Modeling Epidemics on Multiplex Networks: Epidemic Threshold and Basic Reproduction Number

Abstract

Accurate epidemic forecasting requires models that account for the layered and heterogeneous nature of real social interactions. The basic reproduction number ${\mathcal{R}}_0$, as calculated from models that assume homogeneous mixing or single-layer contact structures, has limited applicability to complex social systems. Here, we derive an expression for ${\mathcal{R}}_0$ in the context of multiplex networks, enabling the analysis of disease transmission across multiple social layers. We adapt the Degree-Based Mean-Field (DBMF) SIR model for single-layer complex networks to the multiplex setting, where each layer is characterized by its own degree distribution and infection rate. Using the Next Generation Matrix method, we derive an analytical expression for the basic reproduction number ${\mathcal{R}}_0$. Numerical integration of the multiplex DBMF equations shows that ${\mathcal{R}}_0=1$ marks the epidemic threshold and governs the behavior of key outbreak indicators as expected. In addition to the exact expression for ${\mathcal{R}}_0$, we introduce an approximation, denoted by $\ddot{I}„$, which is simpler to compute and admits a more transparent interpretation in terms of the epidemiological and topological parameters of the system. Stochastic agent-based simulations support these findings, demonstrating a direct correspondence between $\ddot{I}„$ and the average number of secondary infections generated during the early stages of an outbreak, consistent with the epidemiological interpretation of ${\mathcal{R}}_0$. This work provides a robust generalization of ${\mathcal{R}}_0$ for layered contact structures, offering a more realistic basis for epidemic forecasting and the design of intervention strategies.

22 pages, 7 figures