Prophylaxis (PrEP) is considered one of the promising interventions against HIV infection as trials on various groups and sites have reported its significant effectiveness. From the studies done, analysis of the data collected shows that PrEP has the potential to help prevent HIV infections and slow the HIV epidemic. In this paper, a deterministic model that incorporates PrEP is developed to assess the impact of the use PrEP on the transmission of HIV/AIDS. Stability analysis of the model showed that DFE is locally and globally asymptotically stable when R p < 1. The EE exists and is locally and globally asymptotically stable when R p > 1. PrEP effectiveness is achieved if taken effectively, hence lack of adherence to PrEP substantially increases the risk of HIV infection among the high risk individuals.

Alcohol addiction is a phenomenon that has attracted the attention of numerous researchers and academics in a variety of professions due to its serious repercussion on all spheres of human life. Alcoholism is a common addiction in adults throughout the globe. Hence there is need to target efficient preventative and therapeutic measures.The impact of media awareness and treatment on the drinking behavior of various drinker classes is discussed in several mathematical models. Nevertheless, the impact of the exposed class of alcoholics on light and heavy drinkers in the presence of media awareness, has not been addressed. The model was formulated based on a system of differential equations. To perform stability analysis of the model at each equilibrium point, Jacobian matrix method was employed. By use of MATLAB, numerical simulations on the impact of awareness on alcoholism were performed. Secondary data obtained from NACADA and data from rehabilitation centers in Kenya was used to validate the analytical results of the impact of media awareness on the drinking population. Analysis of the model indicated that the Alcohol Free Equilibrium (AFE) point is locally asymptotically stable whenever R0 < 1 and unstable whenever R0 > 1. The Alcohol Endemic Point (AEP) exists and is locally asymptotically stable when R0 > 1. Further increase in media awareness programs reduces alcohol prevalence in the community. The study concluded that maximum media awareness is an ideal measure in curbing alcohol abuse in the community. The findings of this study will provide useful insight to the government and policy makers in targeting suitable media awareness programs in combating alcoholism.

In this study, we model the occurrence and length of wet, medium wet and dry spells by Markov chain that best describes the rainfall pattern of Bungoma County (Western Kenya).This is achieved by Markov chain theory and estimation of probabilities of the chain by MLE. Also computed is the distribution of the length of each spells; wet, medium wet and dry from which the central moments of the rainfall pattern are computed. The model developed is applied to rainfall data from Bungoma meteorological station. A three by three transition matrix is obtained and used to predict the weather pattern. It is observed that if everything remains constant, prediction can be certain at the twelfth year as the matrix show stationarity. The three states are recurrent, non-null and a periodic hence forming an ergodic chain.

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In this paper, a mathematical model for COVID-19 disease incorporating clinical management based on a system of Ordinary Differential Equations is developed. The existence of the steady states of the model are determined and the effective reproduction number derived using the next generation matrix approach. Stability analysis of the model is carried out to determine the conditions that favour the spread of COVID-19 disease in a given population. The Disease Free Equilibrium is show to be locally asymptotically stable when \(R\)e < 1 and the Endemic Equilibrium is locally asymptotically stable when \(R\)e > 1. The Disease Free Equilibrium is shown not to be globally asymptotically stable using a technique by Castillo Chavez and the Endemic Equilibrium is shown to be globally asymptotically stable by means of Lyapunov's direct method and LaSalle's invariance principle. This implies that COVID-19 disease transmission can be kept low or manageable with the incorporation of clinical management. Sensitivity analysis of the model is carried out by use of the normalised forward sensitivity index (elasticity) which shows that the higher the rates of clinical management the lower the rate of infection. Numerical simulations carried out using MATLAB software showed that with high success of clinical management, there is low contact rate and low prevalence rate of the disease in the population.