Departments of Mathematics

Permanent URI for this collectionhttps://etd.hu.edu.et/handle/123456789/99

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AMathematical Model Analysis on the Dynamics of Online GameAddiction with Optimal Control
(HAWASSA UNIVERSITY, 2024-06) Iyasu Kaleb
Excessive playing of online games leads to addiction, which causes academic under achievement and prevents the daily achievement of goals. Symptoms of addiction are spend ing more time gaming online and irritability, while risk factors are low self-worth, anxiety and depression. This thesis study focuses on the mathematical model analysis of the dynam ics of online game addiction with optimal control. We demonstrated the existence, positivity and boundedness of the solution of the dynamical system. The system has two equilibrium points, namely the online game addiction free and endemic equilibrium points, and their stability was determined using the linearization technique, the Castillo-Chavrz theorem and LaSalles’s invariant principle. The basic reproduction number (R0) of the model was cal culated using the principle of the next generation matrix and the sensitivity indices of R0 to the model parameters were investigated. Additionally, bifurcation analysis was performed to verify the backward and forward bifurcations, and from the analysis, we observed that the model system exhibits forward bifurcation at R0 = 1. The optimal control problem of the model was analyzed using Pontryagin’s Maximum Principle and the characterization of the optimal control was constructed. Numerical simulations were performed to verify the accuracy of the analytical results using MATLAB software. From the numerical simula tion of sensitivity analysis, decreasing the contact rate with addicted(β1) and incompletely recovered(β2) individuals, the addiction rate(δ), the re-addiction rate(ω) and the incomplete recovery rate of treated individuals(τ) decreases the reproduction number. Using optimal control, the combined strategy of minimizing the contact rate with addicted individuals and the re-addiction rate of incompletely recovered individuals minimizes the exposed and ad dicted individuals as well as the associated cost
Analysis of Mathematical Model on the Transmission Dynamics of COVID-19 with Protected and Hospitalized individuals.
(HAWASSA UNIVERSITY, 2024-11) Teshale Deboch
Mathematical modelling is important for better understanding of disease dynamics and developing strategies to manage rapidly spreading infectious diseases. In this thesis, we propose a mathematical model to investigate coronavirus diseases (COVID-19) transmission in the presence of protected and hospitalized classes. Analytical and numerical approach is employed to investigate the results. In the analytical study of the model, we have shown the local and global stability of disease-free equilibrium, existence of the endemic equilibrium and its local stability, positivity of the solution, invariant region of the solution and sensitivity analysis of the model is conducted. From these analyses, we found that the disease-free equilibrium is globally asymptotically stable for 𝑅0 < 1 and is unstable for 𝑅0 > 1. A locally stable endemic equilibrium exists for 𝑅0 > 1, which shows the persistence of the disease if the reproduction number is greater than unity. Using sensitivity analysis we establish that 𝑅𝑜 is most sensitive to the rate of Protection of Susceptible individuals 𝜃 and that a high level of protection needs to be maintained as well as hospitalization to control the disease. Finally, we performed numerical simulations using MATLAB software Ode 45 codes to supplement the effectiveness of the analytical findings.
Analysis of Unhealthy Attitude on Marriage and its Impact on the Dynamics of Divorce using Mathematical Model
(HAWASSA UNIVERSITY, 2024-06) Etsehiwot Gobezayewu
Divorce is the dissolution of two partners marriage, which is a serious problem challenging the establishment of the family in a routine manner and causing severe impacts on the emotional and mental health of the individual. In this thesis, we modified and analyzed a mathematical model that describes the spread of unhealthy attitude on marriage and its impact on the dynamics of divorce. The population was divided in to six compartments and analyzed using a system of non-linear ordinary differential equations. The basic reproduction number R0 is calculated using next generation matrix operator. We have found two equilibrium points, namely the unhealthy attitude free equilibrium point and the unhealthy attitude present equilibrium point. We have established the conditions for the stability of equilibrium points. The unhealthy attitude free equilibrium point is both locally and globally stable if R0 < 1, and the unhealthy attitude present equilibrium point is also both locally and globally stable if R0 > 1. The modified model exhibits a forward bifurcation whenever R0 < 1, which indicates the threshold parameter plays an important role in reducing the spread of unhealthy attitude on marriage. Secondary data sources for divorce case were collected from Hawassa first instance court. The data were fitted to the model to estimate some parameter values using Least Square optimization method. Sensitivity analysis was performed to identify parameters which are sensitive to the reproduction number. According to this the contact rate between married and individuals who have unhealthy attitude β1 and the rate of transfer from unhealthy attitude to healthy counseling φ plays an important role in reducing R0 less than unity. Finally, we performed numerical simulations using ODE 45 codes to support the analytical results in agreement with numerical solutions.
MATHEMATICAL MODEL ANALYSIS ON THE TRANSMISSION DYNAMICS OF CHOLERA
(2024-10) Alemayehu Animaw Nigussie
This research focuses on the development and analysis of a mathematical model to un derstand the transmission dynamics of cholera, a waterborne disease caused by Vibrio cholera. The model divides the population into five compartments: Susceptible (S), Symptomatic Infected (Is), Asymptomatic Infected (Ia), Recovered (R), and Contami nated Environment (B), considering both human-to-human and environment-to-human transmission routes. The threshold parameter R0 = kδβ1λψ+ϵ1λeλψ µkδ(µ+τ1) + kδβ2λϕ+ϵ2λeλϕ µkδ(µ+τ2) is cal culated to assess whether cholera will spread in the population. It is found that if R0 is less than 1, the disease-free equilibrium point is stable, suggesting that cholera can be eradicated under these conditions. Stability analysis of both disease-free and endemic equilibrium points shows that cholera can be controlled when certain conditions, such as reduced infection rates and environmental contamination, are met. Simulations using MATLAB’s ODE45 solver confirm the theoretical results, demonstrating that controlling R0 can effectively manage the disease dynamics. The study identifies key parameters in f luencing R0, including the transmission rates and the impact of therapeutic treatments, providing valuable insights for intervention strategies. The model is based on a system of nonlinear ordinary differential equations that describe the dynamics of the disease in the population. It incorporates therapeutic treatment rates, environmental bacte rial concentration, and different infection categories to represent a more realistic cholera transmission scenario. The research concludes that cholera transmission can be reduced by controlling the following conditions: Lowering the contact rates between susceptible individuals and infected individuals (both symptomatic and asymptomatic) significantly impacts R0. Controlling environmental contamination from infected individuals is cru cial in minimizing disease spread. Effective therapeutic treatments for symptomatic cases can reduce the spread and severity of the disease. This study provides a comprehensive framework for understanding the dynamics of cholera and offers actionable insights for public health strategies aimed at controlling and preventing outbreaks.
Mathematical Modeling and Analysis for the Dynamics of a Two Stage Hepatitis B Virus with Vaccination and Vertical Transmission
(HAWASSA UNIVERSITY, 2024-06-27) Fikeradis Nega
In this thesis, we proposed a deterministic mathematical model for acute and chronic hepatitis B virus with vaccination and vertical transmission. We examined the well-posedness of the model by proved its positivity, uniqueness, and boundedness of the solutions to the system equation. We computed both the disease-free (DFE) and endemic equilibrium (EEP) points of the model and analyzed their local and global stability by computing its basic reproduction number. We have shown that when the reproduction number R0 < 1, the illness will die out or go extinct, as the disease-free (DFE) equilibrium point is stable for R0 < 1. The global stability of the disease-free equilibrium point is also proved by applying the castillo-Chavez theorem. The local stability of the endemic equilibrium point was determined using center manifold theory and its bifurcation analy sis, and it was found that the model exhibits a forward bifurcation at R0 = 1. The global stability of the endemic equilibrium point is also determined by using the Lyapounov function. We have also performed a sensitivity analysis of the model with respect to the model parameters. We observed that, the recruitment rate, and the transmission rate of hepatitis B virus have positive correlation with R0, while the vaccination rate of newborns(pediatric vaccination) and vaccination rate have negative correlation with R0. Numerical simulation of the model is also performed using some assumed and literaturized values of the parameters
MATHEMATICAL MODELING AND ANALYSIS OF CRIME WITH MEDIA IMPACT
(HAWASSA UNIVERSITY, 2024-10) ABDURO MAMU DEKEMA
In this thesis, we proposed and analyzed a nonlinear mathematical model that explains the dynamics of crime that includes media coverage. Some fundamental properties of the model including existence and positivity as well as boundedness of the solutions are investigated. Algebraic expression for the basic reproduction number is computed using the next generation matrix method. The model exhibits two equilibria: the crime-free equilibrium and the persistent equilibrium. The analysis shows that the crime-free equilibrium point is locally asymptotically stable if the basic reproduction number is less than unity (R0 < 1) and the persistent equilibrium point is locally asymptotically stable when the basic reproduction number is greater than one (R0 > 1). It is pointed out that the crime dies out if there is effective media coverage and awareness about crime, otherwise, the crime spreads rapidly in the society and it becomes an epidemic. The Numerical simulation is carried out using Ode45 of Matlab, sensitivity analysis of the basic reproduction number is also constructed. Analytical and numerical result shows that effective use of media coverage is the strategy to reduce the dynamics of crime transmission.
Mathematical Modeling and Analysis on the Dynamics of Tobacco Smoking by Considering Occasional and Habitual Smokers
(HAWASSA UNIVERSITY, 2024-06) Genet Endale
This study investigate smoking tobacco considering occasional and habitual smokers using mathematical model. The study aims to understand how to decrease smoking habit. we clas sified the total population in to four compartments namely; susceptible, secondhand smokers, occasional smokers, habitual smokers and Recovered. Positivity of the solutions and bound edness of the model are discovered. Using Routh-Hurwitz criteria ,center manifold theory and Castillo Chavez Theorem, we assed equilibrium point of the model and investigate their sta bility. The basic reproduction number (R0) is computed using next generation matrix method. We have proved that the smoking free equilibrium is locally and globally asymptotically stable when R0 < 1 the smoking habit removed & unstable. Also smoking present equilibrium point is locally asymptotically stable,when R0 > 1 smoking habit persist. Additionally, our model in dicates that a forward bifurcation and sensitive analysis is performed. Using sensitivity analysis the rate of interaction of smokers with susceptible and rate of recruitment has positive impact on the spread of smoking habit and rate of death due to smoking and treatment rate has negative impact on the spread of smoking habit. Finally, we perform numerical simulations using MAT LAB software Ode 45 codes to support the analytical results are in agreement with numerical solutions