# American Institute of Mathematical Sciences

## Dynamics of a stochastic hepatitis C virus system with host immunity

 1 Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, China 2 College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

* Corresponding author: Zhipeng Qiu

Received  September 2018 Revised  December 2018 Published  July 2019

Fund Project: T. Feng is supported by the Scholarship Foundation of China Scholarship Council grant No. 201806840120, the Postgraduate Research & Practice Innovation Program of Jiangsu Province grant No. KYCX18_0370 and the Fundamental Research Funds for the Central Universities grant No. 30918011339, Z. Qiu is supported by the National Natural Science Foundation of China (NSFC) grant No. 11671206, X. Meng is supported by the Research Fund for the Taishan Scholar Project of Shandong Province of China and the SDUST Research Fund (2014TDJH102)

In this paper, stochastic differential equations that model the dynamics of a hepatitis C virus are derived from a system of ordinary differential equations. The stochastic model incorporates the host immunity. Firstly, the existence of a unique ergodic stationary distribution is derived by using the theory of Hasminskii. Secondly, sufficient conditions are obtained for the destruction of hepatocytes and the convergence of target cells. Moreover based on realistic parameters, numerical simulations are carried out to show the analytical results. These results highlight the role of environmental noise in the spread of hepatitis C viruses. The theoretical work extend the results of the corresponding deterministic system.

Citation: Tao Feng, Zhipeng Qiu, Xinzhu Meng. Dynamics of a stochastic hepatitis C virus system with host immunity. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019143
##### References:
 [1] L. Allen, B. Bolker, Y. Lou and A. Nevai, Asymptotic profiles of the steady states for an sis epidemic patch model, SIAM Journal on Applied Mathematics, 67 (2007), 1283-1309. doi: 10.1137/060672522. Google Scholar [2] S. Banerjee, R. Keval and S. Gakkhar, Modeling the dynamics of hepatitis c virus with combined antiviral drug therapy: Interferon and ribavirin, Mathematical Biosciences, 245 (2013), 235-248. doi: 10.1016/j.mbs.2013.07.005. Google Scholar [3] M. Barczy and G. Pap, Portmanteau theorem for unbounded measures, Statistics and Probability Letters, 76 (2006), 1831-1835. doi: 10.1016/j.spl.2006.04.025. Google Scholar [4] B. Berrhazi, M. E. Fatini, T. Caraballo and R. Pettersson, A stochastic siri epidemic model with lévy noise, Discrete and Continuous Dynamical Systems-B, 23 (2018), 2415-2431. doi: 10.3934/dcdsb.2018057. Google Scholar [5] G. Blé, L. Esteva and A. Peregrino, Global analysis of a mathematical model for hepatitis c considering the host immune system, Journal of Mathematical Analysis and Applications, 461 (2018), 1378-1390. doi: 10.1016/j.jmaa.2018.01.050. Google Scholar [6] T. Britton and A. Traoré, A stochastic vector-borne epidemic model: Quasi-stationarity and extinction, Mathematical Biosciences, 289 (2017), 89-95. doi: 10.1016/j.mbs.2017.05.004. Google Scholar [7] Y. Cai, Y. Kang, M. Banerjee and W. Wang, A stochastic sirs epidemic model with infectious force under intervention strategies, Journal of Differential Equations, 259 (2015), 7463-7502. doi: 10.1016/j.jde.2015.08.024. Google Scholar [8] T. Caraballo, M. E. Fatini, R. Pettersson and R. Taki, A stochastic siri epidemic model with relapse and media coverage, Discrete and Continuous Dynamical Systems-B, 23 (2018), 3483-3501. doi: 10.3934/dcdsb.2018250. Google Scholar [9] Z. Chang, X. Meng and T. Zhang, A new way of investigating the asymptotic behaviour of a stochastic sis system with multiplicative noise, Applied Mathematics Letters, 87 (2019), 80-86. doi: 10.1016/j.aml.2018.07.014. Google Scholar [10] N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for internal hiv dynamics, Journal of Mathematical Analysis and Applications, 341 (2008), 1084-1101. doi: 10.1016/j.jmaa.2007.11.005. Google Scholar [11] N. T. Dieu, D. H. Nguyen, N. H. Du and G. Yin, Classification of asymptotic behavior in a stochastic sir model, SIAM Journal on Applied Dynamical Systems, 15 (2016), 1062-1084. doi: 10.1137/15M1043315. Google Scholar [12] N. M. Dixit, J. E. Layden-Almer, T. J. Layden and A. S. Perelson, Modelling how ribavirin improves interferon response rates in hepatitis c virus infection, Nature, 432 (2004), 922-924. doi: 10.1038/nature03153. Google Scholar [13] T. Feng, Z. Qiu and X. Meng, Analysis of a stochastic recovery-relapse epidemic model with periodic parameters and media coverage, Journal of Applied Analysis and Computation, 9 (2019), 1-15. doi: 10.11948/2156-907X.20180231. Google Scholar [14] T. Feng and Z. Qiu, Global dynamics of deterministic and stochastic epidemic systems with nonmonotone incidence rate, International Journal of Biomathematics, 11 (2018), Paper No. 1850101, 24 pp. doi: 10.1142/S1793524518501012. Google Scholar [15] T. Feng and Z. Qiu, Global analysis of a stochastic tb model with vaccination and treatment, Discrete and Continuous Dynamical Systems-B, 24 (2019), 2923-2939. doi: 10.3934/dcdsb.2018292. Google Scholar [16] T. Feng, Z. Qiu, X. Meng and L. Rong, Analysis of a stochastic hiv-1 infection model with degenerate diffusion, Applied Mathematics and Computation, 348 (2019), 437-455. doi: 10.1016/j.amc.2018.12.007. Google Scholar [17] Z. Feng and H. Thieme, Endemic models with arbitrarily distributed periods of infection ii: Fast disease dynamics and permanent recovery, SIAM Journal on Applied Mathematics, 61 (2000), 983-1012. doi: 10.1137/S0036139998347846. Google Scholar [18] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302. Google Scholar [19] S. Jerez, S. Díaz-Infante and B. Chen, Fluctuating periodic solutions and moment boundedness of a stochastic model for the bone remodeling process, Mathematical Biosciences, 299 (2018), 153-164. doi: 10.1016/j.mbs.2018.03.006. Google Scholar [20] J. Jiang and Z. Qiu, The complete classification for dynamics in a nine-dimensional west nile virus model, SIAM Journal on Applied Mathematics, 69 (2009), 1205-1227. doi: 10.1137/070709438. Google Scholar [21] J. Jiang, Z. Qiu, J. Wu and H. Zhu, Threshold conditions for west nile virus outbreaks, Bulletin of Mathematical Biology, 71 (2009), 627-647. doi: 10.1007/s11538-008-9374-6. Google Scholar [22] R. Khasminskii, Stochastic Stability of Differential Equations, Stochastic Modelling and Applied Probability, 66. Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0. Google Scholar [23] D. Li, J. Cui, M. Liu and S. Liu, The evolutionary dynamics of stochastic epidemic model with nonlinear incidence rate, Bulletin of Mathematical Biology, 77 (2015), 1705-1743. doi: 10.1007/s11538-015-0101-9. Google Scholar [24] M. Liu and C. Bai, Analysis of a stochastic tri-trophic food-chain model with harvesting, Journal of Mathematical Biology, 73 (2016), 597-625. doi: 10.1007/s00285-016-0970-z. Google Scholar [25] X. Mao, G. Marion and E. Renshaw, Environmental brownian noise suppresses explosions in population dynamics, Stochastic Processes and their Applications, 97 (2002), 95-110. doi: 10.1016/S0304-4149(01)00126-0. Google Scholar [26] X. Meng, S. Zhao, T. Feng and T. Zhang, Dynamics of a novel nonlinear stochastic sis epidemic model with double epidemic hypothesis, Journal of Mathematical Analysis and Applications, 433 (2016), 227-242. doi: 10.1016/j.jmaa.2015.07.056. Google Scholar [27] A. U. Neumann, N. P. Lam, H. Dahari, D. R. Gretch, T. E. Wiley, T. J. Layden and A. S. Perelson, Hepatitis c viral dynamics in vivo and the antiviral efficacy of interferon-$\alpha$ therapy, Science, 282 (1998), 103-107. Google Scholar [28] Z. Qiu, M. Y. Li and Z. Shen, Global dynamics of an infinite dimensional epidemic model with nonlocal state structures, Journal of Differential Equations, 265 (2018), 5262-5296. doi: 10.1016/j.jde.2018.06.036. Google Scholar [29] L. Rong, R. M. Ribeiro and A. S. Perelson, Modeling quasispecies and drug resistance in hepatitis c patients treated with a protease inhibitor, Bulletin of Mathematical Biology, 74 (2012), 1789-1817. doi: 10.1007/s11538-012-9736-y. Google Scholar [30] I. Rusyn and S. M. Lemon, Mechanisms of hcv-induced liver cancer: What did we learn from in vitro and animal studies?, Cancer Letters, 345 (2014), 210-215. doi: 10.1016/j.canlet.2013.06.028. Google Scholar [31] S. Sengupta, P. Das and D. Mukherjee, Stochastic non-autonomous holling type- prey-predator model with predator's intra-specific competition, Discrete and Continuous Dynamical Systems-B, 23 (2018), 3275-3296. doi: 10.3934/dcdsb.2018244. Google Scholar [32] C. W. Shepard, L. Finelli and M. J. Alter, Global epidemiology of hepatitis c virus infection, The Lancet Infectious Diseases, 5 (2005), 558-567. doi: 10.1016/S1473-3099(05)70216-4. Google Scholar [33] A. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Translations of Mathematical Monographs, 78. American Mathematical Society, Providence, RI, 1989. Google Scholar [34] B. Stephenson, C. Lanzas, S. Lenhart and J. Day, Optimal control of vaccination rate in an epidemiological model of clostridium difficile transmission, Journal of Mathematical Biology, 75 (2017), 1693-1713. doi: 10.1007/s00285-017-1133-6. Google Scholar [35] Q. Yang, D. Jiang, N. Shi and C. Ji, The ergodicity and extinction of stochastically perturbed sir and seir epidemic models with saturated incidence, Journal of Mathematical Analysis and Applications, 388 (2012), 248-271. doi: 10.1016/j.jmaa.2011.11.072. Google Scholar [36] S. Zhang, X. Meng, T. Feng and T. Zhang, Dynamics analysis and numerical simulations of a stochastic non-autonomous predator-prey system with impulsive effects, Nonlinear Analysis: Hybrid Systems, 26 (2017), 19-37. doi: 10.1016/j.nahs.2017.04.003. Google Scholar [37] Y. Zhang, K. Fan, S. Gao and S. Chen, A remark on stationary distribution of a stochastic sir epidemic model with double saturated rates, Applied Mathematics Letters, 76 (2018), 46-52. doi: 10.1016/j.aml.2017.08.002. Google Scholar [38] Y. Zhao, S. Yuan and J. Ma, Survival and stationary distribution analysis of a stochastic competitive model of three species in a polluted environment, Bulletin of Mathematical Biology, 77 (2015), 1285-1326. doi: 10.1007/s11538-015-0086-4. Google Scholar [39] C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM Journal on Control and Optimization, 46 (2007), 1155-1179. doi: 10.1137/060649343. Google Scholar

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##### References:
 [1] L. Allen, B. Bolker, Y. Lou and A. Nevai, Asymptotic profiles of the steady states for an sis epidemic patch model, SIAM Journal on Applied Mathematics, 67 (2007), 1283-1309. doi: 10.1137/060672522. Google Scholar [2] S. Banerjee, R. Keval and S. Gakkhar, Modeling the dynamics of hepatitis c virus with combined antiviral drug therapy: Interferon and ribavirin, Mathematical Biosciences, 245 (2013), 235-248. doi: 10.1016/j.mbs.2013.07.005. Google Scholar [3] M. Barczy and G. Pap, Portmanteau theorem for unbounded measures, Statistics and Probability Letters, 76 (2006), 1831-1835. doi: 10.1016/j.spl.2006.04.025. Google Scholar [4] B. Berrhazi, M. E. Fatini, T. Caraballo and R. Pettersson, A stochastic siri epidemic model with lévy noise, Discrete and Continuous Dynamical Systems-B, 23 (2018), 2415-2431. doi: 10.3934/dcdsb.2018057. Google Scholar [5] G. Blé, L. Esteva and A. Peregrino, Global analysis of a mathematical model for hepatitis c considering the host immune system, Journal of Mathematical Analysis and Applications, 461 (2018), 1378-1390. doi: 10.1016/j.jmaa.2018.01.050. Google Scholar [6] T. Britton and A. Traoré, A stochastic vector-borne epidemic model: Quasi-stationarity and extinction, Mathematical Biosciences, 289 (2017), 89-95. doi: 10.1016/j.mbs.2017.05.004. Google Scholar [7] Y. Cai, Y. Kang, M. Banerjee and W. Wang, A stochastic sirs epidemic model with infectious force under intervention strategies, Journal of Differential Equations, 259 (2015), 7463-7502. doi: 10.1016/j.jde.2015.08.024. Google Scholar [8] T. Caraballo, M. E. Fatini, R. Pettersson and R. Taki, A stochastic siri epidemic model with relapse and media coverage, Discrete and Continuous Dynamical Systems-B, 23 (2018), 3483-3501. doi: 10.3934/dcdsb.2018250. Google Scholar [9] Z. Chang, X. Meng and T. Zhang, A new way of investigating the asymptotic behaviour of a stochastic sis system with multiplicative noise, Applied Mathematics Letters, 87 (2019), 80-86. doi: 10.1016/j.aml.2018.07.014. Google Scholar [10] N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for internal hiv dynamics, Journal of Mathematical Analysis and Applications, 341 (2008), 1084-1101. doi: 10.1016/j.jmaa.2007.11.005. Google Scholar [11] N. T. Dieu, D. H. Nguyen, N. H. Du and G. Yin, Classification of asymptotic behavior in a stochastic sir model, SIAM Journal on Applied Dynamical Systems, 15 (2016), 1062-1084. doi: 10.1137/15M1043315. Google Scholar [12] N. M. Dixit, J. E. Layden-Almer, T. J. Layden and A. S. Perelson, Modelling how ribavirin improves interferon response rates in hepatitis c virus infection, Nature, 432 (2004), 922-924. doi: 10.1038/nature03153. Google Scholar [13] T. Feng, Z. Qiu and X. Meng, Analysis of a stochastic recovery-relapse epidemic model with periodic parameters and media coverage, Journal of Applied Analysis and Computation, 9 (2019), 1-15. doi: 10.11948/2156-907X.20180231. Google Scholar [14] T. Feng and Z. Qiu, Global dynamics of deterministic and stochastic epidemic systems with nonmonotone incidence rate, International Journal of Biomathematics, 11 (2018), Paper No. 1850101, 24 pp. doi: 10.1142/S1793524518501012. Google Scholar [15] T. Feng and Z. Qiu, Global analysis of a stochastic tb model with vaccination and treatment, Discrete and Continuous Dynamical Systems-B, 24 (2019), 2923-2939. doi: 10.3934/dcdsb.2018292. Google Scholar [16] T. Feng, Z. Qiu, X. Meng and L. Rong, Analysis of a stochastic hiv-1 infection model with degenerate diffusion, Applied Mathematics and Computation, 348 (2019), 437-455. doi: 10.1016/j.amc.2018.12.007. Google Scholar [17] Z. Feng and H. Thieme, Endemic models with arbitrarily distributed periods of infection ii: Fast disease dynamics and permanent recovery, SIAM Journal on Applied Mathematics, 61 (2000), 983-1012. doi: 10.1137/S0036139998347846. Google Scholar [18] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302. Google Scholar [19] S. Jerez, S. Díaz-Infante and B. Chen, Fluctuating periodic solutions and moment boundedness of a stochastic model for the bone remodeling process, Mathematical Biosciences, 299 (2018), 153-164. doi: 10.1016/j.mbs.2018.03.006. Google Scholar [20] J. Jiang and Z. Qiu, The complete classification for dynamics in a nine-dimensional west nile virus model, SIAM Journal on Applied Mathematics, 69 (2009), 1205-1227. doi: 10.1137/070709438. Google Scholar [21] J. Jiang, Z. Qiu, J. Wu and H. Zhu, Threshold conditions for west nile virus outbreaks, Bulletin of Mathematical Biology, 71 (2009), 627-647. doi: 10.1007/s11538-008-9374-6. Google Scholar [22] R. Khasminskii, Stochastic Stability of Differential Equations, Stochastic Modelling and Applied Probability, 66. Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0. Google Scholar [23] D. Li, J. Cui, M. Liu and S. Liu, The evolutionary dynamics of stochastic epidemic model with nonlinear incidence rate, Bulletin of Mathematical Biology, 77 (2015), 1705-1743. doi: 10.1007/s11538-015-0101-9. Google Scholar [24] M. Liu and C. Bai, Analysis of a stochastic tri-trophic food-chain model with harvesting, Journal of Mathematical Biology, 73 (2016), 597-625. doi: 10.1007/s00285-016-0970-z. Google Scholar [25] X. Mao, G. Marion and E. Renshaw, Environmental brownian noise suppresses explosions in population dynamics, Stochastic Processes and their Applications, 97 (2002), 95-110. doi: 10.1016/S0304-4149(01)00126-0. Google Scholar [26] X. Meng, S. Zhao, T. Feng and T. Zhang, Dynamics of a novel nonlinear stochastic sis epidemic model with double epidemic hypothesis, Journal of Mathematical Analysis and Applications, 433 (2016), 227-242. doi: 10.1016/j.jmaa.2015.07.056. Google Scholar [27] A. U. Neumann, N. P. Lam, H. Dahari, D. R. Gretch, T. E. Wiley, T. J. Layden and A. S. Perelson, Hepatitis c viral dynamics in vivo and the antiviral efficacy of interferon-$\alpha$ therapy, Science, 282 (1998), 103-107. Google Scholar [28] Z. Qiu, M. Y. Li and Z. Shen, Global dynamics of an infinite dimensional epidemic model with nonlocal state structures, Journal of Differential Equations, 265 (2018), 5262-5296. doi: 10.1016/j.jde.2018.06.036. Google Scholar [29] L. Rong, R. M. Ribeiro and A. S. Perelson, Modeling quasispecies and drug resistance in hepatitis c patients treated with a protease inhibitor, Bulletin of Mathematical Biology, 74 (2012), 1789-1817. doi: 10.1007/s11538-012-9736-y. Google Scholar [30] I. Rusyn and S. M. Lemon, Mechanisms of hcv-induced liver cancer: What did we learn from in vitro and animal studies?, Cancer Letters, 345 (2014), 210-215. doi: 10.1016/j.canlet.2013.06.028. Google Scholar [31] S. Sengupta, P. Das and D. Mukherjee, Stochastic non-autonomous holling type- prey-predator model with predator's intra-specific competition, Discrete and Continuous Dynamical Systems-B, 23 (2018), 3275-3296. doi: 10.3934/dcdsb.2018244. Google Scholar [32] C. W. Shepard, L. Finelli and M. J. Alter, Global epidemiology of hepatitis c virus infection, The Lancet Infectious Diseases, 5 (2005), 558-567. doi: 10.1016/S1473-3099(05)70216-4. Google Scholar [33] A. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Translations of Mathematical Monographs, 78. American Mathematical Society, Providence, RI, 1989. Google Scholar [34] B. Stephenson, C. Lanzas, S. Lenhart and J. Day, Optimal control of vaccination rate in an epidemiological model of clostridium difficile transmission, Journal of Mathematical Biology, 75 (2017), 1693-1713. doi: 10.1007/s00285-017-1133-6. Google Scholar [35] Q. Yang, D. Jiang, N. Shi and C. Ji, The ergodicity and extinction of stochastically perturbed sir and seir epidemic models with saturated incidence, Journal of Mathematical Analysis and Applications, 388 (2012), 248-271. doi: 10.1016/j.jmaa.2011.11.072. Google Scholar [36] S. Zhang, X. Meng, T. Feng and T. Zhang, Dynamics analysis and numerical simulations of a stochastic non-autonomous predator-prey system with impulsive effects, Nonlinear Analysis: Hybrid Systems, 26 (2017), 19-37. doi: 10.1016/j.nahs.2017.04.003. Google Scholar [37] Y. Zhang, K. Fan, S. Gao and S. Chen, A remark on stationary distribution of a stochastic sir epidemic model with double saturated rates, Applied Mathematics Letters, 76 (2018), 46-52. doi: 10.1016/j.aml.2017.08.002. Google Scholar [38] Y. Zhao, S. Yuan and J. Ma, Survival and stationary distribution analysis of a stochastic competitive model of three species in a polluted environment, Bulletin of Mathematical Biology, 77 (2015), 1285-1326. doi: 10.1007/s11538-015-0086-4. Google Scholar [39] C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM Journal on Control and Optimization, 46 (2007), 1155-1179. doi: 10.1137/060649343. Google Scholar
Trajectories of the system (2) and its deterministic system (1)
Density distribution of the system (2)
Trajectories of the system (2) and its deterministic system (1)
Density distribution of the system (2)
Trajectories of the system (2) and its deterministic system (1)
Density distribution of the system (2)
Trajectories of the solution of the system (2) and its deterministic system (1)
Variables and parameters for HCV spread
 Initial values $H_s$ concentration of target cells 1000$\; \rm{mm}^{-3}$ $H_i$ concentration of infected liver cells 0 $V$ concentration of viral load $10^{-2}\; \rm{mm}^{-3}$ $T$ concentration of T killer cells 0 $\beta_s$ produce rate of target cells $20\; \rm{day}^{-1}\; \rm{mm}^{-3}$ $k$ scaled transmission rate between target cells and infected liver cells $3.5\times10^{-5}\; \rm{mm}^{3}\; \rm{day}^{-1}$ $\mu_s$ death rate of target cells $0.03\; \rm{day}^{-1}$ $\delta$ destroy rate of T cells to infected liver cells $2.5\times10^{-5}\; \rm{mm}^{3}\; \rm{day}^{-1}$ $\mu_i$ death rate of infected liver cells $0.02\; \rm{day}^{-1}$ $p$ $0.003\; \rm{day}^{-1}$ $\beta_T$ reproduction rate of T killer cells $3.0\times10^{-5}\; \rm{mm}^{3}\; \rm{day}^{-1}$ $T_{max}$ the maximum of T killer cells in the body $2000\; \rm{mm}^{-3}$ $\mu_T$ death rate of T killer cells $0.01\; \rm{day}^{-1}$
 Initial values $H_s$ concentration of target cells 1000$\; \rm{mm}^{-3}$ $H_i$ concentration of infected liver cells 0 $V$ concentration of viral load $10^{-2}\; \rm{mm}^{-3}$ $T$ concentration of T killer cells 0 $\beta_s$ produce rate of target cells $20\; \rm{day}^{-1}\; \rm{mm}^{-3}$ $k$ scaled transmission rate between target cells and infected liver cells $3.5\times10^{-5}\; \rm{mm}^{3}\; \rm{day}^{-1}$ $\mu_s$ death rate of target cells $0.03\; \rm{day}^{-1}$ $\delta$ destroy rate of T cells to infected liver cells $2.5\times10^{-5}\; \rm{mm}^{3}\; \rm{day}^{-1}$ $\mu_i$ death rate of infected liver cells $0.02\; \rm{day}^{-1}$ $p$ $0.003\; \rm{day}^{-1}$ $\beta_T$ reproduction rate of T killer cells $3.0\times10^{-5}\; \rm{mm}^{3}\; \rm{day}^{-1}$ $T_{max}$ the maximum of T killer cells in the body $2000\; \rm{mm}^{-3}$ $\mu_T$ death rate of T killer cells $0.01\; \rm{day}^{-1}$
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