Atangana-Baleanu Fractional Dynamics of Predictive Whooping Cough Model with Optimal Control Analysis

Abstract

Whooping cough, or pertussis, is an infectious disease that causes serious threats to people of all ages, specifically to infant and young children, all over the world. Due to the severe impact on health, it is necessary to construct a mathematical model that can be used to predict future dynamics of the disease, as well as to suggest strategies to eliminate the disease in an optimal way. For this, we constructed a new Atangana–Baleanu fractional model for whooping cough disease to predict the future dynamics of the disease, as well as to suggest strategies to eliminate the disease in an optimal way. We prove that the proposed model has a unique solution that is positive and bounded. To measure the contagiousness of the disease, we determined the reproduction number R0 and used it to examine the local and global stability at equilibrium points that have symmetry. Through sensitivity analysis, we determined parameters of the model that are most sensitive to R0. The ultimate aim of this research was to analyze different disease prevention approaches in order to find the most suitable one. For this, we included the vaccination and quarantine compartments in the proposed model and formulated an optimal control problem to assess the effect of vaccination and quarantine rates on disease control in three distinct scenarios. Firstly, we study the impact of vaccination strategy and conclude the findings with a graphical presentation of the results. Secondly, we examine the impact of quarantine strategy on whooping cough infection and its possible elimination from society. Lastly, we implement vaccination and quarantine strategies together to visualize their combined effect on infection control. In addition to the study of the optimal control problem, we examine the effect of the fractional order on disease dynamics, as well as the impact of constant vaccination and quarantine rates on disease transmission and control. The numerical results reveal that the optimal control strategy with vaccination and quarantine together would be more effective in reducing the spread of whooping cough infection. The implementation of the Toufik–Atangana-type numerical scheme for the solution of the fractional optimal control problem is another contribution of this article.

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