The hepatitis C virus is hitherto a tremendous threat to human beings, but many researchers have analyzed mathematical models for hepatitis C virus transmission dynamics only in the deterministic case. Stochasticity plays an immense role in pathology and epidemiology. Hence, the main theme of this article is to investigate a stochastic epidemic hepatitis C virus model with five states of epidemiological classification: susceptible, acutely infected, chronically infected, recovered or removed and chronically infected, and treated. The stochastic hepatitis C virus model in epidemiology is established based on the environmental influence on individuals, is manifested by stochastic perturbations, and is proportional to each state. We assert that the stochastic HCV model has a unique global positive solution and attains sufficient conditions for the extinction of the hepatotropic RNA virus. Furthermore, by constructing a suitable Lyapunov function, we obtain sufficient conditions for the existence of an ergodic stationary distribution of the solutions to the stochastic HCV model. Moreover, this article confirms that using numerical simulations, the six parameters of the stochastic HCV model can have a high impact over the disease transmission dynamics, specifically the disease transmission rate, the rate of chronically infected population, the rate of progression to chronic infection, the treatment failure rate of chronically infected population, the recovery rate from chronic infection and the treatment rate of the chronically infected population. Eventually, numerical simulations validate the effectiveness of our theoretical conclusions.
[1] Manns M P, et al. Hepatitis C virus infection. Nature Reviews Disease Primers, 2017, 3: 17006
[2] Rehermann B, Nascimbeni M. Immunology of Hepatitis B virus and Hepatitis C virus infection. Nature Reviews Immunology, 2005, 5: 215–229
[3] Westbrook R H, Dusheiko G. Natural history of hepatitis C. Journal of Hepatology, 2014, 61: S58–S68
[4] Niederau C, et al. Prognosis of chronic hepatitis C: results of a large prospective cohort study. Hepatology, 1998, 28: 1687–1695
[5] Corey K et al. Outcomes and treatment of acute hepatitis C virus infection in a United States population. Clinical Gastroenterology and Hepatology, 2006, 4: 1278–1282
[6] Shi R Q, Cui Y T. Global analysis of a mathematical model for Hepatitis C virus transmissions. Virus Research, 2016, 217: 8–17
[7] Modi A A, Liang T J. Hepatitis C: a clinical review. Oral Diseases, 2008, 14: 10–14
[8] Zein N N. Clinical Significance of Hepatitis C Virus Genotypes. Clinical Microbiology Reviews, 2000, 13: 223–235
[9] Li H C, Lo S Y. Hepatitis C virus: Virology, diagnosis and treatment. World Journal of Hepatology, 2015, 7: 1377–1389
[10] Bukh J. The history of hepatitis C virus (HCV): Basic research reveals unique features in phylogeny, evolution and the viral life cycle with new perspectives for epidemic control. Journal of Hepatology, 2016, 65: S2–S21
[11] Tohme R A, Holmberg S D. Is sexual contact a major mode of hepatitis C virus transmission? Hepatology, 2010, 52: 1497–1505
[12] van de Laar T J W, et al. Increase in HCV incidence among men who have Sex with men in Amsterdam most likely caused by sexual transmission. Journal of Infectious Diseases, 2007, 196: 230–238
[13] vander Meer A J, Berenguer M. Reversion of disease manifestations after HCV eradication. Journal of Hepatology, 2016, 65: S95–S108
[14] Eckels D D, Wang H, Bian T H, Tabatabai N, Gill J C. Immunobiology of hepatitis C virus (HCV) infection: the role of CD4 T cells in HCV infection. Immunological Reviews, 2000, 174: 90–97
[15] Chang Kyong-Mi, et al. Differential CD4+ and CD8+ T-Cell Responsiveness in Hepatitis C Virus Infection. Hepatology, 2001, 33: 267–276
[16] Elbasha E H. Model for hepatitis C virus transmissions. Mathematical Biosciences and Engineering, 2013, 10: 1045–1065
[17] Nazari F, Gumel A B, Elbasha E H. Differential characteristics of primary infection and re-infection can cause backward bifurcation in HCV transmission dynamics. Mathematical Biosciences, 2015, 263: 51–69
[18] Murphy D G, et al. Hepatitis C Virus Genotype 7, a New Genotype Originating from Central Africa. Journal of Clinical Microbiology, 2015, 53: 967–972
[19] Khas’minskii R I. Stochastic Stability of Differential Equations. The Netherlands: Sijthoff & Noordhoff, 1980
[20] World Health Organization. Global health sector strategy on viral hepatitis 2016–2021. https://www.who.int/hepatitis/strategy2016-2021/ghss-hep/en/
[21] Mao X R. Stochastic Differential Equations and Applications. Chichester: Horwood Publishing, 2007
[22] Rajasekar S P, Pitchaimani M. Qualitative analysis of stochastically perturbed SIRS epidemic model with two viruses. Chaos, Solitons & Fractals, 2019, 118: 207–221
[23] Rajasekar S P, Pitchaimani M. Quanxin Zhu, Dynamic threshold probe of stochastic SIR model with saturated incidence rate and saturated treatment function. Physica A: Statistical Mechanics and its Applications, 2019, 535: 122300
[24] Rajasekar S P, Pitchaimani M, Zhu Q X. Progressive dynamics of a stochastic epidemic model with logistic growth and saturated treatment. Physica A: Statistical Mechanics and its Applications, 2020, 538: 122649
[25] Rajasekar S P, Pitchaimani M. Ergodic stationary distribution and extinction of a stochastic SIRS epidemic model with logistic growth and nonlinear incidence. Applied Mathematics and Computation, 2020, 377: 125143
[26] Rajasekar S P, Pitchaimani M, Zhu Q X, Shi K B. Exploring the stochastic host-pathogen tuberculosis model with adaptive immune response. Mathematical Problems in Engineering, 2021, 2021: 8879538
[27] Rajasekar S P, Pitchaimani M, Zhu Q X. Higher order stochastically perturbed SIRS epidemic model with relapse and media impact. Mathematical Methods in the Applied Sciences, 2022, 45: 843–863
[28] Yang J X, Wang L H. Dynamics analysis of a delayed HIV infection model with CTL immune response and antibody immune response. Acta Mathematica Scientia, 2021, 41B: 991–1016
[29] Zhou J L, Ma X S, Yang Y, Zhang T H. A diffusive SVEIR epidemic model with time delay and general incidence. Acta Mathematica Scientia, 2021, 41B: 1385–1404
[30] Huang M Z, Liu S Z, Song X Y, Zou X F. Control strategies for a tumor-immune system with impulsive drug delivery under a random environment. Acta Mathematica Scientia, 2022, 42B: 1141–1159
[31] Ma W J, Luo X H, Zhu Q X. Practical exponential stability of stochastic age-dependent capital system with Levy noise. Systems & Control Letters, 2020, 144: 104759
[32] Yang X T, Zhu Q X. Stabilization of stochastic functional differential systems by steepest descent feedback controls. IET Control Theory & Applications, 2021, 15: 805–813
[33] Cao W, Zhu Q X. Razumikhin-type theorem for pth exponential stability of impulsive stochastic functional differential equations based on vector Lyapunov function. Nonlinear Analysis: Hybrid Systems, 2021, 39: 100983
[34] Kong F, Zhu Q X, Sakthivel R, Mohammadzadeh A. Fixed-time synchronization analysis for discontinuous fuzzy inertial neural networks with parameter uncertainties. Neurocomputing, 2021, 422: 295–313
[35] Gao L, Cao Z, Zhang M, Zhu Q X. Input-to-state stability for hybrid delayed systems with admissible edge-dependent switching signals. Journal of the Franklin Institute, 2020, 357: 8823–8850
[36] Ding K, Zhu Q X. Fuzzy intermittent extended dissipative control for delayed distributed parameter systems with stochastic disturbance: A spatial point sampling approach. IEEE Transactions on Fuzzy Systems, 2021, DOI:https://doi.org/10.1109/TFUZZ.2021.3065524
[37] Zhao Y A, Jiang D Q. The threshold of a stochastic SIS epidemic model with vaccination. Applied Mathematics and Computation, 2014, 243: 718–727
[38] Hu W, Zhu Q X, Karimi H R. Some Improved Razumikhin Stability Criteria for Impulsive Stochastic Delay Differential Systems. IEEE Transactions on Automatic Control, 2019, 64: 5207–5213
[39] Feng T, Qiu Z P, Meng X Z. Dynamics of a stochastic hepatitis C virus system with host immunity. Discrete and Continuous Dynamical Systems — Series B, 2019, 24: 6367–6385
[40] Zhang X H. Global dynamics of a stochastic avian-human influenza epidemic model with logistic growth for avian population. Nonlinear Dynamics, 2017, 90: 2331–2343
[41] Liu Q, Jiang D Q. The dynamics of a stochastic vaccinated tuberculosis model with treatment. Physica A: Statistical Mechanics and its Applications, 2019, 527: 121274
[42] Xin M Z, Wang B G. Stationary distribution and extinction of a stochastic tuberculosis model. Physica A: Statistical Mechanics and its Applications, 2019, 545: 123741
[43] Pitchaimani M, Devi M B. Stochastic dynamical probes in a triple delayed SICR model with general incidence rate and immunization strategies. Chaos, Solitons and Fractals, 2021, 143: 110540
[44] Sabbar Y, Kiouach D, Rajasekar S P, El-idrissi S El A. The influence of quadratic Levy noise on the dynamic of an SIC contagious illness model: New framework, critical comparison and an application to COVID-19 (SARS-CoV-2) case. Chaos, Solitons and Fractals, 2022, 159: 112110. https://doi.org/10.1016/j.chaos.2022.112110
[45] Higham D J. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Review, 2001, 43(3): 525–546
