Stability Analysis and Optimal Control of a Hepatitis B Virus Transmission Model in the Presence of Vaccination and Treatment Strategy
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Published
Aug 17, 2021
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Abstract
In this paper, the dynamics of hepatitis B virus (HBV) infection is studied through a mathematical model. The model includes vaccination class of population, vertical transmission of newborns and treatment of both acute HBV infected individuals and chronic HBV carriers. The stability of equilibria both locally and globally is analyzed. The result shows that the basic reproduction number becomes threshold value i.e. the disease-free equilibrium is globally stable when it is less than unity, and the infection is uniformly persistent and the endemic equilibrium is globally stable when it is greater than one. Further, an optimal control model is developed to seek the strategy to minimize the transmission of HBV. There are three control variables within the model which are prevention by vaccination, treatment of acute infected and treatment of chronic HBV carriers. Our numerical results show that among three controls, treatment of acute infected individuals gives the best impact in reducing the HBV infection. However, with three controls together, they give the best strategy in reducing overall HBV infection.
Keywords: hepatitis B virus, HBV chronic carrier, vaccination, vertical transmission, optimal control
References
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Borgia, G., Carleo, M. A., Gaeta, G. B., & Gentile, I. (2012). Hepatitis B in pregnancy. World Journal of Gastroenterology, 18(34), 4677-4683.
Butler, G., Freedman, H. I., & Waltman, P. (1986). Uniformly persistent systems. Proceedings of the American Mathematical Society, 96(3), 425-430.
Chenar, F. F., Kyrychko, Y. N., & Blyuss, K. B. (2018). Mathematical model of immune response to hepatitis B. Journal of theoretical biology, 447, 98-110. https://doi.org/10.1016/j.jtbi.2018.03.025
Ciupe, S. M., Ribeiro, R. M., Nelson, P.W., Dusheiko, G., & Perelson, A.S. (2007). The role of cells refractory to productive infection in acute hepatitis B viral dynamics. Proceedings of the National Academy of Sciences, 104(12), 5050-5055. https://doi.org/10.1073/pnas.0603626104
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Edmunds, W. J., Medley, G.F., & Nokes, D. J. (1996b). Vaccination against hepatitis B virus in highly endemic areas: waning vaccine-induced immunity and the need for booster doses. Transactions of the Royal Society of Tropical Medicine and Hygiene, 90(4), 436-440. https://doi.org/10.1016/S0035-9203(96)90539-8
Eigenwillig, A. (2008). Real root isolation for exact and approximate polynomials using Descartes' rule of signs. Retrieved from https://publikationen.sulb.uni-saarland.de/handle/20.500.11880/26046
Eikenberry, S., Hews, S., Nagy, J.D., & Kuang, Y. (2009). The dynamics of a delay model of hepatitis B virus infection with logistic hepatocyte growth. Mathematical Biosciences & Engineering, 6(2), 283-299. https://doi: 10.3934/mbe.2009.6.283
Emerenini, B.O., & Inyama, S.C. (2017). Mathematical model and analysis of transmission dynamics of hepatitis B virus. Retrieved from arXiv preprint arXiv:1710.11563
Freedman, H. I., Ruan, S., & Tang, M. (1994). Uniform persistence and flows near a closed positively invariant set. Journal of Dynamics and Differential Equations, 6(4), 583-600. https://doi.org/10.
1007/BF02218848
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Kamyad, A.V., Akbari, R., Heydari, A. A., & Heydari, A. (2014). Mathematical modeling of transmission dynamics and optimal control of vaccination and treatment for hepatitis B virus. Computational and mathematical methods in medicine, 2014(1), 475451. https://doi.org/10.1155/2014/475451
Kimbir, A. R., Aboiyar, T., Abu, O., & Onah, E. S. (2014). Modelling hepatitis B virus transmission dynamics in the presence of vaccination and treatment. Mathematical Theory and Modeling, 4(12), 2224-5804.
Lavanchy, D. (2008). Chronic viral hepatitis as a public health issue in the world. Best Practice & Research Clinical Gastroenterology, 22(6), 991-1008. https://doi: 10.1016/j.bpg.2008.11.002
Li, Y., & Muldowney, J. S. (1993). On Bendixson’s criterion. Journal of Differential Equations, 106(1), 27-39. https://doi.org/10.1006/jdeq.1993.1097
Li, M.Y., & Muldowney, J. S. (1996). A geometric approach to global-stability problems. SIAM Journal on Mathematical Analysis, 27(4), 1070-1083. https://doi.org/10.1137/S0036141094266449
Medley, G.F., Lindop, N. A., Edmunds, W. J., & Nokes, D. J. (2001). Hepatitis-B virus endemicity: heterogeneity, catastrophic dynamics and control. Nature medicine, 7(5), 619-624. https://doi: 10.1038/87953
Mendy, M., Peterson, I., Hossin, S., Peto, T., Jobarteh, M.L., Jeng-Barry, A., … Whittle, H. (2013). Observational study of vaccine efficacy 24 years after the start of Hepatitis B vaccination in two gambian villages: no need for a booster dose. PLOS One, 8(3), e58029. https://doi.org/10.1371/journal.pone.
0058029
Nowak, M. A., Bonhoeffer, S., Hill, A. M., Boehme, R., Thomas, H. C., & McDade, H. (1996). Viral dynamics in hepatitis B virus infection. Proceedings of the National Academy of Sciences, 93(9), 4398-4402. https://doi.org/10.1073/pnas.93.9.4398
Nowak, M. A., & May, R. M. (2000). Viral dynamics. UK: Oxford University Press.
Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., & Mishchenko, E.F. (1986). In L. S. Pontryagin (Ed.), The Mathematical Theory of Optimal Processes. New York: Gordon and BS Publishers.
Thornley, S., Bullen, C., & Roberts, M. (2008). Hepatitis B in a high prevalence New Zealand population: a mathematical model applied to infection control policy. Journal of Theoretical Biology, 254(3), 599-603. https://doi.org/10.1016/j.jtbi.2008.06.022
van den Driessche, P., & Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical biosciences, 180(1-2), 29-48. https://doi.org/10.1016/S0025-5564(02)00108-6
WHO. (2002) Hepatitis B. Retrieved from https://apps.who.int/iris/handle/10665/67746
WHO. (2020). Hepatitis B Fact Sheet. Retrieved from https://www.who.int/en/news-room/fact-sheets/detail/hepatitis-b
Yosyingyong, P., & Viriyapong, R. (2019). Global stability and optimal control for a hepatitis B virus infection model with immune response and drug therapy. Journal of Applied Mathematics and Computing, 60(1-2), 537-565. https://doi.org/10.1007/s12190-018-01226-x
Zhang, T., Wang, K., & Zhang, X. (2015). Modeling and Analyzing the Transmission Dynamics of HBV Epidemic in Xinjiang. China. Journal PLOS One, 10(9), 1-14. https://doi.org/10.1371/journal.
pone.0138765
Zhang, S., & Zhou, Y. (2012). The analysis and application of an HBV model. Applied Mathematical Modelling, 36(3), 1302-1312. https://doi.org/10.1016/j.apm.2011.07.087
Zhao, S., Xu, Z., & Lu, Y. (2000). A mathematical model of hepatitis B virus transmission and its application for vaccination strategy in China. International journal of epidemiology, 29(4), 744-752. https://
doi.org/10.1093/ije/29.4.744
Zou, L., Zhang, W., & Ruan, S. (2010). Modeling the transmission dynamics and control of hepatitis B virus in China. Journal of theoretical biology, 262(2), 330-338. https://doi.org/10.1016/j.jtbi.2009.
09.035
Anderson, R. M., & May, R. M. (1991). Infectious diseases of humans: dynamics and control. UK: Oxford University Press.
Borgia, G., Carleo, M. A., Gaeta, G. B., & Gentile, I. (2012). Hepatitis B in pregnancy. World Journal of Gastroenterology, 18(34), 4677-4683.
Butler, G., Freedman, H. I., & Waltman, P. (1986). Uniformly persistent systems. Proceedings of the American Mathematical Society, 96(3), 425-430.
Chenar, F. F., Kyrychko, Y. N., & Blyuss, K. B. (2018). Mathematical model of immune response to hepatitis B. Journal of theoretical biology, 447, 98-110. https://doi.org/10.1016/j.jtbi.2018.03.025
Ciupe, S. M., Ribeiro, R. M., Nelson, P.W., Dusheiko, G., & Perelson, A.S. (2007). The role of cells refractory to productive infection in acute hepatitis B viral dynamics. Proceedings of the National Academy of Sciences, 104(12), 5050-5055. https://doi.org/10.1073/pnas.0603626104
Edmunds, W. J., Medley, G. F., & Nokes, D. J. (1996a). The transmission dynamics and control of hepatitis B virus in The Gambia. Statistics in medicine, 15(20), 2215-2233. https://doi.org/10.1002/(SICI)
1097-0258(19961030)15:20<2215::AID-SIM369>3.0.CO;2-2
Edmunds, W. J., Medley, G.F., & Nokes, D. J. (1996b). Vaccination against hepatitis B virus in highly endemic areas: waning vaccine-induced immunity and the need for booster doses. Transactions of the Royal Society of Tropical Medicine and Hygiene, 90(4), 436-440. https://doi.org/10.1016/S0035-9203(96)90539-8
Eigenwillig, A. (2008). Real root isolation for exact and approximate polynomials using Descartes' rule of signs. Retrieved from https://publikationen.sulb.uni-saarland.de/handle/20.500.11880/26046
Eikenberry, S., Hews, S., Nagy, J.D., & Kuang, Y. (2009). The dynamics of a delay model of hepatitis B virus infection with logistic hepatocyte growth. Mathematical Biosciences & Engineering, 6(2), 283-299. https://doi: 10.3934/mbe.2009.6.283
Emerenini, B.O., & Inyama, S.C. (2017). Mathematical model and analysis of transmission dynamics of hepatitis B virus. Retrieved from arXiv preprint arXiv:1710.11563
Freedman, H. I., Ruan, S., & Tang, M. (1994). Uniform persistence and flows near a closed positively invariant set. Journal of Dynamics and Differential Equations, 6(4), 583-600. https://doi.org/10.
1007/BF02218848
Hepatitis B Foundation. (n.d.). What Is Hepatitis B?. Retrieved from https://www.hepb.org/what-is-hepatitis-b/what-is-hepb/acute-vs-chronic/
Jonas, M. M. (2009). Hepatitis B and pregnancy: an underestimated issue. Liver International, 29(1), 133-139. https://doi.org/10.1111/j.1478-3231.2008.01933.x
Kamyad, A.V., Akbari, R., Heydari, A. A., & Heydari, A. (2014). Mathematical modeling of transmission dynamics and optimal control of vaccination and treatment for hepatitis B virus. Computational and mathematical methods in medicine, 2014(1), 475451. https://doi.org/10.1155/2014/475451
Kimbir, A. R., Aboiyar, T., Abu, O., & Onah, E. S. (2014). Modelling hepatitis B virus transmission dynamics in the presence of vaccination and treatment. Mathematical Theory and Modeling, 4(12), 2224-5804.
Lavanchy, D. (2008). Chronic viral hepatitis as a public health issue in the world. Best Practice & Research Clinical Gastroenterology, 22(6), 991-1008. https://doi: 10.1016/j.bpg.2008.11.002
Li, Y., & Muldowney, J. S. (1993). On Bendixson’s criterion. Journal of Differential Equations, 106(1), 27-39. https://doi.org/10.1006/jdeq.1993.1097
Li, M.Y., & Muldowney, J. S. (1996). A geometric approach to global-stability problems. SIAM Journal on Mathematical Analysis, 27(4), 1070-1083. https://doi.org/10.1137/S0036141094266449
Medley, G.F., Lindop, N. A., Edmunds, W. J., & Nokes, D. J. (2001). Hepatitis-B virus endemicity: heterogeneity, catastrophic dynamics and control. Nature medicine, 7(5), 619-624. https://doi: 10.1038/87953
Mendy, M., Peterson, I., Hossin, S., Peto, T., Jobarteh, M.L., Jeng-Barry, A., … Whittle, H. (2013). Observational study of vaccine efficacy 24 years after the start of Hepatitis B vaccination in two gambian villages: no need for a booster dose. PLOS One, 8(3), e58029. https://doi.org/10.1371/journal.pone.
0058029
Nowak, M. A., Bonhoeffer, S., Hill, A. M., Boehme, R., Thomas, H. C., & McDade, H. (1996). Viral dynamics in hepatitis B virus infection. Proceedings of the National Academy of Sciences, 93(9), 4398-4402. https://doi.org/10.1073/pnas.93.9.4398
Nowak, M. A., & May, R. M. (2000). Viral dynamics. UK: Oxford University Press.
Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., & Mishchenko, E.F. (1986). In L. S. Pontryagin (Ed.), The Mathematical Theory of Optimal Processes. New York: Gordon and BS Publishers.
Thornley, S., Bullen, C., & Roberts, M. (2008). Hepatitis B in a high prevalence New Zealand population: a mathematical model applied to infection control policy. Journal of Theoretical Biology, 254(3), 599-603. https://doi.org/10.1016/j.jtbi.2008.06.022
van den Driessche, P., & Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical biosciences, 180(1-2), 29-48. https://doi.org/10.1016/S0025-5564(02)00108-6
WHO. (2002) Hepatitis B. Retrieved from https://apps.who.int/iris/handle/10665/67746
WHO. (2020). Hepatitis B Fact Sheet. Retrieved from https://www.who.int/en/news-room/fact-sheets/detail/hepatitis-b
Yosyingyong, P., & Viriyapong, R. (2019). Global stability and optimal control for a hepatitis B virus infection model with immune response and drug therapy. Journal of Applied Mathematics and Computing, 60(1-2), 537-565. https://doi.org/10.1007/s12190-018-01226-x
Zhang, T., Wang, K., & Zhang, X. (2015). Modeling and Analyzing the Transmission Dynamics of HBV Epidemic in Xinjiang. China. Journal PLOS One, 10(9), 1-14. https://doi.org/10.1371/journal.
pone.0138765
Zhang, S., & Zhou, Y. (2012). The analysis and application of an HBV model. Applied Mathematical Modelling, 36(3), 1302-1312. https://doi.org/10.1016/j.apm.2011.07.087
Zhao, S., Xu, Z., & Lu, Y. (2000). A mathematical model of hepatitis B virus transmission and its application for vaccination strategy in China. International journal of epidemiology, 29(4), 744-752. https://
doi.org/10.1093/ije/29.4.744
Zou, L., Zhang, W., & Ruan, S. (2010). Modeling the transmission dynamics and control of hepatitis B virus in China. Journal of theoretical biology, 262(2), 330-338. https://doi.org/10.1016/j.jtbi.2009.
09.035
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How to Cite
YOSYINGYONG, Pensiri; VIRIYAPONG, Ratchada.
Stability Analysis and Optimal Control of a Hepatitis B Virus Transmission Model in the Presence of Vaccination and Treatment Strategy.
Naresuan University Journal: Science and Technology (NUJST), [S.l.], v. 30, n. 2, p. 66-86, aug. 2021.
ISSN 2539-553X.
Available at: <https://www.journal.nu.ac.th/NUJST/article/view/Vol-30-No-2-2022-66-86>. Date accessed: 19 apr. 2024.
doi: https://doi.org/10.14456/nujst.2022.17.