A deterministic mathematical model for optimal control of diphtheria disease with booster vaccination

Diphtheria is an infectious disease caused by a strain of Corynebacterium diphtheria and forms part of the childhood vaccine-preventable diseases. The Diphtheria vaccine is a component of one of the routine vaccines given to children thrice before their first birthday. The protection against diphtheria derived from the diphtheria vaccine in infancy wanes in later childhood, necessitating a booster dose to protect the child as they grow older. To determine the impact of a booster dose of the diphtheria vaccine amidst a contaminated environment, a diphtheria model that incorporates a vaccine booster and a contaminated environment is formulated. The reproduction number R0 is computed and used to prove the local and global stability of the disease-free equilibrium. Global sensitivity analysis is conducted via the application of Latin Hypercube Sampling (LHS) with a Partial Rank Correlation coefficient on the infected humans and the contaminated environment to deduce the most sensitive parameters of the dynamics of diphtheria disease. Then, the model is further extended based on the result of the global sensitivity analysis by introducing four time-dependent controls, disinfection, screening/treatment, booster vaccination, and hygiene practice, to form an optimal control model. The control model is analyzed using Pontryagin's maximum principle. The numerical simulation shows that diphtheria disease will reduce drastically in the community if any control combination involves booster vaccination since the diphtheria vaccine in infancy wanes after ten years. In a situation where there are limited resources to implement all the controls simultaneously, it is recommended to implement any two of the combined controls: disinfection of the environment and administration of booster vaccination or screening/treatment of the asymptomatic infected and administration of booster vaccination. The study shows that the best combination is to disinfect the environment, screen/treat the asymptomatic infected humans, and administer booster vaccination to the community.