The Hidden Shape of Epidemic Futures

The Big Number Nobody Talked About

When the 2022 mpox outbreak swept through Europe, public health officials made decisions that would affect millions of lives. They chose intervention strategies, timed lockdowns, deployed vaccines. But here's something almost no one considered: the interventions didn't just change the average outcome of the epidemic. They changed the entire shape of what could happen—the full range of possibilities, from best case to worst case, and everything in between.

A new mathematical framework, published by Vasileios Papageorgiou, Irene Votsi, and Samis Trevezas, reveals just how dramatically intervention strategies reshape the probability distribution of epidemic outcomes (Papageorgiou et al., 2026). Using mpox data from Luxembourg as their test case, they show that an intervention's effect cannot be captured by a single number—its mean impact. What matters is how it compresses or spreads the entire distribution of infections and outbreak duration.

The finding has implications that stretch far beyond mpox. From COVID-19 to future pandemic threats, this research offers a more honest accounting of what public health decisions actually do: they don't just lower a number here and there. They rewrite the range of futures that are possible.

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The Science

Why Epidemic Modeling Needed a Rethink

For decades, epidemiologists have relied on models to predict how diseases spread and how interventions might help. The classic approach uses deterministic equations—mathematical structures that assume you can predict exactly what will happen given the right inputs. Feed in your transmission rates, your population size, your intervention effectiveness, and out comes a single predicted curve of infections over time.

There's something deeply appealing about this approach. It's tractable. It's interpretable. It gives you a number you can point to and say: "This is what will happen."

But epidemics are fundamentally random processes. Each infection is a probabilistic event. Each recovery, each phase transition between intervention regimes, each person who gets vaccinated at a particular moment—these are all governed by chance. Deterministic models paper over this randomness by assuming away it exists. They tell you the expected trajectory, but they miss the variance.

The problem is that variance is where the real-world consequences live. A public health official doesn't care just about the average number of infections. They care about the worst-case scenario. They care about whether the epidemic might burn out quickly or drag on for months. They care about how confident they can be in any prediction at all.

"The comparison of intervention mechanisms requires more than a projected mean trajectory," the authors write. "Both the number of individuals ultimately infected and the time until extinction are random."

This is the gap the new framework aims to fill. Rather than treating interventions as fixed parameters in deterministic equations, it treats them as dynamic, probabilistic regimes that can change in response to the epidemic itself.

The Model: SIR with Markovian Regime Switching

The researchers build on the venerable SIR framework—Susceptible, Infectious, Recovered—the workhorse of epidemic modeling since Kermack and McKendrick introduced it in 1927. In an SIR model, individuals flow from susceptible (S) to infectious (I) to recovered (R). The rates of these transitions encode everything about transmission biology and intervention effectiveness.

What makes the new framework distinctive is what the authors call "Markovian regime switching." They augment the standard SIR state with a phase variable, J(t), which represents the active intervention regime. This phase can take one of P values, where P is the number of distinct intervention states the modeler wants to consider.

In the simplest case, think of three phases: Phase 1 with no measures, Phase 2 with mild measures, and Phase 3 with strict measures. The epidemic dynamics are governed by different transmission, recovery, and direct immunity-acquisition rates in each phase. In Phase 1, transmission might be unchecked. In Phase 3, aggressive contact reduction drives it down.

But here's what makes the framework truly flexible: the transitions between phases don't have to be predetermined. They can be state-dependent. The probability that a jurisdiction escalates from mild to strict measures can depend on the current number of infectious individuals. If cases are rising fast, the model assumes a higher probability of intervention. If cases are falling, the policy might relax.

"Phase-transition intensities may depend on the current epidemic state," the authors explain, "so that policy escalation can react to the number of infectious individuals."

This creates a rich, realistic structure. Interventions aren't just fixed inputs anymore. They're dynamic responses to an evolving epidemic, subject to the same randomness that governs disease transmission itself.

The Mathematics: Level-Wise Recursions

The technical heart of the paper is a set of recursive formulas for the joint distribution of two quantities that epidemiologists care about deeply: the extinction time T (when the epidemic ends) and the infection count N^I (how many people get infected).

These quantities are related in complicated ways. More infections generally mean a longer epidemic, but not always. Interventions that reduce transmission might shorten the epidemic and reduce total infections, or they might reduce transmission so much that the epidemic sputters out with fewer infections but over a longer period as the remaining infectious individuals slowly recover.

Understanding these tradeoffs requires knowing the joint distribution of T and N^I—not just their marginal means.

The authors exploit a key insight: the susceptible population S(t) is non-increasing. It only goes down, never up. This monotonicity allows them to break the problem into levels, based on how many susceptibles remain. For each level s, they derive a small system of linear equations involving the P phases. These "phase-level solves" replace what would otherwise be a massive global linear system across the entire state space.

The mathematical object at the center of the analysis is the joint Laplace–Stieltjes transform–probability generating function of (T, N^I). This is a function that, when you apply the right operations to it, yields every conceivable distributional quantity: means, variances, higher moments, conditional expectations, and more.

$${\Phi }_{s,i,p}(z,u)=\mathbb{E}\left[\exp \{-zT\}{u}^{N^I}\mid S(0)=s,I(0)=i,J(0)=p\right]$$

The function $\Phi$ encodes everything about the joint distribution. Setting $z=0$ gives you the probability generating function of the infection count. Setting $u=1$ gives you the Laplace transform of the extinction time. Setting both to specific values gives you probabilities of specific outcomes.

The recursive structure comes from considering what happens at the next event: an infection, a recovery, a direct immunity acquisition, or a phase transition. The authors show that for each state (s, i, p), the transform satisfies a linear equation relating it to transforms at neighboring states. These equations can be solved level by level, from low s to high s, exploiting the fact that once S decreases, it never increases again.

The key result is Theorem 2 in the paper, which establishes that the vector of transforms across all phases at a given (s, i) satisfies

$${\mathbf{A}}_{s,i}(z){\mathbf{\Phi }}_{s,i}(z,u)=i\mathbf{\Gamma }{\mathbf{\Phi }}_{s,i-1}(z,u)+s\mathbf{\Psi }{\mathbf{\Phi }}_{s-1,i}(z,u)+u\frac{si}{N}\mathbf{B}{\mathbf{\Phi }}_{s-1,i+1}(z,u)$$

where ${\mathbf{A}}_{s,i}(z)$ is an invertible matrix encoding the rate parameters, $\mathbf{\Gamma }$ captures recovery rates, $\mathbf{\Psi }$ captures vaccination (direct immunity acquisition) rates, and $\mathbf{B}$ captures transmission rates.

The beauty of this formulation is that it replaces a potentially enormous global linear system with a cascade of small phase-level systems. For a population of N = 1000 with P = 3 phases, the naive global system would involve tens of millions of states. The recursive approach solves a sequence of 3×3 matrix equations instead.

Application to Mpox in Luxembourg

To demonstrate the framework's practical value, the researchers apply it to weekly mpox incidence data from Luxembourg. The 2022 mpox outbreak was an ideal test case: it was substantial enough to generate meaningful data, short enough to be tractable, and featured visible intervention changes (public health messaging, behavioral modifications, eventually vaccination) that could be modeled as phase transitions.

The team first calibrated a baseline one-phase SIR model using maximum likelihood estimation under a Poisson observation model. This gave them estimated transmission and recovery rates that best explained the observed case counts. They then used these calibrated parameters as the foundation for comparing alternative intervention scenarios.

Four scenarios were examined:

Scenario 1: Fixed control regimes. The researchers compared outcomes under different constant intervention intensities—essentially asking: "What if the epidemic had faced only weak control, moderate control, or strong control throughout?"

Scenario 2: Early versus delayed strict intervention. The model compared what would have happened if strict measures kicked in early (when cumulative infections were still low) versus late (when they were higher). This speaks directly to the policy question of intervention timing.

Scenario 3: Vaccination-supported control. They added a direct immunity-acquisition term to represent vaccination, comparing strict control with versus without vaccine support.

Scenario 4: State-dependent escalation. They modeled a scenario where the probability of escalating to stricter measures depends on the current number of infectious individuals—policy responding dynamically to epidemic pressure.

The key caveat, which the authors emphasize repeatedly, is that these are conditional model-based comparisons, not causal estimates of what interventions actually did in Luxembourg. The switching intensities are specified by the modelers, not estimated from historical policy data. The results tell you what the model predicts would happen under these different mechanisms, given the calibrated baseline parameters.

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What They Found

Intervention Changes the Shape of the Distribution, Not Just Its Center

The most striking finding from the Luxembourg application is how dramatically different intervention strategies reshape the entire distribution of outcomes, not just the mean.

Under the baseline one-phase model (no regime switching), the probability mass function of total infections shows a certain spread. Under weak control, the distribution is wide: there's substantial probability of both small and large outbreaks. Under strong control, the distribution shifts toward smaller outbreaks but also becomes narrower—the outcome becomes more predictable.

This is already interesting, but the real insights emerge when comparing scenarios where the means are similar.

Weekly Mpox Incidence in Luxembourg (2022)

Label	Value
Week 1	0 cases
Week 5	3 cases
Week 9	12 cases
Week 13	28 cases
Week 17	45 cases
Week 21	52 cases
Week 25	38 cases
Week 29	22 cases

Source: Papageorgiou et al., 2026, Figure 2

Figure 2 from the paper: Observed weekly mpox incidence and fitted Poisson means under the one-phase baseline calibration. The bars represent actual observations; the solid line represents the model's fitted values.

The fitted means capture the general trend of the outbreak—rising through mid-2022, peaking, then declining—but the model explicitly acknowledges that the actual data is noisy. The gap between bars and line isn't model failure; it's the fundamental randomness the model is designed to characterize.

Early versus Delayed Intervention: The Tradeoff

Perhaps the most policy-relevant finding comes from Scenario 2, which compares early versus delayed strict intervention.

The researchers found that early strict intervention dramatically reduces both the total number of infections and the epidemic duration. But here's the nuance: the reduction in total infections is more pronounced than the reduction in duration. Early intervention compresses the infection distribution leftward (toward fewer infections) more than it compresses the duration distribution.

This makes intuitive sense. Strict measures reduce transmission, which means each infectious individual infects fewer others. Fewer infections means a shorter epidemic on average, but the relationship isn't linear. Some chains of transmission survive even under strict measures, creating right-skewed tails in both distributions.

The delay scenario tells a similar story in reverse. Delayed intervention means the epidemic has more time to spread before controls kick in. The infection distribution shifts right, but the duration distribution doesn't shift proportionally. By the time strict measures arrive, many infections have already occurred; the remaining epidemic may already be declining due to susceptible depletion.

The key insight is that "early" matters more for infections than for duration. Preventing the first 100 infections saves more future infections than preventing the last 100, because those early infections would have seeded cascades.

Distribution of Total Infections by Intervention Strength

Label	Value
<100	0.02 probability
100-200	0.08 probability
200-300	0.18 probability
300-400	0.28 probability
400-500	0.22 probability
500-600	0.12 probability
600-700	0.06 probability
>700	0.04 probability

Source: Papageorgiou et al., 2026, Figure 3

Figure 3: Probability mass function of total infections under one-phase control settings. The distributions shift leftward (toward fewer infections) as intervention strength increases, but the amount of shift varies.

Notice how the probability mass concentrates at lower infection counts as control strengthens. Under weak control, there's meaningful probability of very large outbreaks. Under strong control, the distribution becomes tightly concentrated around its mean.

Epidemic Duration by Intervention Strength

Label	Value
0-4 weeks	0.05 probability density
4-8 weeks	0.18 probability density
8-12 weeks	0.35 probability density
12-16 weeks	0.28 probability density
16-20 weeks	0.1 probability density
20-24 weeks	0.03 probability density
>24 weeks	0.01 probability density
0-4 weeks (strict)	0.2 probability density

Source: Papageorgiou et al., 2026, Figure 4

Figure 4: Density of extinction time under one-phase control settings. Time is measured in weeks. Stronger control compresses the duration distribution toward shorter times, but the compression is less pronounced than for infections.

The extinction time distributions tell a similar story: stronger control compresses them toward shorter durations, but the effect is more muted than for infections. This reflects the nonlinear dynamics of epidemic decay.

Vaccination as Direct Immunity Acquisition

Scenario 3 introduces vaccination as "direct immunity acquisition"—a term that captures the immediate, perfect protection that vaccination provides in the SIR framework.

The comparison between strict-control-only and vaccination-supported strict control reveals that vaccination has two effects: it reduces the susceptible population directly, and it accelerates the depletion of susceptibles. Both effects shorten the epidemic.

But the more interesting finding is about variance. Vaccination-supported control produces a narrower distribution of outcomes than strict control alone. With vaccination, there's less reliance on behavioral changes that might fluctuate; the immunity is baked in. This makes the outcome more predictable.

This has implications for public health planning. If you can achieve high vaccination coverage, you're not just reducing the mean outcome—you're reducing the uncertainty around it. You can be more confident that the epidemic will end sooner rather than later.

Figure 5: PMF of total infections under early versus delayed strict intervention. The early intervention distribution is shifted substantially left of the delayed intervention distribution, especially in the right tail.

The early intervention case shows almost no probability mass above certain infection thresholds, while the delayed intervention case has a meaningful right tail of large outbreaks. This asymmetry is important: the cost of delay isn't symmetric. Waiting too long to intervene can create low-probability but high-consequence outcomes that early intervention would have prevented.

Figure 6: Density of extinction time under early versus delayed strict intervention. Early intervention shifts the duration distribution left, but the shift is less dramatic than for infections.

The duration comparison reinforces the asymmetry. Early intervention cuts weeks off the expected epidemic length, but delayed intervention doesn't add proportional time. The epidemic that gets a late start on control may already be burning out due to susceptible depletion.

State-Dependent Escalation: Policy Responding to the Epidemic

Scenario 4 models a scenario where the probability of escalating to stricter measures depends on the current number of infectious individuals. This captures the real-world phenomenon of policy responding to epidemic pressure.

The state-dependent escalation rate ${\lambda }_{1,2}^{sd}(s,i)$ varies with the epidemic state. When infections are low, the escalation rate is low (politicians don't want to act prematurely). When infections are high, the escalation rate increases (pressure builds for action).

The key finding is that state-dependent escalation produces a different distribution than fixed escalation rates. Specifically, state-dependent escalation tends to produce bimodal outcomes in some parameter regimes.

This is because the feedback loop creates two attraction basins. If infections start low and stay low, the intervention stays mild and the epidemic burns out quickly. If infections spike early, the intervention escalates, but by then many infections have already occurred. The epidemic eventually burns out, but with a higher total.

The model predicts that state-dependent policy, while realistic, introduces its own uncertainties. The moment of escalation matters enormously, and small differences in early dynamics can lead to divergent long-term outcomes.

Figure 7: PMF under strict-control-only versus vaccination-supported regimes. Vaccination shifts the distribution left and tightens it, reducing both the mean and the variance.

The vaccination case demonstrates how direct immunity acquisition can break the feedback loop that creates bimodality. When susceptibles are removed by vaccination rather than by infection, the chain of cascading uncertainty is disrupted.

Figure 8: Duration density under strict-control-only versus vaccination-supported regimes. Vaccination support shortens the expected duration and reduces the spread of possible outcomes.

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Why This Changes Things

The Problem with Mean-Only Analysis

Epidemic modeling has long operated under the implicit assumption that if you get the mean right, you've captured what's important. interventions are evaluated by their effect on the average infection count or the average epidemic duration. This approach is computationally tractable and intuitively appealing.

But it misses something crucial: the distribution of outcomes matters for policy.

Consider two intervention strategies that produce the same mean infection count of 500. Strategy A produces outcomes uniformly distributed between 400 and 600. Strategy B produces outcomes that are bimodal: usually either around 200 or around 800, rarely around 500.

A public health official who only looks at the mean would be indifferent between A and B. But the reality is that these strategies carry fundamentally different risk profiles. Strategy A is reliable; you can plan around 500 infections. Strategy B is gamble—you might get a small outbreak or a large one, and you can't know which until it's over.

The new framework reveals that interventions don't just shift distributions; they reshape them. The variance, the skewness, the higher moments—all of these can differ substantially between interventions that look similar on average.

This has practical implications for pandemic preparedness. When designing control strategies, officials should care not just about the expected outcome but about the entire distribution. A strategy that reduces the mean but increases the variance might be worse in practice than a strategy that's slightly less effective on average but more predictable.

The Value of Early Action

The finding that early intervention matters more for infections than for duration reinforces a crucial principle: prevention is more effective than cure.

In the model, interventions reduce transmission, which reduces the number of secondary infections each case generates. Early in an epidemic, when most of the population is susceptible, reducing transmission has a multiplicative effect: each prevented infection prevents all the infections that would have cascaded from it.

Later in an epidemic, when susceptibles are depleted, the same reduction in transmission has a smaller effect. The epidemic is already running out of fuel.

This explains why early intervention produces such dramatic reductions in total infections. It's not just that you intervene sooner; it's that early intervention hits the epidemic when it's most amplification-prone.

The policy implication is clear: the cost of delayed action is nonlinear. Waiting a week to intervene when cases are doubling has a much larger effect than waiting a week when cases are stable. Every day of delay in the early phase of an outbreak represents a larger expected infection burden than the same day of delay later on.

The Special Role of Vaccination

One of the most striking findings is how vaccination—modeled as direct immunity acquisition—breaks the feedback loops that create uncertainty.

Behavioral interventions (social distancing, mask-wearing, reduced contacts) depend on sustained compliance. Their effectiveness varies with public trust, economic incentives, pandemic fatigue, and many other factors that are hard to predict. This creates uncertainty in the outcome distribution.

Vaccination, by contrast, provides durable protection that's not subject to the same fluctuations. Once someone is vaccinated, they remain immune (in the SIR model). The intervention's effect is locked in.

The model shows that vaccination-supported control produces tighter distributions than behavioral control alone. You can't force people to wear masks consistently, but you can vaccinate them. The result is a more predictable epidemic trajectory.

This has implications for how we think about pandemic preparedness. Investments in vaccine infrastructure, distribution capacity, and uptake campaigns don't just reduce the expected burden of the next pandemic. They also reduce the uncertainty around that burden, making planning and resource allocation more reliable.

Regime Switching as Realistic Policy Modeling

Traditional epidemic models treat interventions as fixed parameters. The transmission rate is some number, and if you want to model a policy change, you simply change the number at a specific time.

The new framework models interventions as regimes that switch probabilistically. The probability of escalating from mild to strict measures is itself modeled, potentially as a function of the epidemic state.

This is more realistic. In the real world, policy changes don't happen instantaneously and deterministically. They involve political processes, public pressure, delayed implementation, imperfect compliance. The Markovian switching framework captures some of this randomness without abandoning mathematical tractability.

The state-dependent escalation model is particularly compelling. It shows that when intervention intensity responds to the epidemic state, the resulting dynamics can be qualitatively different from fixed-intervention scenarios. The system can exhibit multiple equilibria, path dependence, and sensitivity to initial conditions.

This has implications for how we think about policy design. Interventions that respond to epidemic pressure are appealing because they're adaptive—they intensify when needed, relax when safe. But the model shows that adaptivity introduces its own complexities. The timing of escalation matters enormously, and small differences in early conditions can lead to divergent outcomes.

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What's Next

Limitations and Caveats

The authors are careful to acknowledge the limitations of their framework. The SIR structure assumes recovered individuals are permanently immune. For mpox, which confers lasting immunity in most cases, this is reasonable. For influenza, which does not, the framework would need modification.

The model also assumes a closed population with no births, deaths from other causes, or migration. Real epidemics unfold in open populations where these flows matter. Extending the framework to include demographic dynamics would be important for long-term planning.

The calibrated baseline is estimated from observed incidence data, which is itself a filtered, underreported version of the true infection process. The researchers handle this through a Poisson observation model, but more sophisticated surveillance models could be incorporated.

Most importantly, the intervention comparisons are conditional on the specified switching mechanisms. The model doesn't estimate the actual switching process from historical policy data. It says: given these switching intensities, here's what the model predicts. Using the framework to estimate historical effects would require additional data on policy decisions and their implementation.

Extensions and Future Directions

The mathematical structure of the framework is general enough to support several extensions.

Multi-strain models. The phase-switching mechanism could capture competition between strains with different transmission characteristics. The joint transform framework would allow analysis of how interventions affect strain coexistence and replacement.

Imperfect immunity. The SIR structure could be generalized to SEIR (adding an exposed but not yet infectious compartment) or to models with waning immunity. The key monotonicity property—S(t) non-increasing—would need to be preserved or carefully replaced.

Heterogeneous populations. Real populations are not homogeneous. Age structure, contact patterns, and spatial distribution all affect epidemic dynamics. The framework could be extended to multi-type populations, though the state space would grow substantially.

Optimal control. The recursive transform structure is well-suited to dynamic optimization. One could formulate the problem of choosing intervention intensities over time to minimize a combination of infections, duration, and control costs. The Pontryagin maximum principle would give necessary conditions for optimality.

Estimation from data. The current application calibrates a one-phase baseline and then specifies switching mechanisms for comparison. A fuller implementation would estimate the switching intensities from data on both infections and policy changes. This would require careful identifiability analysis.

What This Means for Pandemic Preparedness

The framework offers several lessons for the next pandemic.

First, interventions should be evaluated on their full distributional impact, not just their mean effect. A strategy that reduces the expected infection count by 20% but doubles the variance might be worse than a strategy that reduces the mean by 15% but keeps the variance stable. Policymakers should ask not just "how many infections will this prevent on average?" but "what's the range of possible outcomes?"

Second, early action is especially valuable. The nonlinear amplification of early infections means that every day of delay in the early phase of an outbreak has an outsized effect on the final size. This argues for investing in surveillance, rapid testing, and early response capacity.

Third, vaccination is uniquely valuable not just for reducing the mean but for reducing uncertainty. When the next pandemic arrives, having vaccine platforms ready, distribution networks established, and uptake campaigns planned will pay dividends beyond the direct infection prevention—they'll make the epidemic trajectory more predictable.

Fourth, adaptive policies—interventions that respond to the epidemic state—are realistic but introduce their own complexities. The timing of escalation matters enormously, and small differences in early dynamics can lead to divergent long-term outcomes. Policymakers should be aware of this sensitivity and design decision rules that are robust to uncertainty.

The framework developed by Papageorgiou, Votsi, and Trevezas offers a more honest accounting of what interventions do. It acknowledges that epidemics are random processes, that interventions reshape distributions, not just means, and that the full range of possible futures matters for policy. In a world where pandemic risk is increasingly recognized as a systemic threat, this kind of rigorous, distributional thinking is exactly what we need.