Human African trypanosomiasis (HAT), commonly known as sleeping sickness, is a neglected tropical vector-borne disease caused by trypanosome protozoa. It is transmitted by bites of infected tsetse fly. In this paper, we first present the vector-host model which describes the general transmission dynamics of HAT. In the tsetse fly population, the HAT is modelled by three compartments, while in the human population, the HAT is modelled by four compartments. The next-generation matrix approach is used to derive the basic reproduction number, R 0, and it is also proved that if R 0 ≤ 1, the disease-free equilibrium is globally asymptotically stable, which means the disease dies out. The disease persists in the population if the value of R 0 > 1. Furthermore, the optimal control model is determined by using the Pontryagin's maximum principle, with control measures such as education, treatment, and insecticides used to optimize the objective function. The model simulations confirm that the use of the three control measures is very efficient and effective to eliminate HAT in Africa.

Cholera is an infectious intestinal disease which occurs as a result of poor sanitation and lack of basic education in its transmission. It is characterized by profuse vomiting and severe diarrhea when an individual eats food or drinks water contaminated with the Vibrio cholerae. A dynamic mathematical model that explicitly simulates the transmission mechanism of cholera by taking into account the role of control measures and the environment in the transmission of the disease is developed. e model comprises two populations: the human population and bacteria population. e next-generation method is used to compute the basic reproduction number, R 0 . Both the disease-free and endemic equilibria are shown to be locally and globally stable for R 0 values less than unity and unstable otherwise. Necessary conditions of the optimal control problem were analyzed using Pontryagin's maximum principle with control measures such as educational campaign and treatment of water bodies used to optimize the objective function. Numerical values of model parameters were estimated using the nonlinear least square method. e model simulations confirm the significant role played by control measures (education and treatment of water bodies) and the bacteria in the environment in the transmission dynamics as well as reducing the spread of cholera.

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