2.1 The ross–macdonald model in fragmented environments

The Ross–Macdonald model was originally developed for malaria dynamics and describes transmission between hosts and mosquito vectors. Auger et al. [4] generalised this model to n spatial patches, allowing host migration but not vector migration. Let x_i(t) denote the prevalence of infected hosts in patch i and y_i(t) the prevalence of infected vectors. The metapopulation dynamics read:

(1)

(2)

where m is the vector density vector, is the migration matrix of hosts (with ), and are the host-to-vector and vector-to-host transmission rates, c is the host recovery rate, and is the vector mortality rate. The basic reproduction number takes the form:

(3)

where restricted to the p patches carrying vectors, and denotes the spectral radius. The main theorem of [4] establishes:

• if R₀ ≤ 1, the disease-free equilibrium (DFE) is globally asymptotically stable.
• If R₀ > 1, there exists a unique endemic equilibrium that is globally asymptotically stable on the positive orthant.

This structural result reveals the central role of spatial migration encoded in D: increasing connectivity can either accelerate or impede epidemic spread depending on its interaction with of the migration-adjusted system. The spectral bridge between R₀ in (3) and of the contact network will be a key theoretical anchor for our hybrid model (see the Hybrid Theoretical Framework section).

2.2 Structural approaches to epidemic threshold prediction

Let G=(V,E) be an undirected contact network with n = |V| nodes and m = |E| edges. Let A be its adjacency matrix and its Laplacian, where is the degree matrix.

1. Mean-Field (MF).

Under the homogeneous mixing assumption: , where is the mean degree. MF ignores all topological heterogeneity beyond the mean degree.

1. Heterogeneous Mean-Field (HMF).

For uncorrelated networks with degree distribution p(k): . HMF captures degree variance but ignores correlations and clustering.

1. Quenched Mean-Field (QMF).

Introduced by Wang et al. [2], QMF uses the full adjacency spectrum:

(4)

QMF is known to be exact for regular graphs and a lower bound for scale-free networks in the thermodynamic limit [6]. It consistently outperforms MF and HMF, yet systematically underestimates when degree assortativity is positive.

1. KSEL.

Kanyou et al. [3] proposed the K Spectral Energy of Laplacian estimator using the Laplacian energy , where are eigenvalues of L:

(5)

with empirically calibrated constant . KSEL combines information from the diffusion operator L (via LE) and the epidemic contact matrix A (via ), giving it a complementary theoretical foundation compared with QMF.

3.1 Unified feature representation

Our hybrid approach rests on the observation that is a function of both the network structure and the intrinsic disease parameters:

(6)

where is a complex, potentially non-linear function that existing closed-form estimators approximate with different levels of fidelity. We propose to learn f from data while enforcing physical constraints derived from the Ross–Macdonald theory.

For each network G and parameter pair , we construct a feature vector :

1. (a) Spectral features: , (algebraic connectivity), LE(G);
2. (b) Size features: number of nodes n, number of edges m;
3. (c) Topological moments: , , mean clustering coefficient , network diameter ;
4. (d) Epidemiological parameters: , , ratio ;
5. (e) Auxiliary predictors: (used as a feature encoding the linearised spectral constraint).

Note on as a feature versus a benchmark. is included in as an input feature because it encodes the theoretically motivated mean-field baseline in a compact, interpretable form. The benchmarking comparison (see the Results section) evaluates the PGNN against used as a standalone predictor—a fundamentally different role. Concretely, the network receives as one of eleven inputs and learns to correct its systematic bias using additional structural and epidemiological information. This is analogous to residual learning in physics-informed machine learning: the model learns the deviation rather than from scratch, which explains the strong SHAP importance of (see the SHAP Interpretability Analysis section) without implying circular evaluation.

This representation encodes the information used by all existing estimators while enriching it with second-order topological moments and explicit epidemiological parameters.

3.2 Physics-guided neural network architecture

We model f as a multi-layer perceptron (MLP) with architecture illustrated in Fig 1:

• Input layer: , batch-normalised.
• Hidden layers: three fully-connected layers of widths 128, 64, 32 with ReLU activations, Dropout (p = 0.2), and Batch Normalisation between each pair.
• Output layer: single neuron with Softplus activation to guarantee .

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Fig 1. Physics-Guided Neural Network architecture.

Input features are batch-normalised and passed through three hidden layers (128–64–32 neurons, ReLU activation, Dropout p = 0.2). The Softplus output guarantees . Three epidemiologically motivated soft constraints are added to the MSE loss: (QMF lower bound), (monotonicity), and (SIS upper bound). The ablation study (Fig 2) reveals that is a beneficial regulariser while over-regularises in the low-data regime. All elements are original and created by the author.

https://doi.org/10.1371/journal.pone.0344827.g001

1. Hybrid loss function

The training objective combines a data-fidelity term and a physics-consistency penalty:

(7)

where is the mean squared error over training examples, and penalises three types of physical violations:

• Stability constraint: For any network with , the epidemic should be sub-critical (consistency with the QMF lower-bound property of the DFE).
• Monotonicity constraint: must decrease when increases for fixed , i.e., .
• Boundedness constraint: over the training distribution.

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Fig 2. Performance Breakdown by Network Family and PGNN Ablation Study Results.

(A) RMSE breakdown by network topology family for QMF, Random Forest, Gradient Boosting, and PGNN. Tree-based methods dominate across all families. The PGNN advantage is relatively largest on regular graphs (REG), consistent with less heterogeneity reducing the sample-complexity burden. (B) Ablation study: percentage change in RMSE relative to the full PGNN. Green bars indicate that removing the component improves performance (over-regularisation); red bars indicate degradation (component is beneficial). Key findings: is a genuine regulariser (+25.0% RMSE when removed); as an auxiliary feature provides the strongest structural prior (−33.7% RMSE when removed). All elements are original and created by the author.

https://doi.org/10.1371/journal.pone.0344827.g002

The hyperparameter is set by cross-validated grid search. We have verified that increasing beyond 0.5 degrades data-fidelity without improving physical consistency, while values below 0.01 result in constraint violations on out-of-domain networks.

Theoretical basis and scope of validity of the constraints. The three soft constraints are grounded in well-established properties of the SIS epidemic threshold under the QMF approximation [2,6]. Specifically: (i) the stability constraint () encodes the QMF lower bound, which is asymptotically exact for regular graphs and a provable lower bound in the thermodynamic limit for uncorrelated networks; (ii) the monotonicity constraint () follows directly from ; and (iii) the boundedness constraint () follows from the SIS persistence condition . These constraints are theoretically motivated within the regime where the QMF approximation is reliable, namely networks with moderate degree heterogeneity and weak finite-size effects. Outside this regime—and in particular for the ≈18% of networks in our dataset where the stochastic ground truth violates the QMF lower bound due to finite-size effects—the stability constraint acts primarily as a regulariser (preventing extreme predictions) rather than as a theoretically grounded bound. This distinction is consistent with the ablation results of the Results section: is genuinely beneficial while over-regularises precisely because the finite-size regime falls outside the validity of the QMF bound.

4.1 Ethics statement

This study does not involve human participants, animal experiments, or identifiable human data. No ethics approval was required.

4.2 Synthetic network dataset generation

We generated 775 undirected networks with nodes, divided among four families:

1. Erdős–Rényi (G(n,p)): , 194 instances.
2. Barabási–Albert (scale-free, preferential attachment with ): 194 instances.
3. Watts–Strogatz (small-world, ring with initial neighbours and rewiring probability ): 194 instances.
4. Regular and empirical (random d-regular graphs and contact traces from published datasets): 194 instances.

For each network G_i, we sample epidemiological parameters uniformly: , . The true epidemic threshold is estimated via stochastic SIS Monte Carlo simulation: we identify the critical at which the probability of epidemic outbreak transitions from <10% to >90% over 1,000 independent realisations. This produces an accurate continuous ground truth label for each (G_i, , ) triplet.

The dataset is partitioned into train / validation / test splits of 70% / 15% / 15% (542 / 116 / 116 networks), stratified by network family and size decile to ensure balanced coverage.

4.3 Training protocol

All models are implemented in PyTorch and optimised with Adam (, exponential decay factor 0.95 every 20 epochs) for a maximum of 200 epochs with early stopping (patience = 20 epochs). A 5-fold cross-validation on the training set guides hyperparameter selection: layer widths, , and Dropout rate .

We evaluate seven models:

1. (i) QMF: (closed-form).
2. (ii) KSEL: .
3. (iii) Linear regression on .
4. (iv) Random Forest (RF): 200 trees, max depth 10.
5. (v) Gradient Boosting (GB): 200 estimators, lr = 0.05.
6. (vi) Standard MLP: PGNN architecture, .
7. (vii) PGNN (ours): full model, .

To obtain robust performance estimates, we use extbf5-fold cross-validation repeated over 5 independent random seeds (25 (seed, fold) combinations total). Results are reported as mean ± standard deviation over these 25 runs. Statistical significance of pairwise differences is assessed by the Wilcoxon signed-rank test on the 25 paired RMSE values.

5.1 Predictive performance

Table 1 reports test-set performance for all methods. Fig 3 visualises predicted versus true values.

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Fig 3. Comparative performance of seven methods on the 116-network test set.

(A) RMSE: lower is better. Gradient Boosting (RMSE = 0.0731) and Random Forest (RMSE = 0.0744) substantially outperform the PGNN (RMSE = 0.2299) in this data regime. (B) R²: QMF and KSEL yield negative R², confirming that closed-form spectral estimators fail to track the variance of the stochastic ground-truth . The PGNN (R² = 0.093) offers interpretability and physical consistency rather than predictive superiority. All elements are original and created by the author.

https://doi.org/10.1371/journal.pone.0344827.g003

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Table 1. Predictive performance: mean ± std over (seed, fold) combinations of 5-fold cross-validation. Best results in bold; second best underlined. GB and RF dominate in the low-data regime; the PGNN (, 8.7× less stable than GB) exhibits high variance consistent with overfitting (Section 5.2).

https://doi.org/10.1371/journal.pone.0344827.t001

Table 1 reports mean ± std RMSE and R² over 25 (seed, fold) combinations of 5-fold cross-validation. Gradient Boosting achieves the best predictive accuracy (0.0757 ± 0.0463, ), followed closely by Random Forest (0.0784 ± 0.0511, ). Both ensemble methods are highly stable across folds (std ≈0.05). Linear regression performs competitively (), confirming that a substantial fraction of the variance in is linear in the eleven features.

The PGNN shows mean RMSE and . The large standard deviations—a factor 8.7× larger than GB—reveal high instability across folds, the empirical signature of overfitting in a high-dimensional model applied to a small dataset. Section 5.2 provides a principled explanation in terms of the parameter-to-sample ratio and the interaction between physical constraints and fifinite-size effects.

The closed-form baselines QMF () and KSEL () yield negative mean R², confirming that closed-form spectral estimators fail to track the variance of the stochastic ground-truth across diverse topologies. Inference time for all learned models remains ∼1.2 ms per network, orders of magnitude faster than stochastic simulation (∼50–100 ms).

5.2 Why does the PGNN underperform tree-based methods?

The substantial performance gap between the PGNN and tree-based methods (RMSE = 0.448, on the 5-fold cross-validation) deserves a principled explanation beyond the observation that the dataset is small. We identify three complementary mechanisms.

1. (i) Parameter-to-sample ratio. The PGNN contains approximately 12,845 trainable parameters (including BatchNorm statistics), yielding a parameter-to-sample ratio of ≈252 at the typical training fold size of . By contrast, Gradient Boosting uses effective parameters (100 trees × depth-4 nodes), giving a ratio of ≈88. Operating in the regime where parameters substantially exceed examples is a well-known source of high variance [7,8].
2. (ii) Instability under cross-validation. The standard deviation of PGNN RMSE across the (seed, fold) combinations is 0.403—a factor 8.7× larger than that of Gradient Boosting (0.046). This high variance is the empirical signature of overfitting: on some folds the PGNN finds a reasonable solution, on others it collapses to a trivial predictor.
3. (iii) Physical constraint interaction. The ablation study (Table 2) shows that removing reduces RMSE by 28.4%, implying that this constraint is actively harmful in the current regime. The QMF lower bound is a valid asymptotic result but is violated by the stochastic ground truth on a non-negligible fraction of networks (≈18% of our dataset) due to fifinite-size effects. Enforcing it as a hard soft constraint therefore introduces systematic bias in the low-data regime.

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Table 2. Ablation study. = percentage change in RMSE relative to the full PGNN (positive = degradation; negative = improvement, indicating over-regularisation in the current data regime). is the only constraint that is genuinely beneficial; as an auxiliary feature provides the strongest structural prior.

https://doi.org/10.1371/journal.pone.0344827.t002

These findings collectively suggest that the PGNN would become competitive with tree-based methods at training examples, where the parameter-to-sample ratio falls below 10 and regularisation by physical constraints can act as a genuine inductive bias rather than a source of misspecification.

This highlights an important limitation of the present study: hybrid neural approaches may not be the optimal choice when data availability is limited, and their advantages may only emerge at larger scales. Practitioners working with small network datasets (n < 500) should prefer tree-based ensemble methods for pure prediction tasks, and reserve physics-guided models for settings where out-of-distribution safety or theoretical interpretability is a primary requirement.

5.3 Automatic calibration of the KSEL coefficient k

Gradient-based optimisation of k converges to topology-specific values that depart substantially from the empirical constant k = 0.3 of [3]: , , , , and (over the full training set). These values are visualised in Fig 4B.

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Fig 4. SHAP Feature Importance and Topology-Dependent KSEL Calibration.

(A) Mean |SHAP value| per input feature for the PGNN. is the dominant predictor (0.045), confirming that the linearised spectral constraint encodes the most informative structural signal. Epidemiological parameters (0.040) and (0.032) collectively rank second, while ranks tenth (0.011), its information subsumed by . (B) Topology-dependent optimal KSEL coefficient k^* obtained by gradient descent. Values range from to , substantially departing from the universal constant k = 0.3 of prior work. All elements are original and created by the author.

https://doi.org/10.1371/journal.pone.0344827.g004

The topology-dependence is substantial: Watts–Strogatz small-world networks require the lowest k^* (0.627), while Barabási–Albert scale-free and regular graphs require values 4× larger (1.265 and 1.322 respectively). This heterogeneity confirms that a universal constant k is a significant simplification, and that the Laplacian energy captures fundamentally different aspects of diffusion dynamics across graph families. A theoretically rigorous derivation of k(G) as a topology-dependent spectral invariant represents a promising direction for future work.

5.4 SHAP interpretability analysis

Fig 4A shows the mean absolute SHAP values for each input feature of the PGNN. The QMF auxiliary predictor dominates with importance 0.045, confirming that the linearised spectral constraint is the single most informative input. Epidemiological parameters (0.040) and (0.032) jointly account for more importance (0.072) than any individual spectral feature. The Laplacian energy LE(G) (0.022) and algebraic connectivity (0.021) contribute comparably, while the raw ranks tenth (0.011)—its information being largely subsumed by .

This ranking is consistent across the PGNN and Gradient Boosting, validating the feature engineering independently of model class. The convergence on , , and as top predictors provides actionable guidance: epidemic threshold prediction benefits most from combining the spectral structural baseline with epidemiological parameters, rather than from raw spectral features alone.

6.1 Why physics-guided models remain relevant

Although tree-based models outperform the proposed PGNN in our experiments, this does not invalidate the relevance of hybrid approaches. Physics-guided models offer three key advantages that purely data-driven methods cannot provide.

First, structural inductive bias. The physical constraints , , and encode known properties of epidemic dynamics derived from the QMF approximation and SIS compartmental theory. In out-of-distribution settings—network topologies not represented in the training data—these constraints prevent the model from producing predictions that violate fundamental epidemiological laws, a guarantee that Gradient Boosting, as a purely data-driven method, cannot provide.

Second, principled knowledge integration. When new theoretical results become available (e.g., tighter spectral bounds, heterogeneous SIS corrections, age-structured contact patterns), they can be incorporated directly into the PGNN loss function. This modular extensibility is architecturally impossible for tree-based methods, making the PGNN a natural framework for progressive model refinement as domain knowledge grows.

Third, interpretability. Through SHAP attribution, the PGNN provides feature-level insight into how structural and epidemiological factors jointly influence . The convergence of PGNN and Gradient Boosting on the same top predictors (, , ) validates this interpretability independently of model class.

Therefore, rather than competing directly with classical models in terms of raw predictive performance, physics-guided neural networks should be viewed as complementary tools, particularly suited for scientifically grounded modelling in settings where theoretical consistency and extrapolation safety matter as much as in-sample accuracy.

Limitations. Several limitations merit acknowledgment. First, the ground truth is estimated via stochastic simulation, which introduces Monte Carlo noise; the susceptibility-peak estimator used here is robust for the network families considered (median ) but may be sensitive in strongly heterogeneous or periodic regimes; future work could use SIS quasi-stationary distribution methods or multi-criteria estimators (persistence time, final epidemic size, branching factor) for more precise labels. Second, the training set consists of synthetic networks; the model has not been validated on empirical contact networks (workplace, school, hospital settings); such validation is needed before claims of direct applicability can be made, and is identified as a priority for future work. Third, the current model addresses a homogeneous SIS process; extension to SEIR dynamics, age-structured populations, or multiplex networks requires augmenting the feature vector and deriving new physical constraints. Finally, the model assumes and are known, whereas in practice they must be inferred from surveillance data via Bayesian inverse methods.