MST-Department of Mathematics
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Item Analysis of a 3D Heat Transfer of Magneto hydrodynamics Cu-H20 and Al2O3-H20 Nonoflluid over an Exponationally Stretching Plate(Kenyatta University, 2023-08) Ruto, Celestine C.Item Mathematical Modeling and Analysis of Corruption Dynamics in Kenya(Kenyatta University, 2024-04) Muriithi, F. MuthoniCorruption, which can be defined as the abuse of public office for private gains, is a complex and multifaceted problem that has negative impacts on a countryโs economy, development, and governance. In the presence of corrupt practices, the affected countries have witnessed an upsurge in poverty levels, political instability, limited employment opportunities, the proliferation of debts (old and new), and a host of other challenges. Although some countries have made commendable strides trying to combat corruption, others have achieved minimal progress, and regrettably, despite efforts aimed at eradicating corruption, it still remains remains a persistent issue and especially in Kenya. Itโs for this reason that a better understanding of its dynamics is needed to design effective policy interventions to reduce its prevalence and impact. The goal of this study is to use mathematical modeling and analysis to better understand the dynamics of corruption in Kenya, specifically by modeling the spread and dynamics of corruption using an epidemiological approach. The study aims to investigate the existence and stability of the corrupt-free and endemic equilibrium points, determine the parameters that drive corruption, and compute the reproduction number. The methods applied include the use of ordinary differential equations, linearization method by Jacobian Matrix, Lyapunov function, Next Generation Matrix, Normalized forward sensitivity index, and numerical simulation using MATLAB software. The study conducted stability analysis of the equilibrium states by applying linearization, Lyapunov function and Routh-Hurwitz criteria. The findings indicated that the corruption free equilibrium is stable both locally and globally in cases where R0 < 1 as well as the endemic equilibrium being asymptotic stable when R0 > 1. In addition, a sensitivity analysis was conducted to identify the most sensitive parameter that could be strategically manipulated to effectively combat corruption. This study will contribute to a deeper understanding of corruption dynamics in Kenya and inform policy-making and guide anti-corruption efforts. The expected output is to provide insights into the factors that influence the spread and persistence of corruption in a society. The study also identifies strengths and limitations associated with the epidemiological approach to modeling the dynamics of corruption and recommends potential ways of combining different approaches to study this complex and multifaceted problem. The study recommends policies that aim to reduce the benefits of engaging in corruption and increase the costs of engaging in corrupt behavior to effectively address the issue of corruption.Item Water, Ethylene Glycol and Propylene Glycol Based Nanofluids with Copper Oxide and Magnesium Oxide for Optimal Radiator Cooling(Kenyatta University, 2023-11) Kisengese, Hilder MaryAutomakers recognize the importance of coolants in keeping engines running smoothly by eliminating waste heat and preventing corrosion in the cooling system. The automotive industryโs cooling system is a considerable problem in producing efficient and cost-effective engines. Most integrated circuit engines use fluid cooling, which relies on liquid coolants like ethylene glycol and water with poor heat transmission properties. Nanoparticles, which have been shown to improve in thermal conductivity, are another option for enhancing their thermal physical properties. With their improved thermophysical qualities, nanofluids find utility as coolants in various mechanical and engineering contexts, including, but not limited to, the following: electronics, vehicles, transformers, computers, and electrical devices. This study compares nanofluids with three base fluids; water, Propylene glycol, and ethylene glycolโ each containing copper oxide or magnesium oxide nanoparticles in order to establish the optimal coolant for a radiator. The governing equations will be nondimensionalised using appropriate similarity transformation. The resulting equations are solved using numerical method with the Runge-Kutta method of the fourth order. An in-depth discussion is given, along with graphical representations of relevant parameters, for the effects on fluid temperature, skin friction, fluid velocity and rate of heat transfer and the results discussed taking into account industrial applications. The results show that water-based nanofluid has the highest flow temperature and velocity among the three base fluids. At low magnetic field, MgO-water nanofluid has lower skin friction but CuO-water has the lower skin friction at high magnetic field.Item Magnetohydrodynamic Flow in a Rotating System Over Horizontal Parallel Plates with Mass Transfer in a Porous Media.(Kenyatta University, 2023-12) Momanyi,Francis OndusoMHD as a field has got range of applications in real-life situations such as engineering and industries, polymer processing, glass fibre production, metallurgy, paper production glass blowing, purifying molten metal and non-metal inclusion, solar energy harvesting, paper production, cooling the nuclear reactant, and plastic film. This work investigated MHD flow in a rotating system over horizontal parallel plates with mass transfer in a porous media. The flow is considered steady and at the same time, the porous plates are stationary. The sheet along the z direction was considered infinite. The governing equations and the boundary conditions were non dimensionalised. The non-dimensional numbers obtained were; modified Grashof number, magnetic parameter, Rotational number, and permeability parameter. The resulting equations were then transformed using the FD method resulting in algebraic matrices that were solved by computer software (MATLAB) and then analysed for velocity and concentration profile. The numerical data on concentration and velocity profiles were recorded and presented graphically for interpretation and discussion. Some of the results revealed that velocity decreases when the magnetic parameter and Grashof number increase and an increase in Grashof values leads to a decrease in velocity.Item Mathematical Modelling of Underground Water Contamination(Kenyatta University, 2024-04) Kibet, AruasaGroundwater pollution is a major cause of many health hazards in our society. The pollution comes as a result of human indifference to waste disposal, industrial effluents, chemical spills etc. One major contaminant of underground water is nitrogen; which stems from excessive application of nitrogen-containing fertilisers, chemical spillage, etc. Its transport as it percolates through the porous media of ground surface layer to the underground water can be modelled mathematically by the advection-diffusion equation; advection (transport of contaminants by a bulk of the fluid parcel) and diffusion (random movement of the solutes i.e. contaminants, during transport). A linear model that incorporates both source and sinks is formulated and then non-dimensionalised by introducing suitable dimensionless variables and parameters. A finite difference scheme is developed for the problem using the Crank-Nilcoson scheme. The resulting algebraic equations are solved simultaneously at each time step to unravel the effect of the parameters on the concentration of nitrogen at each soil layer. The findings show that raising diffusion increases in concentration of nitrogen but increase in Peclet number leads to decreases in the concentration.Item Determining the Socio-Economic Classes Using Principal Component Analysis Based on Binary Data(Kenyatta University, 2023-01) Nasokho, Ruth Simuli; Leo Odongo; Anthony GichangiAbstractItem Numerical Simulation of Natural Turbulent Convection with Vorticity Vector Formulation(Kenyatta University, 2023-10) FILENTINUS, OTULO ONYANGO; Kennedy AwuorAbstractItem Modeling Jiggersโ Infestation with Incomplete Recoveries Incorporating the Flea Population; A Case Study of Murangโa County, Kenya(Kenyatta University, 2023-09) Agutu, Arthur Omondi; Kennedy AwuorSand flea insect thrives in hot and humid regions full of dust particles. It attacks humans leading to jiggers infection (tungiasis). Spread of jiggers has been recorded in Caribbean, South American and African countries. In Kenya, Murang'a, Homabay and Siaya Counties are among the top regions affected by tungiasis infections. Poverty, lack of sufficient awareness, improper sanitation, and poor control methods are the major reasons for the unending spread of jiggers in Kenya. Prevention and treatment measures have been put in place by the government and NGO's to combat the unending new infections, yet the recoveries are still incomplete. A number of mathematical frameworks have been put in place to unravel the cyclic behavior of this infectious disease. However, a comparative study of the dynamical behavior of the disease in both human and flea population has not been conducted. In this research, we designed a model of jiggers infestation which incorporates the human and sand flea population in Muranga County, Kenya. We derived an ODE system from SEIR-FLA mathematical model to investigate the dynamics of jiggers infestation which incorporates both the human and flea population in Muranga County, Kenya. We used the Mathematica software tools to determine the effective basic reproduction number by applying the next generation matrix method. We applied the MATLAB software to generate the solutions of equations. Results confirmed local stability of JFE when Ro = 4.9827ยข โ 13 as t โ oo for all the Susceptible, Exposed, Infectious, Recovered human compartments and the Egg, Larval, Adult sand flea compartments. All state variables are positive at all times ยข, and numerical analysis of the invariant region reveals that the model is well-posed. These findings confirm that treatment aid in reducing incomplete recoveries of jiggers infestation.Item A Jump Diffusion Model with Fast Mean Reverting Stochastic Volatility for Pricing Vulnerable Options(Kenyatta University, 2023-10) Kalekye, Nthiwa Joy; Ananda Omutokoh Kube; Cyprian Ondieki OmariThe Black-Scholes-Merton option pricing model is a classical approach that assumes the underlying asset prices follow a normal distribution with constant volatility. However, this assumption is often violated in real-world financial markets, resulting in mispricing and inaccurate hedging strategies for options. Such discrepancies may result into financial losses for investors and other related market inefficiencies. To address this issue, this study proposes a jump diffusion model with fast mean-reverting stochastic volatility to capture the impact of market price jumps on vulnerable options. The performance of the proposed model was compared under three different error distributions: Normal, Student-t, and Skewed Student-t and under different market scenarios that consist Bullish, Bearish, and Neutral markets. In a simulation study, the results show that our model under Skewed Student-t distribution performs better in pricing vulnerable options than the rest under different market scenarios. Our proposed model was fitted to S&P 500 Index by maximum likelihood estimation for the mean and volatility processes and Gillespie algorithm for the jump process. The best model was selected based on AIC and BIC. Samples of the simulated values were compared with the S&P 500 values and MSE computed at various sample sizes. Values of MSE at different sample sizes indicate significant decrease to actual MSE values demonstrating it provides the best fit for modeling vulnerable options.Item Enumeration of Sigma Algebras on Sets with at Most Seven Elements(Kenyatta University, 2023-04) Mariga, Hildbrand; Benard KivungeItem Magnetohydrodynamic Nanofluid Flow over Convectively Heated Porous Radially Stretching Sheet(Kenyatta University, 2023-11) Alai, Kulow Alou; Lawrence Njau; Maurine WafulaAny material containing pores is termed a Porous media. Extensive research has been conducted on MHD nanofluid flow through porous materials. Included in these experiments are parallel-rotating plates surrounded by a porous channel and the influence of rotation on unstable couette flow. MHD nanofluid flow across a plate heated by convection atop a stretching sheet immersed in a porous medium has not been taken into account in any of these research. Consequently, the studyโs goal is to investigate the motion of MHD nanofluids across a convectively heated plate superimposed on a radially expanding sheet embedded in a porous media. The model is formulated and non dimensionalised using similarity variables. By employing shooting technique to transform the boundary conditions and, Runge-Kutta scheme in MATLAB bvp4c, the system of ODEs are solved. The results obtained are displayed in graphs and others in tables. The results indicate that with increasing porosity, magnetism and surface rotation, the flow primary velocity decreases while the temperature profile surges. The findings from this study will provide beneficial theoretical insight on what parameters should be varied for maximum profit in a number of sectors like, power engineering sector, aerodynamic combination, drug recovery systems and water solar heating system.Item Sequential Change Point Estimation Using Empirical Likelihood for Time Series Data(Kenyatta University, 2023-09) Machuka, Carolyne Kemunto; Ananda KubeSequential Change point detection has enhanced the reliance on analysis of live data as it streams into the system to support real-time decision-making processes. This has played a key role in the advancement of time series modeling and forecasting in financial time series and risk management. This domain has a growing demand to identify change points precisely and efficiently for development of automated analytical models. In this work, sequential change point estimation based on empirical likelihood test statistic is developed by maximizing the likelihood of the empirical distribution of the data subject to constraints based on the sample moments. Change point is declared when the test statistic exceeds a set threshold. The threshold is set such that it maximises the power of rejecting the null hypothesis. A stop time function is defined based on the null hypothesis. Consistency of the change point estimator has been verified through monte carlo simulations based on different sample sizes to demonstrate the empirical power of the test statistic. As the pre change historical data grew unbounded, the bias of the Mean Absolute Error (MAE) approaches zero. The estimator converges closer to the true values. The estimator was fitted on KES/USD exchange rate data from January 2017 to December 2021.Item Effects of Induced Magnetic Field, Hall Current and Radiative Heat on 3-D Unsteady Hydromagnetic Stagnation Flow of a Casson Fluid(Kenyatta University, 2023) Kyalo, Muli Gabriel; Winifred MutukuIn this study, 3-D stagnation flow of an MHD viscous electrically conducting Casson fluid including effects of induced magnetic field, Hall current and radiative heat is analyzed. The unsteady state model governing the flow is analyzed by first coming up with non-linear partial differential equations. Secondly, similarity transformation is done to change the nonlinear partial differential equations which are non-linear to ordinary differential equations, in order to account for the boundary layer and ease the computation. The ordinary differential equations obtained are then solved numerically using the collocation method via the MATLAB software. Analysis is then done to investigate effects of flow parameters like Hall current, magnetic parameters, temperature field, the Casson parameter and unsteadiness parameter.Item An Investigation of 3d Heat and Mass Transfer in Magnetohydrodynamics Laminar Flow about An Inclined Semi-Infinite Porous Plate(Kenyatta University, 2022) Onyinkwa, Linah Mogirango; ChepkwonyInvestigation of transmission of mass and heat in magnetohydrodynamics (MHD) flow over a plate inclined at an angle past semi-infinite porous plate is carried out in this study. MHD flow model is formulated by combining the electromagnetic laws with Navier-stokes equations. The equations that govern the flow are transformed into their dimensionless form. The numerical scheme in the form of Implicit finite centre difference is used to obtain a numerical solution to the equations in MATLAB. The numerical results are presented graphically by varying the parameters that emerged from the flow. Effects of porosity parameter, magnetic strength parameter, and the inclination angle are studied. Velocity profiles in all directions are reduced with increasing porosity, magnetic field and inclination angle while there is a raise in the temperature profiles with increasing porosity, magnetic field and surface inclination angleItem Numerical Simulation of Natural Turbulent Convection with Vorticity Vector Formulation(Kenyatta University, 2023) Filentinus, Otulo OnyangoTurbulent 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. The performance of a numerical turbulence model k-ฮต 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 = 1.552ร1010, Ra = 9.934ร1011, Ra = 1.552ร1013 and Ra = 2.425ร1014. The Low-Reynolds-number turbulence k-ฮต model was employed in this numerical study to model the non linear relations โยท ฯuโฒ iuโฒ j and โCpTโฒuโฒ i โ 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. 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.Item Mathematical Model for Coinfection of Hiv/Aids and Kaposiโs Sarcoma with Treatment(Kenyatta university, 2023) Juma, Joy Tengโan; Isaac ChepkwonyHIV destroys T-cells in order to target the body's defence mechanism. Without treatment HIV infection advances in stages causing destruction and reduction in T-cells thus, rendering the body incapable of fighting other infections such as respiratory infections, sexually transmitted diseases and some cancers. Kaposiโs sarcoma is the cancer that allows tumour to grow in an HIV-patient and its presence in a patient is an indication that HIV has fully developed to AIDS in the patient. AIDS-related Kaposiโs sarcoma (AIDS-KS) is still one of the most common malignancies in Kenya and sub-Saharan Africa and is associated with high morbidity and mortality. Researches have indicated that AIDS-associated KS was on the rise in sub-Saharan Africa until the introduction of ART. It is of great significance to comprehend the impact of ART used on HIV/AIDS and the coinfection of HIV/AIDS and AIDS-related Kaposiโs sarcoma. Mathematical models have been proven valuable to give a decisive quantitative information about the dynamics and pathogenesis of HIV, responses of the immunity to anti-retroviral treatment and to the study of the coinfection of HIV/AIDS and other opportunistic infections like Malaria, TB, Pneumonia and Hepatitis however the coinfection of HIV/AIDS and AIDS-related Kaposiโs sarcoma has not been considered much incorporating the aspect of treatment. In this study, a mathematical model for the coinfection of HIV/AIDS and KS with treatment is developed and analysed to explore the effect of usage of ART on HIV/AIDS and the coinfection of HIV/AIDS and Kaposiโs sarcoma. The model solution is explored for positivity and boundedness. The next generation matrix is used to derive the basic reproduction number of the model, while the disease free equilibrium point is determined for stability where it was verified that the infection-free equilibrium ๐ธ0 is locally asymptotically stable whenever ๐ 0๐ป<1 and ๐ 0๐พ<1. KS and HIV infection will go to extinction if the reproduction number is < l and persist in the population if it is > 1. Numerical simulations is used to illustrate that by providing treatment to the HIV and the co-infected population immune system is strengthened and thus progression rate to AIDS is reduced.Item Maximum Likelihood Estimation of Parameters for Kumaraswamy Distribution Based on Progressive Type Ii Hybrid Censoring Scheme(Kenyatta university, 2023) Meymuna, Shariff Jaffer; Edward .G. NjengaThe project considers the maximum likelihood estimators for Kumaraswamy distribution based on progressive type II hybrid censoring scheme using the expectation maximization algorithm. A two parameter Kumaraswamy distribution can be applied in natural phenomena that have outcomes with an upper and a lower bound. Kumaraswamy distribution remains of keen consideration in disciplines such as economics, hydrology and survival analysis. The field of survival analysis has advanced over the years and extensive research has been undertaken. Previous studies have considered maximum likelihood estimation for Kumaraswamy distribution based on progressive type II censoring scheme using methods like Newton-Raphson and EM algorithm but none has used progressive type II hybrid censoring scheme and obtained maximum likelihood estimators of Kumaraswamy distribution via EM algorithm. EM algorithm has been utilized in manipulation of missing data as it is a more superior method when handling incomplete data. Comparison of different combinations of censoring schemes with respect to the MSEs and biases at fixed parameters of ๏ก and ๏ข are obtained through simulation. It is observed that in the three censoring schemes, for an increasing sample size, the MSEs and biases are generally decreasing. Eventually, an illustration with real life data set is provided and it illustrates how maximum likelihood estimators works in practice under different censoring schemes.Item Maximum Likelihood Estimation of Parameters for Kumaraswamy Distribution Based on Progressive Type Ii Hybrid Censoring Scheme(Kenyatta University, 2023) Jaffer, Meymuna Shariff; Edward .G. NjengaThe project considers the MLEs for Kumaraswamy distribution centered on PTHCS using the expectation maximization algorithm. A two parameter Kumaraswamy distribution can be applied in natural phenomena that have outcomes with an upper and a lower bound. Kumaraswamy distribution remains of keen consideration in disciplines such as economics, hydrology and survival analysis. The field of survival analysis has advanced over the years and extensive research has been undertaken. Previously, maximum likelihood estimator of various distributions has been done using methods like Newton-Raphson, Bayesian inference and EM algorithm. The application of these techniques in survival analysis is mainly intended to save on costs and duration taken in an experiment. Based on PTHCS, MLEs of Kumaraswamy distribution are obtained via EM algorithm. EM algorithm has been utilized in manipulation of missing data as it is a more superior method when handling incomplete data. Comparison of different combinations of censoring schemes with respect to the MSEs and biases at fixed parameters of ๏ก and ๏ข are obtained through simulation. It is observed that in the three censoring schemes, for an increasing sample size, the MSEs and biases are generally decreasing. Eventually, an illustration with real life data set is provided and it illustrates how MLEs works in practice under different censoring schemes. It is apparent from the observations made that the estimated values of ^๏ก and ^๏ข increases from scheme one to scheme three.Item Analysis of Magnetohydrodynamic Heat and Mass Transfer with Carbon Nanotubes-Graphene Casson Hybrid Nanofluid(Kenyatta University, 2022) Juma, Belindar Atieno; W.N.MutukuFluids such as liquids and gases are amassed into either Newtonian or non-Newtonian groups. Most fluids fall under the non-Newtonian category and, as a result, various models like Casson fluid and Williamson fluid model have been proposed to deal with the non-Newtonian fluid behaviour. Due to its ability to model the flow of blood, Casson fluid model is of major medical importance. A development on ordinary fluids are nanofluids, which posses enhanced thermophysical properties. Hybrid nanofluid, obtained when two non-identical nanoparticles are dispersed in a fluid, is an improvement on the novel nanofluids. It has superior thermal conductivity when compared with nanofluids or ordinary fluids. Multiple research have shown that the shape of the nanoparticles used during the hybridization process has significant impact on the thermal properties of the hybrid formed. Applications of hybrid nanofluid include refrigerators, electronic devices, and cancer treatments. In the study of hybrid nanofluids, the focus has been placed on the dynamic properties and heat transfer rate. In consensus, the superiority of the hybridโs properties are emphasized. Carbon being the most abundant product, hardest, strongest and stable known compound, it is an excellent thermal conductor. CNTs and graphene are allotropes of Carbon. In the HAMT research, no researcher has explored the impact of suspending a combination of CNTs and graphene nanoparticles on a Casson base fluid. To bridge this gap, this study is designed to analyse the HAMT rate of a 2-D magnetohydrodynamic hybrid Casson nanofluid. The nanoparticles are Carbon nanotubes and Graphene. The flow is across a surface stretching exponentially. Volume fraction, nanoparticle size and other pertinent parameters are investigated on the HAMT rate.The governing equations are converted to their non - dimensional form using similarity variables, and subsequently to an ODEs. The RK4 with Shooting Technique is adopted as a method of solution. Simulation of the model and investigation of the HAMT rate is carried out using MATLAB bvp4c. The primary velocity is reduced with Casson fluid parameter but enhanced with the radiation parameter. The temperature profiles boost with Casson fluid parameter, magnetic and radiation parameters. The local skin friction increases with Casson fluid parameter and radiation parameter but decreases with magnetic field strength. HAMT rate is enhanced with increasing Grashof number but decreases with Casson fluid parameter and magnetic field strength.Item Analysis of Vertical Transmission Dynamics of Infectious Hepatitis B Virus: Mathematical Model Involving Vaccination And Treatment in Turkana County, Kenya(Kenyatta University, 2022) Ghai Kuei, Geu; GatotoHepatitis B has been a major global health menace for it is a potentially life-threatening liver disease. Close to 0.25 billion persons are living with this infectious disease across the world. Itโs transmitted by infected individual to uninfected person either vertically (transmission before or during birth by carrier mother to the baby) or horizontally (transmission when the bodily fluid of an infected person comes into contact with the hepatitis B virus-free person). This can happen through the sharing of non-sterilized injection syringes, tattooing objects and through sexual intercourse. This particular project studied a mathematical model that combined both vaccination and treatment as a means to controlling the hepatitis B virus (HBV). In our mathematical model, equations are derived from the flow chart representing the HBV transmission dynamics. The determination of the disease-free equilibrium state (DFE), the endemic equilibrium state (EE) and the basic reproduction number ๐ 0 were made. The stability of these points were determined and the results show that the disease-free equilibrium was both locally and globally asymptotically stable. In other words, ๐ 0 < 1. The stability analysis of endemic equilibrium point also reveals that the point is locally and globally asymptotically stable, that is, ๐ 0 > 1. The basic reproduction number ๐ 0 is computed using the next generation matrix method. The system of ordinary differential equations (ODEs), which is non-linear are solved by numerical simulation. This was achieved by use of Runge-kutta method of order four with the help of MATLAB software. These results show that either of the method, treatment or vaccination, administered is effective in alleviating the spread of HBV disease. However, when both control strategies are combined, the disease is quickly controlled and ultimately brought to eradication.