## Mathematical Modelling of Human Papillomavirus (HPV) Dynamics with Vaccination Incorporating Optimal Control Analysis

Published: 2023-10-16

Page: 36-51

Fednant O. Okware *

Department of Mathematics and Statistics, Kaimosi Friends University, Kenya.

Samuel B. Apima

Department of Mathematics and Statistics, Kaimosi Friends University, Kenya.

Amos O. Wanjara

Department of Mathematics and Statistics, Kaimosi Friends University, Kenya.

*Author to whom correspondence should be addressed.

### Abstract

Human Papillomavirus (HPV) is an infectious illness with complex behavior that has had dangerous consequences in the society. In women, HPV is the leading cause of Cervical Cancer (CC). If not treated early, cervical cancer causes abnormal growth of the cervical walls, which leads to death. It is a threat, with half a million documented cases worldwide resulting in over 200 000 recorded deaths every year. In this research, we develop a mathematical model of HPV dynamics with vaccination and perform optimal control to reduce HPV and CC preventive expenses. The invariant region of the model solution was examined, and it was determined that the model was well posed and biologically meaningful. The feasibility of the model solution was examined, and it was discovered that the solution of the model remained positive in the feasible limited region $$\Omega$$. The disease equilibrium points were shown to exist. The basic reproduction number was examined and discovered to be the biggest eigenvalue of the next generation matrix. The local stability of the equilibrium points was investigated, and it was discovered that the disease free equilibrium and the endemic equilibrium points were asymptotically stable. The model was extended into optimal control, and their optimality system was derived analytically using the Pontryagin Maximum Principle. The optimality system was numerically solved using MATLAB software, and the graphs for various interventions were shown against time. Finally, the outcomes of this study suggest that when the three interventions (awareness, screening and treatment of HPV and CC, and vaccination) are combined, the infection begins to decrease considerably and eventually dies out in the community when the interventions are intensified.

Keywords: HPV and CC, transmission dynamics, optimality system, interventions, local stability, equilibrium points, numerical simulation

#### How to Cite

Okware, Fednant O., Samuel B. Apima, and Amos O. Wanjara. 2023. “Mathematical Modelling of Human Papillomavirus (HPV) Dynamics With Vaccination Incorporating Optimal Control Analysis”. Asian Research Journal of Mathematics 19 (11):36-51. https://doi.org/10.9734/arjom/2023/v19i11751.

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