Bifurcation Analysis on the Dynamics of a Genralist Predator-Prey System

Abstract In this paper a generalist predator prey system is modeled as a two dimensional coupled differential system. We identified three vital parameters stands for the maximum uptake rate of the generalist predator, stands for half saturation value and such that / is the conversion efficiency of the generalist predator where is the intrinsic growth rate of the predator. Using these parameters a novel way to divide the parameter space based on the number of interior equilibrium solutions admitted by the system has been presented. We investigate the considered model is reach in dynamics and identified various bifurcations that are experienced by the considered system from the parameter space. These are Saddle-Node bifurcation, Trans-Critical bifurcation and pitchfork bifurcation. In this study we offer mathematical proof for the incidence of these bifurcations that take place in the considered dynamical system as the parameters move between the regions presented in the parameter space

Spread and Control of the Dynamics of HIV/AIDS-TB Co-infection in Ethiopia: A Mathematical Model Analysis

In this work we considered a nonlinear deterministic dynamical system to study the dynamics of HIV/AIDS-TB co-infection in Ethiopia. We found the system exhibit disease free equilibrium point and endemic equilibrium point. For the reproduction number  the disease-free equilibrium point is locally asymptomatically stable and the endemic equilibrium point is locally asymptomatically unstable. We calculate basic reproduction number of the HIV/AIDS-TB co-infection dynamical system which depends on six parameters. Using real data collected from different sectors in Ethiopia we found that the numerical value of the basic reproduction number is. This shows that HIV/AIDS–TB co-infection spread in the society. Using sensitive analysis, we identify the most influential control parameter is the HIV/AIDS-TB co-infection transmission rate. The HIV/AIDS-TB co-infection transmission rate which numerical value to be 0.021. But the real value of is 0.74, to be 0.74 in to 0.021 by fixing the number of contacts for HIV/AIDS-TB co-infection we decrease the effective number of contacts for HIV/AIDS-TB co-infection 74 to 21.  We also perform numerical simulation based on real data collected from different health sectors in Ethiopia. &nbsp

Impact of Climatic Factors and Intervention Strategies on the Dynamics of Malaria in Ethiopia: A Mathematical Model Analysis

In this work we considered a nonlinear dynamical system to study the impact of temperature and rainfall on the transmission of malaria disease in Ethiopia. We found disease free and endemic equilibrium points and we proved their local and global stability. We calculate the effective reproduction number using real data collected from different health sectors in Ethiopia and we found that the malaria disease spreads in both high risk and low risk areas since the effective reproduction number  is greater than unity. We perform sensitivity analysis to identify the most influential control parameter of the spread of malaria disease. And thus, the most temperature dependent influential control parameter is mosquito biting rate  which can be controlled by insecticide treated net. The most rainfall dependent influential control parameter is larvae development rate  which can be controlled by destruction of mosquitoes breeding sites and regular use of larvicides

An Age Structure Mathematical Model Analysis on the Dynamics of Chronic and Hyper Toxic Forms of Hepatitis B Virus by SVEA_1 A_2 CR Model with Vaccination Intervention Control Strategy in Ethiopia

In our work we considered nonlinear ordinary differential equations to study the dynamics of hepatitis B virus (HBV) epidemics in Ethiopia. We proved that the invariant and bounded ness of the solution of the dynamical system. We used a nonlinear stability analysis method for proving the local and global stability of the existing equilibrium points. We have got that the diseases free equilibrium point and endemic equilibrium point exist for some conditions. We proved that the disease free equilibrium point is locally asymptotically stable and also globally asymptotically stable. We found that the effective reproduction number for the system is   which depends on fifteen parameters. On the other hand, the basic reproduction number is   Using standard parameter estimation we found that the numerical value of the effective reproduction number is and   From this numerical value we conclude that the disease spreads in the community and vaccination intervention strategy reduces the spread. Out of these fifteen parameters we identified five parameters which contribute significant role in control of the disease; and these are the rate of moving from exposed to acutely infected class with age below or equal to 5 years , the rate of moving from exposed to acutely infected class with age above 5 years the proportion of leaving acutely infected class with age below or equal to 5 years and progressing to chronically infected class ,the proportions of vaccinated newborns  and the ratio of vaccinated newborn  to total population H; which influence the effective reproduction number. We also conduct numerical simulations which support the finding in the sensitivity analysis