Deterministic and stochastic model for the hepatitis C with different types of virus genome

Yousef Alnafisah; Moustafa El-Shahed; Yousef Alnafisah; Moustafa El-Shahed

doi:10.3934/math.2022664

AIMS Mathematics

2022, Volume 7, Issue 7: 11905-11918. doi: 10.3934/math.2022664

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Deterministic and stochastic model for the hepatitis C with different types of virus genome

Yousef Alnafisah ^{1
,
,},
Moustafa El-Shahed ²

1.
Department of Mathematics, College of Sciences, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi Arabia
2.
Department of Mathematics, Unaizah College of Sciences and Arts, Qassim University, P.O. Box 3771, Unaizah 51911, Saudi Arabia

Received: 08 March 2022 Revised: 12 April 2022 Accepted: 13 April 2022 Published: 20 April 2022
MSC : 92D25, 92D30, 37C75, 37H30, 37L45

In this paper, a deterministic and stochastic model for hepatitis C with different types of virus genomes is proposed and analyzed. Some sufficient conditions are obtained to ensure the stability of the deterministic equilibrium points. We perform a stochastic extension of the deterministic model to study the fluctuation between environmental factors. Firstly, the existence of a unique global positive solution for the stochastic model is investigated. Secondly, sufficient conditions for the extinction of the hepatitis C virus from the stochastic system are obtained. Theoretical and numerical results show that the smaller white noise can ensure the persistence of susceptible and infected populations while the larger white noise can lead to the extinction of disease. By introducing the basic reproduction number $ R_0 $ and the stochastic basic reproduction number $ R_0^s $, the conditions that cause the disease to die out are indicated. The importance of environmental noise in the propagation of hepatitis C viruses is highlighted by these findings.
- hepatitis C,
- extinction,
- stochastic,
- stability,
- stationary distribution,
- Lyapunov function
Citation: Yousef Alnafisah, Moustafa El-Shahed. Deterministic and stochastic model for the hepatitis C with different types of virus genome[J]. AIMS Mathematics, 2022, 7(7): 11905-11918. doi: 10.3934/math.2022664

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Abstract

In this paper, a deterministic and stochastic model for hepatitis C with different types of virus genomes is proposed and analyzed. Some sufficient conditions are obtained to ensure the stability of the deterministic equilibrium points. We perform a stochastic extension of the deterministic model to study the fluctuation between environmental factors. Firstly, the existence of a unique global positive solution for the stochastic model is investigated. Secondly, sufficient conditions for the extinction of the hepatitis C virus from the stochastic system are obtained. Theoretical and numerical results show that the smaller white noise can ensure the persistence of susceptible and infected populations while the larger white noise can lead to the extinction of disease. By introducing the basic reproduction number $ R_0 $ and the stochastic basic reproduction number $ R_0^s $, the conditions that cause the disease to die out are indicated. The importance of environmental noise in the propagation of hepatitis C viruses is highlighted by these findings.

References

[1]	S. Allegretti, I. M. Bulai, R. Marino, M. A. Menandro, K. Parisi, Vaccination effect conjoint to fraction of avoided contacts for a SARS-CoV-2 mathematical model, Math. Model. Numer. Simul. Appl., 1 (2021), 56–66. https://doi.org/10.53391/mmnsa.2021.01.006 doi: 10.53391/mmnsa.2021.01.006
[2]	Y. Alnafisah, The implementation of milstein scheme in two-dimensional SDEs using the fourier method, Abstr. Appl. Anal., 2018 (2018). https://doi.org/10.1155/2018/3805042 doi: 10.1155/2018/3805042
[3]	Y. A. Alnafisah, Comparison between milstein and exact coupling methods using MATLAB for a particular two-dimensional stochastic differential equation, J. Inform. Sci. Eng., 36 (2020), 1223–1232.
[4]	I. Bashkirtseva, L. Ryashko, T. Ryazanova, Analysis of regular and chaotic dynamics in a stochastic eco-epidemiological model, Chaos Soliton. Fract., 131 (2020), 109549. https://doi.org/10.1016/j.chaos.2019.109549 doi: 10.1016/j.chaos.2019.109549
[5]	B. Berrhazi, M. El Fatini, A. Lahrouz, A. Settati, R. Taki, A stochastic SIRS epidemic model with a general awareness-induced incidence, Physica A, 512 (2018), 968–980. https://doi.org/10.1016/j.physa.2018.08.150 doi: 10.1016/j.physa.2018.08.150
[6]	Y. Cai, Y. Kang, W. Wang, A stochastic SIRS epidemic model with nonlinear incidence rate, Appl. Math. Comput., 305 (2017), 221–240. https://doi.org/10.1016/j.amc.2017.02.003 doi: 10.1016/j.amc.2017.02.003
[7]	Y. Cai, X. Mao, Stochastic prey-predator system with foraging arena scheme, Appl. Math. Model., 64 (2018), 357–371. https://doi.org/10.1016/j.apm.2018.07.034 https://doi.org/10.1080/17442508.2019.1612897
[8]	Z. Chang, X. Meng, X. Lu, Analysis of a novel stochastic SIRS epidemic model with two different saturated incidence rates, Physica A, 472 (2017), 103–116. https://doi.org/10.1016/j.physa.2017.01.015 doi: 10.1016/j.physa.2017.01.015
[9]	J. Cresson, B. Puig, S. Sonner, Stochastic models in biology and the invariance problem, Discrete Cont. Dyn.-B, 21 (2016), 2145. https://doi.org/10.3934/dcdsb.2016041 doi: 10.3934/dcdsb.2016041
[10]	J. Cresson, S. Sonner, A note on a derivation method for sde models: Applications in biology and viability criteria, Stoch. Anal. Appl., 36 (2018), 224–239. https://doi.org/10.1080/07362994.2017.1386571 doi: 10.1080/07362994.2017.1386571
[11]	B. Daşbaşı, Stability analysis of an incommensurate fractional-order SIR model, Math. Model. Numer. Simul. Appl., 1 (2021), 44–55. http://doi.org/10.53391/mmnsa.2021.01.005 doi: 10.53391/mmnsa.2021.01.005
[12]	T. Feng, Z. Qiu, X. Meng, Dynamics of a stochastic hepatitis C virus system with host immunity, Discrete Cont. Dyn.-B, 24 (2019), 6367. https://doi.org/10.3934/dcdsb.2019143 doi: 10.3934/dcdsb.2019143
[13]	C. Gokila, M. Sambath, K. Balachandran, Y. K. Ma, Analysis of stochastic predator-prey model with disease in the prey and Holling type Ⅱ functional response, Adv. Math. Phys., 2020 (2020). https://doi.org/10.1155/2020/3632091 doi: 10.1155/2020/3632091
[14]	Z. Hammouch, M. Yavuz, N. Özdemir, Numerical solutions and synchronization of a variable-order fractional chaotic system, Math. Model. Numer. Simul. Appl., 1 (2021), 11–23. https://doi.org/10.53391/mmnsa.2021.01.002 doi: 10.53391/mmnsa.2021.01.002
[15]	D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525–546. https://doi.org/10.1137/S0036144500378302 doi: 10.1137/S0036144500378302
[16]	J. L. Horsley-Silva, H. E. Vargas, New therapies for hepatitis C virus infection, Gastroenterol. Hepat., 13 (2017), 22.
[17]	R. Ikram, A. Khan, M. Zahri, A. Saeed, M. Yavuz, P. Kumam, Extinction and stationary distribution of a stochastic COVID-19 epidemic model with time-delay, Comput. Biol. Med., 141 (2022), 105115. https://doi.org/10.1016/j.compbiomed.2021.105115 doi: 10.1016/j.compbiomed.2021.105115
[18]	J. Jiang, S. Gong, B. He, Dynamical behavior of a rumor transmission model with Holling-type Ⅱ functional response in emergency event, Physica A, 450 (2019), 228–240. https://doi.org/10.1016/j.physa.2015.12.143 doi: 10.1016/j.physa.2015.12.143
[19]	T. Kar, P. K. Mondal, Global dynamics and bifurcation in delayed SIR epidemic model, Nonlinear Anal.-Real, 12 (2011), 2058–2068. https://doi.org/10.1016/j.nonrwa.2010.12.021 doi: 10.1016/j.nonrwa.2010.12.021
[20]	A. Khan, G. Hussain, A. Yusuf, A. H. Usman, A hepatitis stochastic epidemic model with acute and chronic stages, Adv. Differ. Equ., 2021 (2021), 1–10. https://doi.org/10.1186/s13662-021-03335-7 doi: 10.1186/s13662-021-03335-7
[21]	P. Kumar, V. S. Erturk, Dynamics of cholera disease by using two recent fractional numerical methods, Math. Model. Numer. Simul. Appl., 1 (2021), 102–111. https://doi.org/10.53391/mmnsa.2021.01.010 doi: 10.53391/mmnsa.2021.01.010
[22]	G. Lan, S. Yuan, B. Song, The impact of hospital resources and environmental perturbations to the dynamics of SIRS model, J. Franklin I., 358 (2021), 2405–2433. https://doi.org/10.1016/j.jfranklin.2021.01.015 doi: 10.1016/j.jfranklin.2021.01.015
[23]	D. Lestari, N. Y. Megawati, N. Susyanto, F. Adi-Kusumo, Qualitative behaviour of a stochastic hepatitis C epidemic model in cellular level, Math. Biosci. Eng., 19 (2022), 1515–1535. https://doi.org/10.3934/mbe.2022070 doi: 10.3934/mbe.2022070
[24]	J. Li, K. Men, Y. Yang, D. Li, Dynamical analysis on a chronic hepatitis C virus infection model with immune response, J. Theor. Biol., 365 (2015), 337–346. https://doi.org/10.1016/j.jtbi.2014.10.039 doi: 10.1016/j.jtbi.2014.10.039
[25]	J. Li, M. Shan, M. Banerjee, W. Wang, Stochastic dynamics of feline immunodeficiency virus within cat populations, J. Franklin I., 353 (2016), 4191–4212. https://doi.org/10.1016/j.jfranklin.2016.08.004 doi: 10.1016/j.jfranklin.2016.08.004
[26]	L. Li, W. Zhao, Deterministic and stochastic dynamics of a modified Leslie-Gower prey-predator system with simplified Holling-type Ⅳ scheme, Math. Biosci. Eng., 18 (2021), 2813–2831.
[27]	Q. Li, F. Cong, T. Liu, Y. Zhou, Stationary distribution of a stochastic HIV model with two infective stages, Physica A, 554 (2020), 124686. https://doi.org/10.1016/j.physa.2020.124686 doi: 10.1016/j.physa.2020.124686
[28]	Q. Liu, D. Jiang, T. Hayat, A. Alsaedi, B. Ahmad, A stochastic SIRS epidemic model with logistic growth and general nonlinear incidence rate, Physica A, 551 (2020), 124152. https://doi.org/10.1016/j.physa.2020.124152 doi: 10.1016/j.physa.2020.124152
[29]	P. S. Mandal, M. Banerjee, Stochastic persistence and stationary distribution in a Holling-Tanner type prey-predator model, Physica A, 391 (2012), 1216–1233. https://doi.org/10.1016/j.physa.2011.10.019 doi: 10.1016/j.physa.2011.10.019
[30]	X. Mao, Stochastic differential equations and applications, Elsevier, 2007.
[31]	I. Moneim, M. Al-Ahmed, G. Mosa, Stochastic and monte carlo simulation for the spread of thehepatitis B, Aust. J. Basic Appl. Sci., 3 (2009), 1607–1615.
[32]	I. Moneim, G. Mosa, Modelling the hepatitis C with different types of virus genome, Comput. Math. Method. M., 7 (2006), 3–13. https://doi.org/10.1080/10273660600914121 doi: 10.1080/10273660600914121
[33]	Y. Mu, W. C. Lo, Stochastic dynamics of populations with refuge in polluted turbidostat, Chaos Soliton. Fract., 147 (2021), 110963. https://doi.org/10.1016/j.chaos.2021.110963 doi: 10.1016/j.chaos.2021.110963
[34]	P. A. Naik, Z. Eskandari, H. E. Shahraki, Flip and generalized flip bifurcations of a two-dimensional discrete-time chemical model, Math. Model. Numer. Simul. Appl., 1 (2021), 95–101. https://doi.org/10.53391/mmnsa.2021.01.009 doi: 10.53391/mmnsa.2021.01.009
[35]	F. Özköse, M. Yavuz, Investigation of interactions between COVID-19 and diabetes with hereditary traits using real data: A case study in Turkey, Comput. Biol. Med., 141 (2022), 105044. https://doi.org/10.1016/j.compbiomed.2021.105044 doi: 10.1016/j.compbiomed.2021.105044
[36]	F. Özköse, M. Yavuz, M. T. Şenel, R. Habbireeh, Fractional order modelling of omicron SARS-CoV-2 variant containing heart attack effect using real data from the United Kingdom, Chaos Soliton. Fract., 157 (2022), 111954. https://doi.org/10.1016/j.chaos.2022.111954 doi: 10.1016/j.chaos.2022.111954
[37]	T. A. Phan, J. P. Tian, B. Wang, Dynamics of cholera epidemic models in fluctuating environments, Stoch. Dynam., 21 (2021), 2150011. https://doi.org/10.1142/S0219493721500118 doi: 10.1142/S0219493721500118
[38]	S. Rajasekar, M. Pitchaimani, Qualitative analysis of stochastically perturbed SIRS epidemic model with two viruses, Chaos Soliton. Fract., 118 (2019), 207–221. https://doi.org/10.1016/j.chaos.2018.11.023 doi: 10.1016/j.chaos.2018.11.023
[39]	R. Shi, T. Lu, C. Wang, Dynamic analysis of a fractional-order model for hepatitis B virus with Holling Ⅱ functional response, Complexity, 2019 (2019). https://doi.org/10.1155/2019/1097201 doi: 10.1155/2019/1097201
[40]	G. Song, Dynamics of a stochastic population model with predation effects in polluted environments, Adv. Differ. Equ., 2021 (2021), 1–19. https://doi.org/10.1186/s13662-021-03297-w doi: 10.1186/s13662-021-03297-w
[41]	X. Wang, C. Wang, K. Wang, Extinction and persistence of a stochastic SICA epidemic model with standard incidence rate for HIV transmission, Adv. Differ. Equ., 2021 (2021), 1–17. https://doi.org/10.1186/s13662-021-03392-y doi: 10.1186/s13662-021-03392-y
[42]	X. Wang, Y. Tan, Y. Cai, K. Wang, W. Wang, Dynamics of a stochastic HBV infection model with cell-to-cell transmission and immune response, Math. Biosci. Eng., 18 (2021), 616–642. https://doi.org/10.3934/mbe.2021034 doi: 10.3934/mbe.2021034
[43]	Z. Wang, M. Deng, M. Liu, Stationary distribution of a stochastic ratio-dependent predator-prey system with regime-switching, Chaos Soliton. Fract., 142 (2021), 110462. https://doi.org/10.1016/j.chaos.2020.110462 doi: 10.1016/j.chaos.2020.110462
[44]	C. Wei, J. Liu, S. Zhang, Analysis of a stochastic eco-epidemiological model with modified Leslie-Gower functional response, Adv. Differ. Equ., 2018 (2018), 1–17. https://doi.org/10.1186/s13662-018-1540-z doi: 10.1186/s13662-018-1540-z
[45]	C. Xu, G. Ren, Y. Yu, Extinction analysis of stochastic predator-prey system with stage structure and crowley-martin functional response, Entropy, 21 (2019), 252. https://doi.org/10.3390/e21030252 doi: 10.3390/e21030252
[46]	M. Yavuz, N. Sene, Stability analysis and numerical computation of the fractional predator-prey model with the harvesting rate, Fractal Fract., 4 (2020), 35. https://doi.org/10.3390/fractalfract4030035 doi: 10.3390/fractalfract4030035
[47]	J. Z. Zhang, Z. Jin, Q. X. Liu, Z. Y. Zhang, Analysis of a delayed SIR model with nonlinear incidence rate, Discrete Dyn. Nat. Soc., 2008 (2008). https://doi.org/10.1155/2008/636153 doi: 10.1155/2008/636153
[48]	Q. Zhang, X. Wen, D. Jiang, Z. Liu, The stability of a predator-prey system with linear mass-action functional response perturbed by white noise, Adv. Differ. Equ., 2016 (2016), 1–24. https://doi.org/10.1186/s13662-016-0776-8 doi: 10.1186/s13662-016-0776-8

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