 Research
 Open Access
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
 virus dynamics
 celltocell transmission
 compound matrices
 global stability
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
References
 Marsh, M, Helenius, A: Virus entry: open sesame. Cell 124, 729740 (2006) View ArticleGoogle Scholar
 Mothes, W, Sherer, NM, Jin, J, Zhong, P: Virus celltocell transmission. J. Virol. 84, 83608368 (2010) View ArticleGoogle Scholar
 Sattentau, Q: Avoiding the void: celltocell spread of human viruses. Nat. Rev. Microbiol. 6, 815826 (2008) View ArticleGoogle Scholar
 Zhong, P, Agosto, LM, Munro, JB, Mothes, W: Celltocell transmission of viruses. Curr. Opin. Virol. 3, 4450 (2013) View ArticleGoogle Scholar
 Guidotti, LG, Rochford, R, Chung, J, Shapiro, M, Purcell, R, Chisari, FV: Viral clearance without destruction of infected cells during acute HBV infection. Science 284, 825829 (1999) View ArticleGoogle Scholar
 Lewin, SR, Ribeiro, RM, Walters, T, Lau, GK, Bowden, S, Locarnini, S, Perelson, AS: Analysis of hepatitis B viral load decline under potent therapy: complex decay profiles observed. Hepatology 34, 1011020 (2001) View ArticleGoogle Scholar
 Essunger, P, Perelson, AS: Modeling HIV infection of CD4^{+} Tcell subpopulations. J. Theor. Biol. 170, 367391 (1994) View ArticleGoogle Scholar
 Hattaf, K, Yousfi, N, Tridane, A: Mathematical analysis of a virus dynamics model with general incidence rate and cure rate. Nonlinear Anal., Real World Appl. 13, 18661872 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Hattaf, K, Yousfi, N: A numerical method for a delayed viral infection model with general incidence rate. J. King Saud Univ., Sci. (2015). doi:10.1016/j.jksus.2015.10.003 MATHGoogle Scholar
 Wang, XY, Hattaf, K, Huo, HF, Xiang, H: Stability analysis of a delayed social epidemics model with general contact rate and its optimal control. J. Ind. Manag. Optim. 12(4), 12671285 (2016) MathSciNetView ArticleGoogle Scholar
 Nowak, MA, Bangham, CRM: Population dynamics of immune responses to persistent viruses. Science 272, 7479 (1996) View ArticleGoogle Scholar
 Nowak, MA, Bonhoeffer, S, Hill, AM, Boehme, R, Thomas, HC, McDade, H: Viral dynamics in hepatitis B virus infection. Proc. Natl. Acad. Sci. USA 93, 43984402 (1996) View ArticleGoogle Scholar
 Neumann, AU, Lam, NP, Dahari, H, Gretch, DR, Wiley, TE, Layden, TJ, Perelson, AS: Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferonα therapy. Science 282, 103107 (1998) View ArticleGoogle Scholar
 Perelson, AS: Modelling viral and immune system dynamics. Nat. Rev. Immunol. 2, 2836 (2002) View ArticleGoogle Scholar
 Min, L, Su, Y, Kuang, Y: Mathematical analysis of a basic model of virus infection with application to HBV infection. Rocky Mt. J. Math. 38(5), 15731585 (2008) MathSciNetView ArticleMATHGoogle Scholar
 Huang, G, Ma, W, Takeuchi, Y: Global properties for virus dynamics model with BeddingtonDeAngelis functional response. Appl. Math. Lett. 22, 16901693 (2009) MathSciNetView ArticleMATHGoogle Scholar
 Wang, K, Fan, A, Torres, A: Global properties of an improved hepatitis B virus model. Nonlinear Anal., Real World Appl. 11, 31313138 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Hattaf, K, Yousfi, N: Dynamics of HIV infection model with therapy and cure rate. Int. J. Tomogr. Stat. 16, 7480 (2011) MATHGoogle Scholar
 Zhou, X, Cui, J: Global stability of the viral dynamics with CrowleyMartin functional response. Bull. Korean Math. Soc. 48(3), 555574 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Bonhoeffer, S, Coffin, JM, Nowak, MA: Human immunodeficiency virus drug therapy and virus load. J. Virol. 71, 32753278 (1997) Google Scholar
 Zhang, T, Meng, X, Zhang, T: Global dynamics of a virus dynamical model with celltocell transmission and cure rate. Comput. Math. Methods Med. (2015). doi:10.1155/2015/758362 MathSciNetMATHGoogle Scholar
 Gradshteyn, IS, Ryzhik, IM: RouthHurwitz theorem. In: Tables of Integrals, Series, and Products. Academic Press, San Diego (2000) Google Scholar
 Srivastava, PK, Chandra, P: Modeling the dynamics of HIV and CD4^{+} T cells during primary infection. Nonlinear Anal., Real World Appl. 11, 612618 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Pang, J, Cui, JA, Hui, J: The importance of immune responses in a model of hepatitis B virus. Nonlinear Dyn. 67(1), 723734 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Wang, L, Li, MY: Mathematical analysis of the global dynamics of a model for HIV infection of CD4^{+} T cells. Math. Biosci. 200, 4457 (2006) MathSciNetView ArticleMATHGoogle Scholar
 LaSalle, JP: The Stability of Dynamical Systems. Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1976) View ArticleMATHGoogle Scholar
 Freedman, H, Ruan, S, Tang, M: Uniform persistence and flows near a closed positively invariant set. J. Dyn. Differ. Equ. 6, 583600 (1994) MathSciNetView ArticleMATHGoogle Scholar
 Li, MY, Muldowney, JS: A geometric approach to the globalstability problems. SIAM J. Math. Anal. 27, 10701083 (1996) MathSciNetView ArticleMATHGoogle Scholar
 Martin, RH Jr.: Logarithmic norms and projections applied to linear differential systems. J. Math. Anal. Appl. 45, 432454 (1974) MathSciNetView ArticleMATHGoogle Scholar
 Butler, G, Waltman, P: Persistence in dynamical systems. Proc. Am. Math. Soc. 98, 425430 (1986) MathSciNetView ArticleMATHGoogle Scholar
 Beddington, JR: Mutual interference between parasites or predators and its effect on searching efficiency. J. Anim. Ecol. 44, 331341 (1975) View ArticleGoogle Scholar
 DeAngelis, DL, Goldsten, RA, Neill, R: A model for trophic interaction. Ecology 56, 881892 (1975) View ArticleGoogle Scholar
 Crowley, PH, Martin, EK: Functional responses and interference within and between year classes of a dragonfly population. J. North Am. Benthol. Soc. 8, 211221 (1989) View ArticleGoogle Scholar
 Hattaf, K, Yousfi, N, Tridane, A: Stability analysis of a virus dynamics model with general incidence rate and two delays. Appl. Math. Comput. 221, 514521 (2013) MathSciNetMATHGoogle Scholar
 Zhuo, X: Analysis of a HBV infection model with noncytolytic cure process. In: IEEE 6th International Conference on Systems Biology, pp. 148151 (2012) Google Scholar
 Yousfi, N, Hattaf, K, Tridane, A: Modeling the adaptative immune response in HBV infection. J. Math. Biol. 63, 933957 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Wang, J, Tian, X: Global stability of a delay differential equation of hepatitis B virus infection with immune response. Electron. J. Differ. Equ. 2013, 94 (2013) MathSciNetView ArticleMATHGoogle Scholar