Vaccination and treatment are the most effective ways of controlling the transmission of most infectious diseases. While vaccination helps susceptible individuals to build either a long-term immunity or short-term immunity, treatment reduces the number of disease-induced deaths and the number of infectious individuals in a community/nation. In this paper, a nonlinear deterministic model with time-dependent controls has been proposed to describe the dynamics of bacterial meningitis in a population. The model is shown to exhibit a unique globally asymptotically stable disease-free equilibriumE0, when the effective reproduction numberRVT≤1, and a globally asymptotically stable endemic equilibriumE1, whenRVT>1; and it exhibits a transcritical bifurcation atRVT=1. Carriers have been shown (by Tornado plot) to have a higher chance of spreading the infection than those with clinical symptoms who will sometimes be bound to bed during the acute phase of the infection. In order to find the best strategy for minimizing the number of carriers and ill individuals and the cost of control implementation, an optimal control problem is set up by defining a Lagrangian functionLto be minimized subject to the proposed model. Numerical simulation of the optimal problem demonstrates that the best strategy to control bacterial meningitis is to combine vaccination with other interventions (such as treatment and public health education). Additionally, this research suggests that stakeholders should press hard for the production of existing/new vaccines and antibiotics and their disbursement to areas that are most affected by bacterial meningitis, especially Sub-Saharan Africa; furthermore, individuals who live in communities where the environment is relatively warm (hot/moisture) are advised to go for vaccination against bacterial meningitis.

The novel coronavirus (COVID-19) pandemic continues to be a global health problem whose impact has been significantly felt in South Africa. With the global spread increasing and infecting millions, containment efforts by countries have largely focused on lockdowns and social distancing to minimise contact between persons. Social distancing has been touted as the best form of response in managing a rapid increase in the number of infected cases. In this paper, we present a deterministic model to describe the impact of social distancing on the transmission dynamics of COVID-19 in South Africa. The model is fitted to data from March 5 to April 13, 2020, on the cumulative number of infected cases, and a scenario analysis on different levels of social distancing is presented. The model shows that with the levels of social distancing under the initial lockdown level between March 26 and April 13, 2020, there would be a projected continued rise in the number of infected cases. The model also looks at the impact of relaxing the social distancing measures after the initial announcement of the lockdown. It is shown that relaxation of social distancing by 2% can result in a 23% rise in the number of cumulative cases whilst an increase in the level of social distancing by 2% would reduce the number of cumulative cases by about 18%. The model results accurately predicted the number of cases after the initial lockdown level was relaxed towards the end of April 2020. These results have implications on the management and policy direction in the early phase of the epidemic.

The presence of carriers usually complicates the dynamics and prevention of a disease. They are not recognized as disease cases themselves unless they are screened and they usually spread the infection without them being aware. We argue that this has been one of the major causes of the spread of human immunodeficiency virus (HIV). We propose, in this paper, a model for the heterogeneous transmission of HIV/acquired immunodeficiency syndrome in the presence of disease carriers. The model allows us to assess the role of screening, as an intervention program that can slow the epidemic. A threshold value ψ^*, for the screening rate is obtained. It is shown numerically that if 80% or more of the carrier population is screened, the epidemic can be contained. The qualitative analysis is done in terms of the model reproduction number R. The model has two equilibria, the disease free equilibrium and a unique endemic equilibrium. The disease free equilibrium is globally stable of R < 1 and the endemic equilibrium is is locally stable for R > 1. A detailed discussion of the model reproduction number is given and numerical simulations are done to show the role of some of the important model parameters.