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A Two-Hidden-Layer Radial Basis Neural Surrogate for a Caputo Fractional-Order Schistosomiasis Model

This paper develops a reproducible numerical framework that couples a five-compartment Caputo fractional-order schistosomiasis model with a two-hidden-layer radial basis function neural network to create a highly accurate and computationally efficient surrogate for reference trajectories generated by the Adams–Bashforth–Moulton method.

Original authors: Gilder Cieza Altamirano, Alvaro Salas

Published 2026-09-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Gilder Cieza Altamirano, Alvaro Salas

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 ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed.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

Disease does not always move in a straight line. In the real world, the spread of an infection often depends on history: how long a person has been sick, how long a mosquito has waited to bite, or how the environment has changed over time. Traditional mathematical models often treat these events as if they happen only in the present moment, ignoring the past. However, a newer branch of mathematics allows scientists to build models that remember the past, treating time as a continuous memory rather than a series of disconnected steps. This approach is particularly useful for diseases like schistosomiasis, a parasitic infection that spreads through freshwater snails and affects millions of people. By accounting for the "memory" of the system, researchers can create more realistic pictures of how the disease behaves, how it might fade away, and how long it takes for populations to recover.

In a recent study, researchers set out to improve the way these complex, memory-aware models are calculated and understood. Schistosomiasis is a persistent global health challenge, transmitted when people come into contact with water containing microscopic parasites released by infected snails. To fight it, scientists use computer models to simulate the interaction between humans and snails. These models divide the population into groups: those who are healthy, those who are infected, those who have recovered, and the corresponding groups of snails. The challenge is that when scientists add the "memory" of the disease into these equations, the math becomes incredibly difficult to solve on a computer. It requires immense computing power and time to run the simulations, making it slow to test different scenarios or visualize the results.

The team behind this study, based in Peru and Colombia, tackled this problem by creating a new kind of digital assistant. They first built a highly accurate, detailed simulation of the disease using a rigorous mathematical method known as the fractional Adams method. Think of this method as a slow, painstakingly precise way of calculating the future of the disease, step by tiny step, ensuring that every number respects the rules of biology, such as the fact that populations cannot be negative and that the total number of people and snails stays within realistic limits. They confirmed that their model behaved correctly, showing that if the disease is not strong enough to sustain itself, the number of infected individuals will naturally drop to zero over time.

Once they had these precise, but slow-to-calculate, reference paths, the researchers trained a specialized computer program to learn from them. This program is a type of neural network, a system designed to recognize patterns, specifically one built with two layers of "radial basis" units. In plain terms, the network was shown thousands of examples of how the disease spreads over time under different conditions. It learned to recognize the shape of the curves that describe the rise and fall of infections. The researchers used a specific training technique called scaled conjugate gradient, which is like a smart, efficient way of adjusting the network's internal settings until its predictions matched the slow, precise reference data almost perfectly.

The results were striking. After the training was complete, this neural network could predict the future state of the disease almost instantly, without needing to perform the heavy, time-consuming calculations of the original method. When the researchers tested the network on data it had never seen before, the difference between its predictions and the true, precise values was incredibly small. For the different scenarios they tested, the error was less than one part in a thousand, and the network's ability to match the true patterns was better than 99.99 percent. This means the network successfully captured the complex behavior of the disease, including how the "memory" of the system changes the speed at which infections rise and fall.

The study makes a clear distinction between the two tools they used. The slow, precise method was used to generate the truth, while the fast neural network was built only to mimic that truth. The researchers emphasized that the network is not a new way of solving the underlying math; rather, it is a highly efficient shortcut that reproduces the results of the proven method. This is a crucial difference. The network does not invent new physics or guess at the rules of the disease; it simply learns to draw the correct lines very quickly. This allows scientists to run simulations, test different strategies, or visualize the spread of the disease in real-time, tasks that would be too slow with the traditional method alone.

The researchers also took care to ensure their work could be repeated by anyone. They provided all the data, the code, and the instructions needed to rebuild the entire experiment from scratch. They showed that their model holds up under different conditions, proving that the populations remain realistic and that the disease eventually dies out when the conditions are right. While the specific numbers used in the study were chosen as a standard benchmark rather than a specific real-world outbreak, the method itself is robust. The study demonstrates that by combining rigorous mathematical proofs with modern machine learning, it is possible to create tools that are both scientifically trustworthy and computationally efficient, offering a powerful new way to understand and manage complex infectious diseases.

Technical Summary: A Two-Hidden-Layer Radial Basis Neural Surrogate for a Caputo Fractional-Order Schistosomiasis Model

Problem Statement
Schistosomiasis is a parasitic disease involving a transmission cycle between humans and freshwater snails. While classical compartmental models use ordinary derivatives, fractional derivatives offer a nonlocal alternative where the current rate of change depends on the system's history, potentially capturing heterogeneous waiting times and delayed responses. Previous work by Botmart et al. [9] proposed a five-compartment fractional schistosomiasis model but contained inconsistencies in transfer terms, lacked full specification of the numerical method, and ambiguously presented neural approximations as independent solvers. Furthermore, the prior study lacked the transparency required for reproducible numerical investigation, such as explicit initial conditions, step sizes, and data partitioning.

Methodology
The authors reconstruct the model and numerical procedure into a reproducible framework consisting of three main components:

Mathematical Formulation:
- A five-compartment Caputo fractional-order model is defined with state variables for susceptible humans ( $W$ ), infected humans ( $A$ ), recovered humans ( $Q$ ), susceptible snails ( $E$ ), and infected snails ( $S$ ).
- The equations are written in a population-balanced form to ensure exact balance equations for total human and snail populations.
- Theoretical properties are established, including the nonnegativity of solutions, explicit bounds for total populations using the Mittag-Leffler function, the disease-free equilibrium (DFE), and the basic reproduction number ( $R_0$ ). For the benchmark parameters used, $R_0 \approx 0.6977 < 1$ , implying local asymptotic stability of the DFE.
Reference Solver (Fractional Adams Method):
- Reference trajectories are generated using the Diethelm–Ford–Freed predictor–corrector (PECE) method (a fractional Adams–Bashforth–Moulton scheme).
- Simulations run on the interval $0 \le t \le 10$ with a step size $h=0.02$ .
- A step-refinement study confirms the convergence of the reference solution.
Neural Surrogate Construction:
- A two-hidden-layer Radial Basis Function (RBF) network is trained as a supervised surrogate for the reference trajectories.
- Architecture: The input is normalized time ( $\xi \in [0,1]$ ). The first hidden layer contains 15 Gaussian radial units, and the second hidden layer contains 25 radial units. The output layer produces the five compartment values.
- Training: The network is trained using the Scaled Conjugate Gradient (SCG) algorithm to minimize a regularized least-squares loss function. The hidden centers and widths are fixed based on the training data distribution; only the output weights are optimized.
- Data Partitioning: For each fractional order $\alpha \in \{0.5, 0.7, 0.9\}$ , 501 time points are generated. Using a fixed random seed (20260716), the data is split into 70% training, 15% validation, and 15% testing.

Key Contributions
The paper explicitly lists six specific contributions:

A coherent, population-balanced five-compartment Caputo model correcting previous typographical inconsistencies.
Proofs of positivity, boundedness, and the derivation of the threshold reproduction number $R_0$ .
A complete implementation of the fractional Adams–Bashforth–Moulton predictor–corrector method with step-refinement verification.
The development of a two-hidden-layer RBF surrogate trained via SCG.
Transparent reporting of training, validation, and test metrics for three distinct fractional orders.
Provision of a supplementary package containing LaTeX source, figures, data, and Python code to regenerate all results.

Results
The study evaluates the surrogate's performance on held-out test sets for $\alpha = 0.5, 0.7,$ and $0.9$:

Accuracy: The Root Mean Squared Errors (RMSE) on the test sets are $2.406 \times 10^{-4}$ ( $\alpha=0.5$ ), $2.122 \times 10^{-4}$ ( $\alpha=0.7$ ), and $1.793 \times 10^{-4}$ ( $\alpha=0.9$ ).
Fit: The coefficients of determination ( $R^2$ ) for all test sets exceed $0.99996$.
Behavior: The surrogate accurately reproduces the reference trajectories. The largest discrepancies occur near the initial point ( $t=0$ ), where fractional solutions exhibit high curvature and where random validation splits may lack dense neighboring training points.
Dynamics: The numerical results confirm that for $R_0 < 1$ , infected populations decay while susceptible populations relax toward disease-free levels. Smaller fractional orders produce stronger memory effects and slower long-time relaxation.

Significance and Claims
The authors emphasize a strict methodological distinction: the Adams method generates the validated reference solution, while the neural network serves as a compact surrogate of that solution. The network is not presented as an independent solver for the fractional differential equations, nor does it approximate the Caputo derivative within the loss function (i.e., it is not a physics-informed neural network).

The significance of this work lies in:

Reproducibility: Providing a fully specified, corrected model and numerical pipeline that addresses gaps in previous literature.
Efficiency: Once trained, the RBF surrogate offers a computationally inexpensive alternative to the $O(N^2)$ history summation required by the fractional Adams method, making it suitable for repeated visualization, parameter screening, or initialization for inverse problems.
Clarity: The study clarifies the role of neural networks in this context as trajectory approximators rather than differential equation solvers, avoiding the conflation of discretization errors with learning errors.

The paper remains modest regarding its scope, noting that the parameters are nondimensional benchmarks rather than epidemiological calibrations for specific regions, and that the surrogate is trained separately for each fractional order without generalizing to new parameter vectors. Future work is suggested to include physics-informed training, parameter uncertainty quantification, and the use of fast convolution methods for long simulations.

Technical Summary: A Two-Hidden-Layer Radial Basis Neural Surrogate for a Caputo Fractional-Order Schistosomiasis Model

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