A deep learning approach for solving a fractional order Monkeypox transmission model using a harmonic neural network optimized with SGDM

Abstract

This study investigates the transmission dynamics of Monkeypox disease using a Harmonic neural network (HNN) framework optimized through stochastic gradient descent with momentum (SGDM). The proposed HNN-SGDM approach is applied to a nonlinear Monkeypox model consisting of nine coupled differential equations describing the interactions between human and rodent populations. HNN are employed because traditional non-oscillatory activation functions often struggle to capture the periodic and complex dynamics of disease transmission, whereas harmonic activation functions efficiently approximate such oscillatory patterns. SGDM is chosen to improve convergence and optimization stability in high-dimensional, non-convex search spaces. The proposed solver achieves high precision, with absolute errors ranging from to , confirming its numerical stability and convergence. The robustness and reliability of HNN-SGDM framework are further validated through statistical performance measures, including mean absolute error, root mean square error, and Theil’s inequality coefficient. Graphical analyses, including weight distributions, box plots, histograms, and loss curves, further validate the model’s performance. This approach highlights the effectiveness of deep learning in epidemiological modeling and provides a methodology extendable to other infectious disease frameworks.

Introduction

The zoonotic disease Monkeypox (Mpox) is caused by a double-stranded DNA virus belonging to the genus orthopoxvirus of the Poxviridae family, capable of infecting both humans and animals. In Africa, various animals, including monkeys, serve as efficient disease reservoirs and vectors¹. Mpox has emerged as a significant global health concern, with reported outbreaks outside Africa. In 2003, the United States experienced its first outbreak, involving approximately 70 confirmed cases linked to human-exposed pet prairie dogs. Subsequent cases were documented in the United Kingdom and Singapore in 2018 and 2019, respectively. Between July and November 2021, Mpox cases re-emerged in the United States^2,3, highlighting its potential for international spread and the need for accurate epidemiological modeling.

The World Health Organization (WHO) has highlighted a recent outbreak of Mpox, initially identified in the UK in 2022, involving a patient believed to have traveled from Nigeria.The virus can be transmitted through direct contact with infectious rashes, scabs, or bodily fluids. It can also spread via respiratory secretions during extended face-to-face interactions or other physical contact^4,5. Wild animals, such as African rats and monkeys, are the primary sources of virus transmission to humans. However, human-to-human transmission is also common in most reported cases⁶. Animal-to-human transmission can occur through bites or scratches, handling bush meat, direct contact with bodily fluids, or consuming food contaminated by rodents. The infection can also spread through direct contact with sores or bodily fluids from infected individuals⁷.

Mohbey et al.⁸ employed a convolutional neural network (CNN) with a long short-term memory (LSTM) layer hybrid model for sentiment analysis of Mpox-related tweets, achieving 91 percent accuracy in understanding public perceptions of the outbreak. Environmental factors also play a crucial role in the spread of the virus, as contaminated surfaces and materials can serve as indirect transmission pathways. Neural network-based approaches have been successfully applied to model complex nonlinear systems in areas such as dynamic distribution modeling, engineering prediction, and robust computational frameworks^9–11. Ali et al.¹² proposed a Morlet wavelet neural network (MWNN) optimized to solve nonlinear Van der Pol-Mathieu-Duffing oscillators, demonstrating its accuracy through statistical analyses. Additionally, aerosol transmission, though less common, has been observed in crowded or poorly ventilated settings, increasing the likelihood of outbreaks. Preventative measures, including proper sanitation, protective equipment, and early detection, are essential in minimizing the spread of infection¹³. Additionally, the extensive study of artificial neural network applications across various fields has been conducted, as detailed in references^14–16.

Differential equations have been successfully implemented to model a wide range of real-world problems¹⁷, spanning various applications in all the fields of science and engineering^18–20. In recent years, scientists have increasingly favored fractional differential equation models over conventional differential equation models for several reasons^21–23. Despite the numerous advantages of fractional differential equations, certain challenges may limit their applicability in specific scenarios^24,25. One common difficulty is determining the exact solution to a given problem. Due to this challenge, alternative numerical and approximate methods must be utilized^26–28. Among the various approaches available, the method developed by Adam Bashforth is particularly effective for approximating numerical solutions of fractional differential equations^29–32.

Mathematical modeling has been instrumental in understanding the transmission dynamics of Mpox and evaluating control strategies^33–35. Early models adapted frameworks from similar infectious diseases, such as smallpox, to study Mpox transmission^36–38. Over time, more specialized models have been developed to capture the unique aspects of Mpox epidemiology. Historically, Mpox has received limited attention, resulting in an insufficient understanding of its transmission mechanisms. Despite this, several studies have utilized advanced computational and neural network-based techniques to investigate complex dynamical systems and disease-related associations^39–42. In 2017, Usman et al.⁴³ examined the dynamics of Mpox in human hosts and rodents, including stability analysis. Additional and recent notable contributions to the field include studies by^44,45.

The incorporation of fractional derivatives introduces memory and hereditary properties into the model, which are essential for capturing the temporal dependencies inherent in biological processes. Unlike integer-order models, fractional-order systems account for the complete history of the system’s states, making them particularly effective in representing latency periods and delayed responses in disease progression. This enables a more realistic and accurate modeling of infectious diseases.

In recent years, numerous researchers have conducted extensive studies on fractional mathematical modeling. Peter et al.⁴⁶ explored the transmission dynamics of the Mpox virus using a mathematical approach⁴⁷. Adom-Konadu et al.⁴⁸ investigated a fractional precautionary Mpox model incorporating passenger interactions, employing the fixed-point concept and Newton polynomial interpolation. A.Venkatesh et al.⁴⁹conducted a numerical analysis of a novel fractional Mpox model over time, utilizing Caputo derivatives. Shyamsunder et al.⁵⁰ introduced a new fractal mathematical model to assess the impact of immunization on COVID-19 . Although these studies are primarily based on deterministic techniques, they do not account for critical factors such as isolation periods, hospitalized populations, or the effects of direct interactions between humans and rats^51,52. Zafar et al.⁵³ discussed the epidemiology, genomics, clinical aspects, and treatment strategies of MPXV, drawing links to smallpox and ongoing clinical trials. Elsadany et al.⁵⁴ utilized a discrete fractional model to study the maize streak plant epidemic, highlighting its biological plausibility and control strategies.

The nonlinear dynamical systems exhibit complex wave-like and oscillatory behaviors, making their computational investigation challenging. Standard neural networks with non-oscillatory activation functions such as ReLU or sigmoid often fail to efficiently approximate periodic or wave-like epidemic dynamics⁵⁵. To overcome these limitations, it is necessary to incorporate oscillatory behavior directly into the neural architecture. In this context, there is a growing need for advanced neural networks capable of capturing the intrinsic oscillatory nature of disease transmission. Harmonic activation functions provide a powerful alternative by naturally representing periodic structures within epidemiological systems. When combined with efficient optimization techniques such as stochastic gradient descent with momentum (SGDM), these networks achieve improved convergence, stability, and approximation accuracy for complex nonlinear models.

In this paper, we numerically investigate the fractional-order mathematical model for Monkeypox (Mpox) transmission using a novel neural network architecture, termed the Harmonic Neural Network (HNN). This framework employs the sine function as an activation function, enhancing the network’s capacity to capture complex, oscillatory patterns inherent in biological system dynamics. Compared to non-harmonic activation functions, the sine function excels at modeling periodic behaviors, making it particularly suitable for epidemiological applications. To optimize the HNN, we employed the Stochastic Gradient Descent with Momentum (SGDM) optimizer, which efficiently converges to minima in high-dimensional, non-convex search spaces. By incorporating momentum, SGDM reduces oscillations and accelerates convergence, resulting in more stable and efficient training. This approach allows the HNN to minimize the error between predicted and actual solutions, accurately capturing the nonlinear patterns of Mpox transmission. The proposed HNN-SGDM framework demonstrates robustness and reliability, highlighting its potential for application to a wide range of biological and epidemiological models.

• A novel application of the HNN-SGDM framework is presented to treat the nonlinear Mpox transmission model effectively and demonstrates the capability of this scheme to capture oscillatory behavior in biological systems.

• The precision and active convergence of the HNN-SGDM method are confirmed by comparing its outcomes with benchmark solutions numerical method.

• The proposed HNN-SGDM framework undergoes thorough evaluation using statistical measures such as MAE RMSE, and TIC across multiple test runs, demonstrating its reliability and robustness.

In summary, our proposed HNN-SGDM architecture offers a reliable and precise method for resolving sophisticated nonlinear epidemiological models. It provides an effective replacement for conventional numerical computations by precisely capturing the oscillatory behavior of biological systems. The credibility of the framework is ensured by extensively validating its performance using comparative analysis and statistical measures. While classical numerical solvers such as Runge-Kutta, finite volume and finite element methods are effective for single simulations, they rely on step-by-step time discretization, which can become computationally expensive for repeated simulations and parameter studies^56–58. In contrast, the proposed HNN-SGDM framework adopts a mesh-free global approximation strategy, eliminating the need for time-stepping grids and enabling efficient repeated evaluations once trained. In addition to demonstrating HNN-SGDM’s compatibility for the Mpox transmission model, this work opens the door for its possible extension to a larger class of biological and epidemiological systems.

The remainder of this paper is organized as follows: “Model formulation” Section introduces the fractional-order compartmental model for monkeypox transmission, detailing the compartments and key assumptions. “Preliminaries” Section outlines the mathematical preliminaries of fractional calculus required for the analysis. “Analysis of the dynamics” Section presents the analysis of the model, including positivity, boundedness, and the invariance of the solution domain. “Designed methodology” Section explains the architecture of the HNN designed to approximate the model’s solution. “Optimization procedure” Section discusses the SGDM optimization strategy used to train the network. “Statistical performance metrics” Section defines the statistical performance metrics used to assess model accuracy. Finally, “Results with discussion” Section provides the results along with a detailed discussion, highlighting the effectiveness and reliability of the proposed HNN-SGDM framework.

Model formulation

Manivel at al.⁵⁹ developed a fractional mathematical model using Caputo’s fractional differential equations to describe Mpox virus transmission dynamics. The approach addressed the limitations of integer-order models by incorporating fractional-order derivatives, providing a more accurate framework for understanding the disease. They conducted numerical simulations to predict the impact of isolation measures on disease transmission and created a compartmental model that considered interactions between humans and rodents. The model’s positivity, boundedness, and stability were verified, and fractional derivatives were used to regulate disease dynamics in all compartments. The human population is divided into six compartments: Susceptible humans (), Exposed humans (), Infected humans (), Quarantined humans (), Hospitalized humans (), and Recovered humans (). The total human population is represented as

Similarly, the rodent population () consists of three compartments: Susceptible rodents (), Exposed rodents (), and Infected rodents ().

A time-dependent power-law correlation function is introduced to include memory effects in the Mpox virus transmission model, which guarantees dimensional consistency of both sides of the fractional equations using the caputo derivative.

1

Following are the assumptions of the model:

• Transmission occurs through direct contact between infected and susceptible individuals.

• Fractional derivatives model memory effects in disease progression and transmission.

• The model assumes homogeneous mixing in both human and rodent populations

From an epidemiological perspective, early diagnosis is represented through transition rates that move infected individuals into treatment or isolation compartments. Vaccination is incorporated by transferring susceptible individuals into protected classes via vaccination rate parameters. The timing of vaccination is implicitly captured through these rate coefficients, which influence the speed of population transition between compartments.

Parameters are shown in Table 1. All model parameters used in this study are adopted from established literature sources, where their estimation and sensitivity analyses have already been reported. These parameters are retained to ensure consistency and to focus on validating the proposed HNN-SGDM computational framework.

Table 1.

Parameter definitions, values, and units used in the model.

Parameter	Explanation	Values	Units
	Susceptible human rate	0.029	week
	Susceptible rate for rodent	0.2	week
	Rate of contact among rodents and humans	0.052466	week
	Rate of contact human to human	0.022325	week
	Contact rate of rodent to rodent	0.4	week
ϕ	Exposed to infected rate	0.016744	week
	Quarantine people to again susceptible	0.5	week
	Exposed to quarantine rate	0.06	week
β	Infected to hospitalized rate	0.001	week
γ	Infected to recovery rate	0.0088366	week
\varepsilon	Quarantine to recover rate	0.8	week
ρ	Rate of hospitalized to recovery	0.8	week
	Human mortality rate	0.00303	week
	Rodent mortality rate	0.002	week
π	Rate of rodent exposure to infection	2.0	week
	Human dying rate due to Mpox	0.003286	week
	Rodent dying due to Mpox	0.06	week

Preliminaries

In this section, we present fundamental definitions and mathematical tools used throughout the analysis of the proposed fractional-order model.

Definition 1

(Fractional Integral in the Riemann–Liouville Sense) The fractional integral of order μ in the Riemann–Liouville sense for a function is expressed as:

where is the Gamma function^60,61.

Definition 2

(Caputo Fractional Derivative) The fractional derivative of order μ in the Caputo sense is defined as:

Lemma 3

Let and where s - 1< α < s. Then, the following conditions hold:

1. .

2. .

Analysis of the dynamics

This section analyzes the well-posedness of the proposed fractional-order Mpox transmission model by proving positivity, boundedness, and the existence of an invariant domain.

Theorem 1

Let the initial values of the state variables be specified as

If these initial values are positive, then for all time t > 0, the functions

remain positive. Additionally, we have the following bounds:

Furthermore, if the initial value then will remain bounded by for all t > 0.

The domain for solving the differential equations is defined as:

Thus, the total domain is a subset of and it is positive invariant⁵⁹.

This theorem demonstrates that given positive initial values, the state variables remain positive for all t > 0. It also shows that the human and rodent populations and are bounded above, ensuring the model’s biological feasibility and mathematical stability.

Using Lemma 1, the existence and uniqueness of the solution to the fractional-order Mpox transmission model can be established. This ensures that for the specified initial conditions, the solution remains unique and well-defined within the domain, making the model reliable for simulation and analysis⁵⁹.

Designed methodology

This research formulates the HNN architecture boosted by use of the SGDM algorithm into two distinct phases, each intended to handle the model’s inherent nonlinear behavior:

• iThe cost function is carefully designed to match the HNN’s parameters, ensuring that it accurately reflects the intricacies of the monkey-pox disease model.
• iiMore precise predictions and effective solution search in the non-convex search space arise from a structured approach for effectively optimizing the loss function that utilizes the use of the SGDM optimizer for improved convergence and endurance.

Structure of ANNs

In order to learn and approximate highly complex, nonlinear relationships within the data, artificial neural networks (ANNs) use a multi-layered architecture that is inspired by the structure and function of the neural network in the human brain^62,63. These networks are made up of multiple layers of interconnected neurons, each of which applies mathematical transformations to the data that it receives before sending the processed output to the next layer⁶⁴. ANNs are highly effective at solving a wide range of challenging tasks in fields like image recognition, natural language processing, and predictive modeling because of their capacity to capture abstract patterns and model complex dependencies⁶⁵. We have used the 15 number of neurons in the hidden layer and the model variables are represented in equation (2).

2

3

The model’s first-order derivatives of the variables are provided in equation (3). Weight vectors, acting as the trainable parameters in our system, are initialized as unknowns:

where:

with:

The error function e is given by:

4

where,

5

6

7

8

9

10

11

12

13

The corresponding fitness functions are represented by values in (5), (6), (7), (8),(9), (10), (11), (12), (13) and (14) account for initial conditions of the non-linear system whereas (12) represents total loss.

14

Optimization procedure

SGDM is one of the most commonly used optimization techniques in machine learning, especially for neural network training^66,67. It updates the parameters of the network based on the gradient of the loss function but incorporates momentum to accelerate convergence toward the optimal solution⁶⁸.

Unlike in the standard gradient descent, SGDM incorporates a momentum term in the update rule, which will damp oscillations and therefore help converge faster particularly with objective functions that are noisy or have complex structures. Such properties of SGDM make it highly effective for training deep learning models, especially the neural networks in solving the epidemic model proposed in this work.

SGDM optimizes iteratively by updating the weights of the network using both the current gradient and past updates. This theoretically allows the model to escape local minima and converge more quickly to better solutions. Compared with the basic optimization methods, SGDM improves both the convergence speed and the accuracy of the solutions. In the present study, SGDM considerably improved the training process and rendered reliable results when applied to the non-linear epidemic model⁶⁹.

While traditional gradient descent can get stuck at a point of slow convergence, especially for highly nonlinear systems, SGDM handles such challenges efficiently by providing a more stable and faster path toward optimal solutions. In this study, SGDM helped in training the ANN models accurately and efficiently; thereby, this allowed pinpointing optimal solutions for highly complex and multi-modal optimization problems, usually encountered in epidemic modeling.

Algorithm 1.

Multistage SGD with momentum (SGDM).

The algorithm implements a multistage SGDM in order to improve the optimization by applying momentum at each stage and updating model parameters iteratively by the computed gradients^70,71. This approach fastens the convergence and improves stability by imposing momentum at each stage.

Statistical performance metrics

MAE, RMSE, and TIC are also used as statistical thresholds to assess the strength and predictive power of the model. The mathematical form of these metrics for the compartments of the model are defined as follows:

15

The MAE is a measure of the average size of errors between predicted and actual values. It calculates the absolute difference between these two values and then averages them for the entire dataset. Because it’s simple, the MAE is probably the most common metric used to measure model accuracy. The lower the MAE, the closer the model predictions are to the actual values. However, it doesn’t punish larger errors quite as heavily as some other metrics.

16

To quantify prediction error, RMSE calculates the square root of the mean squared deviations between the actual and estimated values By squaring the errors, RMSE gives higher weight to larger errors and is sensitive to outliers. The RMSE would be particularly important when it becomes critical to reduce large deviations in prediction. As such, this measure is usually preferred if accuracy at high-error points is of great importance.

17

TIC is a normalized measure for forecast accuracy, comparing predicted values to actual values by taking into account their magnitudes. TIC is always between 0 and 1; the smaller the value, the closer it gets to 0, which means greater accuracy of prediction. Unlike MAE and RMSE, TIC is a relative error measure in that it normalizes the error based on the scales of the actual and predicted values. This makes it useful for comparing models across datasets with varying scales.

Results with discussion

This section provides a detailed description of the obtained results by using the proposed HNN framework optimized by the SGDM algorithm. The time domain is discretized using uniformly spaced N collocation points over a fixed time interval chosen to capture both transient and long-term dynamics of Monkeypox transmission. This ensures stable training and accurate approximation of the system behavior.

18

Now, the optimized weights have been created and substituted in the derivative equations. Using the optimized parameters of the proposed HNN model, the following expressions are formulated for each compartment:

We evaluate the model performance using some key metrics such as MAE, RMSE, and TIC. Furthermore, box plots and histograms provide a visual insight into how the model precisely converges. The model was trained using synthetic data generated from the numerical solution of the fractional-order system via Rk4 While real-world datasets were not directly incorporated in this study, the synthetic data was calibrated to reflect realistic transmission patterns derived from existing literature. The use of simulated data allows precise benchmarking of the neural network’s accuracy against known solutions, serving as a crucial step before applying the model to real epidemiological datasets in future work. Box plots and histograms can be used for the validation of the regression-based approach. From the box plots, one could determine the distribution of MAE, RMSE, and TIC in predicting the accuracy of the model. While the interquartile ranges of the medians give an indication that errors are mostly small and very consistent, the presence of outliers would suggest where in fact improvement can be done. The histogram for TIC further shows that most trials get very low values, hence proving the effectiveness of the model. On the histogram, the red curve represents the expected distribution of TIC values. Overall, the box plots demonstrated tight interquartile ranges and a lack of extreme outliers, while the histograms showed symmetric error distributions centered around zero. These patterns suggest that the model’s predictions are stable, reproducible, and not prone to random fluctuations. Although real-world monkeypox data was not directly used in this study, the low variability in error distribution highlights the model’s robustness and its capacity to generalize well , a critical requirement for future applications involving noisy or incomplete epidemic datasets.

The results in Fig. 1 demonstrate an excellent agreement between the HNN framework and the numerical solutions for the susceptible () and exposed () human populations, confirming the accuracy of the proposed model. The dynamics reflected in the results align with the equations governing and , where the susceptible population decreases over time due to infection, quarantine, and removal, while the exposed population captures the transition from susceptibility to exposure, followed by progression or removal. The HNN framework effectively models these dynamics by replicating the nonlinear interactions represented in the equations, as evidenced by the overlap with numerical solutions.

Fig. 1.

The susceptible and exposed populations of the model are compared between the HNN results and reference data.

Figure 2 further supports this by visualizing the optimized weights (m, w, and d) achieved for and . These weights correspond to the parameters used in the equations and indicate how the HNN framework adapts to represent infection rates, transitions, and external influences. The distribution of weights across neurons demonstrates the network’s ability to encode the underlying complexity of the system, ensuring a precise fit to the modeled equations. These results validate the effectiveness of the proposed framework in accurately capturing the epidemiological dynamics of the susceptible and exposed human populations.

Fig. 2.

The graphs depict the optimal weights for the human population compartments and in the model.

Figure 3 presents an analysis of the HNN and numerical solutions for the infected and quarantined human populations. The left subfigure shows the increasing trend of infected individuals, where the HNN predictions closely follow the numerical reference, highlighting the model’s ability to approximate the infection dynamics with high precision. The right subfigure depicts the rapid decline in quarantined individuals before stabilizing, demonstrating the HNN’s effectiveness in capturing both transient and steady-state behaviors in epidemiological modeling.

Fig. 3.

Comparison of the HNN and reference results for the infected and quarantined human population of the model.

The subsequent Fig. 4 visualizes the learned weights associated with these compartments, providing information on how the HNN assigns importance to different factors influencing the infection and quarantine dynamics. The weight distribution reflects the adaptive learning process of the network, capturing the underlying patterns that govern disease spread and containment.

Fig. 4.

The bar charts showing optimal weights of the model’s human population for and .

Figure 5 depicts a close match between the predictions of the HNN framework and the numerical reference values for the hospitalized () and recovered () human populations. The excellent agreement between the two sets of results over time underscores the accuracy and reliability of the HNN in approximating the dynamics of these compartments. The trends observed align well with the governing equations, where the hospitalized population evolves based on infection severity, admission, and discharge rates, while the recovered population grows as individuals transition out of hospitalization or recover directly.

Fig. 5.

Comparison HNN and reference findings for the model’s hospitalized and recovered human population.

Figure 6b presents the best-learned weights (m, w, and d) for these two compartments, as optimized by the HNN framework. These weights reflect the model’s capacity to adapt and represent the nonlinear interactions in the equations, such as recovery rates, hospital admissions, and discharges. The distribution of weights across the neurons showcases the HNN’s ability to accurately encode the underlying dynamics and external influences affecting these compartments.

Fig. 6.

The stacked graphs display the ideal weights of and human population of the model.

Figure 7 compares the HNN framework with numerical solutions for the susceptible () and exposed () rodent populations. The results show that the agreement between the two approaches is very good. The susceptible rodent population, , decreases in time due to the infection spread and removal mechanisms. The exposed population, , increases at the beginning of infections and then decreases as the disease progresses, showing that individuals move on to later stages or out of the system altogether. This is consistent with the equations given above and so demonstrates the ability of the HNN to accurately model complicated nonlinear interactions.

Fig. 7.

Comparison of the HNN and numerical results for susceptible and exposed rodents population of the model.

Figure 8 offers additional verification and shows the learned optimal weights (m, w, and d) for the susceptible and exposed rodent compartments. They have learned parameters optimized by the HNN framework for the best emulation of the epidemiological processes. For the susceptible population, , the weights capture the balance of infection rates, natural removal, and coupling with other compartments. For the exposed population, , similarly, the weights bring out the capability of the framework to model progression and transition rates.

Fig. 8.

Bar graphs of the optimal weights of and of the rodents population model.

Figure 9 gives the comparison between the HNN framework and numerical solutions for the infected rodent population. Obviously, the infected population first undergoes a phase of exponential growth reflecting the transmission dynamics before stabilizing to an equilibrium value as the system tends toward an equilibrium state. This is consistent with the predicted trajectory of the underlying mathematical model and further confirms that HNN is capable of faithfully reproducing the complex nonlinear interactions and transitions of the system.

Fig. 9.

The bar graphs compare HNN and numerical results for infected rodents using the optimal weight values.

Panel (b) of Fig. 9 depicts learned weights by the HNN to simulate the infected rodent population. The weights symbolize those parameters to which the network has been adapted to apprehend the principal infection progression features, such as transmission dynamics, recovery processes, and influences from external forces. Such a spread of these weights throughout the neurons of the network brings out the adaptability of the HNN and its potential in encoding complicated patterns that govern the epidemiological system. The visual enhancement puts emphasis on the strength of the HNN, which goes beyond fitting the numerical solutions but, more so, in its ability to distinguish and model meaningful system dynamics.

These depict the performance of the proposed model in approximating the Susceptible (), Exposed (), and Infected () human populations affected by Mpox. Figure 10 compares the MAE values for , , and over numerous trials, which consistently show reductions in error, thereby further ascertaining the robustness of the model in capturing the dynamics of Mpox transmission. To evaluate the computational efficiency of the proposed HNN-SGDM framework, simulations were conducted using graphical processing unit(GPU). While classical numerical solvers such as Runge–Kutta methods provide faster solutions for single simulations, the proposed approach offers competitive accuracy along with meshfree formulation and global approximation capability, making it suitable for complex nonlinear dynamical systems. Figure 11 gives more ideas through histogram plots, whose results show that in all compartments, the MAE values were mostly concentrated at the smaller ranges, evidencing a high predictive accuracy.

Fig. 10.

Comparison of MAE values for succeptible, exposed and infected human population of the model.

Fig. 11.

Histograms of MAE values representing convergence measures for the succeptible, exposed and infected human population of the model.

The box plots shown in Fig. 12 display MAE values, representing the way the model predictions were stable, given that very low median error occurred in all compartments. Although some of the realizations exhibit outliers, especially for the infected population , deviations are rather minimal and can easily be explained by the complex non-linear nature of the infection dynamics in Mpox disease.

Fig. 12.

Box Plots of MAE values depicting convergence measures for the succeptible, exposed, and infected human population of the model.

Now we test the performance of this model in the estimation of a susceptible, exposed, and infected human population affected by Mpox using the RMSE metric. Figure 13 depicts the comparison of RMSE for the number of trials, showing that the model was able to capture the complex transmission dynamics of Mpox. The curves for , , and show a similar trend, that the RMSE values remain low, indicating the predictive reliability of the model.

Fig. 13.

Comparison of RMSE values for succeptible, exposed and infected human population of the model.

The histograms in Fig. 14 investigate further the distribution of RMSE for , , and . High concentrations of values near the lower ranges indicate that the model has been successful in keeping the error minimal. This is supported by the probability density curves, which have an even tighter spread for and compared to the slightly wider spread for to capture increased uncertainty within the exposed population.

Fig. 14.

Histograms of RMSE values representing convergence measures for the succeptible, exposed and infected human population of the model.

The RMSE of every compartment, as revealed by Fig. 15, identifies the performance of the model across various compartments. The low median RMSE and relatively compact interquartile ranges for all three compartments underline the stability of the model in approximating the populations. Even though there are outliers–more so for the infected population, –these are minimal and expected due to the intrinsic variability in infection progression.

Fig. 15.

The convergence of the susceptible, exposed, and infected human populations is illustrated through box plots of RMSE values.

Figure 16 compares the values of , , and over time, demonstrating consistent accuracy across trials with clear differences among the human compartments.

Fig. 16.

Comparison of TIC values between succeptible, exposed and infected human population of the model.

Figure 17 present histograms of TIC values, all showing high densities in the lower ranges, which is an indication of strong predictive performance for each compartment. Similarly, Figure 18 presents box plots for variability in TIC values; these depict a stable model with low spread, although the outliers do highlight areas possibly requiring further improvement.

Fig. 17.

Histograms of TIC values depicting convergence measures for the succeptible, exposed and infected human population of the model.

Fig. 18.

Box Plots of values of TIC showcasing convergence measures for susceptible, exposed and infected human population of the model.

In Fig. 19, MAE values are compared for the quarantined, hospitalized, and recovered human populations, demonstrating the model’s effectiveness in capturing distinct epidemiological trends. The lower MAE values for the hospitalized () and recovered () compartments indicate that the model consistently predicts their population dynamics with high accuracy. In contrast, the quarantined population () exhibits slightly higher MAE values, reflecting the inherent complexity and variability in the progression of this compartment.

Fig. 19.

Comparison of MAE between quarantined, hospitalized and recovered human population.

Figure 20 presents the histogram of MAE values in multiple trials, highlighting the distribution of errors for each compartment. The sharper peak in the distributions of hospitalized and recovered populations suggests strong predictive stability, whereas the relatively broader spread for the quarantined group underscores the challenges associated with modeling its dynamics. Figure 21, is giving the box plots of MAE values for the three compartments.The quarantined compartment shows a wider range and occasional outliers, expressing potential fluctuations that could arise due to variations in quarantine protocols and external interventions.

Fig. 20.

Histograms of MAE values depicting convergence of quarantined, hospitalized and recovered human population.

Fig. 21.

Box plots illustrating MAE values reveal the convergence behavior for the quarantined, hospitalized, and recovered compartments in the model.

Figure 22 is giving the RMSE comparison, which proves that the model really traces the population dynamics of multiple trials and hence is adaptable. The clear separation of RMSE trends among these three groups focuses on the model’s ability to distinguish between distinct population behaviors. Histograms in Fig. 23 further validate the model performance by showing well-defined distributions of RMSE values. The peaked shapes for the hospitalized () and recovered () groups indicate consistent predictive performance, while the broader distribution for the quarantined () population highlight the unique challenges posed by this group. These histograms, complemented by the fitted probability density functions, gives statistical perspective on model accuracy.

Fig. 22.

A comparison of RMSE values between quarantined, hospitalized and recovered human population of the model.

Fig. 23.

RMSE histograms are displayed to demonstrate the convergence patterns for the quarantined, hospitalized, and recovered compartments of the model.

The box plots Fig. 24 shows that the model achieves strong convergence for hospitalized and recovered populations, as evidenced by the smaller inter-quartile ranges and minimal outliers.Most Importantly, the box plots demonstrate the model’s capacity to manage a wide range of scenarios, making it a versatile tool for understanding population dynamics.

Fig. 24.

Box Plots of RMSE depicting convergence measures for the quarantined, hospitalized and recovered human population of the model.

A study of the TIC value across the hospitalized, recovered, and quarantined compartments is displayed in Fig. 25, which clearly demonstrates the model’s capacity to accurately reflect the unique dynamics of each group. Histograms of TIC values for these compartments are shown in Fig. 26, providing information on the distribution and convergence of the inequality coefficients. The performance of the model and its consistency in capturing population behavior may be seen through the histograms as they depict that quite a large part of TIC values lie within low ranges.

Fig. 25.

Comparison of TIC values between Quarantined, Hospitalized and Recovered human Population of the Model.

Fig. 26.

Histograms of TIC values illustrating convergence measures for the quarantined, hospitalized, and recovered human populations in the model.

Box plots of TIC values are shown in Fig. 27, representing the range and unpredictability seen over numerous trials. With a small number of outliers and a steady range of values, the box plots emphasize how reliable the model is and how well it depicts the underlying population dynamics.

Fig. 27.

Box plots depicting TIC values to represent convergence measures for the quarantined, hospitalized, and recovered populations within the model.

Figure 28 presents the MAE comparison for the susceptible (), exposed (), and infected () rodent populations, demonstrating the model’s ability to capture variations in disease spread. The consistently low MAE values for and indicate stable predictive performance, while slightly higher values for reflect the increased complexity in modeling infection dynamics. Figure 29 provides histograms of MAE values, revealing well-defined distributions for and , reinforcing the model’s precision. The slightly broader spread for suggests more variability in the infected population, which aligns with the expected unpredictability in disease progression. Figure 30 presents box plots of MAE values, confirming the model’s accuracy, as evident from the narrow interquartile ranges and minimal outliers.

Fig. 28.

Comparison of MAE values between succeptible, exposed and infected rodents population of the model.

Fig. 29.

Histograms of values of MAE showing convergence measures for the succeptible, exposed and infected human population of the model rodent population of the model.

Fig. 30.

The convergence behavior of the susceptible, exposed, and infected rodent populations is shown through box plots of MAE values.

Figure 31 provides an evaluation with RMSE for these compartments, showing that model maintains low RMSE values for and , with exhibiting slightly greater variability due to its dynamic nature. Figure 32 provides histograms of RMSE values, which indicate a strong concentration of errors in the lower range for and , demonstrating reliable predictions. In contrast, has a more dispersed distribution, reflecting the natural fluctuations in infection dynamics. Figure 33 presents box plots of RMSE values, highlighting that while errors remain consistently low for most trials, minor outliers exist, particularly in , where small deviations may occur due to model constraints. Finally, Figure 34 compares TIC values for , , and , confirming the model’s ability to capture distinct epidemiological behaviors. Figure 35 presents histograms of TIC values, illustrating both their distribution and convergence patterns, showing that most values remain within a stable range.

Fig. 31.

Comparison of RMSE between succeptible, exposed and infected rodents population.

Fig. 32.

RMSE histograms illustrate the convergence measures for the susceptible, exposed, and infected rodent populations within the model.

Fig. 33.

Box Plots of RMSE values giving convergence for the succeptible, exposed and infected rodent population.

Fig. 34.

Comparison of values of TIC between succeptible, exposed and infected rodents population of the model.

Fig. 35.

Histograms of values of TIC showing convergence measures for the succeptible, exposed and infected rodent population of the model.

Figure 36 displays box plots of TIC values, demonstrating the range and variability across various trials. The limited presence of outliers, combined with the overall consistency of the results, validates the robustness of the model in tracking rodent population dynamics.

Fig. 36.

Box Plots of values of TIC showing convergence for the succeptible, exposed and infected rodent population of the model.

The reported residual errors, ranging from to , are comparable to or better than those obtained using conventional numerical solvers such as the RK4. This level of accuracy indicates that the HNN-SGDM approach not only approximates the solution effectively but also does so with improved generalization capabilities due to its neural network structure. For real-world forecasting, such precision enhances the credibility of predictions, especially in capturing subtle dynamical features like delayed peaks or long-tail recovery phases.

Conclusion

In conclusion, we have used HNN optimized with SGDM to successfully approximate the solution of the non-linear Mpox disease model. With the help of reference solutions, the non-linear Mpox model, which is composed of nine correlated non-linear differential equations, has been accurately solved. Our study has achieved its objective of utilizing a deep learning-driven paradigm to solve the complex and highly non-linear models. The complexities related to the model are successfully addressed by the suggested HNN-SGDM solver. Plotting ideal weight vectors, comparing results, and calculating a reducible absolute error between and , and all these parameters supported the accuracy and stability of the suggested approach. Additionally, the accuracy and consistency of the HNN-SGDM solver are also validated by statistical evaluations from several trials consisting of histograms and box plots. Despite its advantages, the proposed framework has certain limitations, including scalability challenges, stiffness handling in highly nonlinear systems, and lack of real-data calibration. Future work will focus on addressing these issues and extending the framework to more complex and data-driven epidemiological models.

Author contributions

Nimra Shoket contributed to the Methodology, Investigation, analyzing the data curation, designing the experiments, data analysis, computation, Matlab calculation verifications, and wrote the initial draft of the paper. Abdul Mannan contributed to the computation and investigated and editing the final draft of the paper. Jamshaid Ul Rahman contributed to supervision, Methodology, conceptualization, software. Ebraheem Alzahrani contributes to formal analyzing experiments, software and funding resources. Osman Abubakar Fiidow contributes technical feedback, editing and drafting the final version. All authors read and approved the final version.

Funding

There is no funding to support this article.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.