Developing computationally efficient optimal control strategies to eradicate Rubella disease

Abstract

Abstract The threat of Rubella virus disease looms large, posing significant risks to public health and emphasizing the urgent need for comprehensive prevention, control, and awareness strategies. We conducted an extensive analysis of a newly developed SEITR deterministic model for the lethal Rubella virus disease. The main objective of our study is to gain deep insights into the disease dynamics and devise an optimal control strategy for the model, utilizing vaccination and treatment as preventive measures. We employed various mathematical techniques to establish the positivity and bounded nature of solutions. The value of threshold parameter is computed using the next-generation method to anticipate future dynamical behavior of the epidemic. The local and global stability of the equilibrium points was successfully assessed. Additionally, we utilized the well-known Non-Standard Finite Difference (NSFD) method to obtain numerical solutions for the Rubella model. A numerical analysis is carried out to assess the efficacy of a constant treatment strategy, and the results are presented through graphical illustrations. The developed model is subjected to sensitivity analysis and the most sensitive parameters are identified. In addition, the bifurcation nature of the model is examined. Subsequently, an optimal control problem is introduced for the model, aiming to determine the best time-dependent strategies for treatment and vaccination. The main goal is to reduce the number of individuals infected within the human population and the cost of controls. Designed optimal control problem and its corresponding optimality conditions of Pontryagin type have been derived. An important aspect of this study is the utilization of the NSFD method, implemented backward in time, to solve the optimal control problem, as opposed to other conventional methods. Numerical simulations were carried out to assess the impact of the applied controls on the dynamics of all classes, both before and after optimization.