquandle $X,$ its geometric realization $BX$ is discussed, which has the potential to build invariants of links and knotted surfaces. In particular, we demonstrate that the second homotopy group of $BX$ is finitely generated if the biquandle $X$ is finite.

The metapopulation network model is effectively used to study the spatial spread of epidemics with individuals mobility. Considering the time-varying nature of individual activity and the preferences for attractive destinations in population mobility, this paper develops a time-varying network model in which activity of a population is correlated with its attractiveness. Based on the model, the spreading processes of the SIR disease on different correlated networks are studied, and global migration thresholds are derived. It is observed that increasing the correlation between activity and attractiveness results in a reduced outbreak threshold but suppresses the disease outbreak size and introduces greater heterogeneity in the spatial distribution of infected individuals. We also investigate the impact of non-pharmacological interventions (self-isolation and self-protection) on the spread of epidemics in different correlation networks. The results show that the simultaneous implementation of these measures is more effective in negatively correlated networks than in positively correlated or non-correlated networks, and the prevalence is reduced significantly. In addition, both self-isolation and self-protection strategies increase the migration threshold of the spreading and thus slow the spread of the epidemic. However, the effectiveness of each strategy in reducing the density of infected populations varies depending on different correlated networks. Self-protection is more effective in positively correlated networks, whereas self-isolation is more effective in negatively correlated networks. These findings contribute to a better understanding of epidemic spreading in large-scale time-varying metapopulation networks and provide insights for epidemic prevention and control.