Optimal Control of HPV Infection and Cervical Cancer Cells with Beddington–DeAngelis Functional Response

The uncontrollable proliferation of human papillomavirus (HPV) can cause severe cervical cancer. The objective of this work is to study the optimal control of HPV viral replication and its consequence in reducing cervical cancer cells. For this purpose, the dynamics involving healthy cells, HPV, and cervical cancer cells will be modeled by a system of five differential equations. The transmission describing viral infection will be represented by the Beddington–DeAngelis incidence function. In order to reduce the HPV infection, two controls are incorporated to the model. The first one represents the efficacy of drug treatment in blocking new infections, whereas the second serves as the drug effectiveness in inhibiting viral production. The well-posedness of the mathematical model is established in terms of proving the existence, positivity, and boundedness of solution. The optimization problem consists of maximizing a certain objective functional which depends on healthy cells and the two suggested controls. The optimality system is found by applying Pontryagin’s maximum principle. This last system is solved numerically using the forward–backward finite difference numerical approximation scheme. Numerical simulation has shown that the optimal control strategy has an essential role in reducing HPV viral replication. Consequently, the number of cervical cancer cells is reduced considerably by adopting an HPV control strategy specifically in the early stage of cervical cancer.