RP-Department of Mathematics

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Thermal Radiation on Darcy Forchheimmer Flow of a Casson Nanofluid
(2024-06) Loco, Valerie Sasha; Mutuku, Winifred Nduku
Owing to Casson nanofluids’ enhanced thermophysical properties, it has a wide range of applications in the fields of mining, drilling operations, material science, metallurgy, food manufacturing, and nanotechnology and bio-engineering. Casson fluid is frequently modelled since the model has a great agreement with the rheological evidence about human blood. Some of its applications in technological, industrial, mechanical, and scientific disciplines include; grain storage, geothermal energy production, designing warm protectors, artificial dialysis, catalytic converters, circulation of water in reservoirs and fermentation processes. On the other hand, radiation heat transfer mechanism has an immense impact in industries, engineering, technological fields where apparatus operate at extremely high temperatures. In this paper, the effects of thermal radiation on Darcy Forchheimer flow of a two- dimensional, steady, incompressible flow of Casson Nanofluid over a linear stretching surface are studied. The equations governing the fluid flow are formulated, then transformed to a system of ordinary differential equations using similarity variables then, the resulting ordinary differential equations are solved using the fourth order Runge- Kutta Method. The model is simulated using MATLAB bvp4c to demonstrate the impact of pertinent parameters on the temperature, velocity, and concentration of the fluid. It was observed that an increase in the thermophoretic parameter leads to an increase of the temperature at the boundary. The concentration of the fluid decreases with an increase porosity parameter value. Increasing the Brownian Motion parameter leads to an increase in concentration. An increase in Schmidt number leads to a decrease in temperature. Schmidt number increases with a decrease in concentration. The velocity and temperature profiles decrease with an increase in porosity parameter and an increase in thermal radiation leads to an increase of heat energy on the surrounding of a system leading to a decrease in fluid’s flow temperature.
Optimizing Vaccination Strategies to Reduce Conjunctivitis Transmission: Mathematical Modeling Insights from Kenya
(IRJMETS, 2024-05) Muli, Charles Ndambuki; Kimulu, Ancent Makau
Conjunctivitis is a widespread condition with significant public health implications, but its potential impact on transmission patterns due to vaccination programs, particularly in Kenya, remains underexplored. The main aims of this study to investigate the role of vaccination in preventing conjunctivitis spread and related complications. A deterministic mathematical model was developed in an attempt to simulate conjunctivitis incidence, considering factors like population size, contact rates, and vaccination efficacy. The basic reproduction number (R₀) was calculated using the next-generation matrix method. Stability analysis of the disease-free equilibrium (DFE) showed stability will occur when R₀ < 1 and instability when R₀ > 1. Numerical computations using the MATLAB ode45 solver indicated that increased vaccination campaigns reduce the infected population. This implies that vaccination strengthens the immune response against the infection, lowering the risk of severe outcomes like vision loss. This study is vital for understanding the potential impact of effective vaccination programs on conjunctivitis transmission in Kenya, aiding policy-makers and public health practitioners in developing effective disease control measures.
Numerical Comparison of 𝑪𝒖 and 𝑨𝒍𝟐𝑶𝟑 Nanoparticles in an MHD Water-based Nanofluid
(JERR, 2024-05) Rutto, Celestine Chepkemoi; Chepkwony, Isaac; Oke, Abayomi Samuel
In this study, the impact of 𝐶𝑢 and 𝐴𝑙2𝑂3 nanoparticles in a water-based nanofluid are considered. The application of this can be found in biomedical sensors and drug delivery. Specifically, it investigates heat transfer in the MHD flow of two nanofluids (𝐶𝑢-water and 𝐴𝑙₂𝑂₃-water) over an exponentially stretching surface. The study formulates a model and renders it dimensionless using Similarity Transformation. Numerical solutions are obtained using the MATLAB package bvp4c. The focus is on analysing the heat transfer rate variation with nanoparticle volume fraction. Results indicate that 𝐶𝑢-water nanofluid exhibits higher heat transfer rates and lower skin frictions compared to 𝐴𝑙₂𝑂₃-water nanofluid.
Mathematical Modeling of the Transmission Dynamics of Gumboro Disease
(Hindawi, 2024-05) Musaili, J. S.; Chepkwony, I.; Mutuku, W. N.
Gumboro disease is a viral poultry disease that causes immune suppression on the infected birds leading to poor production, mortality, and exposure to secondary infections, hence a major threat in the poultry industry worldwide. A mathematical model of the transmission dynamics of Gumboro disease is developed in this paper having four compartments of chicken population and one compartment of Gumboro pathogen population. The basic reproduction number Rog is derived, and the dynamical behaviors of both the disease-free equilibrium (DFE) and endemic equilibrium are analyzed using the ordinary differential equation theory. From the analysis, we found that the system exhibits an asymptotic stable DFE whenever Rog < 1 and an asymptotic stable EE whenever Rog > 1. The numerical simulation to verify the theoretical results was carried out using MATLAB ode45 solver, and the results were found to be consistent with the theoretical findings.
An optimal control model for Coffee Berry Disease and Coffee Leaf Rust co-infection
(SABA Publishing, 2024-03) Nyaberi, H.O.; Mutuku, W.N.; Malonza, D.M.; Gachigua, G.W.; Alworah, G.O.
In the 1980s, coffee production in Kenya peaked at an average of 1.7 million bags annually. Since then, this production has declined to the current output of below 0.9 million bags annually. Coffee berry disease (CBD) and Coffee leaf rust (CLR) are some of the causes of this decline. This is due to insufficient knowledge of optimal control strategies for CBD and CLR co-infection. In this research, we derive a system of ODEs from the mathematical model for co-infection of CBD and CLR with control strategies to perform optimal control analysis. An optimal control problem is formulated and solved using Pontryagin’s maximum principle. The outcomes of the model’s numerical simulations indicate that combining all interventions is the best strategy for slowing the spread of the CBD-CLR co-infection.
Effects of Activation Energy on Reactive MHD Flows with Joule Heating
(WJRR, 2022-12) Kaingu, Nyundo Stephen; Ngesa, Joel; Nduku, Mutuku Winifred; Okelo, Jeconia; Awuor, Kennedy
-This paper explores the effects of Arrhenius activation energy on a chemically reactive MHD flow through a porous surface while considering the joule heating effect of the two-dimensional free convective flow. The study incorporates the impact of the activation energy on the flow, the effect of the chemical reaction rate parameter on the velocity and temperature profiles as well as the joule heating parameter on the flow. The concentration of the species in the fluid will be investigated and the results discussed. The resulting non-linear partial differential equations are solved using the finite difference methods and results displayed graphically to show the effects of the resulting dimensionless parameters. It is found that the presence of the chemical reaction rate parameter on the flow decreases the concentration of the fluid while the activation energy shows the converse results.
A Mathematical Model of the Dynamics of Coffee Berry Disease
(Hindawi, 2023-09) Nyaberi, H. O.; Mutuku, W. N.; Malonza, D. M.
Coffee berry disease (CBD) is a fungal disease caused by Colletotrichum kahawae. CBD is a major constraint to coffee production to Kenya and Africa at large. In this research paper, we formulate a mathematical model of the dynamics of the coffee berry disease. The model consists of coffee plant population in a plantation and Colletotrichum kahawae pathogen population. We derived the basic reproduction number , and analyzed the dynamical behaviors of both disease-free equilibrium and endemic equilibrium by the theory of ordinary differential equations. Using the MATLAB ode45 solver, we carried out numerical simulation, and the findings are consistent with the theoretical results.
Numerical Investigation of HIV/AIDS Dynamics among the Truckers and the Local Community at Malaba and Busia Border Stops
(SAP, 2023-06) Kimulu, Ancent Makau
Busia and Malaba are the busiest border crossing points for truckers using the Northern Corridor connecting landlocked countries of Uganda, DR Congo, South Sudan and parts of Rwanda to Mombasa port. On average, Malaba and Busia clears 1000 and 600 trucks respectively per day. Delays in clearance of the trucks causes long queues at crossing points and influx of truckers in these respective towns. The truckers spent this time in the company of commercial sex workers and interact with the local communities in these towns, hence this is a conduit of spread of HIV/AIDS in these regions. This paper designed a model to study the HIV/AIDS dynamics in this border towns. The reproduction number, the disease-free equilibrium and endemic equilibrium points determined using the Next Generation Matrix (NGM) method. From the analysis of the model, it is found that delays in clearance time increases the force of infections from females to males. It is found out that also found out that increase in force if infections to males from females increases the male HIV infections. This is due to prolonged time of sexual interactions as the trucks await clearance and also due to higher number of male truckers than females. Furthermore, the analysis shows that delays in clearance time leads to increase in both male and female HIV/AIDS infections which causes an increase in the number of AIDS cases in this border crossing points
Computational Fluid Dynamics (CFD) for Blood Flow in Cardiovascular Medical Devices and Blood Damage Prediction
(NCM, 2023) Niyonkuru, Venant
Background: The hemodynamic performance of cardiovascular medical devices and their potential to cause blood damage are critical factors in ensuring patient safety and device efficacy. Computational Fluid Dynamics (CFD) has emerged as a valuable tool for simulating blood flow within these devices and predicting the risk of blood damage. Objectives: This study aims to utilize CFD simulations to evaluate the local hemodynamic performance of a particular implantable device and to provide precise predictions about likely adverse clinical effects, cutting-edge techniques like laser doppler anemometry (LDA) or particle image velocimetry (PIV) must be accessible. Methods: A patient-specific CFD model of the cardiovascular system and medical devices was developed based on medical imaging data. Hemodynamic parameters such as shear stress and flow recirculation were computed to identify regions of potential blood damage. The simulations were validated against data. Results: The CFD simulations revealed intricate flow patterns and areas of concern within the medical devices. Elevated shear stresses and prolonged residence times were identified in certain regions, indicating a risk of blood damage. By quantifying these parameters, the study provided a comprehensive assessment of potential blood damage locations and severity levels. Conclusion: CFD proved to be a robust approach for evaluating blood flow within cardiovascular devices and predicting potential blood damage. The study highlighted specific design modifications that could mitigate the risk of blood damage, thus contributing to the improvement of device safety. The integration of CFD with patient-specific data offers clinicians and engineers a powerful tool for optimizing cardiovascular device design and minimizing patient risk
Numerical Simulation of Natural Turbulent Convection with Vorticity Vector Formulation
(IOSR-JM, 2023) Otulo, Onyango Filentinus; Awuor, Kennedy Otieno
Background: Turbulent natural convection in an enclosure plays a big part in heat transmission and the building environment. Sophisticated buildings around the world are outfitted with costly heaters and coolers to maintain comfortable temperatures for human existence, manufacturing, and sophisticated farming methods, a scenario that many people cannot afford. Over a time, researchers have consistently developed a number of numerical study models to simulate the natural turbulent flow in these rectangular enclosures to solve complex problems associated with turbulent flows. In spite of several experimental studies and model simulations on the structure of natural turbulence convection, the fundamental mechanism in turbulent phenomena is still incomplete. Significant variations in experimental data and model simulation data in previous studies have been noted. This is because the unknown turbulent correlation coefficients resulting from the nonlinear terms of the turbulent flow control equations make it difficult to accurately determine fluid flow variables such as mean velocity distribution, temperature distribution and kinetic energy in a model simulation. Thus, an accurate numerical prediction of natural turbulence convection is crucial to solving the nonlinear equations for subsequent practical applications. Methodology: The performance of a numerical turbulence model k-ε in estimating the amount of heat transfer that occurs as a result of the naturally occurring turbulent convection that takes place within an air-filled rectangular enclosure is investigated in this work using vorticity vector formulation. The workflow of simulating the heat transfer which results from the action of natural convection within an enclosed rectangular cavity takes into account the effect of turbulence for the Rayleigh numbers Ra = 1.552 × 1010, Ra = 9.934 × 1011, Ra = 1.552 × 1013 and Ra = 2.425 × 1014. The Low-Reynolds-number turbulence. k-ε model was employed in this numerical study to model the non-linear relations 𝛻⋅𝜌𝑢′𝑢′̅̅̅̅̅̅̅̅̅̅̅and 𝜕𝐶𝑝𝑇′𝑢′𝜕𝑥𝑖 in the averaged Navier Stokes equation and energy equation respectively to complete the governing equations. Apart from the hot and cold walls, which are maintained at 308K and 288K, respectively, all of the walls of the enclosure are adiabatic. The vorticity vector formulation allowed the pressure term to be removed from the momentum equation. Finite difference approximations were used in the FLUENT program to solve the vorticity, energy, vector potential, and two resultant equations for each model together with their boundary conditions. Results and Conclusion: The outcomes of the study for the distribution of the velocity and temperature components are presented, demonstrating that the number of contours and vortices increases proportionally with the Rayleigh Number. In addition, a higher Rayleigh number indicates more turbulence, which in turn implies a higher absolute value of the velocity hence increased Kinetic energy.
Cost-Effectiveness Analysis of Optimal Control Strategies for Malaria Transmission in Bubanza Province, Burundi.
(Elsevier, 2022) Niyonkuru, Venant; Mutuku, Winifred
Malaria is a parasitic infection ranked among the leading causes of mortality and morbidity in Sub-Sahara African countries. If recommended interventions measures are well applied, malaria can be prevented and controlled. In many cases, the budget allocated to malaria prevention and treatment project is not enough, using malaria intervention measures properly will guarantee the reduction of infected population while the intervention costs is minimized. This saves the budget and produces the results in economical way. The aim of this article is to understand the cost so that decision makers are well informed when they determine budget allocated to malaria interventions. After ordering different possible strategies from the smallest to the highest, utilizing Incremental Cost-Effectiveness Ratio (ICER), we studied the Cost-effectiveness of each strategy. This study analyses the cost-effectiveness of all possible optimal control measures to identify which is the intervention strategy is going to save available resources and cost-effective. After analysis, this study shows that malaria can be minimized in Bubanza using preventive measures at the most cost effective way.
Development of an Advanced Optimization and Optimal Control Mathematical Model for Energy Efficient Operations of Renewable Energy.
(IJRS, 2023) Niyonkuru, Venant
The potential for developing energy-efficient operations for renewable energy systems has increased with the expansion of renewable energy sources. As a result, creating advanced optimization and optimum control mathematical models will be necessary to handle the difficulties related to their integration into the grid. This study develops an enhanced optimization and optimal control model for the energy-efficient operation of renewable energy systems based on photovoltaic (PV) and hydropower. The created model takes into account the economic element while integrating an optimization method to address the energy management issue of PV systems. PV system operation is optimized taking into account a variety of operational restrictions using economic objectives like profit maximization and cost minimization. To further simplify energy balancing and lower risk, a hydropower energy system is also taken into consideration. Furthermore, by forecasting the ideal PV output and hydropower generation for various economic purposes, a dynamic optimum control approach is created to discover the best operational strategy for the systems. Case studies are used to evaluate this sophisticated optimization and optimum control model, and the findings show how effective it is at lowering operating expenses for the operation of renewable energy systems in an energy-efficient manner.
Description of Minimal Entropy Hellinger Sigma Martingale Density of Order One, Order q and Order Zero
(Scientific Research Publishing Inc, 2021) Mwigilwa, Winfrida Felix; Aduda, Jane; Kube, Ananda Omutokoh
Generally in this paper, we show how the new version of parameter 2 U ∈  in Jacod decomposition will change an expression of entropy-Hellinger process of order one, order q and order zero and consequently an equation of minimal entropy Hellinger sigma martingale density for all orders. This is because even the measurable function W ∈ which is an important parameter of an equation of minimal martingale density changes. In order to get a required parameter W ∈ P , we introduce the function 1 t m f = − ∈  during our calculation for all orders. The result is different to order zero because we failed to get an equation of minimal entropy-Hellinger sigma martingale density of order zero.
Marshall-Olkin Exponentiated Fréchet Distribution
(Scientific Research Publishing Inc, 2023) Niyoyunguruza, Aurise; Odongo, Leo Odiwuor; Nyarige, Euna; Habineza, Alexis; Muse, Abdisalam Hassan
In this paper, a new distribution called Marshall-Olkin Exponentiated Fréchet distribution (MOEFr) is proposed. The goal is to increase the flexibility of the existing Exponentiated Fréchet distribution by including an extra shape parameter, resulting into a more flexible distribution that can provide a better fit to various data sets than the baseline distribution. A generator method introduced by Marshall and Olkin is used to develop the new distribution. Some properties of the new distribution such as hazard rate function, survival function, reversed hazard rate function, cumulative hazard function, odds function, quantile function, moments and order statistics are derived. The maximum likelihood estimation is used to estimate the model parameters. Monte Carlo simulation is used to evaluate the behavior of the estimators through the average bias and root mean squared error. The new distribution is fitted and compared with some existing distributions such as the Exponentiated Fréchet (EFr), Marshall-Olkin Fréchet (MOFr), Beta Exponential Fréchet (BEFr), Beta Fréchet (BFr) and Fréchet (Fr) distributions, on three data sets, namely Bladder cancer, Carbone and Wheaton River data sets. Based on the goodness-of-fit statistics and information criteria values, it is demonstrated that the new distribution provides a better fit for the three data sets than the other distributions considered in the study.
Mathematical Modelling of the Effects Funding on HIV Dynamics among Truckers and Female Sex Workers along the Kenyan Northern Corridor Highway
(science publishing group, 2022-08) Kimulu, Ancent Makau; Mutuku, Winifred Nduku; Mwalili, Samuel Musili; Malonza, David; Abayomi, Samuel Oke
The Southern and Eastern parts of Africa are the most hit by HIV/AIDS in the world and a huge financial commitment is required to control the spread of the disease. Of these countries, Kenya and South Africa have been able to increase prevention and treatment services due to their financial commitment to fighting the epidemic. However, studies have shown that most of the financial commitment comes from private donors and the private sectors are recently becoming reluctant to release funds. It is therefore important to ensure that the available funding is effectively utilised. Studies in 2018 show that infections occurred mostly among the key populations on the Kenyan Northern Corridor highway; such as sex workers and truckers. Moreso, transactional sex which involves cash transfer is the main mode of transmission of HIV/AIDS along the Northern corridor highway in Kenya. In this paper, we study the effect of funding on HIV transmission between truckers and female sex workers. A mathematical model with funding parameters is developed and analysed to determine the effects of funding on the HIV transmission dynamics between truckers and female sex workers. The reproduction number is obtained using the next-generation matrix and the conditions for the stability of the equilibrium points are established. The model is fed into the MATLAB ode45 solver and a numerical simulation is carried out. The results show that increasing circumcision funding reduces the rate of migration from the Susceptible class to the Infected class. Also, increasing treatment funding increases the Treatment class and reduces the overall number of AIDS-related.
Action of the Cyclic Group Cn Acting on the Diagonals of a Regular N–Gon
(iiste, 2016) Olum, Fredrick Odondo; Ireri, Kamuti; Mutie, Kavila; Ochieng’, Raymond Calvin
The main objective of this paper is to investigate the act ion of the cyclic group G=Cn on set,X , the diagonals of a regular - gon. We will first d iscuss the transitivity and primitivity of this action after which we will give useful results regarding the suborbits, subdegrees an d ranks of this action. It is worth mention ing that most of the results here have been given as Lemmas and Theorems.
Moore-Penrose Inverse of Linear Operators in Hilbert Space
(academic journals, 2022-10) Mwanzia, J. M.; Kavila, M.; Khalagai, J. M.
In this paper, we investigate properties of with closed range satisfying the operator equations
Ranks, Subdegrees and Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets
(science publishing group, 2017-02) Gikunju, David Muriuki; Nyaga, Lewis Namu; Rimberia, Jane Kagwiria
Transitivity and Primitivity of the action of the direct product of the symmetric group on Cartesian product of three sets are investigated in this paper. We prove that this action is both transitive and imprimitive for all n ³ 2 . In addition, we establish that the rank associated with the action is a constant 3 2 . Further; we calculate the subdegrees associated with the action and arrange them according to their increasing magnitude.
A Novel Model for Female Population on the Effects of African Stalk Borer on Saccharumofficinarum L. under the Sterile Insect Technology Interventions
(IJAAMM, 2023) Luvaha, Joel Lutumbi; Akanga, Jotham; Chepkwony, Isaac; Wali, Augustus
Sugarcane is an important plant, not only for its economic value but its ecological importance. An infestation of E. Walker Lepidoptera pyralidae, which occurs naturally in wetland habitats and tall grasses, ravages the sugarcane stalk reducing its value. The study proposes a novel model for formulating the dynamics of the E. walker population with SIT. A mathematical analysis of the proposed governing equations representing the model has unique and positive solutions. A basic reproduction number computed based n wild free equilibrium(WFE) points was found to beR0 Ç 1 indicating an eminent wipeout of the wild E. walker population under SIT. The local stability of the WFE indicated that the establishedR0 was locally asymptotically stable andR0 Ç 1. The global stability showed that WFE is globally asymptotically stable when R0 Ç 1. The numerical simulation revealed that the wild E. walker population under Sterile Insect Technology (SIT) will be wiped out after more than 120 weeks, which is unrealistic, considering that the sugarcane matures after approximately 78 weeks. Elasticity analysis of the model parameters based on R0 indicated that a possible control lies in controlling the eggs laid and sex ratio. The effectiveness of the control is indicated in the numerical simulation that showed that the population of the wild E. walker is wiped out after approximately 130 weeks. Future studies into the area need to refocus on the timelines to investigate other strategies to reduce the wild E. walker population below the sugarcane maturity stage.
Modelling the Impact of Spread of Human Papillomavirus Infections under Vaccination in Kenya
((EJ-MATH, 2022) Malia, Miriam; Chepkwony, Isaac; Malonza, David
Human Papillomavirus (HPV), a sexually transmitted virus is a collection of more than 40 types of viruses, some of which are linked to several cancers. HPV type 16 and HPV type 18 are accountable for 70% of cervical cancer cause. Besides cervical cancer, HPV has been linked to several cancers such as anal cancer, oropharyngeal cancer and neck cancer. Mathematical models have been used in the evaluation of control strategies and making of health policies. Very few mathematical models have been developed on HPV awareness in Kenya. In this study we developed a deterministic model on the impact of HPV infection under vaccination. In this model we incorporated an ineffective media awareness. We computed the equilibrium points of the model and local and global stability analysis was conducted on the reproduction number. The numerical simulation results show that the HPV infections continue to stay in the community due to the ineffective mass media awareness. Sensitivity analysis show that the infection contact rate 𝜷_𝟏𝟏 and negative attitudes influencing condom use rate 𝜹𝒄are parameters that contribute to the persistence of HPV infections in the community.

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