×

Optimal control against the human papillomavirus: protection versus eradication of the infection. (English) Zbl 1474.92122

Summary: We investigate the optimal vaccination and screening strategies to minimize human papillomavirus (HPV) associated morbidity and the interventions cost. We propose a two-sex compartmental model of HPV-infection with time-dependent controls (vaccination of adolescents, adults, and screening) which can act simultaneously. We formulate optimal control problems complementing our model with two different objective functionals. The first functional corresponds to the protection of the vulnerable group and the control problem consists of minimizing the cumulative level of infected females over a fixed time interval. The second functional aims to eliminate the infection, and, thus, the control problem consists of minimizing the total prevalence at the end of the time interval. We prove the existence of solutions for the control problems, characterize the optimal controls, and carry out numerical simulations using various initial conditions. The results and properties and drawbacks of the model are discussed.

MSC:

92D30 Epidemiology
92C60 Medical epidemiology
49J15 Existence theories for optimal control problems involving ordinary differential equations
49K15 Optimality conditions for problems involving ordinary differential equations
49N90 Applications of optimal control and differential games
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] WHO, Cervical cancer, 2018, https://www.who.int/cancer/prevention/diagnosis-screening/cervical-cancer/en/
[2] FDA, Fda approves expanded use of gardasil-9 to include individuals 27 through 45 years old, 2018, https://bit.ly/2zVRxa9
[3] Brisson, M.; Laprise, J.-F.; Chesson, H. W.; Drolet, M.; Malagón, T.; Boily, M.-C.; Markowitz, L. E., Health and economic impact of switching from a 4-valent to a 9-valent hpv vaccination program in the united states, Journal of the National Cancer Institute, 108, 1, (2016)
[4] Seto, K.; Marra, F.; Raymakers, A.; Marra, C. A., The cost effectiveness of human papillomavirus vaccines: a systematic review, Drugs, 72, 5, 715-743, (2012)
[5] Garnett, G. P.; Kim, J. J.; French, K.; Goldie, S. J., Chapter 21: Modelling the impact of HPV vaccines on cervical cancer and screening programmes, Vaccine, 24, 3, S178-S186, (2006)
[6] Newall, A. T.; Beutels, P.; Wood, J. G.; Edmunds, W. J.; MacIntyre, C. R., Cost-effectiveness analyses of human papillomavirus vaccination, The Lancet Infectious Diseases, 7, 4, 289-296, (2007)
[7] Elbasha, E. H.; Dasbach, E. J.; Insinga, R. P., Model for assessing human papillomavirus vaccination strategies, Emerging Infectious Diseases, 13, 1, 28-41, (2007)
[8] Kim, J. J.; Goldie, S. J., Cost effectiveness analysis of including boys in a human papillomavirus vaccination programme in the United States, British Medical Journal, 339, article b3884, (2009)
[9] Elbasha, E. H.; Dasbach, E. J., Impact of vaccinating boys and men against HPV in the United States, Vaccine, 28, 42, 6858-6867, (2010)
[10] Smith, M. A.; Lew, J.-B.; Walker, R. J.; Brotherton, J. M. L.; Nickson, C.; Canfell, K., The predicted impact of HPV vaccination on male infections and male HPV-related cancers in Australia, Vaccine, 29, 48, 9112-9122, (2011)
[11] Bogaards, J. A.; Kretzschmar, M.; Xiridou, M.; Meijer, C. J. L. M.; Berkhof, J.; Wallinga, J., Sex-specific immunization for sexually transmitted infections such as human papillomavirus: insights from mathematical models, PLoS Medicine, 8, 12, article e1001147, (2011)
[12] Malik, T.; Reimer, J.; Gumel, A.; Elbasha, E.; Mahmud, S., The impact of an imperfect vaccine and Pap cytology screening on the transmission of human papillomavirus and occurrence of associated cervical dysplasia and cancer, Mathematical Biosciences and Engineering, 10, 4, 1173-1205, (2013) · Zbl 1273.92056
[13] Alsaleh, A. A.; Gumel, A. B., Analysis of risk-structured vaccination model for the dynamics of oncogenic and warts-causing HPV types, Bulletin of Mathematical Biology, 76, 7, 1670-1726, (2014) · Zbl 1300.92049
[14] Guerrero, A. M.; Genuino, A. J.; Santillan, M.; Praditsitthikorn, N.; Chantarastapornchit, V.; Teerawattananon, Y.; Alejandria, M.; Toral, J. A., A cost-utility analysis of cervical cancer screening and human papillomavirus vaccination in the Philippines, BMC Public Health, 15, 1, article 730, (2015)
[15] Saldaña, F.; Barradas, I., The role of behavioral changes and prompt treatment in the control of STIs, Infectious Disease Modelling, 4, 1-10, (2019)
[16] Sharomi, O.; Malik, T., Optimal control in epidemiology, Annals of Operations Research, 251, 1-2, 55-71, (2017) · Zbl 1373.92140
[17] Brown, V. L.; Jane White, K. A., The role of optimal control in assessing the most cost-effective implementation of a vaccination programme: HPV as a case study, Mathematical Biosciences, 231, 2, 126-134, (2011) · Zbl 1214.92047
[18] Malik, T.; Imran, M.; Jayaraman, R., Optimal control with multiple human papillomavirus vaccines, Journal of Theoretical Biology, 393, 179-193, (2016) · Zbl 1343.92496
[19] Huh, W. K.; Joura, E. A.; Giuliano, A. R.; Iversen, O.-E.; de Andrade, R. P.; Ault, K. A.; Bartholomew, D.; Cestero, R. M.; Fedrizzi, E. N.; Hirschberg, A. L.; Mayrand, M.-H.; Ruiz-Sternberg, A. M.; Stapleton, J. T.; Wiley, D. J.; Ferenczy, A.; Kurman, R.; Ronnett, B. M.; Stoler, M. H.; Cuzick, J.; Garland, S. M.; Kjaer, S. K.; Bautista, O. M.; Haupt, R.; Moeller, E.; Ritter, M.; Roberts, C. C.; Shields, C.; Luxembourg, A., Final efficacy, immunogenicity, and safety analyses of a nine-valent human papillomavirus vaccine in women aged 16–26 years: a randomised, double-blind trial, The Lancet, 390, 10108, 2143-2159, (2017)
[20] Muñoz, N.; Méndez, F.; Posso, H.; Molano, M.; Van Den Brule, A. J. C.; Ronderos, M.; Meijer, C.; Muñoz, Á., Incidence, duration, and determinants of cervical human papillomavirus infection in a cohort of Colombian women with normal cytological results, The Journal of Infectious Diseases, 190, 12, 2077-2087, (2004)
[21] Anic, G. M.; Giuliano, A. R., Genital HPV infection and related lesions in men, Preventive Medicine, 53, 1, S36-S41, (2011)
[22] Martcheva, M., An Introduction to Mathematical Epidemiology. An Introduction to Mathematical Epidemiology, Texts in Applied Mathematics, 61, (2015), New York, NY, USA: Springer, New York, NY, USA · Zbl 1333.92006
[23] van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180, 29-48, (2002) · Zbl 1015.92036
[24] Grigorieva, E.; Khailov, E.; Korobeinikov, A., Optimal control for an SEIR epidemic model with nonlinear incidence rate, Studies in Applied Mathematics, 141, 3, 353-398, (2018) · Zbl 1420.35437
[25] Buonomo, B.; Lacitignola, D.; Vargas-De-Leon, C., Qualitative analysis and optimal control of an epidemic model with vaccination and treatment, Mathematics and Computers in Simulation, 100, 88-102, (2014)
[26] Okosun, K. O.; Rachid, O.; Marcus, N., Optimal control strategies and cost-effectiveness analysis of a malaria model, BioSystems, 111, 2, 83-101, (2013)
[27] Camacho, A.; Jerez, S., Bone metastasis treatment modeling via optimal control, Journal of Mathematical Biology, 1-30, (2018)
[28] Fleming, W. H.; Rishel, R. W., Deterministic and Stochastic Optimal Control, 1, (1975), New York, NY, USA: Springer, New York, NY, USA
[29] Lukes, D. L., Differential Equations: Classical to Controlled, (1982), Elsevier
[30] Lenhart, S. M.; Workman, J. T., Optimal Control Applied to Biological Models, (2007), CRC Press
[31] Baussano, I.; Bray, F., Modelling cervical cancer elimination, The Lancet Public Health, 4, 1, e2-e3, (2019)
[32] Pontryagin, L. S., Mathematical Theory of Optimal Processes, (2018), Routledge
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.