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, R0, and it is also proved that if R0≤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 R0>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. The model comprises two populations: the human population and bacteria population. The next-generation method is used to compute the basic reproduction number, R0. Both the disease-free and endemic equilibria are shown to be locally and globally stable for R0 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. The 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.

Marburg virus, like Ebola, causes haemorrhagic disease with high fatality rates. We developed a deterministic SEIRDVT model incorporating vaccination and treatment to study the disease dynamics. Qualitative analysis revealed a backward bifurcation when $R_{0}=1$ , meaning $R_{0} <1$ is insufficient to eradicate the virus. Sensitivity analysis using Latin Hypercube Sampling showed that applying four control measures—screening, prevention, continuous vaccination, and treatment—significantly reduced transmission. The most cost-effective strategy combines prevention, vaccination, and treatment. These findings provide a framework for designing efficient interventions to combat Marburg virus.

Meningitis is an inflammation of the meninges, which covers the brain and spinal cord. Every year, most individuals within sub-Saharan Africa suffer from meningococcal meningitis. Moreover, tens of thousands of these cases result in death, especially during major epidemics. The transmission dynamics of the disease keep changing, according to health practitioners. The goal of this study is to exploit robust mechanisms to manage and prevent the disease at a minimal cost due to its public health implications. A significant concern found to aid in the transmission of meningitis disease is the movement and interaction of individuals from low-risk to high-risk zones during the outbreak season. Thus, this article develops a mathematical model that ascertains the dynamics involved in meningitis transmissions by partitioning individuals into low- and high-risk susceptible groups. After computing the basic reproduction number, the model is shown to exhibit a unique local asymptotically stability at the meningitis-free equilibrium , when the effective reproduction number , and the existence of two endemic equilibria for which and exhibits the phenomenon of backward bifurcation, which shows the difficulty of relying only on the reproduction number to control the disease. The effective reproductive number estimated in real time using the exponential growth method affirmed that the number of secondary meningitis infections will continue to increase without any intervention or policies. To find the best strategy for minimizing the number of carriers and infected individuals, we reformulated the model into an optimal control model using Pontryagin’s maximum principles with intervention measures such as vaccination, treatment, and personal protection. Although Ghana’s most preferred meningitis intervention method is via treatment, the model’s simulations demonstrated that the best strategy to control meningitis is to combine vaccination with treatment. But the cost-effectiveness analysis results show that vaccination and treatment are among the most expensive measures to implement. For that reason, personal protection which is the most cost-effective measure needs to be encouraged, especially among individuals migrating from low- to high-risk meningitis belts.