Sensitivity analysis and optimal treatment control for a mathematical model of Human Papillomavirus infection

Kai Zhang; Yunpeng Ji; Qiuwei Pan; Yumei Wei; Yong Ye; Hua Liu; Kai Zhang; Yunpeng Ji; Qiuwei Pan; Yumei Wei; Yong Ye; Hua Liu

doi:10.3934/math.2020172

AIMS Mathematics

2020, Volume 5, Issue 3: 2646-2670. doi: 10.3934/math.2020172

Previous Article Next Article

Research article Special Issues

Sensitivity analysis and optimal treatment control for a mathematical model of Human Papillomavirus infection

1.
School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030, P. R. China
2.
Biomedical Research Center, Northwest Minzu University, Lanzhou, P. R. China
3.
Department of Gastroenterology and Hepatology, Erasmus MC-University Medical Center, Rotterdam, The Netherlands
4.
Department of Genetics, Inner Mongolia Maternal and Child Care Hospital, Hohhot, Inner Mongolian Autonomous Region, P. R. China
5.
Experimental Teaching Department, Northwest Minzu University, Lanzhou 730030, P. R. China

Received: 07 December 2019 Accepted: 02 March 2020 Published: 16 March 2020
MSC : 92C60, 92D30

Human papillomavirus (HPV) is one of the most common sexually transmitted viruses, and is a causal agent of cervical cancer. We aimed to develop a mathematical model of HPV natural history and qualitatively analyzed the stability of disease-free equilibrium, non-existence of limit cycle and existence of forward bifurcation. We performed sensitivity analysis to identify key epidemiological parameters. The Partial Rank Correlation Coefficient (PRCC) values for basic reproduction number shows that controlling contact rate plays an important role in disturbing equilibrium of HPV infection. Moreover, the increase of medical level is the most effective measure to prevent new HPV infections. Optimal treatment problem is solved and theoretical analysis is verified by numerical simulation.
- HPV-disease,
- forward bifurcation,
- sensitivity analysis,
- optimal control,
- numerical simulations
Citation: Kai Zhang, Yunpeng Ji, Qiuwei Pan, Yumei Wei, Yong Ye, Hua Liu. Sensitivity analysis and optimal treatment control for a mathematical model of Human Papillomavirus infection[J]. AIMS Mathematics, 2020, 5(3): 2646-2670. doi: 10.3934/math.2020172

Related Papers:

Abstract

Human papillomavirus (HPV) is one of the most common sexually transmitted viruses, and is a causal agent of cervical cancer. We aimed to develop a mathematical model of HPV natural history and qualitatively analyzed the stability of disease-free equilibrium, non-existence of limit cycle and existence of forward bifurcation. We performed sensitivity analysis to identify key epidemiological parameters. The Partial Rank Correlation Coefficient (PRCC) values for basic reproduction number shows that controlling contact rate plays an important role in disturbing equilibrium of HPV infection. Moreover, the increase of medical level is the most effective measure to prevent new HPV infections. Optimal treatment problem is solved and theoretical analysis is verified by numerical simulation.

References

[1]	J. G. Baseman, L. A. Koutsky. The epidemiology of human papillomavirus infections, J. Clin. Virol., 32 (2005), 16-24.
[2]	F. X. Bosch, A. Lorincz, N. Munoz, et al. The causal relation between human papillomavirus and cervical cancer, J. Clin. Pathol., 55 (2002), 244-265. doi: 10.1136/jcp.55.4.244
[3]	D. Parkin, F. Bray, J. Ferlay, et al. Estimating the world cancer burden: Globocan 2000, Int. J. Cancer, 94 (2001), 153-156. doi: 10.1002/ijc.1440
[4]	D. M. Parkin, F. Bray, J. Ferlay, et al. Global cancer statistics, 2002, CA: A Cancer Journal for Clinicians, 55 (2005), 74-108. doi: 10.3322/canjclin.55.2.74
[5]	L. Bruni, M. Diaz, X. Castellsagué, et al. Cervical human papillomavirus prevalence in 5 continents: meta-analysis of 1 million women with normal cytological findings, The Journal of Infections Disease, 202 (2010), 1789-1799. doi: 10.1086/657321
[6]	D. Foreman, C. de Martel, C. J. Lacey, et al. Global burden of human papillomavirus and related diseases, Vaccine, 30 (2012), F12-F23.
[7]	F. X. Bosch, A. Lorincz, N. Munoz, et al. The causal relation between human papillomavirus and cervical cancer, J. Clin. Pathol., 55 (2002), 244-265. doi: 10.1136/jcp.55.4.244
[8]	J. M. Walboomers, M. V. Jacobs, M. M. Manos, et al. Human papillomavirus is a necessary cause of invasive cervical cancer worldwide, The Journal of Pathology, 189 (1999), 12-19. doi: 10.1002/(SICI)1096-9896(199909)189:1<12::AID-PATH431>3.0.CO;2-F
[9]	K. Syrjanen, M. Hakama, S. Saarikoski, et al. Prevalence, incidence, and estimated life-time risk of cervical human papillomavirus infections in a nonselected Finnish female population, Sex. Transm. Dis., 17 (1990), 15-19.
[10]	G. Y. F. Ho, R. Bierman, L. Beardsley, et al. Natural history of cervicovaginal papillomavirus infection in young women, New Engl. J. Med., 338 (1998), 423-428. doi: 10.1056/NEJM199802123380703
[11]	A. B. Mosciki, S. Shiboski, J. Broering, et al. The natural history of human papillomavirus infection as measured by repeated DNA testing in adolescent and young women, Journal of Pediatrics, 2 (1998), 277-284.
[12]	S. B. Cantor, E. N. Atkinson, M. Cardenas-Turanzas, et al. Natural history of cervical intraepithelial neoplasia: a meta-analysis, Acta Cytol., 49 (2005), 405-415. doi: 10.1159/000326174
[13]	H. W. Chesson, J. M. Blandford, T. L. Gift, et al. The estimated direct medical costs of sexually transmitted diseases among American youth, 2000, Perspect Sex Reprod Health, 6 (2004), 11-19.
[14]	H. N. Coleman, W. W. Greenfield, S. L. Stratton, et al. Human papillomavirus type 16 viral load is decreased following a therapeutic vaccination, Cancer Immunol Immunother, 65 (2016), 563-573. doi: 10.1007/s00262-016-1821-x
[15]	H. Gemma, H. Karin, D. Lucy, Therapeutic HPV vaccines, Best Practice & Research Clinical Obstetrics and Gynaecology, 47 (2018), 59-72.
[16]	A. Omame, R. A. Umana, D. Okuonghae, et al. Mathematical analysis of a two-sex Human Papillomavirus (HPV) model, Int. J. Biomath., 7 (2018), 43.
[17]	Elamin H, Elbasha, E. J. Dasbach, R. P. Insinga, A Multi-Type HPV Transmission Model, B. Math. Biol., 70 (2008), 2126-2176. doi: 10.1007/s11538-008-9338-x
[18]	Elamin H, Elbasha, Global Stability of Equilibria in a Two-Sex HPV Vaccination Model, B. Math. Biol., 70 (2008), 894-909. doi: 10.1007/s11538-007-9283-0
[19]	A. Mo'tassem, S. Robert, An age-structured model of human papilloma virus vaccination, Math. Comput. Simulat., 82 (2011), 629-652. doi: 10.1016/j.matcom.2011.10.006
[20]	E. B. M. Bashier, K. C. Patidar, Optimal control of an epidemiological model with multiple time delays, Appl. Math. Comput., 292 (2017), 47-56.
[21]	X. Wang, H. Peng, S. Yang, et al. Optimal vaccination strategy of a constrained time-varying SEIR epidemic model, Commun. Nonlinear Sci., 67 (2019), 37-48. doi: 10.1016/j.cnsns.2018.07.003
[22]	T. K. Kar, AshimBatabyal, Stability analysis and optimal control of an SIR epidemic model with vaccination, Biosystems, 104 (2011), 127-135. doi: 10.1016/j.biosystems.2011.02.001
[23]	Y. Yang, S. Y. Tang, X. H. Ren, et al. Global stability and optimal control for a tuberculosis model with vaccination and treatment, Discrete and continuous dynamical systems series B, 21 (2016), 1009-1022. doi: 10.3934/dcdsb.2016.21.1009
[24]	F. X. Bosch, A. Lorincz, N. Munoz, et al. The causal relation between human papillomavirus and cervical cancer, J. Clin. Pathol., 55 (2002), 244.
[25]	S. Lakshmikantham, S. Leela, A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker, New York. 1989.
[26]	H. W. Hethcote, The mathematics of infectious diseases, Siam Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907
[27]	P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6
[28]	O. Sharomi, C. N. Podder, A. B. Gumel, et al. Modelling the Transmission Dynamics and Control of Novel 2009 Swine Influenza (H1N1) Pandemic, B. Math. Biol., 73 (2011), 515-548. doi: 10.1007/s11538-010-9538-z
[29]	L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 1996.
[30]	CIA World Factbook, South Africa Demographics Profile, 2016.
[31]	M. T. Malik, J. Reimer, A. B. Gumel, et al. The impact of an imperfect vaccine and pap cytology screening on the transmission of human papillomavirus and occurrence of associated cervical dysplasia and cancer, Math. Biosci. Eng., 10 (2013), 1173-1205. doi: 10.3934/mbe.2013.10.1173
[32]	A. A. Alsaleh, A. B. Gumel, Dynamics of a vaccination model for HPV transmission, J. Biol. Syst., 22 (2014), 555-599. doi: 10.1142/S0218339014500211
[33]	A. A. Alsaleh, A. B. Gumel, Analysis of a risk-structured vaccination model for the dynamics of oncogenic and warts-causing HPV types, B. Math. Biol., 76 (2014), 1670-1726. doi: 10.1007/s11538-014-9972-4
[34]	H. Huo, L. X. Feng, Global stability for an HIV/AIDS epidemic model with different latent stages and treatment, Appl. Math. Model., 37 (2013), 1480-1489. doi: 10.1016/j.apm.2012.04.013
[35]	M. Simeone, B. H. Ian, J. R. Christian, et al. A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol., 254 (2008), 178-196. doi: 10.1016/j.jtbi.2008.04.011
[36]	S. M. Blower, H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model, as an example, Int. Stat. Rev., 2 (1994), 229-243.
[37]	R. Jan, Y. N. Xiao, Effect of partial immunity on transmission dynamics of dengue disease with optimal control, Mathematical Methods in the Applied Sciences, 42 (2019), 1967-1983. doi: 10.1002/mma.5491
[38]	W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2016), 58-71.
[39]	M. H. A. Biswas, L. T. Paiva, M. Do Rosario de Pinho, A SEIR model for control of infectious diseases with constraints, Math. Biosci. Eng., 11 (2014), 761-784. doi: 10.3934/mbe.2014.11.761
[40]	H. Peng, X. Wang, B. Shi, et al. Stabilizing constrained chaotic system using a symplectic psuedospectral method, Commun. Nonlinear Sci., 56 (2018), 77-92. doi: 10.1016/j.cnsns.2017.07.028
[41]	X. Wang, H. Peng, S. Zhang, et al. A symplectic pseudospectral method for nonlinear optimal control problems with inequality constraints, ISA T., 68 (2017), 335-352. doi: 10.1016/j.isatra.2017.02.018
[42]	X. Wang, J. Liu, Y. Zhang, et al. A unified symplectic pseudospectral method for motion planning and tracking control of 3D underactuated overhead cranes, International Journal of Robust Nonlinear Control, 29 (2019), 2236-2253. doi: 10.1002/rnc.4488
[43]	X. Wang, H. Peng, D. Jiang, et al. Optimal Path Planning of Two-Wheeled Mobile Robots in the Presence of Dynamic Obstacles, The 36th Chinese Control Conference, IEEE, 2017.
[44]	J. Liu, W. Han, X. Wang, et al. Research on Cooperative Trajectory Planning and Tracking Problem for Multiple Carrier Aircraft on the Deck, IEEE System Journal, 2019.
[45]	C. Castillo-Chavez, B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 2 (2004), 361-404.
[46]	J. Carr, Applications Centre Manifold Theory, New York: Springer, 1981.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)