Large deviations in non-Markovian stochastic epidemics

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This is an AI-generated explanation of a preprint that has not been peer-reviewed.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 you are trying to predict how a rumor spreads through a crowded room, or how a flu virus moves through a city. For decades, scientists have used a specific type of math to do this, assuming that people act like dice: every second, there is a fixed, random chance they will get sick or get better. This is called a "Markovian" model. It assumes that your past doesn't matter; only the present moment counts.

But in real life, people aren't dice. If you catch a cold, you don't have a random chance of recovering every second. You usually get sick, feel terrible for a few days, and then slowly get better. The time it takes to recover follows a pattern, not a random roll. Similarly, the time it takes to infect someone else isn't random; it depends on how long you've been sick and how many people you've met.

This paper, written by Matan Shmunika and Michael Assaf, says: "Stop treating epidemics like rolling dice. Let's treat them like real life with memory."

Here is a simple breakdown of what they did and why it matters, using some everyday analogies.

1. The "Memory" Problem

In the old models (Markovian), if you are sick, the model assumes you could get better right now or in 100 years with equal probability, as long as the average time is correct.

In the new model (Non-Markovian), the virus has a "memory."

The Analogy: Think of a popcorn kernel.
- Old Model: The kernel pops at a random time. It could pop in 1 second or 10 seconds. It doesn't care how long it's been heating up.
- New Model: The kernel heats up. It doesn't pop until it reaches a certain temperature. The longer it heats, the more likely it is to pop soon. The history of heating matters.

The authors realized that real infections work more like the popcorn kernel. The time between getting sick and infecting someone, or getting sick and recovering, follows a specific curve (like a bell curve or a skewed hill), not a flat random line.

2. The "Shape" of the Outbreak

The authors focused on a specific mathematical shape called the Gamma distribution. Think of this as the "personality" of the disease's timeline. They have a knob called $\alpha$ (alpha) that controls the shape of this timeline.

Low Alpha (Broad/Chaotic): Imagine a crowd where some people get better instantly, while others stay sick for weeks. The timeline is messy and unpredictable.
High Alpha (Narrow/Predictable): Imagine a crowd where everyone gets sick and recovers at almost the exact same time. The timeline is tight and synchronized.

The Big Discovery: Changing this "knob" ( $\alpha$ ) changes everything about the outbreak, even if the average speed of the disease stays the same.

In the SIR Model (Disease that ends): If the timeline is "messy" (low alpha), the outbreak can be much bigger or much smaller than predicted. It's like a fire: if the fuel burns at random intervals, the fire might die out quickly or rage out of control.
In the SIS Model (Disease that lingers, like the common cold): If the timeline is "messy," the disease is more likely to die out completely. If the timeline is "tight" (high alpha), the disease can stick around longer, but the number of sick people fluctuates wildly.

3. Why "Averaging" Doesn't Work

The paper shows a major flaw in how we usually predict epidemics. Scientists often try to fix the "dice" models by just changing the average speed. They say, "Okay, the virus is slower, so let's just make the dice roll slower."

The authors say: "No, that doesn't work."

The Analogy: Imagine you are driving to work.

Scenario A (Old Model): You drive at a steady 60 mph the whole time.
Scenario B (New Model): You drive at 100 mph for 5 minutes, then stop for 5 minutes, then drive 100 mph. Your average speed is still 60 mph.

If you are trying to predict when you will arrive, the average speed tells you the same thing for both. But if you are trying to predict traffic jams or accidents (the "large deviations" or rare events), the two scenarios are totally different. The stop-and-go traffic (Non-Markovian) creates different risks than the steady driving.

The paper proves that simply adjusting the average speed in old models fails to predict the rare, huge outbreaks or the sudden disappearance of a disease. You have to account for the pattern of the waiting times.

4. The "Ghost" of the Future

The authors used a clever mathematical trick (called WKB approximation, which sounds like a wizard's spell but is just a way to find the most likely path) to calculate the "action" of the disease.

Think of the disease spreading as a hiker trying to cross a mountain range.

The valley is the normal, steady state of the disease.
The mountain peak is the disease dying out completely.
The height of the mountain is how hard it is for the disease to die out.

The authors found that changing the "memory" (the $\alpha$ knob) changes the height of that mountain.

If the waiting times are very variable, the mountain gets lower, making it easier for the disease to die out (or for a massive outbreak to happen).
If the waiting times are very predictable, the mountain gets higher, making the disease harder to kill off.

Why Should You Care?

This isn't just abstract math. This framework helps us understand real-world pandemics like Flu or COVID-19.

We know that elderly people recover differently than young people.
We know that some people are "super-spreaders" who infect others quickly, while others take a long time.

By using this new "memory-aware" math, public health officials can get better estimates of:

How big an outbreak might get (Will it be a small wave or a tsunami?).
How long a disease will last (Will it disappear next month or stick around for years?).
When to intervene (Is it time to lock down, or is the disease about to die out on its own?).

The Bottom Line

The old way of modeling diseases was like watching a movie in black and white with a static camera. It gave you the general plot, but it missed the details.

This paper introduces color and motion. It tells us that the timing of events is just as important as the speed of events. By understanding the "rhythm" and "memory" of an infection, we can predict the wild, rare, and dangerous swings of an epidemic much better than before. It paves the way for smarter, more accurate predictions for the next pandemic.

1. Problem Statement

Traditional epidemiological models (SIR and SIS) often rely on the Markovian assumption, which posits that infection and recovery rates are constant and that waiting times (WTs) between events follow exponential distributions. However, empirical data for diseases like influenza and COVID-19 indicate that transmission and recovery processes are often time-dependent, with WTs better described by non-exponential distributions (e.g., Gamma, Weibull, or Log-normal).

While mean-field analyses of non-Markovian epidemics exist, there is a significant gap in understanding large deviations (rare events and fluctuations) in these systems. Specifically, it is unclear how the shape of the inter-event time distribution affects:

The distribution of final outbreak sizes in SIR models.
The quasi-stationary distribution (QSD) and mean time to extinction (MTE) in SIS models.
Whether rescaling Markovian rates can accurately capture the stochastic fluctuations of non-Markovian systems.

2. Methodology

The authors develop a theoretical framework combining Continuous-Time Random Walk (CTRW) formalism with the WKB (Wentzel-Kramers-Brillouin) approximation to study stochastic epidemics beyond the mean-field limit in well-mixed populations.

Model Setup: They analyze both SIR (Susceptible-Infected-Recovered) and SIS (Susceptible-Infected-Susceptible) models.
Waiting Time Distributions: They utilize Gamma distributions for infection and recovery times as a test case. The shape parameter $\alpha$ $α$ controls the memory of the system:
- $\alpha = 1$ : Exponential (Markovian).
- $\alpha < 1$ : Broad distribution (strong memory).
- $\alpha > 1$ : Narrow distribution (weak memory).
Derivation of Effective Dynamics:
- They derive asymptotic late-time master equations by calculating effective memory kernels ( $M_1, M_2$ ) from the Laplace transforms of the WT distributions.
- Using the Final Value Theorem, they obtain time-independent effective rates that govern the system's long-term behavior.
WKB Analysis:
- They apply the WKB ansatz ( $P \sim e^{-NS}$ ) to the master equations to derive a Hamilton-Jacobi equation.
- This allows them to calculate the action function $S$ , which determines the probability of rare events (large deviations).
- They compute the mean, variance, and full probability distribution for outbreak sizes (SIR) and disease lifetime/extinction (SIS).

3. Key Contributions

Derivation of Non-Markovian Master Equations: The paper provides a systematic method to derive effective late-time master equations with memory kernels for general WT distributions, moving beyond simple mean-field approximations.
Analytical Predictions for Large Deviations:
- SIR Model: Derives an implicit relation for the final outbreak size distribution and its variance, showing how the shape parameter $\alpha$ alters the risk and size of outbreaks.
- SIS Model: Derives the QSD and the Mean Time to Extinction (MTE) analytically, identifying how non-Markovianity shifts the metastable state and extinction barriers.
Validation of Rescaling Limitations: The authors demonstrate that while rescaling Markovian rates can sometimes match the mean behavior of non-Markovian systems, it fails to capture the fluctuations (variance and tail behavior) of the distributions.
Extension to Dual Non-Markovianity: The framework is extended to scenarios where both infection and recovery are non-Markovian, showing complex interplay between the two shape parameters.

4. Key Results

SIR Model (Outbreak Size)

Mean Outbreak Size: As the shape parameter $\alpha$ increases (narrower WT distribution), the typical infection time lengthens, causing the mean outbreak size to decrease.
Fluctuations: The standard deviation of the outbreak size increases with $\alpha$ .
Distribution Shape: The full distribution of outbreak sizes is highly sensitive to $\alpha$ . The WKB approximation accurately predicts the distribution when the action is large, but accuracy degrades as the mean outbreak size approaches zero.
Comparison: Simulations using a modified next-reaction method confirm the theoretical predictions for various $\alpha$ values.

SIS Model (Metastability and Extinction)

Endemic State: The effective basic reproduction number becomes $R_0^{eff} = \alpha R_0 (2^{1/\alpha} - 1)$ $R_{0}^{e f f} = α R_{0} (2^{1/ α} - 1)$ .
- For $\alpha < 1$ , a subcritical epidemic ( $R_0 < 1$ ) can still sustain a non-zero endemic state.
- For $\alpha > 1$ , the endemic prevalence decreases.
Quasi-Stationary Distribution (QSD): The variance of the infected fraction increases significantly as $\alpha$ increases. This implies that disease clearance becomes more likely in systems with narrow waiting time distributions (high $\alpha$ ) due to larger fluctuations.
Mean Time to Extinction (MTE): The MTE is exponentially dependent on the action barrier $S(0)$ . The paper provides an analytical expansion for $S(0)$ near the Markovian limit ( $\alpha \approx 1$ ), showing how deviations from Markovianity drastically alter the disease lifetime.

General Findings

Failure of Rescaling: A Markovian model with rescaled rates can reproduce the mean endemic state or outbreak size but fails to reproduce the variance and the tails of the distribution. This highlights that non-Markovianity introduces intrinsic noise structures that cannot be mimicked by simple rate adjustments.
Empirical Relevance: Using empirical Gamma shape parameters ( $1 < \alpha < 6$ ) for diseases like COVID-19, the authors show that the coefficient of variation (COV) of outbreak sizes can significantly exceed Markovian predictions, particularly when infection progression is synchronized (high $\alpha$ ).

5. Significance and Future Outlook

Theoretical Advancement: This work bridges the gap between non-Markovian stochastic processes and large deviation theory, providing a rigorous analytical toolset for epidemic modeling.
Public Health Implications: The findings suggest that ignoring the shape of waiting time distributions can lead to severe underestimation of outbreak variability and disease extinction risks. This is crucial for predicting the likelihood of eradication and the potential for large, rare outbreaks.
Future Directions: The authors propose extending this formalism to structured populations on degree-heterogeneous networks. They argue that combining non-Markovian dynamics with network topology will provide a more accurate representation of real-world epidemic thresholds, outbreak sizes, and the efficacy of intervention measures.

In summary, the paper establishes that non-Markovianity is not just a correction to rates but a fundamental driver of stochastic fluctuations in epidemic dynamics, necessitating new analytical frameworks to accurately predict rare but critical epidemic events.