Dynamics of a stochastic hepatitis C virus system with host immunity

Tao Feng; Zhipeng Qiu; Xinzhu Meng

doi:10.3934/dcdsb.2019143

Article Contents

2019, Volume 24, Issue 12: 6367-6385. Doi: 10.3934/dcdsb.2019143

This issue Previous Article On the $ L^p $ regularity of solutions to the generalized Hunter-Saxton system Next Article A hybrid method for stiff reaction–diffusion equations

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
^* Corresponding author: Zhipeng Qiu

Received: September 2018

Revised: December 2018

Early access: July 2019

Published: September 2019

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)

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Abstract

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.

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

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

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Figure 2. Density distribution of the system (2)

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

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Figure 4. Density distribution of the system (2)

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

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Figure 6. Density distribution of the system (2)

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

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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} $

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