📊 epidemiology

State-dependent non-identifiability of the reproduction number under adaptive behavior: an empirical characterization from COVID-19 mobility

This paper demonstrates that the basic reproduction number ( $R_0$ ) is fundamentally non-identifiable from epidemic trajectories alone because it conflates pathogen biology with adaptive human behavior, yet empirical analysis of US COVID-19 mobility data reveals that while this behavioral component is structurally characterizable as state-dependent, its practical impact is modest due to a negative correlation between risk responsiveness and behavioral saturation across jurisdictions.

Original authors: Sanchez, F.

Published 2026-07-21

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Sanchez, F.

Original paper licensed under CC BY 4.0 (https://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 ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed.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 you are trying to predict how a crowd of people will move through a maze. In the world of disease modeling, scientists use a special number called the basic reproduction number, or $R_0$ (pronounced "R-naught"). Think of $R_0$ as a scorecard for a virus: it tells you how many new people one sick person will infect if everyone else in the crowd is healthy and has no idea a virus is coming. For a long time, scientists treated this number like a fixed property of the virus itself, similar to how a car's top speed is a fixed feature of its engine.

But here's the catch: people aren't cars. When a virus starts spreading, humans notice. They get scared, they stay home, they wear masks, and they stop hugging. This is called adaptive behavior. The big question for scientists has been: If people change their behavior because of the virus, does that change the virus's "score"? Does the $R_0$ number we calculate today still work if the virus changes tomorrow? This paper dives into that exact puzzle, using the massive amount of data from the COVID-19 pandemic to see if we can actually separate the virus's biology from our own human reactions.

The Great "Who Did It?" Mystery

Imagine you are watching a video of a car slowing down as it approaches a stop sign. You want to know: Did the driver hit the brakes because they saw the sign, or did the car just run out of gas? In the world of epidemics, the "car slowing down" is the number of new infections dropping. The "brakes" are people staying home (behavior), and the "gas" is the virus's natural ability to spread (biology).

For years, scientists tried to figure out how much of the slowdown was the virus losing steam and how much was people hitting the brakes. But they were stuck. They could see the car slowing down, but they couldn't see the driver's foot on the pedal. They had to guess.

This paper, however, had a superpower: Google Mobility Reports. During the pandemic, these reports tracked exactly how much people were moving around in real-time. It was like having a camera inside the car showing the driver's foot. The author, Fabio Sanchez, used this data to finally watch the "foot" move.

The Big Discovery: The "Magic Trick" of the Data

The paper's main finding is a bit of a mind-bender. It turns out that if you only look at the history of the epidemic (the "factual" path), you cannot tell the difference between two very different stories.

Story A: The virus is super strong, but people are terrified and staying home, which is why cases dropped.
Story B: The virus is weak, and people didn't change their behavior at all, which is why cases dropped.

Both stories fit the data perfectly. They produce the exact same curve of infections. This means that the "true" $R_0$ is not identified. You can't pin it down just by looking at the past. The number we usually calculate (the "apparent $R_0$ ") is actually a mix of the virus's strength and our fear, and the data can't tell us which part is which.

The paper calls this an "observational equivalence class." Imagine a magician pulling a rabbit out of a hat. If you only see the rabbit, you don't know if the magician actually had a rabbit, or if they used a trick. The data is the rabbit; the trick is the hidden mix of biology and behavior.

What Happens When We Change the Rules?

So, if we can't tell the difference in the past, does it matter? The paper says yes, but only if we try to predict the future.

The author ran simulations to see what would happen if the virus changed—say, if it became 40% more contagious (a "counterfactual" scenario).

If we assume people don't change their behavior (the "exogenous" view), the model predicts a huge, terrifying wave of infections.
If we assume people react to the new danger (the "endogenous" view), they stay home even more, and the wave is much smaller.

The gap between these two predictions is what the paper calls "what $R_0$ deletes." It's the part of the future that the standard $R_0$ number hides. The paper found that this hidden gap is state-dependent. It's biggest when the epidemic is at a "medium" level of severity.

If the epidemic is tiny, people don't react, so the gap is small.
If the epidemic is already a total disaster, people are already staying home as much as humanly possible (they are "saturated"), so they can't react any more, and the gap is small again.
The gap is biggest in the middle, where people are scared enough to react but haven't hit the "floor" of staying home yet.

The Surprising Twist: Why the Gap Was Small in Reality

You might think, "Okay, so the gap is real, but how big is it in the real world?" The author checked this across all 50 US states and Washington D.C.

Here is the surprising part: The gap was actually quite small in practice.

Why? Because of a weird coincidence in human behavior. The paper found that the two things needed to make the gap huge—high sensitivity to risk (people reacting strongly) and high "headroom" (people having room to move less)—never happened at the same time.

In places where people were very sensitive to the risk (they reacted strongly), they had already reduced their movement to the absolute minimum. They had no "headroom" left to react further.
In places where people still had "headroom" (they were still moving around a lot), they simply didn't react much to the risk.

It's like a thermostat that is either already turned all the way down, or it's broken and won't turn down at all. You never get the situation where the thermostat is sensitive and has room to go lower. Because these two conditions didn't overlap, the "hidden" error in the $R_0$ number turned out to be modest for the first wave of COVID-19.

The Takeaway for the Future

So, what does this mean for us?

Describing the past is safe: If you just want to say how bad the virus was during the time we watched it, the standard $R_0$ number is fine.
Predicting the future is tricky: If you want to know what happens if a new, stronger variant shows up, the standard $R_0$ is incomplete. It assumes people won't change their behavior, which is rarely true.
The "Peak" matters most: The paper suggests that while the total number of people who get sick (the "attack rate") might not change much based on behavior, the height and timing of the peak (how many people get sick at once) changes a lot. This is crucial for hospitals. If people react to a new wave, the peak might be lower and later, giving hospitals a chance to cope. If they don't react, the peak could be a crushing wall.

In short, the paper teaches us that the virus's "score" isn't just a number written in stone; it's a conversation between the virus and our fear. We can't always tell who is speaking louder just by looking at the history, but if the conversation changes, the outcome changes too. And luckily for the first wave of COVID-19, the conversation didn't get too chaotic because people either had nothing left to give or didn't have much to give in the first place.

Technical Summary: State-Dependent Non-Identifiability of the Reproduction Number under Adaptive Behavior

Problem Statement
The basic reproduction number ( $R_0$ ) is traditionally defined as a scalar representing the expected number of secondary infections in a fully susceptible population. However, this definition implicitly assumes that contact rates are fixed properties of the population. In reality, contact behavior is endogenous; individuals adapt their behavior in response to perceived epidemic risk. As noted by prior epidemiological-economic theory, $R_0$ is a reduced-form quantity that conflates pathogen biology with adaptive host behavior. Consequently, estimates derived under one behavioral regime may not describe transmission under another. The central empirical gap identified in this work is the inability to directly observe the behavioral response function (how contact changes with risk) and the resulting inability to decompose observed transmission into biological and behavioral components. While previous literature has used mobility data as an exogenous input to fit transmission models, this approach fails to address counterfactual questions (e.g., "what if a variant were more transmissible?") because it assumes behavior would follow the same path regardless of the altered epidemic trajectory.

Methodology
The study utilizes daily case, hospitalization, and death data from New York City and US states, paired with Google Community Mobility Reports as a proxy for contact. The methodology proceeds in four stages:

Behavioral Response Estimation: The authors estimate a behavioral response function, $R(x)$ , which maps perceived risk (7-day average reported deaths) to contact levels ( $C(t)$ ). They fit a saturating function of the form $C(t) = C_{min} + \frac{C_0 - C_{min}}{1 + x/K}$ using nonlinear least squares, where $x$ is lagged risk. This provides an empirical analog to the forward-looking optimization models proposed in economic epidemiology.
Observational Equivalence Construction: The authors construct a one-parameter family of models indexed by $\lambda \in [0, 1]$ $λ \in [0, 1]$ .
- $\lambda = 0$ represents a fully exogenous model where contact is frozen at the factual path.
- $\lambda = 1$ represents a fully endogenous model where contact re-responds to counterfactual risk.
- Intermediate values represent partial endogeneity.
  Crucially, for the factual epidemic trajectory ( $f=1$ ), all models in this family produce identical incidence curves and identical apparent $R_0$ values because the counterfactual risk term vanishes.
Non-Identifiability Proof: The authors prove that the parameter $\lambda$ (the fraction of endogeneity) is not identified from the observed data $(y(t), C(t))$ on the factual window. While the composite transmission rate $\beta C(t)$ is identified, the decomposition into biological susceptibility ( $\beta$ ) and behavioral response is not.
Counterfactual Analysis and Cross-Jurisdictional Panel: The authors simulate counterfactual scenarios (varying transmissibility $f$ ) to measure the "identified set span"—the divergence in outcomes (peak height, final size) across the $\lambda$ family. They then analyze a panel of 51 US jurisdictions to test whether the conditions required for a large behavioral correction (high risk-responsiveness and non-saturated behavior) co-occur.

Key Contributions

Empirical Characterization of the Response Function: The study provides a direct empirical estimate of the contact-risk response function using mobility data, confirming a monotone decline with saturation, consistent with forward-looking behavioral theories.
Proof of State-Dependent Non-Identifiability: The paper demonstrates that the decomposition of transmission into biology and behavior is fundamentally non-identifiable from a single epidemic trajectory. The "constant-contact" $R_0$ is merely one endpoint of an observational equivalence class that fits the data perfectly but yields divergent counterfactual predictions.
Characterization of "What $R_0$ Deletes": The authors quantify the behavioral content omitted by standard $R_0$ estimates. They find this "deleted" content is state-dependent and unimodal: it is negligible when the epidemic is mild (behavior remains at baseline) or severe (behavior is already saturated), and maximal at intermediate severity where behavior is on the steep part of the response curve.
Empirical Explanation for Modest Corrections: Through a cross-jurisdictional analysis, the paper reveals a negative correlation ( $r = -0.57$ ) between risk-responsiveness and behavioral non-saturation. Jurisdictions with high responsiveness had already saturated their mobility reductions, while those with headroom showed weak responsiveness. This structural confounding explains why the empirical correction to $R_0$ is modest in the specific context of the COVID-19 first wave.

Results

Behavioral Response: In NYC, contact declined from a baseline of 0.857 to a floor of 0.347 as perceived risk increased, with a half-saturation point at 7.3 deaths/day. The fit was robust across different risk proxies and time periods.
Apparent vs. Corrected $R_0$ : The "apparent" $R_0$ (assuming constant contact) and the "corrected" $R_0$ (accounting for baseline behavior) were nearly identical ( $\approx 2.7$ ) because early transmission occurred before significant behavioral adaptation. The bias is not in the scalar estimate itself but in the unidentifiable decomposition.
Counterfactual Divergence: Under a counterfactual transmissibility reduction ( $f=0.8$ ), the identified set for the peak hospitalization ratio spanned approximately 0.18 (a ~20% swing), while the final attack rate remained largely invariant (span < 2.5%). This indicates that adaptive behavior significantly reshapes the timing and height of the surge but has limited impact on the total cumulative incidence.
Cross-State Confounding: The analysis of US states showed that the "large-deletion" quadrant (high responsiveness + high non-saturation) was empty. Consequently, the empirical magnitude of the behavioral correction was small across the board.

Significance and Claims
The paper claims to reframe the warning that $R_0$ "hides" adaptive behavior. The authors argue that $R_0$ is not necessarily "biased" in its scalar value for the observed regime, but rather "incomplete" because its decomposition is unidentifiable. This incompleteness is invisible in-sample but consequential for out-of-sample counterfactuals.

The significance of the work lies in:

Structural Characterization: Providing a precise structural definition of the behavioral content omitted by $R_0$ , rather than a vague warning.
Practical Delimitation: Demonstrating that while the behavioral effect is real and structurally characterizable, it is empirically modest in the COVID-19 setting due to the specific coupling of responsiveness and saturation.
Operational Guidance: Offering rules of thumb for practitioners: description of the current wave is safe using exogenous models, but counterfactual planning (especially regarding peak surge capacity) requires acknowledging the identified set span or explicitly estimating the response function. The authors caution that the small empirical correction found here is specific to the first wave of a severe pandemic and may not hold for milder pathogens or different behavioral regimes.