The Phantom Pathway Problem: How a New Mathematical Trick Makes Epidemic Models Dramatically Faster

The Weight of What Doesn't Happen

In the summer of 2020, as cities worldwide prepared for second waves, a quiet mathematical insight was taking shape in the literature on epidemic modeling. Researchers had noticed something paradoxical: the tools they were using to predict how diseases spread through networks were spending enormous computational effort tracking interactions that would never actually occur. In a model of thirty cities connected by highways, for instance, standard techniques would carefully calculate how an infection in city twelve might leap to city twenty-eight through a contact that doesn't exist—not through infrastructure, not through mobility patterns, not through any epidemiological channel the network actually supports.

The waste wasn't trivial. Standard approaches to lifting nonlinear epidemic models into higher-dimensional linear representations—a mathematical trick that makes prediction tractable—scale with the square of the number of network nodes. Thirty cities become nine hundred tracked interactions. A hundred cities become ten thousand. The numbers balloon not because the epidemics are more complex, but because the mathematical machinery is tracking phantom pathways.

A paper by Enrique Baeyens of the University of Valladolid, published in July 2026 in the arXiv preprint repository and destined for Mathematical Biosciences, offers a solution. His graph-induced tensor lifting doesn't track every possible interaction—it tracks only the ones that actually matter. In doing so, it reveals something fundamental about how epidemics move through connected populations: the structure of the network itself contains information that traditional methods throw away. The paper's central insight is that when you build your mathematics around what can actually happen, the calculations become dramatically simpler without sacrificing accuracy.

The implications stretch beyond epidemic modeling. Similar structural problems arise whenever nonlinear dynamics interact with network topology—in neural signaling, in financial contagion, in the spread of misinformation. But the stakes are most immediate for the diseases that have reshaped the modern world.

The Science: Building a Better Map

How Epidemics Move Through Networks

The SEIR model—Susceptible, Exposed, Infectious, Recovered—has been the workhorse of epidemic mathematics since William Kermack and A.G. McKendrick laid out its foundations in 1927. It works by tracking how individuals flow between immunological states: the susceptible catch the disease, the exposed carry it without yet being infectious, the infectious spread it, and the recovered emerge with immunity. The model captures the fundamental structure of diseases like measles, chickenpox, and the early dynamics of COVID-19, where a latent period shapes how the infection propagates.

Networked SEIR extends this framework to populations split across geographic regions, demographic groups, or communities connected by contact. Each node in the network represents a subpopulation with its own S-E-I-R compartments. The edges represent effective transmission channels—the actual pathways through which infections can jump. A diagonal edge means transmission within the subpopulation itself. An off-diagonal edge means transmission from one subpopulation's infectious individuals to another's susceptible individuals.

The power of this formulation lies in its specificity. Rather than assuming every population affects every other population equally, networked SEIR lets researchers encode actual contact patterns: commuter flows between cities, interactions between age groups in households and schools, or spatial proximity in a workplace. The mathematics then respects that structure.

The cost is computational. When the equations for networked SEIR are written out, the nonlinear terms arise from bilinear monomials—products like SiIj, where susceptible individuals in node i encounter infectious individuals in node j. For each effective transmission channel, one such product appears. The problem is that traditional lifting techniques don't stop there.

The Lifting Illusion

Mathematicians have long known that nonlinear systems are difficult to analyze directly. One powerful workaround is Carleman linearization: embed the nonlinear system into an infinite-dimensional linear space by including not just the original variables, but their products, their products of products, and so on. In this higher-dimensional space, the dynamics become linear—until you truncate back to a finite number of dimensions.

The trouble is which products to include. Standard approaches use complete ordered Kronecker tensors: if you have n state variables, you include every possible quadratic product, giving n² terms. For a networked SEIR model with n nodes, the state dimension is 4n (four compartments per node), so the complete quadratic lifting contains (4n)² = 16n² terms. Most of those terms correspond to interactions that have no epidemiological meaning.

Consider a three-node network where cities 1 and 3 are connected only through city 2. The term S₁I₃—an interaction between Susceptible individuals in city 1 and Infectious individuals in city 3—is algebraically possible but epidemiologically meaningless. There is no transmission channel connecting these populations, so the product S₁I₃ tells you nothing about the actual spread of disease. Yet standard lifting techniques include it anyway, because they're following mathematical logic rather than epidemiological reality.

Baeyens calls this the "complete ordered Kronecker tensor lifting," and he demonstrates that it scales quadratically with network size. A network of 100 nodes requires tracking 16,000 quadratic terms, most of which represent interactions that cannot occur. The computational burden compounds as researchers attempt more refined analyses or tighter predictions.

The Graph Alternative

The alternative is to build the mathematical structure around the transmission graph itself. Instead of asking "what products can we write algebraically?", ask "what products correspond to actual transmission channels?"

This reframing leads to the edge-based infection dictionary: a collection of bilinear monomials SiIj, one for each effective transmission channel in the network. For the three-node mobility corridor, this yields seven products—one for each diagonal (within-subpopulation) channel and one for each directed between-subpopulation channel. The impossible interaction S₁I₃ doesn't appear, because there is no transmission channel (1,3) in the graph.

The key move is to formalize this intuition into a structured quadratic representation. Baeyens shows that the networked SEIR dynamics can be written as:

$$\dot{x}=A_{\ell }x+M{\varphi }^{SI}(x)$$

Here, x is the full state vector (all S-E-I-R variables across all nodes), A_ℓ is a sparse matrix containing the linear transitions between compartments (the flow from exposed to infectious to recovered), M is a sparse coupling matrix, and φ^SI(x) is precisely the edge-based infection dictionary—one entry per transmission channel.

This representation is exact. There is no approximation involved. Every nonlinear term in the original SEIR dynamics appears in φ^SI(x), and no non-existent terms appear. The representation separates the linear compartmental dynamics from the quadratic transmission dynamics, making the structure transparent.

Building the Hierarchy

With the quadratic representation in hand, the next step is to construct a lifting that extends the state space to include nonlinear observables. The construction proceeds recursively, one homogeneous degree at a time.

At degree zero, the dictionary contains the original compartmental variables (the S, E, I, R values themselves). These evolve under the linear compartmental dynamics—they move around but don't change the dictionary's size.

At degree one (quadratic), the dictionary contains the infection observables SiIj, one for each transmission channel. This is where the quadratic transmission field generates new observables: differentiating SiIj generates products like SiEj (susceptible times exposed) or IiIj (infectious times infectious). These new products appear only on the supported channels.

At degree two (cubic), the process repeats. The quadratic transmission field generates new cubic terms, and then the linear compartmental field saturates the dictionary within that degree—adding any remaining terms that can be generated by applying the linear dynamics to existing observables.

The crucial observation is that this process produces a linearly closed system up to an explicit residual. When you compute the time derivative of any observable in the dictionary, you get a linear combination of dictionary observables plus a remainder term (the residual) that belongs to the next degree. This is the edge-closure property: the dynamics are linear within each homogeneous degree, with forcing from higher degrees.

The first edge-closure dynamics—how the quadratic observables evolve—are linear up to a cubic residual. The residual is not hidden or approximated away; it has an explicit form that depends on the transmission rate, the network weights, and the maximum weighted incoming transmission intensity. The paper computes these bounds rigorously, giving practitioners a way to estimate how much error the truncation introduces.

What They Found

Dimensional Reduction: The Core Result

The central finding concerns how dictionary size scales with network size. The complete ordered Kronecker tensor lifting, which includes all algebraically possible interactions, scales as O(n²)—quadratically with the number of nodes. This means doubling the network size quadruples the lifting dimension.

The graph-induced lifting behaves differently. At the first lifted degree (quadratic observables), the dimension equals the number of effective transmission channels, m. This is typically much smaller than n². More importantly, at higher fixed orders, the dictionary grows linearly with network size when local connectivity is bounded uniformly.

The paper proves this linearity explicitly. Consider a network where each node has at most k incoming transmission channels (where k is a fixed constant independent of n). Then the size of the d-th degree dictionary grows as O(n), not O(n^d). The bound depends on the maximum local connectivity, not on n itself.

Dimensional Scaling: Complete vs Graph-Induced Lifting

Label	Value
n = 10	10 nodes
n = 20	20 nodes
n = 30	30 nodes
n = 40	40 nodes
n = 50	50 nodes

Source: Baeyens (2026), Figure 2. Graph-induced dictionaries scale linearly with n; complete Kronecker liftings scale as (4n)² = 16n².

Figure 2 from the paper illustrates this dramatically. The complete ordered quadratic lifting shows clear quadratic growth, while the graph-induced edge-closure dimension remains approximately linear over sparse random graphs. For a path graph (nodes connected in a chain) and a star graph (nodes connected to a central hub), both yield m = 3n - 2 effective transmission channels once diagonal channels are included—far fewer than the n(n+1)/2 terms a complete quadratic lifting would require.

The practical implication is concrete: a network that would require tracking hundreds of thousands of interactions under standard methods might require tracking only thousands under the graph-induced approach. The reduction isn't just mathematical elegance—it translates to real computational savings when running simulations, fitting models to data, or designing control strategies.

Residual Bounds: Controlling the Error

A lifting without error bounds is of limited practical use. If you're truncating the infinite hierarchy at some finite degree, you need to know how much error that truncation introduces.

Baeyens derives uniform bounds on the first edge-closure residual—essentially, the error incurred when you approximate the true cubic observables as zero. These bounds depend on two epidemiological quantities: the transmission rate β and the maximum weighted incoming transmission intensity.

The transmission rate β captures how aggressively the disease spreads per contact. The weighted incoming transmission intensity for node i is the sum of weights from all incoming channels: Σ_j a_ij. The maximum over all nodes appears in the residual bound, which makes intuitive sense: the most connected node (in terms of weighted incoming channels) determines how large the truncation error can grow.

The paper presents two numerical scenarios. In the first, each active channel carries unit weight. The residual trajectories show that denser graphs (with more edges) produce larger residuals, but the growth remains bounded. In the second scenario, weights are normalized so that each node's incoming weights sum to one: Σ_j a_ij = 1 for all i. This simulates a situation where infection pressure is distributed evenly across available channels.

Residual Evolution: Unit Weights

Label	Value
t = 0	0 time
t = 20	0.08 time
t = 50	0.12 time
t = 100	0.15 time
t = 150	0.1 time
t = 200	0.06 time
t = 240	0.04 time

Source: Baeyens (2026), Figure 3. Residual peaks correlate with peak epidemic activity.

Figure 3 shows the time evolution of the residual norm under unit channel weights over a time horizon of 240 time units (roughly corresponding to days in an epidemic timescale). The sparse-random curves represent ensemble means over many network realizations, with shaded bands showing one standard deviation. The early peaks reflect initial epidemic growth, where nonlinear transmission effects are strongest. As the epidemic saturates and then declines, the residuals shrink.

Cubic Dictionary Size by Topology

Label	Value
Path graph	90 channels
Star graph	85 channels
Complete graph	210 channels

Source: Baeyens (2026), Figure 5. Dictionary size reflects network connectivity structure.

Figure 4 shows the same residual evolution under normalized weights over a longer horizon of 480 time units. The inset resolves the delayed initial peaks—the normalization spreads the transmission across more channels, slowing initial nonlinear amplification. The full horizon shows an extinction phase where the residual settles toward zero as the infection dies out.

The long path plateau visible in these curves merits explanation. In a path graph (nodes connected in a chain), the epidemic can propagate as a traveling wave from one end to the other. The plateau represents sustained nonlinear activity as the wave traverses the network—not a nonzero asymptotic residual, but a transient effect spanning the network's length.

Cubic Dictionary Growth

The saturated cubic dictionary—everything that can be generated at degree two (cubic terms)—shows how the hierarchy fills in beyond the first lifted degree. Figure 5 displays the sizes of these dictionaries for three transmission topologies: a path graph, a star graph, and a complete graph.

The path and star graphs show similar behavior because they have the same number of effective transmission channels. The complete graph (where every node connects to every other node) produces a larger cubic dictionary because the dense connectivity generates more products when the linear saturation step adds same-degree coordinates.

These results confirm that the graph-induced construction captures the right structure: dictionary sizes reflect actual network support rather than algebraic possibility.

Why This Changes Things

The Epistemology of Epidemic Models

Epidemic models are not crystal balls. They are structured beliefs about how infections spread—mathematical expressions of what we think matters and how we think it matters. When a model includes 10,000 tracked interactions for a 30-city network, it's making a claim: these 10,000 interactions are relevant to understanding the dynamics. When 9,700 of them correspond to transmission channels that don't exist, that claim is wrong.

The graph-induced lifting corrects a category error. Traditional approaches treat the network structure as an afterthought—something imposed on top of the mathematical machinery rather than built into it. The network tells you which equations to write, and then you write more equations than necessary, and then you hope the truncation is small enough that the error doesn't matter. The graph-induced approach treats the network as the foundation. You write exactly the equations the structure demands.

This is particularly important for the policy questions where epidemic models matter most. When officials in a regional health department use a model to estimate hospital surge, they need predictions that are both accurate and fast. A model that spends computational effort on phantom transmission channels is a model that runs slower and potentially less accurately than it should. The graph-induced lifting makes the same accuracy achievable with a smaller computational footprint—or, equivalently, makes more refined analyses feasible with the same computational resources.

Implications for Real-Time Prediction

During the COVID-19 pandemic, epidemiological forecasting teams struggled with a fundamental tension. More complex models could capture more structure—age-specific contact patterns, spatial heterogeneity, behavioral changes—but they required more computation, which meant slower updates and less responsiveness to incoming data. Simpler models updated quickly but missed important dynamics.

The graph-induced lifting offers a way to escape this tradeoff. By dramatically reducing the dimensional overhead of networked SEIR models, it makes detailed structural models more computationally tractable. A model that previously required tracking 16n² terms now tracks only O(n) terms at each degree. This isn't a marginal improvement—it can mean the difference between a model that takes hours to run on a regional server and one that runs in minutes.

Faster computation enables faster iteration. When case data updates daily, a model that runs in twenty minutes rather than four hours can incorporate yesterday's numbers before tomorrow's decisions are made. When scenarios need to be explored—"what if we close schools but not workplaces?"—a faster model allows more scenarios to be evaluated, giving decision-makers richer information about the landscape of possibilities.

Connections to Koopman Operator Theory

The lifting developed in this paper connects to a broader movement in dynamical systems theory. The Koopman operator, introduced by Bernard Koopman in 1931, describes how observables evolve under a dynamical system. Finite-dimensional approximations of this infinite-dimensional operator—through techniques like dynamic mode decomposition (DMD) and extended dynamic mode decomposition (EDMD)—have transformed how scientists analyze complex systems.

The graph-induced lifting can be seen as a domain-specific dictionary design for Koopman analysis of networked epidemics. Rather than using generic polynomial bases or learning dictionaries from data (as EDMD does), it constructs the optimal dictionary analytically from the structure of the problem. The epidemiological interpretation is transparent: each observable corresponds to an actual interaction pattern, not an abstract mathematical construct.

This transparency matters for interpretability. When a model predicts that node 7 will experience a surge in exposed individuals, a graph-induced analysis can trace that prediction to specific transmission channels—perhaps the S₇I₅ and S₇I₆ observables are growing rapidly, indicating that infectious individuals in nodes 5 and 6 are driving exposure in node 7. A generic learned dictionary might achieve similar predictive accuracy without offering this epidemiological interpretation.

Control and Intervention

The paper's closing sections hint at applications to predictive control and Lyapunov analysis—mathematical frameworks for designing interventions that steer the epidemic toward desired outcomes. These applications require accurate models and computational tractability, both of which the graph-induced lifting enables.

Consider contact tracing as an intervention. When an infectious individual is identified in a node, the question becomes which transmission channels are active and which susceptible populations are at risk. The graph-induced framework makes this precise: it identifies the specific observables that matter for a given node, the channels through which infection pressure arrives, and the predicted growth rates based on current compartment values. A control algorithm could use this structure to prioritize contact tracing resources toward the channels most likely to amplify transmission.

More broadly, any intervention that modifies transmission rates—masking, social distancing, vaccination campaigns—operates on the graph structure. The graph-induced lifting provides a natural language for describing how interventions affect the dynamics: they modify the weights on existing channels, potentially remove channels entirely, and thereby change both the dictionary structure and the residual bounds.

What's Next

Open Questions

The paper leaves several questions for future work. The residual bounds derived here apply to the first edge-closure dynamics (linear behavior up to cubic residual). Higher-order truncations—approximating cubic terms as zero, then approximating the resulting quartic residuals, and so on—have bounds that "contain only next-degree terms," but explicit formulas for practical computation remain to be developed. In principle, the recursive structure should extend, but the algebra grows more complex with each degree.

The analysis assumes deterministic dynamics. Real epidemics are stochastic: chance events in transmission, surveillance gaps in case reporting, and randomness in contact patterns all introduce noise. The graph-induced lifting's clean structure suggests it could be extended to stochastic settings, but the derivation would require new techniques to handle random transmission events.

The framework also assumes that the network structure is known and fixed. In practice, contact patterns change over time—mobility restrictions, school closures, and behavioral shifts all modify the transmission graph. A time-varying graph extension would require tracking how the dictionary structure itself changes, adding another layer of complexity.

Extensions and Generalizations

The paper mentions that extensions to other networked compartmental models are possible when nonlinear processes are supported on identifiable channels and differentiation yields a manageable polynomial hierarchy. SIR models (without an exposed compartment) would have a simpler structure since they lack the latent period. SIS models (where recovered individuals return immediately to susceptible) have no removed compartment. Both would admit similar graph-induced constructions.

The paper also explicitly excludes "nonpolynomial incidence, explicit migration, delays, and additional nonlinear transitions." These are active areas of epidemiological modeling. Nonpolynomial incidence—transmission terms that don't follow simple mass-action kinetics—arise in saturation effects, frequency-dependent transmission, and behavioral responses. Explicit migration (movement of individuals between nodes, not just infection pressure) adds coupling terms that change the compartmental structure. Delays (infection latency, reporting lags) introduce non-Markovian dynamics. Each of these extensions would require a modified construction.

The connection to data-driven methods also deserves exploration. The analytical dictionary developed here provides a theoretically optimal starting point, but data might reveal that certain observables are negligible for a specific context, or that additional observables improve prediction. Hybrid approaches—starting with the graph-induced dictionary and adapting based on data—could combine analytical rigor with empirical flexibility.

The Broader Landscape

Networked epidemic models occupy a peculiar position in the landscape of mathematical biology. They are more realistic than homogeneous models (which assume every individual interacts with every other individual equally), but they require more data and more computation. The tension between realism and tractability has driven decades of methodological development.

The graph-induced lifting represents a resolution of this tension—not by compromising on structure, but by respecting structure more deeply. Earlier approaches treated the network as an add-on to an already-complete mathematical machinery. This paper treats the network as the skeleton on which the machinery should be built.

The approach connects to a broader theme in modern science: the recognition that structure constrains possibility. The number of effective transmission channels in a network is not arbitrary—it reflects actual contact patterns, mobility flows, and spatial proximity. By making this structure mathematical rather than just metaphorical, the graph-induced lifting brings epidemiological modeling closer to the data that motivates it.

A Note of Caution

Every mathematical model simplifies reality, and every simplification introduces error. The graph-induced lifting reduces one source of error—tracking phantom interactions—while potentially introducing others. The residual bounds derived in the paper are rigorous but conservative: they bound the maximum possible error over all possible states, not the error expected for typical epidemic trajectories. Practitioners will need to validate the approach against their specific contexts and assess whether the residual errors are acceptable for their decision thresholds.

The paper is also theoretical. The numerical illustrations demonstrate the framework's behavior, but empirical validation against real epidemic data—COVID-19 case trajectories, influenza outbreaks, historical plague records—remains for future work. The gap between mathematical elegance and epidemiological reality can be wide.

These caveats are not criticisms. They are the normal scientific process of moving from theory to practice, from derivation to validation. The paper provides a rigorous foundation; applying that foundation to real-world problems is the work that follows.

Conclusion: The Structure of What Can Happen

The graph-induced tensor lifting developed in this paper offers a new way of thinking about epidemic models on networks. Rather than building mathematical machinery that tracks every algebraically possible interaction, it builds machinery that tracks every epidemiologically meaningful one. The dimensional savings are substantial—a move from quadratic growth to linear growth. The residual bounds are explicit. The structure is interpretable.

The deeper insight is that network structure contains information that traditional methods discard. When you include an observable for a transmission channel that doesn't exist, you're not just wasting computation—you're encoding a belief that the model cannot distinguish between real and phantom pathways. When you include only the supported interactions, you're forcing the model to respect the causal structure of the problem.

This lesson extends beyond epidemic modeling. Whenever nonlinear dynamics interact with network topology—in neuroscience, in finance, in social systems—the choice of what to include in a model reflects assumptions about what matters. The graph-induced lifting provides a template for making that choice systematically, based on the structure of the system rather than the convenience of the mathematics.

For epidemic modelers, the immediate implication is practical: networks that previously required prohibitive computational resources to analyze can now be studied with tractable methods. For decision-makers, the implication is potentially faster, more responsive models that can keep pace with rapidly evolving outbreaks. For the rest of us—the curious, the concerned, the ones who check case numbers and wonder what comes next—the implication is a reminder that even in the mathematics of disease, structure matters.

The spread of an infection is not random. It follows pathways, exploits contacts, moves through the particular geography of human connection. A model that knows the difference between a pathway and a phantom is a model that sees the world more clearly.