According to   , approximately 200 million people worldwide are persistently infected with the hepatitis C virus (HCV) and are at risk of developing chronic liver disease, cirrhosis, and hepatocellular carcinoma. HCV infection therefore represents a significant global public health problem. HCV establishes chronic hepatitis in 60% - 80% of infected adults . A vaccine against infection with HCV does not exist yet, and standard treatment with interferon-α and ribavirin has produced sustained virological response rates of approximately 50%, with no effective alternative treatment for nonresponders to this treatment protocol. A model of human immunodeficiency virus infection was adapted by Neumann et al.  to study the kinetics of chronic HCV infection during treatment and some mathematical analysis was done by . Since then viral kinetics modeling has played an important role in the analysis of HCV RNA decay during antiviral therapy (see Perelson  , Perelson et al.  ). The original Neumann et al. model for HCV infection included three differential equations representing the populations of target cells, productively infected cells, and virus. In this paper we are going to study global dynamics of an HCV infection mathematical model with full logistic terms, antivirus treatments and homeostasis phenomenon. A similar work has been done by A. Nangue  concerning a mathematical intracellular HCV infection model with therapy.
1.1. The Compartmental Model
There are too many mathematical models of HCV dynamics amongst those, the original model or model of Newmann  and its extended models as those in   for example. Each model can be represented by a compartmental scheme. A compartmental scheme is a scheme for estimating the variation in the number of individuals in each compartment over time. Figure 1 is the schematic representation of the extended model, which we will study, of HCV with cellular proliferation and spontaneous healing designed by T. C. Reluga et al. . This model expands the viral dynamics of the original model of infection and the disappearance of HCV by incorporating the proliferation and death density dependence. In addition to cell proliferation, the number of uninfected hepatocytes may increase through immigration or differentiation of hepatocyte precursors that develop into hepatocytes at a constitutive rate of s or by spontaneous infected
Figure 1. Schematic representation of HCV infection models. T and I represent target and infected cells, respectively, and V represents free virus. The parameters shown in the figure are defined in the text. The original model of Neumann et al.  assumed that there is no proliferation of target and infected cells (i.e. ) and no spontaneous cure (i.e. ). The extended model of Dahari, Ribeiro, and Perelson  , which was used for predicting complex HCV kinetics under therapy, includes target and infected cell proliferation without cure ( , and ). A model including both proliferation and the spontaneous cure of infected cells (dashed line; ) was used to explain the kinetics of HCV in primary infection in chimpanzees .
hepatocyte healing by a non-cytolytic process at the rate q.
The model proposed by Dahari and coworkers   expands on the standard HCV viral-dynamic model  of infection and clearance by incorporating density-dependent proliferation and death. Uninfected hepatocytes or noninfected hepatocytes, T, are infected at a rate per free virus per hepatocyte. Infected cells, I, produce free virus at rate p per cell but also die with rate . Free virus, V, is cleared at rate c by immune and other degradation processes. Besides infection processes, hepatocyte numbers are influenced by homeostatic processes. Uninfected hepatocytes die at rate . Both infected and uninfected hepatocytes proliferate logistically with maximum rates and , respectively, as long as the total number of hepatocytes is less than . Besides proliferation, uninfected hepatocytes may increase in number through immigration or differentiation of hepatocyte precursors that develop into hepatocytes at constitutive rate s, or by spontaneous cure of infected hepatocytes through a noncytolytic process at rate q. Treatment with antiviral drugs reduces the infection rate by a fraction and the viral production rate by a fraction . It should be noted that and are parameters which values are non-negative and less than one. A further comprehensive survey on the description of the model is given in   . Given the meanings of and , the term represents the mass action principle; is the rate of infection of healthy hepatocytes T by interaction with virus V.
1.2. The Mathematical Model
According to Reluga et al.  and more precisely according to the schematic representation of HCV infection model in Figure 1 we have the following dynamics:
· the variation of the healthy hepatocytes or uninfected hepatocytes, T, is expressed by the following equation:
· the variation of infected hepatocytes, I, is expressed by the following equation:
· the variation of free virions or virus, V, is expressed by the following equation:
Thus, the phenomenon described above is governed by the following mathematical model (4), which is a system of three differential autonomous equations:
To analyse the system (4) we need the following initial conditions:
For biological significance of the parameters, three assumptions are employed. (a) Due to the burden of supporting virus replication, infected cells may proliferate more slowly than uninfected cells, i.e. . (b) To have a physiologically realistic model, in an uninfected liver when is reached, liver size should no longer increase, i.e. . (c) Infected cells have a higher turnover rate than uninfected cells, i.e. . The interpretations and biologically plausible values of other parameters and a further comprehensive survey on the description of (4) is given in . Besides HCV infection, the similar model of (4) is also used to describe the dynamics of HBV or HIV infection, in which the full logistic terms mean the proliferation of uninfected/infected hepatocytes    , or the mitotic transmission of uninfected/infected CD4+ T cells.
Our goal is therefore to analyze the stability of an extended model of HCV infection in a patient with cell proliferation and spontaneous healing given by (4) to reveal significant information on pathogenesis and dynamics of this virus. The paper is organized as follows: In Section 1, first focuses on some properties of the solutions of the model, then we calculate the basic reproduction ratio , which is an indispensable element in the study and analysis of the models. We theoretically analyze the local stability where we widely use the works of A. Nangue et al.  in Section 2. In Section 3 we theoretically analyze with some assumptions the global stability of the model by constructing appropriate Lyapounov’s functions.
2. Properties of Solutions to the Initial Value Problem (4), (5)
2.1. Positivity, Global Solutions and Asymptotic Behaviour
Theorem 1. Let . There exists and functions continuously differentiable such that is a solution of system (1) satisfying (4).
Theorem 2. Let be a solution of the system (1) over an interval such that et .
If are positive, then , and are also positive for all .
Proof. We are going to prove by contradiction. so suppose there is such that or or .
Let also be the smallest of all t in the interval such that , , and for a certain i.
Then each of the equations of the system (4) can be written where is a non negative function and any function. As a consequence and , . A contradiction. ,
Theorem 3.  The solutions of the Cauchy problem (4), (5), with positive initial data, exist globally in time in the future that is on .
Theorem 4. For any positive solution of system (4), (5) we have:
Proof. Summing Equations (6) and (7), we get:
thus since .
Let , , , and let us solve the following equation
Coupled to Equation (6) the initial condition:
The solving of the problem (6), (7) gives for all ,
As for all , , it follows that:
Since T and I are positive and , so it follows that and .
Equation (3), according to Gromwall inequality, leads to:
This completes the proof of theorem 4. ,
2.2. Basic Reproduction Ratio , Invariant Set of the Model and Equilibria
Proposition 5. The uninfected equilibrium point of the system (4) is given by
We use the method proposed in   to compute the basic reproduction number .
Proposition 6. The expression of the basic reproduction number associated to the system (4) is given by:
Remark 1. denotes the overall effectiveness rate of the drug.
Remark 2. Henceforth, we will let and .
Theorem 7. Let and be a maximal solution of the Cauchy problem (1), (4) ( ). If and then the set:
is a positively invariant set by system (4).
When it exists, the infected equilibrium point is given by: where , and are positive constants that we are going to determine.
Lemma 1.  exists if and only if
Lemma 2.  When it exists, is defined by:
The combination of the lemma 1 and the lemma 2 leads to the following theorem:
Theorem 8. The model (4) admits a unique infected equilibrium if and only if , where
When the unique equilibrium is the uninfected equilibrium point or the infection-free steady state .
3. Local Stability Analyses
3.1. Case of the Uninfected Equilibrium Point or Infection-Free Steady State
Theorem 9. The infection-free steady state of model (4) is locally asymptotically stable if and unstable if .
Proof. See the appendice of . ,
3.2. Case of Infected Equilibrium Point
We start this section by this lemma where the proof can be found in .
Lemma 3 The characteristic equation of the Jacobian matrix of the system (4) at is given by the following cubic equation:
Proof. See . ,
According to lemma 3 combined with the Routh-Hurwitz criterion  , we have the following results where the proofs can be found in .
Theorem 10. For model (4), when is valid, the unique endemic equilibrium is locally asymptotically stable if and unstable if .
Especially, we have:
Corollary 1. The infected steady state during the therapy of the model (4) is locally asymptotically stable if and unstable if .
4. Global Stability Analyses
The global stability analysis of a dynamical system is usually a very complex problem. One of the most efficient methods to solve this problem is Lyapunov’s theory. To build the functions of Lyapunov we will follow the method proposed by A. Korobeinikov   . In the proofs of the results that follow, to simplify the writings, we can use differently or for the derivation with respect to time.
4.1. Case of Infection-Free Steady State
Theorem 11. The infection-free steady state of the model (4) is globally asymptotically stable if the basic reproduction number and unstable if .
Proof. Consider the Lyapunov function:
L is defined, continuous and positive definite for all , , . Also, the global minimum occurs at the infection free equilibrium . Further, function L, along the solutions of system (4), satisfies:
hence, Further collecting terms, we have:
Since and , we have and if and only if and simultaneously.
Therefore, the largest compact invariant subset of the set
is the singleton . By the Lasalle invariance principle  , the infection-free equilibrium is globally asymptotically stable if . We have seen previously that if , at least one of the eigenvalues of the Jacobian matrix evaluated at has a positive real part. Therefore, the infection-free equilibrium is unstable when . This completes the proof of the theorem.
Remark 3. The Lyapunov function defined in the proof of theorem 11 has been obtained following the general form giving by Korobonikov    for the dynamic virus fondamental model.
4.2. Case of Infected Equilibrium Point
Remark 4. The infected equilibrium point satisfies:
Now we are stating and demonstrating one of the most important results of this work.
Theorem 12. Suppose that , and . Then the infected steady state during therapy of model (4) is globally asymptotically stable as soon as it exists.
Proof. Consider the Lyapunov function defined by:
Let us show that and if and only if , , simultaneously.
The time derivative of L along the trajectories of system (1) is:
Collecting terms, and canceling identical terms with opposite signs, yields:
Reporting equalities (9), (10) and (11) of the remark 4 into (12), we have:
By hypothesis, this leads to:
since the geometric mean is less than or equal to the arithmetic mean.
It should be noted that and holds if and only if take the steady states values . Therefore, By the Lasalle invariance principle  , the infected equilibrium point is globally asymptotically stable. This completes the proof of this theorem. ,
5. Concluding Remark
To understand the dynamics of HCV infection and its infectious processes, mathematical models are present as an important and unavoidable tool. Global stability analysis has been done, by the technique of Lyapunov, to the model of HCV infection with proliferation cell and spontaneous healing, for revealing significant information for making good decision for the fighting against hepatitis C. This work is a starting point to many interesting other future investigations. We plan to extend our study by focusing on more realistic models such as: 1) mathematical models with delay which involve delay ordinary differential equations. 2) mathematical models taking into account space which involve Partial differential equations. 3) mathematical models taking into account random phenomena which evolve stochastic differential equations. We also plan to focus on others methods of studying global stability like the geometric method that can provide results with fewer hypotheses on mathematical model (4).
We thank the Editor and the referee for their comments. We are grateful to Professor Alan Rendall for valuable and tremendous discussions about this paper. We wish to thank him for introducing us to Mathematical Biology and to its relationship with Mathematical Analysis. We also thank the Higher Teachers’ Training College of the University of Maroua were this paper were initiated.
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