Jurnal Diferensial

A Hybrid Semi-Analytical Technique for the Homogeneous Space Fractional Damped Wave Equation with Gaussian White Noise

2026-05-16T04:56:45+00:00

This paper addresses the severely ill-posed final value problem for the homogeneous space fractional damped wave equation subject to Gaussian white noise. Unlike the well-posed forward problem, recovering the initial state from noisy final data is unstable, as high-frequency noise components are amplified exponentially. We propose the Laplace-Residual Power Series Method (LRPSM), a semi-analytical iterative technique, to solve this problem. By transforming the backward problem into a time-reversed initial value problem, we construct a series solution in the Laplace domain. We provide a rigorous theorem and proof regarding the convergence of the method for exact data and discuss its regularizing properties via series truncation for noisy data. A numerical example is presented to illustrate the accuracy and stability of the proposed method compared to standard Fourier truncation techniques.

Modeling the Impact of Mint–Leaf Therapy and Sanitation on Malaria Transmission among Infants

2026-09-01T13:36:59+00:00

Malaria transmission among infants remains a persistent concern to health care practitioners in regions with this disease equilibrium, motivated by the assessment of complementary intervention alongside conventional treatment strategies can help submerge this prevalence. This study investigate the combined effects of mint–leaf (Mentha piperita) herbal therapy and environmental hygiene on malaria dynamics within infant population. The human population is partitioned into susceptible, exposed, infectious, mint–treated and recovered compartments, while the vector dynamics are represented by the infected mosquito population. To enhance epidemiological realism, mosquito-to-human transmission is described by a linear saturated force of infection that incorporates sanitation effects and restricts unbounded growth of infection pressure at elevated vector densities. Two time-dependent control functions are embedded in the model: mint–leaf therapy, which augments treatment and recovery rates and sanitation measures, which attenuate bidirectional transmission between humans and mosquitoes. The model further incorporates newborn recruitment, inflow of infected infants, natural mortality, waning immunity and mosquito population turnover. Qualitative analysis confirms the well-possedness of the model by establishing positivity and boundedness of solutions within a positively invariant region. Numerical simulation was carried out using the Laplace adomian decomposition method to demonstrate the effectiveness of the control measures. Simulation results has a significant reduction in infectious infants and infected mosquitoes under enhanced herbal treatment and sanitation coverage. This model provides a robust means to evaluate malaria control strategies through mint leaf and offers a foundation for further stability, sensitivity and optimal control analysis of this therapy.

A Hermite Collocation Framework for Nonlinear SEIMAR Modeling of COVID-19 Transmission Dynamics

2026-09-01T13:41:21+00:00

Background: The Coronavirus-related infection requires advanced mathematical models to grasp the nature of the transmission and analyze therapeutic interventions. Multi-compartment epidemic models with their nonlinear complexities are difficult to tackle traditionally by numerical means, especially when treatment strategies are considered.

Purpose: The present work designs and validates the Hermite Collocation Method (HCM) to solve the SEIMAR (Susceptible-Exposed-Infected-Masked-Antiviral-Recovered) COVID-19 epidemic model and compares the performance of the method to the already existing Laplace Adomian Decomposition Method (LADM).

Methods: A six-compartment deterministic SEIMAR model had been developed that includes monoclonal antibody therapy and antiviral treatment compartments in addition to the traditional epidemiological states. HCM was used as the solver of the model because it uses orthogonal Hermite polynomials as the basis functions and transforms the system of nonlinear ordinary differential equations to algebraic equations based on collocation points. The estimation of the parameters was done using the data provided by Nigeria Centre of Disease Control and initial conditions are based on the real-world situation of Plateau State in Nigeria.

Findings: The HCM converged better than LADM and had spectral accuracy as it approximates all six compartments. Numerical experiments showed that the susceptible population dynamics were under the exponential assumptions of the baseline conditions and the errors of convergence to the correct values declined systematically with the succeeding values of the Hermite polynomials. This approach was able to effectively model the nonlinear interactions of disease transmission and therapeutic intervention showing a 40 percent improvement in computational efficiency as compared to the traditional methods. The stability and reliability of HCM solutions was verified by error analysis and absolute errors between successive approximations dropped by orders of magnitude.

Conclusions: Hermite Collocation Method is a powerful, computationally inexpensive approach to the modelling of complex COVID 19 dynamics of transmission within multiple treatment compartments. HCM is better in the convergent properties and spectral accuracy and is thus well-suited to policy decision-making situations when there is a need to have an accurate epidemic projection. Such a strategy has great benefits in terms of assessing therapeutic methods and can be easily applied to other multi-compartment epidemiological models

An Integrated Mathematical Model for Detecting Known and Unknown Fraud Patterns with Optimal Resource Allocation in the Nigerian Social Security System

2026-09-01T13:54:23+00:00

Social security systems in developing countries face substantial fraud losses threatening system
sustainability, yet resource-constrained agencies can investigate only a small fraction of claims.
We present a comprehensive mathematical framework integrating Bayesian logistic regression for
known fraud patterns, Gaussian mixture model-based anomaly detection for novel schemes, and
optimal resource allocation under capacity constraints. Temporal adaptation through exponential
smoothing maintains detection performance as fraud patterns evolve quarterly. Our greedy allo-
cation policy, derived through marginal benefit analysis, provably minimizes expected cost under
investigation capacity constraints. Empirical validation on 10,000 pension claims from Nigerian state
bureaus (Lagos, Rivers, Kano) collected over 18 months demonstrates operational effectiveness. The
integrated framework achieves area under ROC curve of 0.811 with precision of 84% and recall of
67% on realistic data exhibiting class overlap, label noise (3%), and missing values (5%). Greedy
resource allocation detects 7–11 times more fraud than random selection across all capacity levels,
translating to estimated annual savings of 165 million ($122,000 USD) for Lagos bureau alone and
2 billion ($1.5 million USD) nationally. Sample complexity analysis reveals that 1,000–2,000 inves-
tigated claims suffice for reliable deployment, achievable within 13–18 months for most Nigerian
bureaus. Temporal adaptation with exponential smoothing parameter α = 0.70 maintains stable
performance despite quarterly fraud pattern evolution, preventing the 13-point AUC degradation
observed in static models over 12-month horizons.