Anthrax is caused by the bacterium Bacillus anthracis and is considered a zoonotic condition, meaning it affects both animals and humans. The pathogen can persist in soil for long periods, making it a persistent threat to farming communities. Animals can contract the bacteria by inhaling spores, coming into direct contact with infected livestock, or consuming contaminated plants or water. The disease manifests in several clinical forms, including cutaneous, pulmonary, gastrointestinal, and septicemic anthrax, with symptoms ranging from fever and exhaustion to bleeding and difficulty breathing. Because the spores can even germinate roughly a week after an animal dies, rapid assessment, confinement, and vaccination protocols are considered essential to preventing outbreaks.

The research team framed the anthrax infection in animals, abbreviated AIA, as a system of nonlinear differential equations dividing the population into four compartments: susceptible, infected, recovered, and vaccinated animals. This compartmental structure, built on an epidemiological framework originally established by Githire and colleagues, tracks how each group changes over time under the influence of parameters such as infection rates, vaccination rates, recovery rates, and natural mortality. Solving such nonlinear systems analytically is often impossible, which is why computational approaches are so valuable to epidemiologists seeking to understand outbreak behavior.

At the heart of the new method is a two-hidden-layer neural network, or TLNN, an architecture the authors argue outperforms single-layer networks at capturing intricate and nonlinear input-output relationships. The first hidden layer contains 15 neurons and the second contains 25, both using sigmoid activation functions that squash inputs into values between zero and one. According to the researchers, this layered design allows the network to acquire hierarchical data representations: the first layer extracts fundamental features from the input, while the second layer combines them into higher-level features that improve performance on complicated systems. The sigmoid function, with its smooth S-shaped curve, is differentiable everywhere, which makes it well suited to gradient-based optimization techniques.

Training the network relies on the Levenberg–Marquardt backpropagation algorithm, a hybrid optimization scheme that combines the strengths of the Gauss-Newton method and the steepest descent technique. The algorithm adaptively regulates both the step size and the direction of parameter updates to minimize the sum of squared errors between actual and predicted outputs. The authors note that this makes LMBP particularly appropriate for nonlinear systems, offering rapid convergence and reduced complexity in selecting the learning rate. The technique has recently been applied to problems ranging from coronavirus detection in robotic systems to typhoid fever modeling and bioconvective nanofluid dynamics, underscoring its versatility across scientific domains.

To generate the training data, the team first produced reference solution trajectories using the Euler numerical method. These trajectories were then randomly divided into three subsets: 69 percent for training, 15 percent for validation, and 16 percent for testing. The authors emphasize an important nuance of their approach: the neural network does not directly minimize the residuals of the governing differential equations during training. Instead, it learns a nonlinear correspondence between the independent variable and the Euler-generated state trajectories, functioning as a supervised trajectory approximation method. The researchers are careful to stress that the trained network operates as a trajectory-specific emulator for each fixed set of initial conditions and parameters, rather than as a generic solution operator capable of generalizing to unencountered configurations.

The team evaluated their framework across three computational test cases, each using the same parameter values but different initial conditions for the four population compartments. The authors caution that these cases serve purely as numerical test setups for evaluating trajectory-approximation performance and should not be interpreted as epidemiologically calibrated outbreak scenarios, since the model was not formally nondimensionalized and the initial values are not normalized animal populations.

The results demonstrated impressive accuracy. Gradient performance values were recorded as 3.5541 × 10^−2, 1.2438 × 10^−4, and 2.3666 × 10^−2 at iterations 97, 1500, and 69 for the three cases, with smaller gradients signaling convergence toward an error minimum. Optimal validation values of 1.4477 × 10^−2, 2.9452 × 10^−2, and 4.2935 × 10^−2 were achieved at the same iterations, indicating strong generalization to unseen data without overfitting. Error histograms of 3.986 × 10^−3, 2.062 × 10^−3, and 2.806 × 10^−2 further revealed minimal bias and outliers, while regression analyses across all cases showed near-ideal modeling of the target trajectories.

Comparisons between the network’s outputs and the reference Euler solutions revealed overlapping curves and small absolute errors for all four compartments—susceptible, infected, recovered, and vaccinated populations—confirming the correctness of the TLNN-LMBP scheme. The authors highlighted several advantages of their design: the Euler-generated trajectories provide a flexible and reliable reference dataset, the two-layer sigmoid structure offers a versatile nonlinear model for the AIA trajectories, and the LMBP optimizer enables efficient minimization of the estimation error. Once trained, the network delivers a straightforward approximation of the solution trajectories without requiring repeated numerical integration.

The study arrives at a moment when machine learning is increasingly deployed against infectious disease challenges, from diabetes regulation models to monkeypox transmission systems and computer virus epidemics. Anthrax itself carries significance beyond veterinary medicine, informing medical and policy decisions about biological warfare hazards and One Health strategies that protect both animal and human health. While the authors are measured in their claims—noting that the outcomes are confined to their specific 15-and-25-neuron architecture and that no declaration of architectural excellence is made—the work demonstrates how supervised neural networks can serve as fast, accurate emulators of complex epidemic dynamics. The code that generates the reference trajectories, trains the networks, and reproduces all tables and figures is available from the authors, and the study received support from Prince Sattam bin Abdulaziz University.

Subject of Research: Neural network approximation of a nonlinear anthrax epidemic model in animals

Article Title: A two-hidden-layer Levenberg–Marquardt neural network for supervised trajectory approximation of a nonlinear anthrax in animals epidemic model

Article References: Panthi, V., Kashyap, N., Gupta, M., Alenizi, F. A., Suanpang, P., & Nanthaamornphong, A. (2026). A two-hidden-layer Levenberg–Marquardt neural network for supervised trajectory approximation of a nonlinear anthrax in animals epidemic model. Discover Artificial Intelligence, 6(1), Article 1234. https://doi.org/10.1007/s44163-026-02326-5

Image Credits: AI Generated

DOI: 10.1007/s44163-026-02326-5

Keywords: anthrax, neural network, Levenberg-Marquardt backpropagation, epidemiological modeling, sigmoid activation function, machine learning, livestock disease, Bacillus anthracis, nonlinear dynamical systems, trajectory approximation, Euler method, zoonotic disease

The motivation behind the work stems from a sobering global picture. The World Health Organization estimates that approximately 1.4 million people die each year from waterborne diseases, and bacterial contamination of drinking water remains among the most consequential water quality threats, with implications spanning public health, environmental stability, and national security. Conventional analytical techniques for detecting dipicolinic acid include high-performance liquid chromatography, mass spectrometry, surface-enhanced Raman spectroscopy, electrochemical methods, and immunoassays. While several of these approaches achieve impressive sensitivity, they typically demand sophisticated instrumentation, specialized substrates or electrodes, and laborious sample preparation, all of which limit their usefulness for rapid, high-throughput screening in the field or in resource-constrained settings.

Fluorescent metal-organic frameworks, or MOFs, have emerged as a compelling alternative. These crystalline materials are built from metal nodes connected by organic linkers, forming porous architectures whose photoluminescence can be switched off, or quenched, when a target molecule binds inside. Many previously reported DPA sensors have relied on lanthanide metals such as europium and terbium, which are expensive and often require intricate multi-step syntheses. The research team, led by Mohammad Rezvani Ghalhari with Kamyar Yaghmaeian and Ghader Ghanizadeh, chose cobalt instead, citing its greater availability, high stability in aquatic environments, and suitability for fluorescence-based pollutant detection.

The synthesis itself was deliberately straightforward. Cobalt nitrate hexahydrate and 2,2′-diamino-4,4′-biphenyldicarboxylic acid were dissolved in a water and dimethylformamide mixture, sonicated, sealed in a Teflon-lined stainless-steel autoclave, and heated at 150 degrees Celsius for six hours under autogenous pressure. The resulting crystalline powder was collected by centrifugation, washed repeatedly to remove unreacted precursors, and vacuum dried. A battery of characterization techniques confirmed the material’s quality. Field-emission scanning electron microscopy revealed well-defined polyhedral particles with triangular to hexagonal prismatic shapes, smooth flat surfaces, and sharp edges, evidence of controlled nucleation and growth. Energy-dispersive X-ray spectroscopy verified the incorporation of cobalt into the organic framework without detectable impurities.

Structural analysis reinforced the picture of a well-formed sensor platform. Powder X-ray diffraction showed sharp peaks at 2-theta values of approximately 10, 19, and 24 degrees, corresponding to the (100), (200), and (210) crystal planes of a highly ordered porous framework, with the absence of amorphous halos indicating high phase purity. Fourier-transform infrared spectroscopy identified the functional groups of the aminated biphenyl dicarboxylic acid linker and, crucially, a band at 630 wavenumbers assigned to Co-O stretching, direct spectroscopic evidence that cobalt centers had coordinated with the carboxylate oxygen atoms. Thermogravimetric analysis showed the framework remained stable up to about 300 degrees Celsius, far beyond the ambient conditions of any water-sensing application.

With the material characterized, the team optimized the sensing conditions. Fluorescence measurements, taken with excitation at 332 nanometers and emission monitored at 425 nanometers, peaked at neutral pH 7, which conveniently matches the natural pH of drinking water. Under acidic conditions, DPA becomes protonated and the framework’s carboxylate linkers risk partial protonation, both of which suppress the response. The optimal sensor concentration proved to be 50 milligrams per liter; below that, signals were weak because too few fluorescent centers were available, while above it the intensity plateaued as the system saturated. Most strikingly, the interaction between DPA and the probe reached equilibrium within just one minute of contact, faster than the three-minute reaction time reported for comparable europium-based sensors.

Under these optimized conditions, the sensor delivered strong analytical performance across a linear range of 0 to 100 micromolar, with a correlation coefficient of 0.9891. The limit of detection was 0.089 micromolar and the limit of quantification 0.271 micromolar, figures competitive with many lanthanide-based MOF probes reported previously. At the highest tested DPA concentration, the emission intensity fell to roughly 55 percent of its original value, a pronounced turn-off response. Selectivity testing against common drinking water ions, including potassium, sodium, magnesium, calcium, chloride, phosphate, and sulfate, as well as structural analogs such as picolinic acid, 2,4-pyridinedicarboxylic acid, and benzoic acid, produced essentially no quenching, confirming that the sensor responds specifically to DPA. Stored in the dark at room temperature, the material retained the vast majority of its luminescence over 90 days.

The team also dissected the photophysical mechanism behind the quenching. Förster resonance energy transfer was ruled out because DPA absorbs only in the deep ultraviolet, below 280 nanometers, while the MOF emits at 425 nanometers, leaving no meaningful spectral overlap. The inner filter effect was likewise excluded because the excitation wavelength falls outside DPA’s absorption envelope. Instead, the evidence points to coordination-induced static quenching, potentially accompanied by photoinduced electron transfer. DPA is an effective tridentate ligand, chelating the open cobalt(II) sites through its pyridine nitrogen and two carboxylate groups to form a stable ground-state complex. Its electron-deficient character then allows it to act as an electron sink, drawing photoexcited electrons away from the framework and suppressing radiative recombination, which manifests as rapid fluorescence loss.

Practical validation followed two tracks. In spiked real water samples, the sensor’s recoveries ranged from 97.31 to 101.68 percent with relative standard deviations between 0.13 and 1.49 percent, statistically indistinguishable from results obtained by high-performance liquid chromatography, whose recoveries spanned 99.1 to 100.27 percent. Calculated p-values all exceeded 0.05, confirming no significant difference between the two methods. In a diagnostic validation involving 48 cultured water samples, 24 inoculated with Bacillus anthracis and 24 with Escherichia coli as negatives, the sensor achieved 92 percent sensitivity, 95.6 percent specificity, a positive predictive value of 95.8 percent, a negative predictive value of 91.6 percent, and an overall accuracy of 93.75 percent. The authors note that high sensitivity is critical for biosecurity, where missed detections carry severe consequences, while high specificity minimizes false alarms that could trigger unnecessary emergency responses.

The researchers conclude that the cobalt-based framework offers a rapid, cost-effective, and reliable tool for anthrax biomarker monitoring in real water samples, combining a one-minute response, a detection limit rivaling lanthanide systems, and stability suitable for extended practical use. By swapping rare and costly metals for abundant cobalt and a simple hydrothermal recipe, the study lowers a significant barrier to deploying fluorescent MOF sensors outside specialized laboratories. As concerns about water safety and biodefense continue to intersect, sensors of this kind could become a routine first line of screening, flagging contaminated supplies in minutes and reserving slower, instrument-heavy confirmatory methods for the samples that truly warrant them.

Subject of Research: A cobalt-based metal-organic framework fluorescent sensor for rapid detection of dipicolinic acid as a Bacillus anthracis biomarker in drinking water

Article Title: A Co-based MOF for rapid and facile fluorescent detection of dipicolinic acid as an anthrax biomarker

Article References: Ghalhari, M. R., Yaghmaeian, K., & Ghanizadeh, G. (2026). A Co-based MOF for rapid and facile fluorescent detection of dipicolinic acid as an anthrax biomarker. Results in Chemistry, 30, Article 103826. https://doi.org/10.1016/j.rechem.2026.103826

Image Credits: AI Generated

DOI: 10.1016/j.rechem.2026.103826

Keywords: metal-organic framework, cobalt, dipicolinic acid, anthrax, Bacillus anthracis, fluorescence sensing, water safety, biosensor, bioterrorism detection, drinking water, nanosensor, fluorescence quenching