Mathematical Modeling on the Transmission Dynamics of Diphtheria with Optimal Control Strategies

Diphtheria is an acute bacterial infection caused by Corynebacterium diphtheriae, characterized by the formation of a pseudo-membrane in the throat, which can lead to airway obstruction and systemic complications. Despite the availability of effective vaccines, diphtheria remains a significant public health concern in many regions, particularly in areas with low immunization coverage. In this study, we formulated and rigorously analyzed a deter ministic epidemiological mathematical model to gain insight into the transmission dynamics of Diphtheria infection, incorporating the concentration of Corynebacterium Diphtheriae in the environment. The analysis of the model begins with the computation of the basic reproduction number and the examination of the local stability of the disease-free equilibrium using the Routh-Hurwitz criterion. An in-depth analysis of the model reveals that the model undergoes the phenomenon of backward bifurcation. This characteristic poses significant hurdles in effectively controlling Diph theria infection within the population. However, under the assumption of no re-infection of Diphtheria infection after recovery, the disease-free equilibrium point is globally asymptotically stable whenever the basic reproduction num ber is less than one. Furthermore, the sensitivity analysis of the basic reproduction number was carried out in order to determine the impact of each of the model basic parameters that contribute to the transmission of the disease. Utilizing the optimal control theory to effectively curb the spread of Diphtheria, We introduced two time dependent control measures, to mitigate the spread of Diphtheria. These time dependent control measures represent preventive actions, such as public enlightenment campaign to sensitize and educate the general public on the dynamics of Diph theria and proper personal hygiene which includes regular washing of hands to prevent susceptible individuals from acquiring Diphtheria, and environmental sanitation practices such as cleaning of surfaces and door handle to reduced the concentration of Corynebacterium diphtheriae in the environment. The results from the numerical simulations reveal that Diphtheria infection can successfully be controlled and mitigated within the population if we can increase the vaccination rate and the decay rate of Corynebacterium Diphtheriae in the environment, as well as properly and effectively implementing these optimal control measures simultaneously.