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Variance of the $SIS$ Epidemic on Networks: A Diffusion Approximation

This paper develops a tractable variance approximation for Markovian $SIS$ epidemics on configuration-model networks by combining Gleeson's approximate master equation with a van Kampen system-size expansion, yielding a closed-form diffusion matrix that accurately predicts fluctuations around the mean-field trajectory across various network topologies.

Original authors: Lucija Nora Farkaš, Sebastian Morel Balbi, Hrvoje Štefančić, István Zoltán Kiss, Vinko Zlatić

Published 2026-07-07

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CC BY 4.0

Original authors: Lucija Nora Farkaš, Sebastian Morel Balbi, Hrvoje Štefančić, István Zoltán Kiss, Vinko Zlatić

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine a disease spreading through a city. Scientists have long been good at predicting the average path of an epidemic: "On average, 10% of people will get sick by Tuesday, and 20% by Wednesday." This is like predicting the average height of a crowd; it gives a solid, reliable center point.

But in the real world, no two outbreaks are exactly the same. Sometimes the disease fizzles out early; other times, it explodes unexpectedly. This paper is about predicting the size of the wobble around that average. It asks: How much can the actual number of sick people swing up or down from the prediction?

Here is a breakdown of what the authors did, using simple analogies:

1. The Problem: The "Average" Lie

Think of the average prediction as a smooth, straight road. In a perfect, infinite world, everyone would drive exactly on that road. But in a real, finite city (a network of people), drivers swerve.

The Old Way: Scientists had a great map for the "SI" model (where once you get sick, you stay sick forever, like a zombie). They knew exactly how much the "zombies" would wobble off the road.
The New Challenge: Most real diseases are "SIS" (Susceptible-Infected-Susceptible). You get sick, you recover, and you go back to being healthy (and potentially getting sick again).
The Mess: When you recover, you don't just disappear; you re-enter the crowd with a specific history. Your neighbors might still be sick, or they might have just recovered. This "history" breaks the clean math used for the zombie model. The authors say, "We need a new way to measure the wobble for diseases that come and go."

2. The Solution: A "Traffic Flow" Simulation

The authors built a new mathematical tool to estimate this wobble. They combined two existing ideas:

The "Master Plan" (AME): This is a detailed map that tracks not just how many people are sick, but who they are connected to. It knows that a person with 10 friends is different from a person with 2 friends.
The "Noise" (Langevin/Lyapunov): This is the math that measures the chaos. Imagine a river flowing smoothly (the average). The "noise" is the ripples and splashes caused by individual raindrops (individual infections and recoveries).

The authors created a simplified "traffic report" that only tracks three things:

How many people are healthy.
How many "healthy-sick" handshakes are happening (edges between a healthy person and a sick one).
How many "healthy-healthy" handshakes are happening.

By tracking just these three numbers, they could calculate the "ripples" (variance) without needing to simulate every single person in the city.

3. The Big Guess (The "Shuffle" Analogy)

To make the math work, the authors had to make one big assumption.

The Scenario: Imagine a person who just got better. In the old "zombie" math, their neighbors were a fixed, known group. But in this "recovery" game, their neighbors might have gotten sick or healed since the last time they checked.
The Assumption: The authors guessed that because people recover and get re-infected often enough, the specific history of who was sick next to you gets "shuffled" and forgotten. It's like shuffling a deck of cards: after enough shuffles, you can't remember where the Ace of Spades started; you just know it's somewhere in the deck.
The Result: They treated the neighborhood of a healthy person as a random draw from the current pool of the whole city, rather than a specific history. This is an approximation, not a perfect law, but it makes the math solvable.

4. Did It Work? (The Test Drive)

The authors tested their new "wobble calculator" against massive computer simulations (running thousands of fake epidemics to see what actually happened).

The Good News: For most networks (like a city where everyone has a similar number of friends), their math matched the simulations almost perfectly. It correctly predicted when the "wobble" would be biggest (usually right when the epidemic is peaking) and how big it would get.
The Bad News: The math got a little sloppy in two specific situations:
1. Very Sparse Networks: If people have very few friends (like a small village where everyone knows everyone), the "shuffling" assumption breaks down because one neighbor's history matters too much.
2. Super-Hubs: If the network has "super-spreaders" (people with hundreds of friends), the math struggled to predict the exact height of the wobble. The "random shuffle" idea doesn't work well when a few specific people dominate the traffic.

5. Why This Matters

The authors aren't saying this will cure diseases or tell doctors exactly what to do tomorrow. Instead, they are providing a faster, smarter way to guess the uncertainty.

Old Way: To know the uncertainty, you had to run a computer simulation 1,000 times to see the spread. This takes a long time.
New Way: Their math solves a tiny set of equations (like a 3x3 grid) in a split second to give you a very good estimate of that same uncertainty.

In short: They built a shortcut to measure how much an epidemic might surprise us, acknowledging that while the "average" path is predictable, the "wobble" depends on the messy, specific connections between people. Their shortcut works great for most cities, but needs a little more work for very small villages or cities with a few massive super-spreaders.

Technical Summary: Variance of the SIS Epidemic on Networks via Diffusion Approximation

Problem Statement
Mathematical models of epidemics on networks typically rely on Functional Laws of Large Numbers (FLLNs) to describe the mean-field trajectory of infection spread as the network size $N$ grows. While these deterministic limits (Ordinary Differential Equations) accurately predict the average behavior, they fail to capture the stochastic fluctuations inherent in finite populations. These fluctuations are critical for quantifying forecast uncertainty, estimating extinction probabilities, and designing risk-aware control strategies.

Crucially, the variance of an epidemic is governed by different network statistics than the mean. While the mean prevalence in early growth depends on the first two moments of the degree distribution, the variance depends on the first three moments (including skewness). Although a rigorous Functional Central Limit Theorem (FCLT) exists for the Susceptible-Infected (SI) process on configuration-model graphs, no analogous closed-form variance framework exists for the Susceptible-Infected-Susceptible (SIS) model. The SIS process is more complex because recovery reintroduces vertices into the susceptible pool with partially known, history-dependent neighborhoods, breaking the "clean" neighborhood distribution assumptions that allow for exact FCLT derivations in the SI case. Existing stochastic analyses of SIS have largely focused on the quasi-stationary distribution in the endemic regime rather than the full temporal evolution of fluctuations.

Methodology
The authors develop a tractable variance approximation for the Markovian SIS process on configuration-model graphs by combining two frameworks:

Approximate Master Equation (AME): Utilizing Gleeson's framework to track the deterministic evolution of degree- and neighborhood-resolved state variables (specifically, the fractions of susceptible and infected vertices with $k$ degree and $m$ infected neighbors).
Van Kampen System-Size Expansion: Applying a diffusion approximation where the stochastic count vector $X(t)$ is decomposed into a deterministic mean trajectory $\phi(t)$ and a fluctuation term $\zeta(t)$ scaled by $\sqrt{N}$ :
$X(t) = N\phi(t) + \sqrt{N}\zeta(t)$

The authors define a reduced three-dimensional state vector tracking the counts of susceptible vertices ( $X_S$ ), susceptible-infected edges ( $X_{SI}$ ), and susceptible-susceptible edges ( $X_{SS}$ ). The evolution of the covariance matrix $C(t)$ of the fluctuations is governed by a linear matrix differential equation (Langevin/Lyapunov equation):
$\dot{C} = JC + CJ^T + B$
where $J$ is the Jacobian of the deterministic drift and $B$ is the diffusion (noise) matrix derived from the rates of infection and recovery events.

Key Approximations and Contributions
To close the system and express the drift and diffusion matrices solely in terms of the reduced variables ( $X_S, X_{SI}, X_{SS}$ ), the paper introduces specific approximations:

Neighborhood Sampling Closure: The authors approximate the distribution of infected neighbors for a susceptible vertex, $P(m|S, k)$ , using a hypergeometric expression. This assumes that due to frequent recovery and reinfection events, the local correlations of a susceptible vertex's neighborhood decorrelate from its infection history, effectively behaving as a fresh random draw from the global pool of half-edges. This mirrors the exact condition for SI but is an approximation for SIS.
Structural Factors ( $\kappa$ ): The closure introduces structural factors $\kappa^{(r)}_S$ and $\kappa^{(r)}_I$ , which represent normalized expected numbers of $r$ -star motifs centered on susceptible or infected vertices. These factors explicitly incorporate higher-order moments (up to the third moment) of the degree distribution into the variance calculation.
Neglect of Derivatives: The authors treat the susceptible-centered wedge factor $\kappa^{(2)}_S$ as a time-dependent coefficient supplied by the AME solution, neglecting its derivatives with respect to the reduced variables in the Jacobian. Validation on SI processes suggests this derivative contribution is negligible.

Results
The proposed AME-Langevin approximation was validated against Gillespie stochastic simulations across various network topologies and epidemiological parameters:

Poisson Networks: The approximation accurately reproduces the transient peak and long-time variance of the susceptible fraction across a wide range of infection rates ( $\beta$ ). Discrepancies are minimal, with slight underestimation near the epidemic threshold.
Network Size: The method performs well for network sizes $N=500$ to $N=4000$ , with absolute agreement improving as $N$ increases, consistent with the asymptotic nature of the linear-noise approximation.
Regular Networks: For $k$ -regular networks, the approximation is exact regarding the $\kappa^{(2)}_S$ derivative term. Agreement is generally good, though the theory slightly overestimates the transient variance peak for sparse networks ( $k=4$ ), likely due to stronger history-dependent correlations in low-degree environments.
Power-Law (Heterogeneous) Networks: In networks with high degree heterogeneity (scale-free), the method captures the timing of the variance peak but fails to accurately predict its height. The authors attribute this to the "crudeness" of the neighborhood-sampling closure when hubs are involved, as infection/recovery events on hubs generate correlations not well represented by global pool sampling.

Significance and Claims
The paper claims to provide the first tractable, time-dependent variance approximation for the SIS process on configuration-model graphs that extends beyond the early growth phase.

Information Content: The framework explicitly incorporates third-order degree moments (via $\kappa^{(3)}_S$ ) into the diffusion matrix, addressing the gap where mean-field models discard structural information relevant to variance.
Computational Efficiency: Solving the resulting $3 \times 3$ Langevin ODE system is computationally efficient, offering a significant advantage over generating thousands of Gillespie simulations to estimate variance empirically.
Limitations: The authors modestly note that the method is an approximation, not a rigorous FCLT. Its accuracy degrades in regimes where the "decorrelation" assumption fails: specifically in sparse, low-degree networks and in highly heterogeneous networks with hubs, where local history dependence is strong.

The work positions itself as a useful intermediate description between deterministic mean-field theory and large ensembles of direct stochastic simulations, providing a tool for quantifying uncertainty in epidemic forecasts on networks.