This paper proposes a novel mathematical model based on a dynamic mixture of Erlang and exponential distributions to capture complex degradation processes in heterogeneous systems. The core feature of the model is the use of sigmoidal weighting functions, defined as ω(x) = 1 / (1 + exp(−r(x − a))), where r > 0 and a > 0. This weighting mechanism allows the model to naturally switch dominance between mixture components, mimicking realistic physical stages of defect nucleation, propagation, and accelerated wear—a critical aspect for effective Prognostics and Health Management (PHM) and remaining useful life prediction.

The scientific novelty lies in the development of an analytical method for calculating the normalization constant C of the dynamic mixture. Unlike traditional approaches that rely solely on numerical integration, the proposed method yields a closed-form expression for the normalization constant using the Lerch transcendent function. Specifically, C = (λ/r) * Φ(−exp(ra), 1, λ/r) + 1 − (μ^k / r^k) * Φ(−exp(ra), k, μ/r). This provides a precise analytical representation, significantly increasing calculation speed and stability in most scenarios. For cases where the Lerch function exhibits computational instability in modern CAS, an approximate method based on the Sommerfeld expansion for Fermi–Dirac type integrals is proposed. The error of this method is controlled by subsequent expansion terms, ensuring the precision required for engineering applications.

Model parameters are estimated via Maximum Likelihood Estimation with gradient-based optimization. To ensure reliable convergence, a dedicated initialization procedure based on the method of moments is proposed to establish robust initial values.

The proposed model was tested on synthetic data with known parameters and on real degradation trajectories from the NASA C-MAPSS dataset. The results indicate accurate parameter recovery in simulations and superior adaptability over conventional static mixtures when modeling nonlinear degradation in real heterogeneous systems.

The complex and non-linear dependencies inherent in emerging financial markets necessitate advanced filtering and allocation techniques to ensure portfolio resilience. This study proposes a novel network-tuned asset allocation model for the PRIBUMI Bursa Malaysia Zakat Index by integrating the Triangulated Maximally Filtered Graph (TMFG) approach with Shapley value-based optimization. Unlike traditional Mean-Variance models that often fail under extreme market correlations, this framework leverages network topology to enhance stock selection and weight distribution. The methodology is structured into four distinct phases. First, we process daily closing prices of Pribumi-indexed stocks to compute log returns and construct a high-dimensional correlation matrix. Second, the TMFG algorithm is applied to filter the network, preserving the most significant hierarchical dependencies while removing noisy links. Third, we analyze the resulting graph using centrality measures specifically degree, closeness, and betweenness alongside a calculated Peripheral Index to identify assets with high diversification potential. Finally, the selection is fed into a Shapley value framework, a cooperative game theory method that assigns portfolio weights based on each stock's marginal contribution to the collective risk-adjusted performance. The findings indicate that the TMFG-filtered network effectively isolates core influential stocks from peripheral diversifiers, with the latter often providing superior stability during periods of market stress. By utilizing the Shapley value for final allocation, the model ensures a fair and mathematically rigorous distribution of weights that accounts for the multifaceted interactions between assets. This research provides a robust tool for institutional investors and policymakers to mitigate contagion risk and optimize returns within the Malaysian equity market, offering a sophisticated alternative to conventional diversification strategies.

Introduction: Influenza remains a major public health concern due to rapid transmission and seasonal recurrence, straining healthcare systems. Mathematical models help understand disease dynamics and evaluate interventions. Vaccination is highly effective, yet resource constraints require optimally designed policies. This study develops a controlled SEIR model to investigate efficient vaccination strategies for influenza.

Methods: A time-dependent vaccination rate is introduced, representing the proportion of susceptible individuals immunized over time. The model incorporates a vaccine efficacy parameter, accounting for imperfect protection. Fundamental properties, including positivity and boundedness of solutions, are established. The basic reproduction number R₀ was derived using the next generation matrix approach. An optimal control problem was formulated to minimize infectious individuals and vaccination costs over a fixed horizon. The objective functional uses a quadratic control cost, reflecting nonlinear costs and diminishing returns in public health interventions. Pontryagin's Maximum Principle provides necessary optimality conditions, solved numerically via forward-backward sweep with fourth order Runge–Kutta discretization.

Results: Using influenza-calibrated parameters with R₀ approximately 1.8, optimal vaccination significantly suppresses epidemic spread. Simulations show peak infection reduced by 62% and total cumulative cases by 47% compared to uncontrolled scenarios. The optimal strategy also achieves 23% lower cumulative vaccination costs than constant policies while maintaining strong control.

Conclusions: Dynamically adjusted vaccination strategies substantially outperform static interventions for influenza. This study demonstrates the value of optimal control frameworks, incorporating vaccine efficacy and nonlinear costs, to guide efficient public health policies under resource constraints.

This work establishes a complete mathematical framework for determining the optimal switching frequency in pulse width modulated systems, resolving a longstanding dimensional inconsistency in the literature. Two competing objectives govern these systems: efficiency, which decreases with switching frequency due to switching losses, and harmonic distortion, which also decreases with frequency through improved waveform reconstruction. This monotonic conflict in a single independent variable forms the core mathematical challenge. Two key insights are introduced. First, dimensional analysis shows that the harmonic distortion coefficient has physical units of frequency raised to the power 0.65 rather than being dimensionless as commonly assumed. The correct formulation expresses it as a dimensionless constant multiplied by nominal frequency to the same exponent. Second, the optimal switching frequency scales with rated power of 2500 watts rather than instantaneous power because switching loss coefficients are defined at nominal conditions. These corrections yield a dimensionally consistent closed-form solution that satisfies the necessary optimality condition, and the associated fixed-point iteration is proven to converge geometrically with contraction rate 0.39. Validation across five pulse width modulation controllers, including Constant Frequency, Variable Frequency, Hysteresis Band, Optimal Normalized Efficiency Total Harmonic Distortion Product, and Dual Mode, shows zero theoretical error after calibrating a single dimensionless coefficient to 0.001. A secondary contribution, the Normalized Efficiency Total Harmonic Distortion Product, unifies both objectives into one scalar metric with a frequency-independent upper bound of 0.711. The infinite dimensional optimization reduces to evaluation at three load points at 10%, 50%, and 100%, reducing computations from 10000 to 3 while preserving rigorous error bounds.

We introduce a coefficient-degenerate analogue of the three-parameter Mittag-Leffler function by replacing the classical rising factorial $(\gamma)_n$ in the numerator with the degenerate Pochhammer symbol $(\gamma)_{\lambda,n}$ associated with the degenerate gamma function $\Gamma_\lambda$, while keeping the standard scaling $\Gamma(\alpha n+\beta)$ in the denominator. Assuming an explicit nonresonance (pole-avoidance) condition that guarantees finiteness of $(\gamma)_{\lambda,n}$ for all $n\ge0$, we prove that the defining power series converges for every $z\in\C$ and hence defines an entire function $E^{[\lambda]}_{\alpha,\beta;\gamma}(z)$. The deformation depends continuously on $\lambda$ and recovers the Prabhakar function as $\lambda\to0^{+}$. We represent $E^{[\lambda]}_{\alpha,\beta;\gamma}$ as a Fox–Wright function and derive coefficient bounds and sharp growth estimates, including the order of the resulting entire function. For the special case $\alpha=1$, we obtain a closed form in terms of a degenerate confluent hypergeometric function, which is convenient for symbolic manipulation and computation. Using these representations, we establish shift relations for Riemann–Liouville fractional integrals and derivatives that parallel the classical Prabhakar calculus. A Laplace transform pair is proved and used to construct a one-parameter, closed-form deformation of the Havriliak–Negami transfer function, yielding analytically tractable relaxation kernels for non-Debye linear response in complex media. Finally, we develop a numerically stable evaluation based on rigorous a posteriori truncation bounds and illustrate the impact of coefficient degeneracy through computational experiments.

In this study, we examine the Hyers–Ulam stability of a fractal–fractional mathematical model for the transmission dynamics of the Zika virus. The model is formulated using the Atangana–Baleanu fractional operator with a Mittag–Leffler kernel, thereby incorporating both memory effects and fractal properties commonly observed in biological systems. This approach provides a more realistic representation of disease spread than classical integer-order models. First, the existence and uniqueness of solutions are established to ensure the well-posedness of the proposed system. Subsequently, fixed-point theory is applied to derive sufficient conditions for Hyers–Ulam stability, demonstrating that the model remains stable under small perturbations. This stability property is essential for confirming the model's reliability and robustness in practical applications. In addition, numerical simulations are performed to support theoretical findings and illustrate the qualitative behavior of the Zika virus transmission process. The obtained results provide valuable insights into the influence of memory and fractal effects on epidemic dynamics. In general, the study highlights the effectiveness of the fractal–fractional framework in capturing complex biological phenomena and confirms its potential for future research in mathematical epidemiology. The proposed methodology can also be extended to analyze other infectious diseases, contributing to the development of more accurate and reliable predictive models.

In this study, we explore a stochastic SIRDS epidemic model that is influenced by fractional Brownian motion (fBm) with a Hurst index H ∈ (1/6, 1/2). By integrating stochastic fluctuations into the transmission rate, we develop a four-compartment framework, which is represented through two ordinary differential equations and two stochastic differential equations driven by fBm. Utilising the symmetric stochastic integral, along with a version of Itô’s formula tailored for fBm and a suitably constructed random Lyapunov function, we establish the existence and uniqueness of a global, positive solution. We subsequently derive sufficient conditions that guarantee the extinction of the disease. For the numerical simulations, we generate sample paths of fBm using the Fast Fourier Transform method (FFT) and apply a modified Euler scheme to manage the resulting fBm increments. The model parameters are calibrated based on empirical epidemiological data, enabling us to evaluate how well the simulated infection paths align with the observed dynamics by identifying the parameters that best fit our context. We specifically compared two simulated infection trajectories corresponding to H = 0.38 and H = 0.5. This comparison indicated that the Hurst index has a significant impact on the model's dynamics. By employing error metrics to support our conclusions, we found that H = 0.38 corresponds more closely with the data than H = 0.5, thereby making it the most appropriate choice for our scenario.

In the rapidly evolving digital era, smart devices play a critical role in daily life, yet efficient and automated tracking of device usage remains a significant challenge. Inspired by the operational principles of vehicle speedometers and odometers, this study proposes a mathematical modelling framework for cumulative usage tracking in intelligent electronic devices. The research investigates how dynamic measurement systems, traditionally used in vehicles to record distance as the time integral of velocity, can be adapted to model real-time usage behaviour in smart gadgets.

The proposed model represents device activity as a time-dependent function, where cumulative usage is formulated as an integral of activity rate over time. Using concepts from applied mathematics, dynamical systems, and signal modelling, this study develops a generalized mathematical structure capable of capturing continuous and discrete usage patterns. The framework incorporates sensor-driven input signals, temporal data accumulation, and algorithmic tracking mechanisms to simulate speedometer-like behaviour in modern devices.

Furthermore, the model explores potential applications in smart technology, including smartphones, IoT systems, wearable devices, and automated monitoring platforms, where accurate usage quantification is essential for performance optimization and resource management. The proposed approach contributes to the interdisciplinary intersection of mathematical modelling, computational intelligence, and intelligent system design.

This research aims to establish a theoretical foundation for automated cumulative tracking mechanisms in next-generation smart devices, offering a scalable and mathematically rigorous solution aligned with emerging trends in artificial intelligence and digital instrumentation.

HIV remains a major public health concern in urban areas of Indonesia, including Bandung City. Understanding its transmission dynamics is essential for designing effective control strategies. This study develops a mathematical model of HIV transmission incorporating real epidemiological data from Bandung City. The population is divided into several compartments representing susceptible individuals, HIV-infected individuals, and individuals under treatment. A nonlinear treatment function is introduced to reflect limited medical resources and treatment saturation effects. Model parameters are estimated using reported HIV case data through a data fitting procedure based on a Genetic Algorithm (GA), which minimizes the difference between model simulations and observed data. This approach allows efficient exploration of the parameter space and provides reliable parameter estimates for the transmission and treatment processes. Based on the estimated parameters, the basic reproduction number is derived analytically and its estimated value is 2.56, indicating that the infection can persist in the population. Stability analysis of the disease-free and endemic equilibrium points is performed using linearization techniques. Furthermore, bifurcation analysis is conducted to investigate qualitative changes in system behavior with respect to key parameters, particularly the transmission rate and treatment effectiveness. Numerical simulations are presented to validate the analytical results and to illustrate the impact of nonlinear treatment and parameter variations on HIV transmission dynamics. The results show that nonlinear treatment significantly influences the stability of equilibria and may lead to backward bifurcation under certain conditions. This study provides a realistic mathematical framework for analyzing HIV transmission in Bandung City and offers valuable insights for optimizing HIV control and treatment policies.

Introduction: Microsporidia MB are naturally occurring, vertically and horizontally transmitted symbionts of mosquitoes that can influence reproductive outcomes. Understanding their invasion dynamics is essential for clarifying their ecological role and interaction with conventional vector control strategies. This study presents a mathematical framework to examine the establishment and persistence of microsporidia MB in a mosquito population exposed to chemical interventions.

Methods: We formulate and analyse a deterministic compartmental model structured by infection status, sex (male and female), and developmental stage (eggs, larvae, and adults). The model incorporates vertical transmission, horizontal transmission through mating, and reduced egg viability associated with specific mating combinations. Larval and adult control measures are represented through compliance and efficacy parameters that increase stage-specific mortality. The basic reproduction number is derived using the next-generation matrix method. Local stability is determined via eigenvalue analysis of the Jacobian matrix, while global stability is established using a quadratic Lyapunov function. Numerical simulations explore the influence of key biological and control parameters on infection prevalence and overall population dynamics.

Results: The analysis establishes threshold conditions governing invasion and persistence. The infection-free, complete-infection, and coexistence equilibria are shown to be locally stable under appropriate reproduction number conditions. Vertical transmission efficiency and egg viability parameters are critical for long-term persistence, whereas horizontal transmission accelerates spread after introduction. Chemical control significantly affects both infection prevalence and mosquito abundance, with outcomes depending on coverage and efficacy levels.

Conclusions: The model provides a mechanistic basis for understanding microsporidia invasion in structured mosquito populations and offers quantitative insight into their interaction with chemical control within integrated vector management.