NSFD scheme and dynamic consistency of a delayed diffusive humoral immunity viral infection model
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NSFD scheme and dynamic consistency of a delayed diffusive humoral immunity viral infection model Xiaosong Tang1
· Tao Yu1 · Zhiyun Deng1 · Dengyu Liu1
Received: 16 February 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract In this paper, we establish a delayed diffusive humoral immunity viral infection model with nonlinear incidence rate and capsids subject to the homogeneous Neumann boundary conditions. By constructing appropriate Lyapunov function, we show that the global threshold dynamics for the original continuous model. Meanwhile, nonstandard finite difference (NSFD) scheme for the original continuous model is also proposed by utilizing Micken’s method. Then, using the theory of M-matrix, it is shown that the discrete model is well-posedness. Additionally, the global stability for the steady states is investigated by constructing discrete Lyapunov function. These results imply that the NSFD scheme may preserve the dynamical properties of solutions for the original continuous model efficiently. Furthermore, some numerical simulations to illustrate the theoretical analysis are carried out. Keywords Virus infection model · Humoral immunity · NSFD scheme · Dynamic consistency · Global stability · Lyapunov function Mathematics Subject Classification 92D30 · 35K57 · 34D23
1 Introduction Nowadays, more and more people in the world are dying of various diseases such as AIDS, avian influenza, cholera, Ebola, Zika virus and so on. To explore the mechanisms of these diseases, scientists have proposed many mathematical models describing the transmission of disease such as infectious disease model (SI, SIR, SEI, SIS) and within-host virus model (HBV, HCV, HIV, Ebola, Zika). As we all know, Hepatitis B virus (HBV) infection is a disease of huge threat for public health at a universal level and it is a hepatic condition resulting from infection of the hepatocytes (or the
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Xiaosong Tang [email protected] School of Mathematics and Physics, Jinggangshan University, Ji’an 343009, China
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X. Tang et al.
liver cells) [1]. According to the data of World Health Organization (WHO), about 2 billion people are infected with HBV in their life and 0.78 million people chronically infected die annually [2], which results in HBV infection a subject of serious research. Thus, plenty of mathematical models have been proposed to explore mechanisms and dynamical behaviours of such infectious viruses as HBV in the past several decades, see [3–17]. However, we have to point out a fact that some HBV models [3–10] do not include the intracellular HBV DNA-containing capsids [11–14]. Based on this reason, in a recent article, Manna and Chakrabarty [11] proposed the following diffusive model with intracellular HBV DNA-containing capsids: ⎧ ∂U = λ − μU (x, t) − aU (x, t)V (x, t), ⎪ ⎪ ⎨ ∂∂tW ∂t = aU (x, t)V (x, t) − δW (x, t), ∂D 2 ⎪ ⎪ ∂t = d1 ∇ D + kW (x, t) − (β + δ)D(x, t), ⎩ ∂V 2 ∂t = d2 ∇ V + β D(x, t) − cV (x, t).
(1.1)
They investigated the global dynamics of model (1.1) by constructing
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