Abstract
In this paper, a hepatitis B virus (HBV) model with an incubation period and delayed state and control variables is firstly proposed. Furthermore, the combination treatment is adopted to have a longer-lasting effect than mono-therapy. The equilibrium points and basic reproduction number are calculated, and then the local stability is analyzed on this model. We then present optimal control strategies based on the Pontryagin’s minimum principle with an objective function not only to reduce the levels of exposed cells, infected cells and free viruses nearly to zero at the end of therapy, but also to minimize the drug side-effect and the cost of treatment. What’s more, we develop a numerical simulation algorithm for solving our HBV model based on the combination of forward and backward difference approximations. The state dynamics of uninfected cells, exposed cells, infected cells, free viruses, CTL and ALT are simulated with or without optimal control, which show that HBV is reduced nearly to zero based on the time-varying optimal control strategies whereas the disease would break out without control. At last, by the simulations, we prove that strategy A is the best among the three kinds of strategies we adopt and further comparisons have been done between model (1) and model (2).
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References
World Health Organization (2008) Hepatitis B [EB/OL] (Revised August 2008). [2010-12-9] http://www.who.int/mediacentre/factsheets/fs204/en/
Nowak M, Bonhoeffer S, Hill A et al (1996) Viral dynamics in hepatitis B virus infection. Proc Natl Acad Sci USA 93:4398–4402
Min L, Su Y, Kuang Y (2008) Mathematical analysis of a basic model of virus infection with application to HBV infection. Rocky Mt J Math 38:1573–1585
Medley GF, Lindop NA, Edmunds WJ, Nokes DJ (2007) Hepatitis-B virus endemicity: heterogeneity, catastrophic dynamics and control. Nat Med 7:619–624
Zhang TL, Teng ZD (2007) On a nonautonomous SEIRS model in epidemiology. Bull Math Biol 69:2537–2559
Blayneh KW, Gumel AB, Lenhart S, Clayton T (2010) Backward bifurcation and optimal control in transmission dynamics of West Nile virus. Bull Math Biol 72:1006–1028
Guo BZ, Cai LM (2011) A note for the global stability of a delay differential equation of hepatitis B virus infection. Math Biosci Eng 8:689–694
Zhang SX, Zhou YC (2014) Dynamic analysis of a hepatitis B model with three-age-classes. Commun Nonlinear Sci Numer Simul 19:2466–2478
Hattaf K, Yousfi N (2012) Optimal control of a delayed HIV infection model with immune response using an efficient numerical method. Int Sch Res Netw Biomath 11:7
Mouofo PT, Tewa JJ, Mewoli B, Bowong S (2013) Optimal control of a delayed system subject to mixed control-state constraints with application to a within-host model of hepatitis virus B. Annu Rev Control 37:246–259
Kamyad AV, Akbari R, Heydari AA, Heydari A (2014) Mathematical Modeling of transmission dynamics and optimal control of vaccination and treatment for hepatitis B virus. Comput Math Methods Med 2014(1):475451. https://doi.org/10.1155/2014/475451
Sheng YJ, Liu JY, Tong SW, Hu HD, Zhang DZ, Hu P, Ren H (2011) Lamivudine plus adefovir combination therapy versus entecavir monotherapy for lamivudine-resistant chronic hepatitis B: a systematic review and meta-analysis. Virol J 8:393–403
Chen X, Min LQ, Sun QL (2013) Dynamics analysis and numerical simulation of an amended HBV infection model. J Biomath 28:278–284
Su YM, Sun DS (2015) Optimal control of anti-HBV treatment based on combination of traditional Chinese medicine and Western Medicine. Biomed Signal Process Control 15:41–48
EIhia M, Rachik M, benlahmar E (2013) Optimal control of an SIR Model with delay in state and control variables. ISRN Biomath 2013(8):1–7. https://doi.org/10.1155/2013/403549
Lau GKK, Tsiang M, Hou JL, Yuen ST (2000) Combination therapy with lamivudine and famciclovir for chronic hepatitis B–infected Chinese patients: a viral dynamics study. Hepatology 32:394–399
Pachpute G, Chakrabarty SP (2013) Dynamics of hepatitis C under optimal therapy and sampling based analysis. Commun Nonlinear Sci Numer Simul 18:2202–2212
Su YM, Zhao L (2013) Analysis and simulation of an adefovir anti-HBV infection therapy immune model with ALT. Inst Eng Technol 7:205–213
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We would like to thank the support of the National Natural Science Foundation of China (no. 61273226).
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Sun, D., Liu, F. Modeling and Control of a Delayed Hepatitis B Virus Model with Incubation Period and Combination Treatment. Interdiscip Sci Comput Life Sci 10, 375–389 (2018). https://doi.org/10.1007/s12539-017-0275-y
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DOI: https://doi.org/10.1007/s12539-017-0275-y