Correlation between transition probability and network structure in epidemic model

TL;DR

The paper addresses whether transition probabilities in SIS dynamics depend on network structure, especially over large time intervals. It develops nonlinear rate-to-probability mappings for SIS on both annealed and static networks, showing that quantities like $1-(1-\beta'\Theta)^k$ and $\mu'$ depend on degree $k$ or on the number of infected neighbors. The authors validate these mappings using heterogeneous mean-field theory, the effective degree approach, and Monte Carlo simulations, demonstrating that the nonlinear relations reproduce continuous-time behavior in discrete-time updates and revealing network-structure constraints. This work highlights the crucial role of topology in transition dynamics and provides a framework for applying nonlinear mappings to temporal or higher-order networks in the future.

Abstract

In discrete-time dynamics, it is frequently assumed that the transition probabilities (e.g., the recovery probability) are independent of the network structure. However, there is a lack of empirical evidence to support this claim in large time intervals. This paper presents the nonlinear relations between the rates (in continuous-time dynamics) and probabilities of the susceptible-infected-susceptible model on annealed and static networks. It is shown that the transition probabilities are affected not only by the rates and the time interval, but also by the network structure. The correctness of the nonlinear relations on networks is verified based on theoretical calculation and Monte Carlo simulation.

Figures (4)

• Figure 1: Mapping from continuous-time SIS model to discrete-time SIS model on annealed networks. (a)-(b) Epidemic prevalence $\rho$ as a function of infection rate $\beta$. (c) Distribution of infected individuals by degree $k$. The degree distribution $P(k)$ adopts a Poisson distribution with $\langle k \rangle=6$ in (a) and a scale-free distribution with $k_{\min}=5$ and $\gamma=2.5$ in (b) and (c), as illustrated in the inset panels. The yellow lines represent the results of continuous-time dynamics, while the other lines represent the results of discrete-time dynamics using nonlinear relations. Parameters: $\mu = 1$, the fraction of individuals initially infected $\rho_0=0.01$, $\Delta t = 1$ in discrete-time dynamics, $\beta=0.1$ in (c), $N = 10^6$ in (a) and $N = 10^5$ in (b)-(c). The dash and dot lines are used to enhance the intuitiveness of overlapping data.
• Figure 2: (a) The probabilities $\mu'_k$, $1 - \left(1-\beta'_k\Theta\right)^{k}$, and $\beta'_k$ as a function of degree $k$ when the system reaches steady state. (b) The fraction of infected individuals as a function of the time for discrete-time SIS model on annealed networks. The results in (a) are obtained from Eq. \ref{['e5']} and Eq. \ref{['e8']}, and the results in (b) are obtained from Eq. \ref{['e5']} and Eq. \ref{['e1']} (the linear relations). The degree distribution $P(k)$ is consistent with Fig. \ref{['fig1']}(b). Parameters: $\mu = 1$, $\Delta t = 1$.
• Figure 3: Mapping from continuous-time SIS model to discrete-time SIS model on static networks. (a) Epidemic prevalence $\rho$ as a function of infection rate $\beta$. (b) The fraction of infected individuals as a function of the time for discrete-time SIS model on static networks. The results in (b) are obtained from Eq. \ref{['e14']}, Eq. \ref{['e16']}, Eq. \ref{['e17']}, and Eq. \ref{['e1']} (the linear relations). The static scale-free networks is generated from the uncorrelated configuration model PhysRevE.71.027103 with power-law degree distributions $P(k)\sim k^{-\gamma}$ where $\gamma=2.5$ and $k_{\min}=5$. Parameters: $\mu = 1$, $\rho_0=0.001$, $\Delta t = 1$ in discrete-time dynamics, $N = 10^5$.
• Figure 4: Monte Carlo simulations for SIS model on annealed and static networks. Epidemic prevalence $\rho$ as a function of infection rate $\beta$ for different update mode and mapping relations. The static network and annealed network are consistent with Fig. \ref{['fig3']} and Fig. \ref{['fig1']}(b), respectively. Parameters: $\mu = 1$, $N = 10^5$, $\rho_0=0.001$, $\Delta t = 1$ in synchronous update.