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.

A deterministic model for the transmission dynamics of two-strains Herpes Simplex Virus (HSV) is developed and analyzed. Following the qualitative analysis of the model, reveals a globally asymptotically stable disease free equilibrium whenever a certain epidemiological threshold known as the reproduction number (R 0 ), is less than unity and the disease persist in the population whenever this threshold exceed unity. However, it was shown that the endemic equilibrium is globally asymptotically stable for a special case. Numerical simulation of the model reveals that whenever R 1 < 1 < R 2 , strain 2 drives strain 1 to extinction (competitive exclusion) but when R 2 < 1 < R 1 , strain 1 does not drive strain 2 to extinction. Finally, it was shown numerically that super-infection increases the spread of HSV-2 in the model.

Human African Trypanosomiasis (HAT) commonly known as sleeping sickness, is a neglected tropical vector&nbsp;borne disease caused by trypanosome protozoa. It is transmitted by bites of infected tsetse fly. In this paper we&nbsp;first present the vector-host model which describes the general transmission dynamics of HAT. In the tsetse fly&nbsp;population, the HAT is modelled by three compartments while in the human population, the HAT is modelled by&nbsp;four compartments. The next generation matrix approach is used to derive the basic reproduction number, R0,&nbsp;and also it is proved that if&nbsp;R0&nbsp;&le; 1 the disease free equilibrium is globally asymptotically stable, which means&nbsp;the disease dies out. The disease persist in the population if the value of R0&nbsp;&gt; 1. Furthermore, the optimal control&nbsp;model is determined by using the Pontryagin&rsquo;s maximum principle with control measures such as education,&nbsp;treatment and insecticides used to optimize the objective function. The model simulations confirm that the use of&nbsp;the three control measures are very efficient and effective to eliminate HAT in Africa.