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

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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References

WHO/ICO. Human Papillomavirus and related Cancers in Kenya. Summary Report 2010.

[internet]2010. Available: who.int/hpvcentre.

Saldana F, ACamacho-Guterre J, Villavicencio-Pulido G, Velasco-Hernandez J. Modelling the transmission dynamics and vaccination strategies for HPV infection: An Optimal control Approach. Baque centre for Applied mathematics, Bilbao, Spain; 2022.

Hong AM, Grulich AE, Jones D, Lee CS, Garland SM, Dobbins TA, Clark JR, Harnett GB, Milross CG, OBrien C.J. Squamous Cell Carcinoma of the oropharynx in australian males induced by human papillomavirus vaccine targets. Vaccine. 2010;28(19):3269-3272.

Ndii, Murtono, Sugiyanto, Mathematical modelling of cervical cancer treatment using Chemotherapy drug. Biology, medicine and natural product chemistry. 2019;8(1):11 - 15.

Centres for Disease Control and Prevention. Human Papillomavirus (HPV) Questions and Answers; 2018. Available:https://www.cdc.gov/parents/questions - answers. html. Accessed August 21, 2018.

Goshu M and Abebe, Mathematical modelling of cervical cancer vaccination and treatment effectiveness. Authorea preprints; 2022.

Nogueira-Rodrigues, A. Human Papillomavirus vaccination in Latin America: global challenges and feasible solutions. American Society of Clinical Oncology Educational Book. 2019;39:e45-52.

Geoffrey PG, Jane JK, Katherine F, Sue JG. Modelling the impact of Human Papillomavirus vaccines on cervical cancer and screening programs. Science Direct-Elsevier, Vaccine. 24S3, S3/178-S3/186; 2006.

Lee SL, Tameru MA. A mathematical model of Human Papillomavirus (HPV) in the united states and its impact on cervical cancer. Journal of cancer. 2012;3:262 - 268.

Miriam Malia, Isaac Chepkwony, David Malonza, Modelling the Impact of spread of Human Papillomavirus infections under vaccination in Kenya. EJ-MATH, European Journal of mathematics and statistics. 2022;3(4):17-26.

Tokose DD. Mathematical model of cervical cancer due to Human Papillomavirus dynamics with vaccination in case of Gamo zone Arbaminch Ethiopia. Msc Thesis, Haramaya University, Haramaya; 2022.

Zhang K, Ji Y, Pan Q, Wei Y, Ye Y, Liu H. Sensitivity Analysis and optimal treatment control for a mathematical model of HPV infection." AIMS Mathematics. 2020;5(3):2646 -2670.

Kai Zang, Xinwei Wang, Hua Liu, Yunpeng Ji, Quiwei Pan, Yumei Wei, Ming Ma. Mathematical analysis of a human papillomavirus transmission model with vaccination as screening. Mathematical Biosciences and Engineering. 2020;5:5449 - 5476.

Karam Allali, Stability analysis and optimal control of HPV infection model with early-stage cervical cancer. Laboratory of mathematics and Applications; 2020.

Olaniyi S, Obabiyi OS. Qualitative analysis of malaria dynamics with non-linear incidence function. Applied Mathematical Sciences. 2014;8(78):3889-3904.

Lukes DL. Di erential equations: Classical to Controlled. Academic Press, New York, USA; 1982.