This paper considered the formulation of continuous third derivative trigonometrically fitted method for the solution of oscillatory first order initial value problems using the technique of interpolation and collocation of the approximate solution by combining polynomial and trigonometric functions. Solving for the unknown parameters and substituting the results into the approximate solution yielded a continuous linear multistep method, which was evaluated at some selected grid points where two cases were considered at equal intervals to give the discrete schemes which are implemented in block form. The blocks are convergent and stable. Numerical experiments show that the methods compete favorably with existing method.This paper considered the formulation of continuous third derivative trigonometrically fitted block method for the solution of stiff and oscillatory problems. The development of the technique involved the interpolation and collocation of the approximate solution which is the combination of polynomial and trigonometric functions. Solving for the unknown parameters and substituting the results into the approximate solution yielded a continuous linear multistep method, which is evaluated at some selected grid points where two cases were considered at equal intervals to give the discrete schemes which are implemented in block form. The blocks are convergent and stable. Numerical experiments show that the methods compete favorably with existing method and efficient for the solution of stiff and oscillatory problems.

HIV/AIDS is a serious health problem that continues to present a significant health concern in underdeveloped nations and may be mostly brought on via unprotected sex. This study is designed and analyzed using a dynamic modeling approach to investigate the dynamic of HIV/AIDS model with vertical transmission and the impact of knowledge on its treatment. Our proposed model exhibit disease free and the endemic equilibrium. The uniqueness and the exactness of the model were investigated and the basic reproduction number using next generation matrix was obtained, Stability analysis was also carried out. The model analysis shows that the disease free equilibrium is locally asymptomatically stable (LAS) when Ro < 1. Our research suggests that treatment and awareness campaigns, when combined with other crucial control measures, may help keep the HIV/AIDS virus from spreading.