A generalized virus dynamics model with celltocell transmission and cure rate
 Khalid Hattaf^{1}Email author and
 Noura Yousfi^{2}
https://doi.org/10.1186/s1366201609063
© Hattaf and Yousfi 2016
Received: 1 May 2016
Accepted: 22 June 2016
Published: 1 July 2016
Abstract
Viruses can be spread and transmitted through two fundamental modes, one by virustocell infection and the other by direct celltocell transmission. In this paper, we propose a new generalized virus dynamics model, which incorporates both modes and takes into account the cure of infected cells. We first show mathematically and biologically the wellposedness of our model. Further, an explicit formula for the basic reproduction number \(R_{0}\) of the model is determined. By analyzing the characteristic equations we establish the local stability of the diseasefree equilibrium and the chronic infection equilibrium in terms of \(R_{0}\). The global behavior of the model is investigated by constructing an appropriate Lyapunov functional for diseasefree equilibrium and by applying geometrical approach to chronic infection equilibrium. Moreover, mathematical virus models and results presented in many previous studies are generalized and improved.
Keywords
1 Introduction
Many viruses infect humans and cause different infectious diseases such as human immunodeficiency virus (HIV), hepatitis B virus (HBV), hepatitis C virus (HCV), Ebola virus, and more recently Zika virus. They are often transmitted in body by two fundamentally distinct modes, either by virustocell infection through the extracellular space or by celltocell transmission involving direct celltocell contact [1–4]. During both infection modes, a part of infected cells returns to the uninfected state by loss of all covalently closed circular DNA (cccDNA) from their nucleus [5–7]. To model viral infection dynamics, several mathematical models have been proposed and developed. Most of these models are based on the assumption that healthy cells can only be infected by viruses, and so they consider only the virustocell infection mode. However, there are few virus dynamics models in the literature with both modes of transmission and taking into account the cure of infected cells.
 (H_{0}):

\(g(0,y)=0\) for all \(y\geq0\), \(\frac{\partial g}{\partial x}(x,y)\geq0\) (or \(g(x,y)\) is a strictly increasing function with respect to x when \(f\equiv0\)), and \(\frac{\partial g}{\partial y}(x,y)\leq0\) for all \(x\geq0\) and \(y \geq0\).
 (H_{1}):

\(f(0,y,v)=0\) for all \(y\geq0\) and \(v\geq0\),
 (H_{2}):

\(f(x,y,v)\) is a strictly increasing function with respect to x (or \(\frac{\partial f}{\partial x}(x,v,y)\geq0\) when \(g(x,y)\) is a strictly increasing function with respect to x) for any fixed \(y\geq0\) and \(v\geq0\),
 (H_{3}):

\(f(x,y,v)\) is a decreasing function with respect to y and v, that is, \(\frac{\partial f}{\partial y}(x,y,v)\leq0\) and \(\frac{ \partial f}{\partial v}(x,y,v)\leq0\) for all \(x\geq0\), \(y\geq0\), and \(v\geq0\).
The rest of this paper is organized as follows. The next section deals with some preliminary results about the positivity and boundedness of solutions, the basic reproduction number, and the existence of equilibria. In Section 3, we establish the local stability of the diseasefree equilibrium and the chronic infection equilibrium. The global stability of both equilibria is investigated in Section 4. The paper ends with some applications of our results in Section 5.
2 Preliminary results
In this section, we first show that the solutions of system (1) with nonnegative initial conditions remain nonnegative and bounded for all \(t\geq0\).
Let \(\mathbb{R}^{3}_{+}=\{(x,y,v)\in\mathbb{R}^{3} : x\geq0, y\geq 0, v\geq0\} \). We have the following result.
Theorem 2.1
Proof
Obviously, \(\mathbb{R}^{3}_{+}\) is positively invariant with respect (1). It remains to prove that all solutions of system (1) are uniformly bounded.
This theorem shows mathematically and biologically the wellposedness of our model (1). Now, we discuss the existence of equilibria.
From the biological point of view, the factor \(\frac{1}{u}\) is the average life expectancy of viruses, and \(\frac{1}{a+\rho}\) is the average life expectancy of infected cells, which is less than \(\frac{1}{a}\) because a part of infected cells returns to the uninfected state by loss of all cccDNA from their nucleus at a rate ρ. Since the viruses are produced by infected cells at a rate ky, \(\frac{k}{a+\rho}\) denotes the amount of viruses generated from living infected cell. Further, the number of susceptible cells at beginning of the infection is \(\frac{\lambda}{d}\), which means that \(f(\frac{\lambda}{d},0,0)\) and \(g(\frac{\lambda}{d},0)\) are the values of both incidence functions when all cells are uninfected. Therefore, \(R_{01}\) is the basic reproduction number corresponding to virustocell infection mode, whereas \(R_{02}\) is the basic reproduction number corresponding to celltocell transmission mode.
The previous discussions can be summarized in the following result.
Theorem 2.2
 (i)
If \(R_{0} \leq1\), then system (1) has a unique diseasefree equilibrium of the form \(E_{f}(\frac{\lambda}{d}, 0, 0)\).
 (ii)
If \(R_{0}>1\), then the diseasefree equilibrium is still present, and system (1) has a unique chronic infection equilibrium of the form \(E^{*}(x^{*}, y^{*}, v^{*})\) with \(x^{*}\in(0, \frac{\lambda}{d})\), \(y^{*}>0\), and \(v^{*}>0\).
3 Local stability
Theorem 3.1
The diseasefree equilibrium \(E_{f}\) is locally asymptotically stable if \(R_{0}<1\) and becomes unstable if \(R_{0}>1\).
Proof
Next, we study the local stability of the chronic infection equilibrium \(E^{*}\). Note that the equilibrium \(E^{*}\) does not exist if \(R_{0}<1\) and \(E^{*}=E_{f}\) when \(R_{0}=1\).
Theorem 3.2
If \(R_{0}>1\), then the chronic infection equilibrium \(E^{*}\) is locally asymptotically stable.
Proof
4 Global stability
In this section, we investigate the global stability of the diseasefree equilibrium \(E_{f}\) and the chronic infection equilibrium \(E^{*}\). For the global stability of \(E_{f}\), we assume that \(a\geq d\). Biologically, this assumption is often satisfied because a represents the death rate of infected cells and includes the possibility of death by bursting of infected cells. Further, this assumption is considered by many authors; see, for example, [23–25]. Therefore, we have the following result.
Theorem 4.1
If \(R_{0}\leq1\), then the diseasefree equilibrium \(E_{f}\) is globally asymptotically stable.
Proof
Now, we will investigate the global dynamics of system (1) when \(R_{0}>1\). Firstly, we need the following lemma.
Lemma 4.2
If \(R_{0}>1\), then system (1) is uniformly persistent.
Proof
This result follows from an application of Theorem 4.3 in [27] with \(X=\mathbb{R}^{3}\) and \(E=\Gamma\). The maximal invariant set M on the boundary ∂Γ is the singleton \(\{E_{f}\}\), and it is isolated. From Theorem 4.3 in [27] we can see that the uniform persistence of system (1) is equivalent to the instability of the diseasefree equilibrium \(E_{f}\). On the other hand, we have proved in Theorem 3.1 that \(E_{f}\) is unstable if \(R_{0}>1\). Thus, system (1) is uniformly persistent when \(R_{0}>1\). □
Theorem 4.3
Assume that \(R_{0}>1\) and (H_{4}) hold. Then the chronic infection equilibrium \(E^{*}\) is globally asymptotically stable.
Proof
To prove the global stability of \(E^{*}\), we will apply the geometrical approach developed by Li and Muldowney [28].
5 Applications
The aim of this section is to apply our main results to some special cases of our model (1).
Example 1
Corollary 5.1
Example 2
Corollary 5.2
Declarations
Acknowledgements
The authors would like to express their gratitude to the editor and the anonymous referees for their constructive comments and suggestions, which have improved the quality of the manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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