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

• * Corresponding author: Zhipeng Qiu

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.

Mathematics Subject Classification: Primary: 92B05, 92D30; Secondary: 60H10.

 Citation: • • Figure 1.  Trajectories of the system (2) and its deterministic system (1)

Figure 2.  Density distribution of the system (2)

Figure 3.  Trajectories of the system (2) and its deterministic system (1)

Figure 4.  Density distribution of the system (2)

Figure 5.  Trajectories of the system (2) and its deterministic system (1)

Figure 6.  Density distribution of the system (2)

Figure 7.  Trajectories of the solution of the system (2) and its deterministic system (1)

Table 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 $0.01\; \rm{day}^{-1}$
• Figures(7)

Tables(1)

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