Spatial correlations in SIS processes on regular random graphs

TL;DR

This work addresses the mismatch between mean-field predictions and SIS epidemic dynamics on networks by incorporating spatial correlations through a shell-based, ODE hierarchy for pairwise infections. It develops corrections tailored to regular random graphs by deriving dynamic equations for distance-\ell pair correlations, computing shell transition probabilities from a Gompertz shell-size distribution, and obtaining a closed-form steady-state correction to endemic infection density. The results show exponential decay of correlations with distance and provide expansions in $1/k_0$ that reduce endemic prevalence relative to mean-field predictions, improving forecasting on RRGs. The approach offers a robust analytical framework for understanding how network structure shapes infectious dynamics and highlights directions for extending corrections to more heterogeneous or intermediate network topologies.

Abstract

In network-based SIS models of infectious disease transmission, infection can only occur between directly connected individuals. This constraint naturally gives rise to spatial correlations between the states of neighboring nodes, as the infection status of connected individuals becomes interdependent. Although mean-field approximations are commonly invoked to simplify disease forecasting on networks, they fail to account for these correlations by assuming that infectious individuals are well-mixed within a population, leading to inaccurate predictions of infection numbers over time. As such, the development of mathematical frameworks that account for spatially correlated infections is of great interest, as they offer a compromise between accurate disease forecasting and analytic tractability. Here, we use existing corrections to mean-field theory on the regular lattice to construct a more general framework for equivalent corrections on regular random graph topologies. We derive and simulate a system of ordinary differential equations for the time evolution of the spatial correlation function at various geodesic distances on random networks, and use solutions to this hierarchy of ordinary differential equations to predict the global infection density as a function of time, finding good agreement with corresponding numerical simulations. Our results constitute a substantial development on existing corrections to mean-field theory for infectious individuals in SIS processes and provide an in-depth characterization of how structural randomness in networks affects the dynamical trajectories of infectious diseases on networks.

Figures (5)

• Figure 1: (a) Schematic of the SIS model on networks. Infected nodes recover with rate $\gamma$(conventionally, time is rescaled as $t \to \gamma t$, implying $\gamma = 1$) and infect their nearest neighbors, each at rate $\beta$. (b)-(e) Snapshots of SIS model simulations on RRGs in the quasi-steady state. For lower values of average node degree $k_0$, the clustering of infectious nodes (orange) is more pronounced than on networks with a higher node degree. Moreover, increasing the infection rate, $\beta$, also decreases the clustering of infectious nodes but increases the overall infection density in the steady state. (f) Steady state infection density on the RRG as a function of average node degree, $k_0$, for simulations on RRGs with $N=10^4$ nodes, $\beta k_0 = 1.7$, and $\gamma = 1$. The dashed line is the expected mean-field value of $1-1/(\beta k_0) \simeq 0.41$. The initial conditions of all simulations are an initial density of $x_{I} = 10^{-2}$ with $100$ infective individuals placed randomly on the network, and all results are averaged over 100 simulations.
• Figure 2: Average infection density, $x_{I}$ (top row), and the distance-one pairwise correlation function, $F_{II}^{(1)}\left(t\right)$ (bottom row). Here, SIS model simulations on the RRG (red) are compared with numerical solutions of the corrections to mean-field theory derived in Section \ref{['RRGCorrectionsSection']} (blue), and with mean-field theory (black). Simulations were executed on networks with $N=10^4$ nodes, with an initial infection density of $10^{-2}$ such that 100 infective individuals were placed randomly on the network, and with $\gamma=1$. The first two columns correspond to a node degree of $k_0=4$ with infection rates of $\beta k_0=1.7$ and $2.0$, while the last two columns correspond to $k_0=6$ with $\beta k_0=1.4$ and $1.7$, respectively. Results in each plot are averaged over $100$ simulations.
• Figure 3: Relaxation time and steady state values for the SIS process ($\gamma = 1, N=10^{4}, I_0=10^{-2}$) versus $k_0\beta$ and $p$. Figures (a) and (c) show log relaxation times, which we define as the time between the peak ($t_{\rm{peak}}$) and half-maximum ($t_{1/2}$) values of $F_{II}^{(1)}(t)$, whilst (b) and (d) show the value of $\ln(\bar{F}_{II}^{(1)} - 1)$ for $k_0=4$ and $6$, respectively. The results of the parameter sweep show that increasing $p$ and increasing $k_0\beta$ both reduce the steady state value of $F_{II}^{(1)}\left(t\right)$ and the relaxation time for the process.
• Figure 4: (a) A two-dimensional regular lattice with its origin at $(0,0)$. The purple, green, blue, and red circles represent the first, second, third, and fourth shells, respectively. (b) RRG shells with node degree $k_0=4$. The central node is black, the first shell nodes are colored purple, while the second shell nodes are colored green.
• Figure 5: (a)-(b) The steady-state pairwise correlation function, $\bar{F}_{II}^\ell$, versus the network distance $\ell$, on linear and semi-logarithmic scales, respectively. Circles indicate values obtained from numerical simulations, while solid lines represent exponential fits based on Eq. \ref{['eq: F_ss']}. (c)-(d) The steady-state pairwise correlation function $\bar{F}_{II}^1$ and steady-state infected fraction $\bar{x}_{I}$, respectively, versus the average node degree $k_0$. Numerical solutions (circles) and simulation values (triangles) show strong agreement. Additionally, power series approximations of the second order of $k_0^{-1}$ (solid green lines) converge toward the numerical and simulation results as $k_0$ increases.