Papers by Hailay Weldegiorgis Berhe
Journal of Applied Mathematics, Feb 12, 2019
In this paper, dysentery diarrhea deterministic compartmental model is proposed. The local and gl... more In this paper, dysentery diarrhea deterministic compartmental model is proposed. The local and global stability of the diseasefree equilibrium is obtained using the stability theory of differential equations. Numerical simulation of the system shows that the backward bifurcation of the endemic equilibrium exists for 𝑅 0 > 1. The system is formulated as a standard nonlinear least squares problem to estimate the parameters. The estimated reproduction number, based on the dysentery diarrhea disease data for Ethiopia in 2017, is 𝑅 0 = 1.1208. This suggests that elimination of the dysentery disease from Ethiopia is not practical. A graphical method is used to validate the model. Sensitivity analysis is carried out to determine the importance of model parameters in the disease dynamics. It is found out that the reproduction number is the most sensitive to the effective transmission rate of dysentery diarrhea (𝛽 ℎ ). It is also demonstrated that control of the effective transmission rate is essential to stop the spreading of the disease.
Chaos Solitons & Fractals, Sep 1, 2020
Cholera is an acute, diarrheal infection of the intestine which spreads by the ingestion of conta... more Cholera is an acute, diarrheal infection of the intestine which spreads by the ingestion of contaminated water or food. It is an important health problem, especially in counties with poor sanitation and hygiene. In this paper, the Cholera dynamics model with continuous controls is theoretically investigated using optimal control theory. First, the parameters of the assumed model are estimated using a real data of Cholera from Ethiopia's Oromia Region. Then the sensitivity analyses of the parameters of the deterministic Cholera model is investigated to see the relative importance of parameters. It is observed that the intake rate of Vibrio Cholerae from infected humans ( β 2 ) and the cholera recovery rate ( γ ) are the most sensitive parameters. The system is then extended into an optimal control problem to investigate how treatment and sanitation should be carried out to reduce the number of infected and pathogenic populations while minimizing the total cost of implementation. To this end, we have designed a control functional having two controls: treatment and sanitation. Next, we have shown that the optimal controls exist for the system. Then the optimal controls are represented in terms of the solutions of the optimality system, and then the uniqueness of the optimality system is proved. The average cost-effectiveness ratio and the incremental cost-effectiveness ratio are used and compared to examine the most cost-effective strategy. The use of treatment as control is found out to be the most cost-effective strategy using both methods.
African Journal of Mathematics and Computer science Research, Nov 30, 2012
The purpose of the study was to investigate how effectively the penalty function methods are able... more The purpose of the study was to investigate how effectively the penalty function methods are able to solve constrained optimization problems. The approach in these methods is to transform the constrained optimization problem into an equivalent unconstrained problem and solved using one of the algorithms for unconstrained optimization problems. Algorithms and matrix laboratory (MATLAB) codes are developed using Powell's method for unconstrained optimization problems and then problems that have appeared frequently in the optimization literature, which have been solved using different techniques compared with other algorithms. It is found out in the research that the sequential transformation methods converge to at least to a local minimum in most cases without the need for the convexity assumptions and with no requirement for differentiability of the objective and constraint functions. For problems of non-convex functions it is recommended to solve the problem with different starting points, penalty parameters and penalty multipliers and take the best solution. But on the other hand for the exact penalty methods convexity assumptions and second-order sufficiency conditions for a local minimum is needed for the solution of unconstrained optimization problem to converge to the solution of the original problem with a finite penalty parameter. In these methods a single application of an unconstrained minimization technique as against the sequential methods is used to solve the constrained optimization problem.
Machine Learning for Cactus Beles Diseases Detection
University, 2019
On the exponential stability of a stochastic model for transmission dynamics of antimicrobial‐resistant infections
Mathematical methods in the applied sciences, Feb 1, 2024
Applied mathematics & information sciences, Nov 1, 2018
In this paper, optimal control theory is applied to a system of ordinary differential equations r... more In this paper, optimal control theory is applied to a system of ordinary differential equations representing a dysentery diarrhea epidemic. Optimal control strategies are proposed to reduce the number of infected humans and the cost of interventions. The Pontryagin's maximum principle is employed to find the necessary conditions for the existence of the optimal controls. Runge-Kutta forward-backward sweep numerical approximation method is used to solve the optimal control system. The incremental costeffectiveness analysis technique is used to determine the most cost-effective strategy. We observe that the control measure implementing sanitation and education campaign is the most efficient and cost-effective.
In this study, a new SIVS epidemic model for human papillomavirus (HPV) is proposed. The global d... more In this study, a new SIVS epidemic model for human papillomavirus (HPV) is proposed. The global dynamics of the proposed model are analyzed under pulse vaccination for the susceptible unvaccinated females and males. The threshold value for the disease-free periodic solution is obtained using the comparison theory for ordinary differential equations. It is demonstrated that the disease-free periodic solution is globally stable if the reproduction number is less than unity under some defined parameters. Moreover, we found the critical value of the pulse vaccination for susceptible females needed to control the HPV. The uniform persistence of the disease for some parameter values is also analyzed. The numerical simulations conducted agreed with the theoretical findings. It is found out using numerical simulation that the pulse vaccination has a good impact on reducing the disease.
Modelling and Stability Analysis of the Dynamics of Measles with Application to Ethiopian Data
Measles is a highly contagious airborne disease that is endemic in many developing countries with... more Measles is a highly contagious airborne disease that is endemic in many developing countries with low levels of vaccination coverage. In this paper, we propose a deterministic compartmental mathematical model for measles disease dynamics. We establish the global stability conditions for the disease-free and endemic equilibria using the Lyapunov function stability method. \textcolor{red}{Using arbitrary parameters it is obtained that the proposed model exhibits forward bifurcation}. Numerical integration using MATLAB software was used to simulate the model solution of the forward problem. We calibrate the proposed model to estimate the parameters with a 95\% confidence interval (CI) using real data from Ethiopia by formulating an inverse problem. We notice that the model fits well with the real data of Ethiopia. The estimated basic reproduction number ($R_0$) is found to be $R_0=1.3973$ which proves that the disease is endemic. It is also obtained from the local sensitivity analysis ...
Computational and Mathematical Methods in Medicine
We proposed a deterministic compartmental model for the transmission dynamics of COVID-19 disease... more We proposed a deterministic compartmental model for the transmission dynamics of COVID-19 disease. We performed qualitative and quantitative analysis of the deterministic model concerning the local and global stability of the disease-free and endemic equilibrium points. We found that the disease-free equilibrium is locally asymptotically stable when the basic reproduction number is less than unity, while the endemic equilibrium point becomes locally asymptotically stable if the basic reproduction number is above unity. Furthermore, we derived the global stability of both the disease-free and endemic equilibriums of the system by constructing some Lyapunov functions. If R 0 ≤ 1 , it is found that the disease-free equilibrium is globally asymptotically stable, while the endemic equilibrium point is globally asymptotically stable when R 0 > 1 . The numerical results of the general dynamics are in agreement with the theoretical solutions. We established the optimal control strategy b...
The approach in these methods is that to transform the constrained optimization problem into an e... more The approach in these methods is that to transform the constrained optimization problem into an equivalent unconstrained problem or into a problem with simple constraints, and solved using one (or some variant) of the algorithms for unconstrained optimization problems. Algorithms and MATLAB codes are developed using Powell’s method for unconstrained optimization problems for barrier function methods and then problems that have appeared frequently in the optimization literature which have been solved using different techniques are solved and compared amongst themselves and with other algorithms.
Results in Physics, 2021
The epidemic of the coronavirus disease 2019 (COVID-19) has been rising rapidly and life-threaten... more The epidemic of the coronavirus disease 2019 (COVID-19) has been rising rapidly and life-threatening worldwide since its inception. The lack of an established vaccine for this disease has caused millions of illnesses and hundreds of thousands of deaths globally. Mathematical models have become crucial tools in determining the potential and seriousness of the disease and in helping the types of strategic intervention measures to be taken to prevent and control the intensity of the spread of the disease. In this study, a compartmental epidemic model of COVID-19 is proposed and analyzed to predict the transmission dynamics of the disease in Ethiopia. Analytically, the basic reproduction number is determined. To observe the dynamics of the system, a detailed stability analysis of the disease-free equilibrium (DFE) of the proposed model is carried out. Our result shows that the DFE is stable if the basic reproduction number is less than unity and unstable otherwise. Also, the parameters of the assumed model are estimated using the actual data of COVID-19 from Ethiopia reported for three months between March and June 2020. Furthermore, we performed a sensitivity analysis of the basic reproductive number and found that reducing the rate of transmission is the most important factor in achieving disease control. Numerical simulations demonstrate the suitability of the proposed model for the actual COVID-19 data in Ethiopia. In particular, the numerical simulation shows an increase in the rate of transmission leads to a significant increase in the infected individuals. Thus, results of the numerical simulations are in agreement with the sensitivity results of the system. The possible implication of this is that declining the rate of transmission to the desired level could enable us to combat the disease. Numerical simulations are also performed to forecast the disease prevalence in the community.
BackgroundCOVID-19 has affected most countries and declared as pandemic. Most countries have impl... more BackgroundCOVID-19 has affected most countries and declared as pandemic. Most countries have implemented some social restrictions to control it. In this work we will use mathematical modelling to assess the current social restrictions in controlling the spread of the disease.MethodsWe formulate a simple susceptible-infectious-recovery (SIR) model to describe the spread of the coronavirus under social restrictions. The transmission rate in this model is considered variable to catch social restrictions impact. We analyze this model, then fit the model to 160 days induced death data in Italy, Iran, USA, Germany, France, India, Spain and China. we estimate some factors that help in understanding not only the spread of the disease but also assess the current social restriction in controlling this disease.ResultsWe find a formula for the basic reproduction function (R(t)) and the maximum number of daily infected people. Then estimate the model’s parameters with 95% confidence intervals in...
Deterministic modeling of dysentery diarrhea epidemic under fractional Caputo differential operator via real statistical analysis
Chaos, Solitons & Fractals, 2020
Abstract Non-Markovian characteristics, possessing memory effects and hereditary properties, play... more Abstract Non-Markovian characteristics, possessing memory effects and hereditary properties, play a vital role when it comes to the transmission dynamics of a disease or an epidemic within human society over the course of time. As a result, non-local operators from the field of fractional calculus are the most suitable choices to comprehend dynamics of the disease transmission. This research study is related with formulation of dysentery diarrhea dynamical nonlinear autonomous model via fractional Caputo differential operator with order τ ∈ (0, 1). Using fixed point theory, its solutions are determined to have properties satisfying existence and uniqueness conditions under the Caputo operator. In addition, the non-negative hyperoctant R + 4 is positively invariant region of the proposed fractional dysentery diarrhea model. Biological parameters of the classical and the Caputo fractional model are estimated under nonlinear parameter estimation technique and the optimized value of the fractional order parameter τ in the Caputo dysentery diarrhea model is computed to be 9.995e-01. The sensitivity of the basic reproduction number R 0 is thoroughly investigated for the most and the least sensitive parameter. Numerical simulations are found to be in good agreement with real statistical data and the theoretical predictions about the disease.
Journal of Biological Dynamics, 2019
In this paper, the dysentery dynamics model with controls is theoretically investigated using the... more In this paper, the dysentery dynamics model with controls is theoretically investigated using the stability theory of differential equations. The system is considered as SIRSB deterministic compartmental model with treatment and sanitation. A threshold number R 0 is obtained such that R 0 ≤ 1 indicates the possibility of dysentery eradication in the community while R 0 > 1 represents uniform persistence of the disease. The Lyapunov-LaSalle method is used to prove the global stability of the disease-free equilibrium. Moreover, the geometric approach method is used to obtain the sufficient condition for the global stability of the unique endemic equilibrium for R 0 > 1. Numerical simulation is performed to justify the analytical results. Graphical results are presented and discussed quantitatively. It is found out that the aggravation of the disease can be decreased by using the constant controls treatment and sanitation.
Applied Mathematics and Computation, 2019
In this paper we propose a co-dynamics deterministic system for measles and dysentery diarrhea di... more In this paper we propose a co-dynamics deterministic system for measles and dysentery diarrhea diseases in a single host population. Using center manifold theory, we show that the co-dynamics model may exhibit a backward bifurcation for some parameter values. Numerical simulations of the system show that the two diseases always coexist if R 0 > 1. The system is extended to include time-dependent control-variables: vaccination, treatment and sanitation of the environment, to minimize the number of infected humans and the cost of implementation of the controls. The Pontryagin Maximum Principle was employed to find the necessary conditions for the existence of the optimal controls. The numerical simulations show that the effective controls could reduce the diseases in the community. The incremental cost-effectiveness ratio was used to quantify the cost-effectiveness analysis. It is found that the control strategy which implements vaccination, treatment of dysentery diarrhea and sanitation of the environment is the most cost-effective.
Nonlinear Dynamics, 2021
In this study, a new SIVS epidemic model for human papillomavirus (HPV) is proposed. The global d... more In this study, a new SIVS epidemic model for human papillomavirus (HPV) is proposed. The global dynamics of the proposed model are analyzed under pulse vaccination for the susceptible unvaccinated females and males. The threshold value for the disease-free periodic solution is obtained using the comparison theory for ordinary differential equations. It is demonstrated that the disease-free periodic solution is globally stable if the reproduction number is less than unity under some defined parameters. Moreover, we found the critical value of the pulse vaccination for susceptible females needed to control the HPV. The uniform persistence of the disease for some parameter values is also analyzed. The numerical simulations conducted agreed with the theoretical findings. It is found out using numerical simulation that the pulse vaccination has a good impact on reducing the disease.
In this study, a new SIVS epidemic model for human papillomavirus (HPV) is proposed. The global d... more In this study, a new SIVS epidemic model for human papillomavirus (HPV) is proposed. The global dynamics of the proposed model are analyzed under pulse vaccination for the susceptible unvaccinated females and males. The threshold value for the disease-free periodic solution is obtained using the comparison theory for ordinary differential equations. It is demonstrated that the disease-free periodic solution is globally stable if the reproduction number is less than unity under some defined parameters. Moreover, we found the critical value of the pulse vaccination for susceptible females needed to control the HPV. The uniform persistence of the disease for some parameter values is also analyzed. The numerical simulations conducted agreed with the theoretical findings. It is found out using numerical simulation that the pulse vaccination has a good impact on reducing the disease.
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Papers by Hailay Weldegiorgis Berhe