http://41.89.164.27:8080/xmlui/handle/123456789/258 2026-09-22T12:01:46Z http://41.89.164.27:8080/xmlui/handle/123456789/2820 Mathematical Optimisation of Smart Irrigation Scheduling Using Climate and Soil-moisture Data: A Simulation Study for Potato Production Kiplimo, Selah; Bitok, Jacob; Rotich, Titus Background: Efficient irrigation scheduling requires simultaneous consideration of soil-water status, atmospheric demand and crop response. Sensor-based monitoring and optimisation methods can support this integration, but model outputs must be interpreted according to the quality and provenance of their inputs. Objective: This study develops a five-state compartmental framework for potato irrigation and evaluates threshold-based and optimisation-based irrigation scenarios using a 2025 rainfall forcing series associated with Elgeyo-Marakwet County, Kenya. Methods: The model represents soil water, root-zone water, plant tissue water, biomass and an environmental water compartment through ordinary differential equations. Soil-water retention follows the van Genuchten relation, reference evapotranspiration is represented with the Penman-Monteith formulation, and irrigation is formulated as a constrained quadratwas represented by an SLSQP-based implementation. The analysis is treated as a simulation study rather than as a field- validation study because the underlying station metadata, raw meteorological file and independent crop-calibration dataset were not available for verification. Results: For the reported loam-soil scenario, total available water was 120 mm m-1 , rooting depth was 0.6 m and the adopted depletion fraction was 0.35, giving root-zone total available water of 72 mm and readily available water of 25.2 mm. The supplied simulations produced a rain-fed yield-equivalent output of 3.1 kg m-2 and a threshold-controlled output of 6.3 kg m-2 , corresponding to a 103% increase within the model scenario. Conclusion: The simulations illustrate how climate and soil-moisture information can be incorporated into an irrigation-control framework, but the quantitative outputs should be interpreted as scenario results pending calibration and independent field validation. 2026-09-01T00:00:00Z http://41.89.164.27:8080/xmlui/handle/123456789/2805 Mathematical Modelling of Malaria Transmission Dynamics in Kenya: The Role of Seasonality, Drug Resistance, and Human Movement Ashibambo, Nancy; Shichika, Julius; Bii, Albert Background: Although there has been a remarkable improvement in controlling malaria, the clinical problem is still of public health importance in Kenya and especially in areas where there is climatic variation, which affects the transmission pattern. Seasonal rainfall has been explored as a central factor in the breeding of mosquitoes and the ensuing outbreaks of malaria, but most models have not included these ecological forces together with drug resistance and human mobility. Objective: The purpose of the study is to formulate and examine a seasonally driven malaria transmission model to represent the interaction of the dynamics of mosquito infection, antimalarial drug resistance, and human mobility in the Kenyan setting. Methods: We developed a compartmental model with drug-susceptible and drug-resistant parasite strains, categorised by stages of human infections, and mosquitoes. The model proposes seasonalforcing in a sinusoidal representation, which is staggered with the Kenyan rainfall pattern, which drives the mosquito recruitment. Inter-regional human migration is thought to take the form of a toggling migration parameter. The deSolve package in R was then used to simulate the model within a 2-year horizon. Monthly averages were then used to determine the peaks of the infection rates, and these were equated to the long (March to May) and short (October to December) rainy seasons in Kenya. Results: It was found that the results of simulations identified clear infection spikes closely corresponding to two periods of rainfall in Kenya (bimodal). The post-rainy periods when mosquitoes infected with the malaria parasites reach their peak (Q), as well as when humans are infected, were consistent, making a difference to resistant infections, which do not drop as fast as susceptible infections. The circulation of human movements enhanced the continuation and propagation of resistant infections. This indicated the environmental drivers of seasonal forcing that explained the time and magnitude of outbreaks. Conclusion: Noting the inclusion of seasonality, drug resistance, and movement in the models of malaria transmission increases their reality and predictability to a great extent. The syncing of the most significant infection-containing seasons with rainy seasons necessitates climate-tactful surveillance and intervention time. In Kenya, where the mobility of the population is high based on trade, labour migration, and between urban and rural regions, the modelling of such mobility is very important in gaining an understanding of the management of the epidemic. The model can be of great benefit in optimising vector control, deploying drugs and allocating resources regionally to malaria-endemic countries such as Kenya. 2025-08-01T00:00:00Z http://41.89.164.27:8080/xmlui/handle/123456789/2803 Navier-stokes Based Modelling of Airflows in Forest Canopies and Its Influence on Local Climate Dynamics Ruto, Faith; Bii, Albert; Shichikha, Maremwa Background: Forest canopies strongly influence atmospheric airflow, turbulence generation, heat exchange, water transport, and carbon dioxide distribution, thereby regulating the local climate. However, accurately representing airflow dynamics within forests remains challenging because of vegetation drag and turbulent mixing. Aims: This study developed a mathematical model based on the Navier–Stokes equations to investigate airflow behaviour within forest canopies and assess its influence on local climate dynamics. Study Design: This was a computational fluid dynamics (CFD)-based modelling study employing the Reynolds-averaged Navier–Stokes (RANS) equations coupled with the standard k–ε turbulence model.Place and Duration of Study: Department of Mathematics and Computer Science, University of Eldoret, Kenya, between July 2025 and April 2026. Methodology: The incompressible Navier–Stokes equations were used to model airflow within and above forest canopies. Vegetation effects were represented using a canopy drag-force term based on leaf area density. Turbulence was simulated using the standard k–ε model, while additional transport equations described temperature, water vapour, and carbon dioxide dynamics. The governing equations were discretised using the Finite Volume Method (FVM) and solved numerically in MATLAB. Simulations were performed for dense, medium, and sparse canopy configurations over a 30 m computational domain. Results: Airflow velocity increased with height in all canopy configurations, with dense canopies showing the greatest attenuation. At canopy height, velocities were approximately 1.45 m/s, 1.95 m/s, and 2.65 m/s for dense, medium, and sparse canopies, respectively. Turbulent kinetic energy (TKE) peaked near the canopy top, reaching approximately 66 m2/s2, 44 m2/s2, and 22 m2/s2, respectively. Temperature increased with height, while moisture and carbon dioxide concentrations decreased because of enhanced turbulent mixing. Dense canopies retained higher moisture and carbon dioxide levels than medium and sparse canopies. Conclusion: Forest canopy density significantly influenced airflow structure, turbulence production, and scalar transport. Dense canopies provided stronger microclimatic regulation through enhanced momentum attenuation, moisture retention, and carbon storage. The developed modelling framework provides a useful tool for studying canopy–atmosphere interactions and local climate dynamics. 2026-08-01T00:00:00Z http://41.89.164.27:8080/xmlui/handle/123456789/2802 Hydrodynamic Modelling of Mixing Efficiency and Optimal Bio-methane Production in Anaerobic Digesters Using a Two-Dimensional Navier–Stokes Framework Obande, Ruth; Kandie, Joseph; Bii, Albert Background: Anaerobic digestion (AD) is a proven technology for renewable bio-methane production, but digester efficiency is often limited by poor hydrodynamic mixing rather than by microbial kinetics alone; most existing models, however, assume idealised, fully homogeneous reactors. Objective: This study investigates the influence of hydrodynamics on mixing efficiency and bio-methane production potential in anaerobic digesters using mathematical modelling. Methods: A two-dimensional incompressible Navier–Stokes model was coupled with a tracer advection– diffusion equation to simulate slurry flow and mixing behaviour. The governing equations were non- dimensionalised using the Reynolds and Péclet numbers, discretised using the finite difference method, and solved numerically in MATLAB. An optimisation framework that treated inlet velocity as the control variable, together with an adjoint sensitivity analysis, was used to evaluate and improve mixing efficiency. Results: At a Reynolds number of 2100, the flow exhibited transitional characteristics, with a dead zone fraction of approximately 35.1%. Velocity contours revealed limited circulation, whereas the tracer distribution showed a non-uniform concentration pattern across the domain. The dead zone fraction declined exponentially as Re increased, with values above 4000 projected to reduce it below 15%. At Pe = 10,000, transport was strongly advection-dominated, and the adjoint sensitivity analysis identified the inlet/impeller region as offering the greatest leverage over mixing performance. Conclusion: Hydrodynamic conditions play a critical role in determining mixing efficiency and, consequently, bio-methane production potential. The developed model provides a computationally efficient framework for analysing and optimising anaerobic digester performance and offers a foundation for future integration with biochemical reaction models. 2026-08-01T00:00:00Z http://41.89.164.27:8080/xmlui/handle/123456789/2800 On the Norm of Jordan Elementary Operators in Tensor Product of C ∗ -Algebras Kegwaro, Winnie; King’ang’i, Denis; Meli, Collins Elementary operators have been studied over years, with their norms being of significant interest in operator theory. The study includes the derivation of formulas that describe norms in terms of their coefficient operators, which is traced back to Stampfli`s Theorem that used the property of Numerical range as a foundation for the study of norms. Other properties of Elementary operators have been studied ever since, but little is known on Jordan Elementary operators in tensor products of C ∗ - algebras. This paper aims to extend the determination of norms of Jordan Elementary operators in the tensor product of C ∗ - algebra by determining the lower bound of the norm using the maximal numerical range by employing the technique of tensor product, finite rank operator and inner product. A lower bound of Jordan elementary operator in tensor product of C ∗algebras is obtained. 2026-06-01T00:00:00Z http://41.89.164.27:8080/xmlui/handle/123456789/2564 Detection and classification of cervical cancer disease among women using machine learning technique Model in Western Kenya. Murere, JF; Wangila, S.; Koech, J. Cervical cancer is the leading cause of cancer related deaths among Kenyan women, claiming approximately the lives of 3,200 women annually. This is primarily due to the low screening uptake (16%) and late diagnosis. The aim of this study was to develop a machine leaning based model to enhance early detection of cervical cancer in Western Kenya, a region in Kenya with limited healthcare resources. Demographic, reproductive, and clinical characteristics data were collected from 968 women across health facilities in western Kenya (MTRH and Kakamega Referral hospital) utilizing a cross sectional study design. The dataset was divided into training set (70%) and testing set (30%). The training set was used to develop the five machine learning model: Logistic Regression, Random Forest, Decision Tree, Support Vector Machine (SVM), and Artificial Neural Network (ANN). The testing set was used to evaluate the models. The machine learning model were trained to classify the cervical cancer cases, addressing the class imbalances using class weighting method for SVM, decision tree, random forest and logit model and synthetic minority oversampling class technique (SMOTE) for ANN. The random forest model demonstrated the superior performance compared to the other four models as it achieved the highest accuracy (94.33%) and specificity (98.37%) making it to be highly effective at ruling out negative cases. It however had a sensitivity of 20% which indicated that it had challenges in detecting positive cases. The logistic regression model excelled in sensitivity (70%) making it suitable for initial screening. ANN model showed the lowest precision (10%). The findings from this study suggested that a two-step approach which combine both Logistic Regression for screening and Random Forest for confirmation of cervical cancer cases which will go a long way in improving early detection and reduce cervical cancer mortality in resource-constrained settings like Western Kenya. 2025-06-01T00:00:00Z http://41.89.164.27:8080/xmlui/handle/123456789/2474 Traffic Flow Modelling Around Roundabouts Using Navier–Stokes and Advection–Diffusion Equations Krifix, Momanyi Mogire; Maremwa, Shichikha; Kandie, Joseph; Ngetich, Lucy Jerop Urban roundabouts are safer than signalised junctions yet remain prone to congestion under unbalanced demand and operational incidents. Existing microscopic and macroscopic approaches rarely capture, within a single framework, the tightly coupled evolution of traffic velocity and density needed for robust design and control. We address this gap with a coupled Navier–Stokes and Advection–Diffusion (NS–AD) model on an annular domain, adopting a barotropic closure 𝒑 = 𝒂𝝆 and a conservative incident force 𝒌𝐨𝐛𝐬𝐫/∥ 𝐫 ∥. Convection is discretised with a QUICK/TVD flux (MUSCL with a van–Leer limiter), while diffusion, sources and viscosity are advanced by a semi-implicit Crank–Nicolson step, enabling high-resolution fronts without spurious oscillations over long horizons. Simulations reproduce two robust signatures: a persistent annular congestion ridge in density coincident with the circulating ring, and a co-located speed amplification that exhibits an upstream/downstream braking/acceleration asymmetry governed by −𝒂𝛁𝐥𝐧𝝆. Probe histories display a negative (|𝒗|, 𝝆) phase slope during the transient, and a quasi-1D azimuthal reduction explains the observed structures and yields a falsifiable ridge-thickness law 𝜹 ∼ 𝑫/𝒖. Difference maps between incident and baseline scenarios isolate the causal footprint of the incident as outward mass migration to the ring and momentum gain along preferred circumferential paths. We conclude that a composite potential 𝒂𝐥𝐧𝝆+ 𝚽 succinctly explains outward push, pressure support and viscous/diffusive regulation, while the numerical pairing QUICK + TVD with Crank–Nicolson provides a stable, accurate basis for design studies. For policy and practice, the results support rapid incident clearance near the central island, targeted entry metering on feeder approaches, and low-cost geometric/control provisions, such as short bypasses and advisory speeds, that reduce the effective barrier 𝒂𝐥𝐧𝝆 and add diffusion pathways. Future work should calibrate (𝒂, 𝑫, 𝝂, 𝑺(𝜽), 𝒌𝐨𝐛𝐬) against field data using ridge width, phase slope, relaxation time and entry/exit counts, extend to multi-lane and multi-class settings, assimilate real-time data for model-predictive control, and couple to emissions and safety surrogates to assess broader impacts. 2025-01-01T00:00:00Z http://41.89.164.27:8080/xmlui/handle/123456789/2473 Mathematical Modelling of Traffic Flow Dynamics Around Roundabouts Using Navier– Stokes and Advection–Diffusion Equations Krifix, Momanyi Mogire; Maremwa, Shichikha; Kandie, Joseph; Ngetich, Lucy Jerop Traffic congestion is a persistent challenge in urban transport systems, particularly around roundabouts where nonlinear merging and circulating flows create instability and delays. Traditional microscopic, mesoscopic, and macroscopic models often neglect explicit treatment of priority rules, capacity drops, and queue spillback, limiting their ability to predict roundabout-induced congestion. To address this gap, this study develops a coupled Navier–Stokes–advection–diffusion framework for simulating traffic dynamics at roundabouts. The governing equations are discretized using the finite volume method with a QUICK scheme for spatial accuracy and advanced in time with a Crank–Nicolson integration for stability. Non-dimensionalization ensures general applicability across different traffic environments. Simulation results reproduce fundamental traffic relations, identify optimal densities for maximum throughput, and reveal oscillatory stop-and-go waves consistent with empirical observations. Scenario analysis shows that increased diffusion enhances stability but reduces flow capacity, capturing real-world trade-offs between efficiency and control. The study concludes that embedding yield laws and circulating feedback into continuum equations provides a rigorous foundation for analyzing roundabout performance. Policy recommendations include adopting geometry- and flow-calibrated entry controls, while future research should incorporate stochastic demand and adaptive control strategies to refine predictive accuracy. This contribution provides both theoretical innovation and practical insights for sustainable traffic management. 2025-01-01T00:00:00Z http://41.89.164.27:8080/xmlui/handle/123456789/2472 A Novel Poiseuille-Based Mathematical Model for Carotid Artery Blood Flow: Modelling Geometry Interruptions and Vascular Stress during Accidents Ngetich, Lucy Jerop; Maremwa, Shichikha; Kandie, Joseph; Krifix, Momanyi Mogire Carotid artery injuries during accidents are a significant contributor to trauma-related morbidity and mortality, necessitating accurate models for predicting vascular stress and flow disruption. Existing approaches either oversimplify flow and underestimate wall shear stress, neglect trauma-induced geometric interruptions, or require computationally intensive methods unsuitable for emergency use. To address these limitations, this study develops a hybrid Poiseuille–Womersley model that integrates distributed and localized pressure loss terms, a geometric penalty factor for lumen constriction, and pulsatile corrections to capture transient flow dynamics. Analytical derivations supported by simulation reveal that accident-induced reductions in carotid lumen diameter cause disproportionate declines in volumetric flow rate, sharp decreases in wall shear stress, and alterations in velocity fields that cannot be captured by steady-state assumptions. The model thus extends classical hemodynamic formulations to accident scenarios, providing an efficient yet physiologically consistent framework. These results confirm geometry as a primary driver of vascular stress under trauma conditions. The study concludes that lightweight analytical models can complement diagnostic and emergency care tools, offering rapid assessment capability. Policy makers and clinicians are encouraged to incorporate such trauma-informed hemodynamic tools into stroke prevention and emergency response strategies, while future work should focus on clinical validation, patient-specific adaptation, and integration with real-time Doppler imaging. 2025-01-01T00:00:00Z http://41.89.164.27:8080/xmlui/handle/123456789/2471 A Novel Poiseuille Equation Framework for Pulsatile Flow Dynamics in Accident-Induced Carotid Artery Constrictions Ngetich, Lucy Jerop; Maremwa, Shichikha; Kandie, Joseph; Krifix, Momanyi Mogire Stroke and carotid artery injury remain significant contributors to global morbidity, with accident-induced disruptions in blood flow presenting acute clinical risks. Existing modelling approaches are either overly simplistic, relying on steady Poiseuille theory, or computationally demanding, as in fluid–structure interaction simulations, which limits their use in real-time diagnostic settings. This paper proposes a novel Poiseuille-based mathematical model that incorporates accident-like geometry interruptions and pulsatility via a nine-point stencil, a geometrypenalty factor, and a Womersley-inspired correction term. Using physiologically validated parameters for viscosity, density, and pressure gradients, the model reproduces key hemodynamic markers including volumetric flow, velocity fields, and wall shear stress under both normal and obstructed conditions. The findings show that even modest reductions in lumen radius lead to sharp declines in flow and shear, while pulsatility modifies waveform oscillations without altering the magnitude of disruption. The study concludes that geometry remains the primary driver of hemodynamic collapse, while pulsatility governs temporal detail. By providing a computationally efficient and clinically interpretable surrogate, the model bridges the gap between oversimplified analytical solutions and resource-intensive CFD. It is recommended that such reduced-order frameworks be integrated into clinical risk screening and trauma diagnostics, with future work directed toward validation against patient-specific data and incorporation of non-Newtonian blood properties. 2025-01-01T00:00:00Z