New applications related to hepatitis C model

Nauman Ahmed; Ali Raza; Ali Akgül; Zafar Iqbal; Muhammad Rafiq; Muhammad Ozair Ahmad; Fahd Jarad; Nauman Ahmed; Ali Raza; Ali Akgül; Zafar Iqbal; Muhammad Rafiq; Muhammad Ozair Ahmad; Fahd Jarad

doi:10.3934/math.2022634

AIMS Mathematics

2022, Volume 7, Issue 6: 11362-11381. doi: 10.3934/math.2022634

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New applications related to hepatitis C model

1.
Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan
2.
Department of Mathematics, Govt. Maulana Zafar Ali Khan Graduate College Wazirabad, 52000, Punjab Higher Education Department (PHED), Lahore, 54000, Pakistan
3.
Siirt University, Art and Science Faculty, Department of Mathematics, 56100 Siirt, Turkey
4.
Department of Mathematics, University of Management and Technology, Lahore, Pakistan
5.
Department of Mathematics, Faculty of Science, University of Central Punjab, Lahore, Pakistan
6.
Department of Mathematics, Cankaya University, Etimesgut 06790, Ankara, Turkey
7.
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
8.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Received: 21 December 2021 Revised: 16 March 2022 Accepted: 24 March 2022 Published: 12 April 2022
MSC : 65P99, 92D25, 92-10

The main idea of this study is to examine the dynamics of the viral disease, hepatitis C. To this end, the steady states of the hepatitis C virus model are described to investigate the local as well as global stability. It is proved by the standard results that the virus-free equilibrium state is locally asymptotically stable if the value of $ R_0 $ is taken less than unity. Similarly, the virus existing state is locally asymptotically stable if $ R_0 $ is chosen greater than unity. The Routh-Hurwitz criterion is applied to prove the local stability of the system. Further, the disease-free equilibrium state is globally asymptotically stable if $ R_0 < 1 $. The viral disease model is studied after reshaping the integer-order hepatitis C model into the fractal-fractional epidemic illustration. The proposed numerical method attains the fixed points of the model. This fact is described by the simulated graphs. In the end, the conclusion of the manuscript is furnished.
- hepatitis C model,
- fractional derivatives,
- stability analysis,
- numerical simulations
Citation: Nauman Ahmed, Ali Raza, Ali Akgül, Zafar Iqbal, Muhammad Rafiq, Muhammad Ozair Ahmad, Fahd Jarad. New applications related to hepatitis C model[J]. AIMS Mathematics, 2022, 7(6): 11362-11381. doi: 10.3934/math.2022634

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Abstract

The main idea of this study is to examine the dynamics of the viral disease, hepatitis C. To this end, the steady states of the hepatitis C virus model are described to investigate the local as well as global stability. It is proved by the standard results that the virus-free equilibrium state is locally asymptotically stable if the value of $ R_0 $ is taken less than unity. Similarly, the virus existing state is locally asymptotically stable if $ R_0 $ is chosen greater than unity. The Routh-Hurwitz criterion is applied to prove the local stability of the system. Further, the disease-free equilibrium state is globally asymptotically stable if $ R_0 < 1 $. The viral disease model is studied after reshaping the integer-order hepatitis C model into the fractal-fractional epidemic illustration. The proposed numerical method attains the fixed points of the model. This fact is described by the simulated graphs. In the end, the conclusion of the manuscript is furnished.

References

[1]	A. B. Pitcher, A. Borquez, B. Skaathun, N. K. Martin, Mathematical modeling of hepatitis c virus (HCV) prevention among people who inject drugs, J. Theor. Biol., 481 (2019), 194–201. http://dx.doi.org/10.1016/j.jtbi.2018.11.013 doi: 10.1016/j.jtbi.2018.11.013
[2]	H. Dahari, J. E. Layden-Almer, E. Kallwitz, R. M. Ribeiro, S. J. Cotler, T. J. Layden, et al., A mathematical model of hepatitis C virus dynamics in patients with high baseline viral loads or advanced liver disease, Gastroenterology, 136 (2009), 1402–1409. http://dx.doi.org/10.1053/j.gastro.2008.12.060 doi: 10.1053/j.gastro.2008.12.060
[3]	H. Dahari, M. Major, X. Zhang, K. Mihalik, C. M. Rice, A. S. Perelson, et al., Mathematical modeling of primary hepatitis C infection: non-cytolytic clearance and early blockage of virus production, Gastroenterology, 128 (2005), 1056–1066. http://dx.doi.org/10.1053/j.gastro.2005.01.049 doi: 10.1053/j.gastro.2005.01.049
[4]	W. Jia, J. Weng, C. Fang, Y. Li, A dynamic model and some strategies on how to prevent and control hepatitis c in mainland China, BMC Infect. Dis., 19 (2019), 724. http://dx.doi.org/10.1186/s12879-019-4311-x doi: 10.1186/s12879-019-4311-x
[5]	H. Dahari, R. M. Ribeiro, C. M. Rice, A. S. Perelson, Mathematical modeling of sub genomic hepatitis C virus replication in Huh-7 cells, J. Virol., 81 (2007), 750–760. http://dx.doi.org/10.1128/JVI.01304-06 doi: 10.1128/JVI.01304-06
[6]	S. Mushayabasa, C. P. Bhunu, Mathematical analysis of hepatitis C model for intravenous drug misusers: impact of antiviral therapy, abstinence and relapse, Simulation, 90 (2014), 487–500. http://dx.doi.org/10.1177/0037549714528388 doi: 10.1177/0037549714528388
[7]	S. D. Lombardo, S. Lombardo, Global stability for a mathematical model of hepatitis C: Impact of a possible vaccination with DAAs therapy, AIP Conference Proceedings, 2159 (2019), 030020. http://dx.doi.org/10.1063/1.5127485 doi: 10.1063/1.5127485
[8]	T. C. Reluga, H. Dahari, A. S. Perelson, Analysis of hepatitis C virus infection models with hepatocyte homeostasis, SIAM J. Appl. Math., 69 (2009), 999–1023. http://dx.doi.org/10.1137/080714579 doi: 10.1137/080714579
[9]	M. Kalemera, D. Mincheva, J. Grove, G. J. Illingworth, Building a mechanistic mathematical model of hepatitis C virus entry, PLoS Comput. Biol., 15 (2019), e1006905. http://dx.doi.org/10.1371/journal.pcbi.1006905 doi: 10.1371/journal.pcbi.1006905
[10]	R. M. Ribeiro, A. Lo, A. S. Perelson, Dynamics of hepatitis B virus infection. Microbes Infect., 4 (2002), 829–835. http://dx.doi.org/10.1016/s1286-4579(02)01603-9 doi: 10.1016/s1286-4579(02)01603-9
[11]	L. J. Durfee, Bio-mathematics: Introduction to the mathematical model of the hepatitis C virus, Electronic Theses, Projects, and Dissertations, 2016,428.
[12]	D. Echevarria, A. Gutfraind, B. Boodram, M. Major, S. Del Valle, S. J. Cotler, et al., Mathematical modeling of hepatitis C prevalence reduction with antiviral treatment scale-up in persons who inject drugs in metropolitan Chicago, PLoS ONE, 10 (2015), e0135901. http://dx.doi.org/10.1371/journal.pone.0135901 doi: 10.1371/journal.pone.0135901
[13]	E. H. Elbasha, Model for hepatitis C virus transmissions, Math. Biosci. Eng., 10 (2014), 1045–1065. http://dx.doi.org/10.3934/mbe.2013.10.1045 doi: 10.3934/mbe.2013.10.1045
[14]	F. Laporte, G. Tap, A. Jaafar, K. Saune-Sandres, N. Kamar, L. Rostaing, et al., Mathematical modeling of hepatitis C virus transmission in hemodialysis, Americal Journal of Infection Control, 37 (2009), 403–407. http://dx.doi.org/10.1016/j.ajic.2008.05.013 doi: 10.1016/j.ajic.2008.05.013
[15]	M. D. Miller-Dickson, V. A. Meszaros, S. Almagro-Moreno, O. C. Brandon, Hepatitis C virus modelled as an indirectly transmitted infection highlights the centrality of injection drug equipment in disease dynamics, J. R. Soc. Interface, 16 (2019), 20190334. http://dx.doi.org/10.1098/rsif.2019.0334 doi: 10.1098/rsif.2019.0334
[16]	A. Cousien, V. C. Tran, S. Deuffic-Burban, M. Jauffret-Roustide, J. S. Dhersin, Y. Yazdanpanah, Dynamic modelling of hepatitis C virus transmission among people who inject drugs: a methodological review, J. Viral Hepatitis, 22 (2015), 213–229. http://dx.doi.org/10.1111/jvh.12337 doi: 10.1111/jvh.12337
[17]	R. Avendano, L. Esteva, J. A. Flores, J. F. Allen, G. Gomez, J. Lopez-Estrada, A mathematical model for the dynamics of hepatitis C, Comput. Math. Method. Med., 4 (2002), 461260. http://dx.doi.org/10.1080/10273660290003777 doi: 10.1080/10273660290003777
[18]	A. Heffernan, G. S. Cooke, S. Nayagam, M. Thursz, T. B. Hallett, Scaling up prevention and treatment towards the elimination of hepatitis C, The Lancet, 393 (2019), 1319–1329. http://dx.doi.org/10.1016/S0140-6736(18)32277-3 doi: 10.1016/S0140-6736(18)32277-3
[19]	H. A. Hadi, A mathematical model of hepatitis C virus infection incorporating immune responses and cell proliferation, M.S. Thesis of The University of Texas, 2017.
[20]	P. Aston, K. Cranfield, H. O'Farrell, A. Cassenote, C. J. Mendes-Correa, A. Segurado, et al., Hepatitis C viral dynamics using a combination therapy of interferon, ribavirin, and telaprevir: mathematical modeling and model validation, In: Hepatitis C–From infection to cure, IntechOpen, 2018. http://dx.doi.org/10.5772/intechopen.75761
[21]	A. Raza, A. Ahmadian, M. Rafiq, S. Salahshour, M. Ferrara, An analysis of a nonlinear susceptible-exposed-infected-quarantine-recovered pandemic model of a novel coronavirus with delay effect, Results Phys., 21 (2021), 103771. http://dx.doi.org/10.1016/j.rinp.2020.103771 doi: 10.1016/j.rinp.2020.103771
[22]	N. Ghorui, A. Ghosh, S. P. Mondal, M. Y. Bajuri, A. Ahmadian, S. Salahshour, et al., Identification of dominant risk factor involved in spread of COVID-19 using hesitant fuzzy MCDM methodology, Results Phys., 21 (2021), 103811. http://dx.doi.org/10.1016/j.rinp.2020.103811 doi: 10.1016/j.rinp.2020.103811
[23]	M. Zamir, K. Shah, F. Nadeem, M. Y. Bajuri, A. Ahmadian, S. Salahshour, et al., Threshold conditions for global stability of disease free state of COVID-19, Results Phys., 21 (2021), 103784. http://dx.doi.org/10.1016/j.rinp.2020.103784 doi: 10.1016/j.rinp.2020.103784
[24]	S. Ahmad, A. Ullah, K. Shah, S. Salahshour, A. Ahmadian, T. Ciano, Fuzzy fractional-order model of the novel coronavirus, Adv. Differ. Equ., 2020 (2020), 472. http://dx.doi.org/10.1186/s13662-020-02934-0 doi: 10.1186/s13662-020-02934-0
[25]	A. Ahmadian, N. Senu, F. Larki, S. Salahshour, M. Suleiman, M. S. Islam, A legendre approximation for solving a fuzzy fractional drug transduction model into the bloodstream, In: Recent advances on soft computing and data mining, Cham: Springer, 2014, 25–34. http://dx.doi.org/10.1007/978-3-319-07692-8_3
[26]	H. M. Srivastava, V. P. Dubey, R. Kumar, J. Singh, D. Kumar, D. Baleanu, An efficient computational approach for a fractional-order biological population model with carrying capacity, Chaos Soliton. Fract., 31 (2020), 109880. http://dx.doi.org/10.1016/j.chaos.2020.109880 doi: 10.1016/j.chaos.2020.109880
[27]	J. Singh, Analysis of fractional blood alcohol model with composite fractional derivative, Chaos Soliton. Fract., 140 (2020), 110127. http://dx.doi.org/10.1016/j.chaos.2020.110127 doi: 10.1016/j.chaos.2020.110127
[28]	E. Bonyah, R. Zarin, Fatmawati, Mathematical modeling of Cancer and Hepatitis co-dynamics with non-local and non-singular kernel, Commun. Math. Biol. Neurosci., 2020 (2020), 91. http://dx.doi.org/10.28919/cmbn/5029 doi: 10.28919/cmbn/5029
[29]	R. E. Gutierrez, J. M. Rosario, J. T. Machado, Fractional order calculus: basic concepts and engineering applications, Math. Probl. Eng., 2010 (2010), 375858. http://dx.doi.org/10.1155/2010/375858 doi: 10.1155/2010/375858
[30]	J. T. Machado, Fractional calculus: fundamentals and applications, In: Acoustics and vibration of mechanical structures–VMS-2017, Cham: Springer, 2018, 3–11. http://dx.doi.org/10.1007/978-3-319-69823-6_1
[31]	T. A. Biala, A. Q. M. Khaliq, A fractional-order compartmental model for the spread of the COVID-19 pandemic, Commun. Nonlinear Sci. Numer. Simulat., 98 (2021), 105764. http://dx.doi.org/10.1016/j.cnsns.2021.105764 doi: 10.1016/j.cnsns.2021.105764
[32]	Z. Iqbal, N. Ahmed, D. Baleanu, W. Adel, M. Rafiq, M. Aziz-ur Rehman, et al., Positivity and boundedness preserving numerical algorithm for the solution of fractional nonlinear epidemic model of HIV/AIDS transmission, Chaos. Soliton. Fract., 134 (2020), 109706. http://dx.doi.org/10.1016/j.chaos.2020.109706 doi: 10.1016/j.chaos.2020.109706
[33]	Z. Iqbal, N. Ahmed, D. Baleanu, M. Rafiq, M. S. Iqbal, M. Aziz-ur Rehman, Structure preserving computational technique for fractional order Schnakenberg model, Comp. Appl. Math., 39 (2020), 61. http://dx.doi.org/10.1007/s40314-020-1068-1 doi: 10.1007/s40314-020-1068-1
[34]	M. Alqhtani, K. M. Saad, Numerical solutions of space-fractional diffusion equations via the exponential decay kernel, AIMS Mathematics, 7 (2022), 6535–6549. http://dx.doi.org/10.3934/math.2022364 doi: 10.3934/math.2022364
[35]	M. Alqhtani, K. M. Saad, Fractal-fractional Michaelis-Menten enzymatic reaction model via different kernels, Fractal Fract., 6 (2022), 13. http://dx.doi.org/10.3390/fractalfract6010013 doi: 10.3390/fractalfract6010013
[36]	K. M. Saad, J. F. Gomez-Aguilar, A. A. Almadiy, A fractional numerical study on a chronic hepatitis C virus infection model with immune response, Chaos Soliton. Fract., 139 (2020), 110062. http://dx.doi.org/10.1016/j.chaos.2020.110062 doi: 10.1016/j.chaos.2020.110062
[37]	K. M. Saad, M. Alqhtani, Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinear, AIMS Mathematics, 6 (2021), 3788–3804. http://dx.doi.org/10.3934/math.2021225 doi: 10.3934/math.2021225
[38]	A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Chaos Soliton. Fract., 102 (2017), 396–406. http://dx.doi.org/10.1016/j.chaos.2017.04.027 doi: 10.1016/j.chaos.2017.04.027
[39]	N. Sene, SIR epidemic model with Mittag-Leffler fractional derivative, Chaos Soliton. Fract., 137 (2020), 109833. http://dx.doi.org/10.1016/j.chaos.2020.109833 doi: 10.1016/j.chaos.2020.109833
[40]	P. Liu, M. Rahman, A. Din, Fractal fractional based transmission dynamics of COVID-19 epidemic model, Computer Methods in Biomechanics and Biomedical Engineering, 2022, in press. http://dx.doi.org/10.1080/10255842.2022.2040489
[41]	J. Zhou, S. Salahshour, A. Ahmadian, N. Senu, Modeling the dynamics of COVID-19 using fractal-fractional operator with a case study, Results Phys., 33 (2022), 105103. http://dx.doi.org/10.1016/j.rinp.2021.105103 doi: 10.1016/j.rinp.2021.105103
[42]	M. Toufik, A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models, Eur. Phys. J. Plus, 132 (2017), 444. http://dx.doi.org/10.1140/epjp/i2017-11717-0 doi: 10.1140/epjp/i2017-11717-0

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