Mathematical modeling and optimal control of the transmission dynamics of Bacterial Meningitis Population in Ghana

Abstract

This research presents two compartmental models on the transmission dynamics of Bacterial Meningitis, which best describes the scenario in real life. The first model is made up of seven (7) mutually exclusive epidemiological compartments. The quantitative analysis of the model is conducted and the criteria for local and global stabilities of the disease-free equilibrium is established. The simulation results show that getting people vaccinated is crucial to the control of the disease. This leads to the novel two-strain vaccination control model denoted by nine (9) mutually exclusive epidemiological compartments. The model is used to analyze the impact of vaccination and early treatment on the population, especially on the recovered populations. It is ascertained that Bacterial Meningitis will not spread in the population if 25% of the population is immune to the disease. Numerical simulations of the model are carried out by implementing the MATLAB ODE45 algorithm to visualize the effects of the various model parameters on each compartment of the developed model. The two strain model is then extended to include control by the introduction of five control mechanisms; effective human personal protection (such as wearing face or surgical masks), vaccination for strains 1 and 2, timely and delayed diagnosis treatments of the infection. An optimal control problem is formulated and the existence of its solution is established. The characterization of the controls is performed using the Pontryagin’s Maximum Principle. The Forward Backward Sweep (FBS) method is implemented and used to solve the optimal control problem and its corresponding adjoint equations. In order to determine the impact of combination of the control strategies on the different model compartments, numerical simulations of the model are performed using real life data from Ghana Center for Disease Control. It was established that the most efficient and cost-effective control strategy is the strategy involving all the five control variables. This is followed by Strategy C which is only the effective human personal protection (such as face or surgical masks) control, uP (t). Based on the findings of this research, necessary recommendations are made for the applications of the model to an endemic area.