SIS Epidemic Modelling on Homogeneous Networked System: General Recovering Process and Mean-Field Perspective

TL;DR

This work extends the classic SIS epidemic model by introducing a general recovering process (grp-SIS) that allows arbitrary recovery time distributions on a homogeneous network. By deriving a mean-field PDE for the infected-age distribution $\rho_I(t;\tau)$ and incorporating recovery via the survival function $F_0(t)$ and pdf $w(t)$, the authors show how non-Poisson recovery shapes transient and steady-state dynamics, infection-time distributions, and the epidemic threshold (which remains $\tau_c = 1/\langle k\rangle$). They provide analytic forms for $f_{T(\infty)}(\tau)$, $\mathbb{E}[T(\infty)]$, and the steady-state infection density, and validate the theory with node-centric simulations for exponential, lognormal, and power-law recoveries. The results highlight that recovery-time distributions significantly influence disease prevalence and infection durations, offering guidance for epidemiological control and network security applications. Potential extensions include arbitrary infection processes, quasistationary methods to address absorption, and accommodating heterogeneous networks.

Abstract

Although we have made progress in understanding disease spread in complex systems with non-Poissonian activity patterns, current models still fail to capture the full range of recovery time distributions. In this paper, we propose an extension of the classic susceptible-infected-susceptible (SIS) model, called the general recovering process SIS (grp-SIS) model. This model incorporates arbitrary recovery time distributions for infected nodes within the system. We derive the mean-field equations assuming a homogeneous network, provide solutions for specific recovery time distributions, and investigate the probability density function (PDF) for infection times in the system's steady state. Our findings show that recovery time distributions significantly affect disease dynamics, and we suggest several future research directions, including extending the model to arbitrary infection processes and using the quasistationary method to address deviations in numerical results.

Figures (7)

• Figure 1: Schematic diagram of the grp-SIS model. Red nodes represent infected nodes, green nodes indicate susceptible nodes, and gray nodes denote nodes whose states are arbitrary. The focal nodes for each process are highlighted with a darker color. Dashed edges represent connections where the nodes at both ends do not influence the ongoing process of the focal node, whereas solid directed edges from red to green nodes indicate the potential transmission of infection along that link. The diagram (a) illustrates the infection process of a susceptible node in the grp-SIS model. Initially, at $t = t_0$, a susceptible node (S) is surrounded by both infected (I) and susceptible (S) nodes, with directed edges indicating potential transmission from the infected nodes. The middle section depicts the probability density function (PDF) of the infection waiting time, with labeled points $\tau_1$, $\tau_2$, and $\tau_3$ representing individual times required for transmission attempts from different infected neighbors. At $t = t_0 + \tau_1$, the susceptible node becomes infected, altering the network's state and the potential interactions. The diagram (b) shows the recovery process of an infected node in the grp-SIS model. At $t = t_0$, an infected node (I) is connected to surrounding nodes with no significant interaction for recovery. The middle section includes a PDF representing the distribution of recovering waiting times, with the labeled point $\tau$ denoting the specific recovery time for the infected node. At $t = t_0 + \tau$, the node transitions to a susceptible state, depicted by a change in color, indicating recovery and potential reinfection by neighboring nodes.
• Figure 2: Schematic diagram of state transitions in nodes under different scenarios. The timeline illustrates the changes in node states over time, with green segments denoting the susceptible state and red segments indicating the infected state. Dashed lines demonstrate potential future states. In the timeline, black vertical lines mark the boundaries of the time interval $(t, t+\text{d}t]$ considered in the derivation, while yellow vertical lines indicate the actual moments of state transitions. Therefore, the timeline segments on both sides of a black vertical line have the same color, while those on both sides of a yellow vertical line have different colors, reflecting the state change at that moment. The top timeline (a) shows the most likely scenario when the current node state is susceptible. The middle timeline (b) depicts the most probable scenario when the current node state is infected. The bottom timeline (c) illustrates other possible scenarios when the current node state is infected, excluding the scenario shown in the middle timeline (b).
• Figure 3: Evolution of infection and susceptibility densities over time for three different recovering waiting time distributions. The three subfigures show the evolution of infection density $\rho_I(t)$ (red) and susceptibility density $\rho_S(t)$ (green) over simulation time $t$ on a uniform regular graph with degree 10. Subfigure (a) corresponds to an exponential distribution ($\beta = 0.26, \mu = 0.5$), (b) to a lognormal distribution ($\beta = 0.33, \mu = 0, \sigma = 1$), and (c) to a power law distribution ($\beta = 0.3, \lambda = 4, t_0 = 1$). The solid lines represent the average simulation results from 50 independent runs, while the dashed horizontal lines indicate the theoretically predicted steady-state densities for each case. The semi-transparent bands represent the numerical solution of the dynamical equation, with the blue band showing the infection density and the purple band the susceptibility density. The theoretical steady-state infection densities are approximately 0.808, 0.816, and 0.778 for the exponential, lognormal, and power law distributions, respectively. In the figure, it can be seen that the infection density starts at an initial value of 0.3 for all cases and stabilizes near the theoretical steady-state infection density (represented by the horizontal dashed line) around simulation time $t = 10$. Among the three distributions, the exponential distribution reaches the steady-state the fastest and stabilizes, followed by the lognormal distribution. The power law distribution, however, exhibits oscillations around the theoretical steady-state value before stabilizing. The numerical solution of the dynamical equation closely matches the simulation results, though the infection density in the simulation is slightly lower.
• Figure 4: Infection Density Evolution and Standard Deviation Comparison for Different Recovery Time Distributions This figure consists of three subplots, each showing the average infection density over 50 simulations over time, along with the corresponding 95% confidence interval band (shaded region). The x-axis represents the simulation time, while the y-axis represents the infection density, using a logarithmic scale. From top to bottom, the subplots correspond to simulations using exponential distribution, lognormal distribution, and power law distribution, with the following parameters: Exponential distribution ($\beta = 0.26, \mu = 0.5$), Lognormal distribution ($\beta = 0.33, \mu = 0, \sigma = 1$), and Power law distribution ($\beta = 0.3, \lambda = 4, t_0 = 1$). From the figure, it can be observed that the upper bound of the 95% confidence interval is approximately 0.07. Additionally, each subplot contains an embedded standard deviation plot in the upper right corner, where the x-axis represents simulation time and the y-axis represents the standard deviation of 50 simulation results at each time point. The results show that, except for the initial period, the standard deviation of the simulation results is mostly below 0.01, indicating low variability in the data.
• Figure 5: The steady-state infection density $\rho_I^\infty$ as a function of the transmission rate $\beta$ for three different recovering waiting time distributions. Subfigure (a) shows results for an exponential distribution ($\mu = 0.5$, with 25 values of $\beta$ ranging from 0.02 to 0.5), subfigure (b) for a lognormal distribution ($\mu = 0, \sigma = 1$, with 20 values of $\beta$ ranging from 0.03 to 0.6), subfigure (c) for a power law distribution ($\lambda = 4, t_0 = 1$, with 22 values of $\beta$ ranging from 0.03 to 0.66), and subfigure (d) presents a summary of the three distributions for the case of average degree $\langle k \rangle = 10$. Each subfigure includes results for regular graphs with degrees $k = 4$, $k = 6$, and $k = 10$. The solid lines denote average values over 50 independent simulations, and the vertical dashed lines indicate the theoretical epidemic thresholds, which are approximately 0.05, 0.067, and 0.061 for (a), (b), and (c) respectively. Subfigure (d) combines the results from subfigures (a), (b), and (c) for $\langle k \rangle = 10$ and includes 68% error bars for each data point, with vertical dashed lines indicating the epidemic thresholds. The plots show that the infection density remains close to zero before reaching the theoretical threshold and then starts increasing as $\beta$ increases, with the growth rate first slow, then fast, and finally slow again. For each distribution, the steady-state infection density is higher for graphs with a larger average degree. In subfigure (d), the error bars at each data point are short, never exceeding 0.04 in length. The error bars are longest during the fastest growth phase and shortest when the infection density growth slows down.

Theorems & Definitions (12)

• Theorem 1
• proof
• Remark 1
• Remark 2
• Remark 3
• Theorem 2
• proof
• Remark 4
• Corollary 1
• Remark 5