Mathematical Modelling of the Dynamics and Control of Communicable Diseases with Emphasis on Cholera Epidemics

Abstract

Cholera remains a disease of public health importance, associated with high morbidity and mortality. This research provides a mathematical approach on the effective control and possible eradication of Vibrio cholera causing menace. This was done using a deterministic SPIRB mathematical model, were we defined S-susceptible, P-protected, I-infected, R-recovered and B-bacteria. We carried out mathematical analyses on the model, such as the invariant region analysis, non-negativity of solutions, equilibrium points, basic reproduction number, stability analysis, sensitivity analysis and bifurcation analysis. All solutions of the SPIRB model are positive and there exists an invariant region for the model. Although, the disease-endemic equilibrium of the model is globally asymptomatically stable, the disease-free equilibrium of the model is not globally stable but locally asymptomatically stable. It was also revealed that the recruitment rate into the susceptible class, the probability that each contact is effective enough to cause infection, the contact rate with contaminated environment, the average contribution of each infected individual to the pathogen population of Vibrio cholera, and the progression rate of protected individuals to the susceptible class are the most sensitive parameters. We showed via the bifurcation analysis that if the progression rate of protected individuals to the susceptible class can be kept very close to zero, say between zero and unity, then the disease-free equilibrium can be kept stable. Numerical simulations were conducted to show the effects of some parameters on the classes.