Mathematics and Computer Science: Research Updates Vol. 11

Analytical and Computational Analysis of the HIV/AIDS Ordinary Differential Equations Model with Optimal Controls

O. A. Odebiyi, A.W. Ayanrinola, A.O. Adeboye, O.A. Olajide, E.O. Elijah — Mon, 25 May 2026 00:00:00 +0000

The global fight against Human Immune Deficiency Syndrome (HIV/AIDS) has been ongoing for decades, with significant progress made in understanding the virus, improving treatment options, and expanding access to care. Mathematical modelling has played a crucial role in understanding the dynamics of HIV/AIDS transmission and informing effective control strategies. Previous HIV transmission has not fully explored the existence and optimal control strategies for minimising the spread of HIV. Therefore, this study aims to identify effective interventions that can be tailored to specific contexts, thereby contributing to global efforts to combat HIV/AIDS and improve health outcomes. The study extends the HIV/AIDS model, investigating the existence and Optimal control of the Human Immunodeficiency Virus (HIV) and Acquired Immune Deficiency Syndrome (AIDS). This study formulates a deterministic mathematical model incorporating three time-dependent optimal control strategies. A comparative analysis of three control strategies was incorporated, namely: awareness campaigns, treatment with ART, and targeted outreach programs. Using Pontryagin's Maximum principle, the study characterises the optimal control strategies and proves the existence of solutions. Numerical simulations using MATLAB software illustrate the effectiveness of the proposed control strategies, highlighting the importance of tailored interventions and sustained investment in HIV/AIDS control programs. The results show that combined implementation of awareness campaigns, treatment with ART, and targeted outreach significantly reduces AIDS cases and suppresses AIDS-related outcomes, while treatment and targeted outreach effectively manage the disease. Overall, the findings provide valuable insights into HIV/AIDS dynamics and optimal intervention strategies, offering important guidance for public health policy, resource allocation, and the design of effective HIV prevention and control programs.

A Multivariate Discriminant Analysis of Factors Influencing Secondary School Performance in Simiyu Region, Tanzania

Maradona Martine, Ramkumar T. Balan — Mon, 25 May 2026 00:00:00 +0000

Aims: The primary objective of the study was to cluster the secondary schools of the Simiyu Region of Tanzania according to their performance.

Study Design: The study analyzed data from 44 secondary schools in the Simiyu Region, including 34 government and 10 private schools, with 18/28 urban schools and 26/133 rural schools represented.

Methodology: The performance of schools was analyzed by applying multivariate tools, descriptive statistics, discriminant function, and Mahalanobis distance. The performance of schools considered five variables: Number of students who sat for the examination (Sat), number of students who passed the examination (Pass), Grade Point Average (GPA), regional ranking (RR), and national ranking (NR) of schools.

Results: Linear discriminant function 1 was the most appropriate for classifying schools by performance. Stacked histogram 1 effectively distinguished three groups of schools without duplication or omission. GPA and RR were the most influencing factors leading to the performance of schools. Most private schools were classified as high-performing, while two-thirds of government schools were lacking in moderate performance. The mean GPA for government schools was 3.6342, compared to 2.3178 for private schools, indicating lower performance in government schools. The RR for government schools averaged 48%, whereas private schools averaged 95.36%, showing that private schools generally held higher regional positions. The number of students enrolled and sitting for examinations also affected performance rates.

Conclusion: The GPA contributes a higher part to the assessment of performance. Among the 44 schools under consideration, 14 were classified as high-performing. The authorities and education institutes can adopt strategies to improve GPA and RR to secure high rankings in performance.

Deriving Mixture Distributions Using Moment Generating Functions: A Hierarchical Model Approach

Subhash Bagui, Jia Liu, Shen Zhang — Mon, 25 May 2026 00:00:00 +0000

Mixture distributions are important because they model data from multiple underlying subpopulations, allowing us to capture heterogeneity that a single distribution can’t explain. Generally, mixture distributions arise as marginal distributions of hierarchical mixture models. In this chapter, we use moment-generating functions (mgfs) to derive the densities of mixture distributions from hierarchical models. When the mgf of a mixture distribution doesn’t exist, the approach can be extended to characteristic functions to derive the mixture density. This chapter uses a result from Villa and Escobar (2006). The present work complements Villa and Escobar’s (2006) article with many new examples.

Relationships among Radio, Clique and Chromatic Numbers

S. Vimalajenifer — Mon, 25 May 2026 00:00:00 +0000

Graph labelling is an assignment of nonnegative integers, sometimes called colours, to the vertices, edges or both. Radio labelling is a well-known concept in graph theory, motivated by practical problems such as frequency assignment in communication networks.

Let (V, Ε) be a simple, connected and undirected graph. A radio labeling of G, ψ: V → {1,2,3,… } is a function satisfying the condition for any two distinct vertices u and v that: d(u,v) + |ψ(u) - ψ(v)| ≥ 1 + diam(G) where d(u,v) denotes the distance between the vertices u and v and diam(G) denotes the diameter of the graph G. The span of a radio labelling is the maximum integer that assigns to a vertex, and radio number, rn(G), is the minimum span taken overall radio labellings of G. This paper presents some bounds connecting radio number with clique number and the chromatic number. In addition, the possible constructions of simple connected graphs with radio number as the algebraic sum of the clique number and a non-negative integer. Also, a graph with radio number equal to the algebraic sum of chromatic number and a non-negative integer.

Dynamics of Prey-Predator and Amensal Model

A. V. Paparao, T. S. N. Murthy — Mon, 25 May 2026 00:00:00 +0000

Predator-prey interactions are among the most common and crucial ecological phenomena in nature. Over the course of long-term evolution, prey populations have developed various anti-predation strategies to cope with the threat of predators, with population dispersal being one of the most common strategies. In traditional ecological models, the prey population is typically constrained by direct predation. The study aims to investigate how time delay (τ) affects the stability and dynamics of a predator–prey–amensal ecological system modelled by delay differential equations. The influence of time lag(τ) on the predator species in the prey-predator and amensal model was studied. In this model, the first species x represents the predator, the second species y represents the prey, and the third species z is amensal to the prey species. The mathematical model is formulated as a system of delay differential equations. A hyperbolic equilibrium point of the system is identified, and its stability analysis was carried out at τ = 0 and τ >0. The hyperbolic equilibrium point is asymptotically stable at τ = 0. However, for τ >0 the system loses its stability and undergoes Hopf bifurcation for some critical values of 'τ'. Further, the global stability analysis was carried out, and the sufficient condition for the existence of Hopf bifurcation is derived. The critical values of 'τ' are identified using simulation for the proposed system. These values are correlated with the analytical finding.

Improving the Precision of Population Mean Estimator in Two-Phase Sampling

Archana Panigrahi, Sanjit Kumar Mohanty — Mon, 25 May 2026 00:00:00 +0000

The use of auxiliary information at the estimation stage improves the precision of estimators in sampling survey methods. The ratio estimators (product estimators) are recommended when there exists a positive correlation (negative correlation) between the study variable and auxiliary variable. When the linear relationship is not very strong, it is preferred to use exponential estimators. The study aims to develop improved product-type estimators for estimating the finite population mean in a two-phase sampling scheme. A product estimator is more efficient than the mean per unit estimator when there exists a negative correlation between the study variable and auxiliary variable. When the population mean of the auxiliary variable is not known in advance, one can use a two-phase sampling scheme or a double sampling scheme. It has been considered that the second-phase sample is directly selected from the population of size N and is independent of the first-phase sample. Some improved product type estimators of finite population mean \(\bar{Y}\) under simple random sampling without replacement a two-phase sampling scheme has been proposed. The efficiency of the proposed product type estimators is compared with the mean per unit estimator, the conventional two-phase product estimator, and the two-phase product type exponential estimator suggested theoretically and empirically.

A Generalised Quadrature Using Clenshaw-Curti’s Quadrature Rule

Sanjit Kumar Mohanty — Mon, 25 May 2026 00:00:00 +0000

Among the various classical quadrature rules, some of the most widely used are the Trapezoidal Rule, Simpson’s Rule, Newton–Cotes Rules, Clenshaw–Curtis Quadrature Rule, and Gauss–Legendre Quadrature Rule. Each of these methods approximates the given integral by replacing the integrand with a simpler function, such as a polynomial, whose integral can be evaluated easily. This paper presents a generalised quadrature rule SM₁₄ (f) of degree of precision seven by combining two well-known quadrature formulas, namely the Clenshaw–Curtis five-point quadrature rule and the Gauss–Legendre three-point quadrature rule, each possessing precision five. The proposed rule is constructed using the generalised quadrature technique to achieve higher accuracy with improved computational efficiency. The theoretical superiority of the developed rule over its constituent formulas is established through rigorous error analysis. Furthermore, the performance and dominance of the proposed quadrature rule are numerically verified by considering different types of test integrals. The proposed rule also demonstrates exactness for certain test integrals where the base rules fail or show larger errors. Hence, the developed generalised quadrature rule provides a more efficient, stable, and accurate technique for numerical integration, making it a valuable contribution to computational mathematics.

Dynamic Transmission Range–Based Connectivity Modeling in VANET Environments

Kenneth Okello, Elijah Mwangi, Ahmed H. Abd El-Malek — Mon, 25 May 2026 00:00:00 +0000

Vehicular Ad Hoc Networks (VANETs) are critical for inter-vehicle communication, enabling timely information exchange for ‘safety ‘and emergency applications. However, dynamic network topology and environmental obstructions pose persistent challenges, causing intermittent connectivity. This study proposes a generalized statistical channel model to characterize vehicle transmission ranges under realistic urban conditions, incorporating wireless fading (Weibull, Nakagamim, Rayleigh, and lognormal) and large-vehicle obstructions. Connectivity probability is evaluated for two traffic models—free-flow and synchronized Gaussian Unitary Ensemble (GUE)—capturing multi-lane vehicle dynamics. Results show that for persistent connectivity between vehicle nodes, an inter-vehicle distance less 50m is required for vehicular network design in the fading channels considered. Big vehicle obstructions is shown to be the principle cause of path-loss in all the environments. Furthermore, Nakagami-m fading yields the largest mean transmission range (≈ 220 m), outperforming Rayleigh (≈ 180 m) and Weibull (≈ 195 m) scenarios, while lognormal shadowing due to large vehicles can reduce effective range by up to 30%, lowering connectivity probability under dense traffic. Incorporating inter-vehicle safety gaps in synchronized traffic further decreases link reliability by 10%. These findings quantify the combined effects of fading, vehicle density, and obstructions on VANET performance, providing guidance for robust network design under diverse traffic scenarios.