^{1}

^{*}

^{2}

^{1}

The aim of this work is to analyse the global dynamics of an extended mathematical model of Hepatitis C virus (HCV) infection in vivo with cellular proliferation, spontaneous cure and hepatocyte homeostasis. We firstly prove the existence of local and global solutions of the model and establish some properties of this solution as positivity and asymptotic behaviour. Secondly we show, by the construction of appropriate Lyapunov functions, that the uninfected equilibrium and the unique infected equilibrium of the mathematical model of HCV are globally asymptotically stable respectively when the threshold number
_{} and when
_{}.

According to [

There are too many mathematical models of HCV dynamics amongst those, the original model or model of Newmann [

hepatocyte healing by a non-cytolytic process at the rate q.

The model proposed by Dahari and coworkers [

According to Reluga et al. [

· the variation of the healthy hepatocytes or uninfected hepatocytes, T, is expressed by the following equation:

d T d t = s + r T T ( 1 − T + I T max ) − d T T − ( 1 − η ) β V T + q I . (1)

· the variation of infected hepatocytes, I, is expressed by the following equation:

d I d t = r I I ( 1 − T + I T max ) − d I I − ( 1 − η ) β V T − q I . (2)

· the variation of free virions or virus, V, is expressed by the following equation:

d V d t = ( 1 − ε ) p I − c V . (3)

Thus, the phenomenon described above is governed by the following mathematical model (4), which is a system of three differential autonomous equations:

{ d T d t = s + r T T ( 1 − T + I T max ) − d T T − ( 1 − η ) β V T + q I ; d I d t = r I I ( 1 − T + I T max ) − d I I − ( 1 − η ) β V T − q I ; d V d t = ( 1 − ε ) p I − c V (4)

To analyse the system (4) we need the following initial conditions:

T 0 = T ( t 0 ) , I 0 = I ( t 0 ) and V 0 = V ( t 0 ) where t 0 ∈ [ 0 , + ∞ [ . (5)

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. r I ≤ r T . (b) To have a physiologically realistic model, in an uninfected liver when T max is reached, liver size should no longer increase, i.e. s ≤ d T T max . (c) Infected cells have a higher turnover rate than uninfected cells, i.e. d I ≥ d T . The interpretations and biologically plausible values of other parameters and a further comprehensive survey on the description of (4) is given in [^{4+} 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 R 0 , 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. [

Theorem 1. Let T 0 , I 0 , V 0 ∈ ℝ . There exists t 1 > 0 and functions T , I , V : [ t 0 ; t 1 [ → ℝ continuously differentiable such that ( T , I , V ) is a solution of system (1) satisfying (4).

Theorem 2. Let ( T , I , V ) be a solution of the system (1) over an interval [ t 0 , t 1 [ such that T ( t 0 ) = T 0 , I ( t 0 ) = I 0 et V ( t 0 ) = V 0 .

If T 0 , I 0 , V 0 are positive, then T ( t ) , I ( t ) and V ( t ) are also positive for all t ∈ [ t 0 , t 1 [ .

Proof. We are going to prove by contradiction. so suppose there is t ∈ [ t 0 , t 1 [ such that T ( t ) = 0 or I ( t ) = 0 or V ( t ) = 0 .

Let x = ( x 1 , x 2 , x 3 ) = ( T , I , V )

Let also t ∗ be the smallest of all t in the interval [ t 0 , t 1 [ such that x i ( t ) > 0 , ∀ t ∈ [ t 0 , t ∗ [ , ∀ t ∈ [ t 0 , t ∗ [ and x i ( t ∗ ) = 0 for a certain i.

Then each of the equations of the system (4) can be written x ˙ i = − h i ( x ) + g i ( x ) where g i is a non negative function and h i any function. As a consequence d x i ( t ) d t ≥ − x i f ( x ) and x i ( t ) > 0 , ∀ t ∈ [ t 0 , t ∗ [ . A contradiction. ,

Theorem 3. [

Theorem 4. For any positive solution ( T , I , V ) of system (4), (5) we have:

T ( t ) ≤ T ˜ 0 , I ( t ) ≤ T ˜ 0 and V ( t ) ≤ λ 0

where

T ˜ 0 = T max 2 r I ( ( r T − d T ) 2 + 4 s r I T max + r T − d T ) , λ 0 = λ 0 = max { V 0 , 1 − ε c p T ˜ 0 } .

Proof. Summing Equations (6) and (7), we get:

d d t ( T + I ) = s + ( 1 − T + I T max ) ( r T T − r I I ) − d T T − d I I = s + ( r T − d T ) T + ( r I − d I ) I − T + I T max ( r T T + r I I ) ≤ s + ( r T − d T ) ( T + I ) − T + I T max ( r T T + r I I ) since r T − d T ≥ r I − d I ,

thus d d t ( T + I ) ≤ s + ( r T − d T ) ( T + I ) − r I T max ( T + I ) 2 since r I ≤ r T .

Let N 1 = T + I , a = s > 0 , b = ( r T − d T ) > 0 , d = − r I T max < 0 and let us solve the following equation

d N 1 d t = a + b N 1 + d N 1 2 (6)

Coupled to Equation (6) the initial condition:

N 1 ( t 0 ) = N 1 0 . (7)

The solving of the problem (6), (7) gives for all t ∈ [ t 0 , + ∞ [ ,

N 1 ( t ) = − 1 2 d [ tanh ( 1 2 − 4 a d + b 2 − 1 2 t 0 − 4 a d + b 2 − arctan ( 2 N 1 0 + b − 4 a d + b 2 ) ) − 4 a d + b 2 ] − b 2 d .

As for all x ∈ ℝ , − 1 ≤ tanh x ≤ 1 , it follows that:

N 1 ( t ) ≤ T max 2 r I ( ( r T − d T ) 2 + 4 s r I T max + r T − d T ) .

Let

T ˜ 0 = T max 2 r I ( ( r T − d T ) 2 + 4 s r I T max + r T − d T ) .

Therefore

T + I ≤ T ˜ 0 .

Since T and I are positive I ≤ T + I and T ≤ T + I , so it follows that T ( t ) ≤ T ˜ 0 and I ( t ) ≤ T ˜ 0 .

Equation (3), according to Gromwall inequality, leads to:

V ( t ) ≤ λ 0 .

where

λ 0 = max { V 0 , 1 − ε c p T ˜ 0 } .

This completes the proof of theorem 4. ,

Proposition 5. The uninfected equilibrium point E 0 of the system (4) is given by

E 0 = ( T 0 , 0 , 0 )

where:

T 0 = T max 2 r T ( r T − d T + ( r T − d T ) 2 + 4 r T s T max ) .

We use the method proposed in [

Proposition 6. The expression of the basic reproduction number R 0 associated to the system (4) is given by:

R 0 = r I d I + q ( 1 − T 0 T max ) + ( 1 − θ ) β T 0 p c ( d I + q ) . (8)

where

1 − θ = ( 1 − ε ) ( 1 − η ) .

Remark 1. θ ∈ ] 0,1 [ denotes the overall effectiveness rate of the drug.

Remark 2. Henceforth, we will let δ = d I + q and 1 − θ = ( 1 − ε ) ( 1 − η ) .

Theorem 7. Let ( t 0 , S 0 = ( T 0 , I 0 , V 0 ) ) ∈ R × R + 3 and ( [ t 0 , T [ , S = ( T , I , V ) ) be a maximal solution of the Cauchy problem (1), (4) ( T ∈ ] t 0 , + ∞ [ ). If T ( t 0 ) + I ( t 0 ) ≤ T ˜ 0 and V ( t 0 ) ≤ λ 0 then the set:

Ω = { ( T , I , V ) ∈ ℝ ; 0 < T + I ≤ T ˜ 0 ; 0 < V ≤ λ 0 } ,

where:

T ˜ 0 = T max 2 r I ( ( r T − d T ) 2 + 4 s r I T max + r T − d T ) and λ 0 max { V 0 , 1 − ε c p T ˜ 0 } ,

is a positively invariant set by system (4).

When it exists, the infected equilibrium point is given by: E ∗ = ( T ∗ , I ∗ , V ∗ ) where T ∗ , I ∗ and V ∗ are positive constants that we are going to determine.

Lemma 1. [

s + q T max r I ( r I − δ ) > 0.

Lemma 2. [

T ∗ = 1 2 ( − D H + ( D H ) 2 + F + 4 s T max r T H )

where:

D = A T max ( 1 r T ( 1 + d T + q A ) − δ r I ( 1 r T + 1 A ) − q r T r I ) ;

F = 4 A q T max 2 H 2 r T 2 r I 2 ( A ( δ − r I ) − d I ( r I − r T ) − r I ( q − r I − r T ) + r T q ) ;

and

H = A 2 r I r T + A r I − A r T ; A = ( 1 − θ ) β p T max c

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 E ∗ = ( T ∗ , I ∗ , V ∗ ) if and only if R 0 > 1 , where

T ∗ = 1 2 ( − D H + ( D H ) 2 + F + 4 s T max r T H ) ,

I ∗ = T ∗ ( A r I − 1 ) + T max ( 1 − δ r I ) ,

V ∗ = ( 1 − ε ) p I ∗ c ;

When R 0 ≤ 1 the unique equilibrium is the uninfected equilibrium point or the infection-free steady state E 0 = ( T 0 , 0 , 0 ) .

Theorem 9. The infection-free steady state E 0 = ( T 0 , 0 , 0 ) of model (4) is locally asymptotically stable if R 0 ≤ 1 and unstable if R 0 > 1 .

Proof. See the appendice of [

We start this section by this lemma where the proof can be found in [

Lemma 3 The characteristic equation of the Jacobian matrix J ( E ∗ ) of the system (4) at E ∗ is given by the following cubic equation:

λ 3 + A 1 λ 2 + A 2 λ + A 3 = 0 ;

where:

A 1 = c + s T ∗ + r T T ∗ + r I I ∗ + A T ∗ T max + q I ∗ T ∗ ,

A 2 = c s T ∗ + c r T T ∗ + s A + c r I I ∗ T max + q I ∗ T ∗ ( r I − δ ) + s r I I ∗ T ∗ T max + r T A T ∗ ( T ∗ + I ∗ ) T max 2 + c q I ∗ T ∗ + q I ∗ T max ,

A 3 = c s r I I ∗ T ∗ T max + c A 2 I ∗ T ∗ T max 2 − c A r I I ∗ T ∗ T max 2 + c A r T I ∗ T ∗ T max 2 + q c I ∗ T ∗ ( r I − δ ) .

Proof. See [

Now let:

Δ 2 = | A 1 1 A 3 A 2 |

According to lemma 3 combined with the Routh-Hurwitz criterion [

Theorem 10. For model (4), when R 0 > 1 is valid, the unique endemic equilibrium E ∗ is locally asymptotically stable if Δ 2 > 0 and unstable if Δ 2 < 0 .

Especially, we have:

Corollary 1. The infected steady state during the therapy E ∗ of the model (4) is locally asymptotically stable if R 0 > 1 and unstable if R 0 > 1 .

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 [

Theorem 11. The infection-free steady state E 0 = ( T 0 , 0 , 0 ) of the model (4) is globally asymptotically stable if the basic reproduction number R 0 < 1 − q δ and unstable if R 0 > 1 − q δ .

Proof. Consider the Lyapunov function:

L ( T , I , V ) = T − T 0 − T 0 ln T T 0 + I + ( 1 − η ) β T 0 c V .

L is defined, continuous and positive definite for all T > 0 , I > 0 , V > 0 . Also, the global minimum L = 0 occurs at the infection free equilibrium E 0 . Further, function L, along the solutions of system (4), satisfies:

d L d t = ∂ L ∂ T d T d t + ∂ L ∂ I d I d t + ∂ L ∂ V d V d t = ( 1 − T 0 T ) T ˙ + I ˙ + ( 1 − η ) β T 0 c V ˙ = ( T − T 0 ) T ˙ T + I ˙ + ( 1 − η ) β T 0 c V ˙ = ( T − T 0 ) ( s T + r T − r T ( T + I ) T max − d T − ( 1 − η ) β V + q I T ) + ( 1 − η ) β V T + r I I ( 1 − T + I T max ) − δ I + 1 − θ c β p T 0 T − ( 1 − η ) β T 0 V

= ( T − T 0 ) ( s T + r T − d T − r T ( T + I ) T max + q I T ) − T ( 1 − η ) β V + T 0 ( 1 − η ) β V + ( 1 − η ) β V T + r I I ( 1 − T + I T max ) − δ I + 1 − θ c β p T 0 I − ( 1 − η ) β T 0 V ,

i.e.

d L d t = ( T − T 0 ) ( s T + r T − d T − r T ( T + I ) T max ) + q I − q I T 0 T + r I I ( 1 − T + I T max ) + ( 1 − θ c β p T 0 − δ ) I ;

yet

r T − d T = r T 0 T max − s T 0 ;

hence, Further collecting terms, we have:

d L d t = ( T − T 0 ) ( s T + r T 0 T max − s T 0 − r T ( T + I ) T max ) + r I I ( 1 − T + I T max ) + ( 1 − θ c β p T 0 − δ ) I = ( T − T 0 ) ( − s T T 0 ( T − T 0 ) − r T T max ( T − T 0 ) − r T I T max ) + r I I ( 1 − T + I T max ) + ( 1 − θ c β p T 0 − δ ) I + q I − q I T 0 T

= − s T T 0 ( T − T 0 ) 2 − r T T max ( ( T − T 0 ) 2 + ( T − T 0 ) I ) + r I I − r I I T T max − r I I 2 T max + ( 1 − θ c β p T 0 − δ ) I + q I − q I T 0 T

= − s T T 0 ( T − T 0 ) 2 − r T T max ( ( T − T 0 ) 2 + ( T − T 0 ) I + r I I T r T + r I I 2 r T ) + r I I + ( 1 − θ c β p T 0 − δ ) I + q I − q I T 0 T

= − s T T 0 ( T − T 0 ) 2 − r T T max [ ( T − T 0 ) 2 + ( T − T 0 ) I + r I I T r T + r I I 2 r T + r I r T I T 0 − r I r T I T 0 ] + r I I + ( 1 − θ c β p T 0 − δ ) I + q I − q I T 0 T = − s T T 0 ( T − T 0 ) 2 − r T T max ( ( T − T 0 ) 2 + ( T − T 0 ) I + r I I T r T ( T − T 0 ) + r I I 2 r T + r I r T I T 0 ) + r I I + ( 1 − θ c β p T 0 − δ ) I + q I − q I T 0 T

= − s T T 0 ( T − T 0 ) 2 − r T T max ( T + I − T 0 ) ( T + r I r T I − T 0 ) − r I T max I T 0 + r I I + ( 1 − θ c β p T 0 − δ ) I + q I − q I T 0 T = − s T T 0 ( T − T 0 ) 2 − r T T max ( T + I − T 0 ) ( T + r I r T I − T 0 ) + δ I ( r I δ − r I T 0 δ T max 1 − θ c δ β p T 0 − 1 ) + q I − q I T 0 T .

Furthermore,

R 0 = 1 − θ c δ β p T 0 + r I δ ( 1 − T 0 T max ) ,

hence

d L d t = − s T T 0 ( T − T 0 ) 2 − r T T max ( T + I − T 0 ) ( T + r I r T I − T 0 ) − q I T 0 T + δ I ( R 0 − 1 ) + q I = − s T T 0 ( T − T 0 ) 2 − r T T max ( T + I − T 0 ) ( T + r I r T I − T 0 ) − q I T 0 T + δ I ( R 0 − 1 + q δ ) .

Since r I ≤ r T and R 0 < 1 − q δ , we have d L d t ≤ 0 and d L d t = 0 if and only if T = T 0 and I = 0 simultaneously.

Therefore, the largest compact invariant subset of the set

M = { ( T , I , V ) ∈ Ω : d L d t = 0 }

is the singleton { E 0 } . By the Lasalle invariance principle [

Remark 3. The Lyapunov function defined in the proof of theorem 11 has been obtained following the general form giving by Korobonikov [

We recall:

Remark 4. The infected equilibrium point E ∗ = ( T ∗ , I ∗ , V ∗ ) satisfies:

r T − d T = − s T ∗ + ( 1 − η ) β V ∗ + r T T max ( T ∗ + I ∗ ) − q I ∗ T ∗ , (9)

r I − δ = − ( 1 − η ) β V ∗ T ∗ I ∗ + r I T max ( T ∗ + I ∗ ) , (10)

c = ( 1 − ε ) p I ∗ V ∗ . (11)

Now we are stating and demonstrating one of the most important results of this work.

Theorem 12. Suppose that r I = r T , s = d T T max and δ = d T . Then the infected steady state during therapy E ∗ of model (4) is globally asymptotically stable as soon as it exists.

Proof. Consider the Lyapunov function defined by:

L ( T , I , V ) = T − T ∗ − T ∗ ln T T ∗ + I − I ∗ − I ∗ ln I I ∗ + ( 1 − η ) β T ∗ V ∗ ( 1 − ε ) p I ∗ ( V − V ∗ − V ∗ ln V V ∗ ) .

Let us show that d L d t ≤ 0 and d L d t = 0 if and only if T = T ∗ , I = I ∗ , V = V ∗ simultaneously.

The time derivative of L along the trajectories of system (1) is:

d L d t = ∂ L ∂ T d T d t + ∂ L ∂ I d I d t + ∂ L ∂ V d V d t = ( 1 − T ∗ T ) T ˙ + ( 1 − I ∗ I ) I ˙ + ( 1 − η ) β T ∗ V ∗ ( 1 − ε ) p I ∗ ( 1 − V ∗ V ) V ˙ = ( T − T ∗ ) T ˙ T + ( I − I ∗ ) I ˙ I + ( 1 − η ) β T ∗ V ∗ ( 1 − ε ) p I ∗ ( V − V ∗ ) V ˙ V .

Collecting terms, and canceling identical terms with opposite signs, yields:

d L d t = ( T − T ∗ ) ( s T + r T − r T ( T + I ) T max − d T − ( 1 − η ) β V + q I T ) + ( 1 − η ) β T ∗ V ∗ ( 1 − ε ) p I ∗ ( V − V ∗ V ) ( ( 1 − ε ) p I − c V ) + ( I − I ∗ ) ( ( 1 − η ) β V T I + r I ( 1 − T + I T max ) − δ ) . (12)

Reporting equalities (9), (10) and (11) of the remark 4 into (12), we have:

d L d t = ( T − T ∗ ) [ s T − s T ∗ + ( 1 − η ) β V ∗ + r T T max ( T ∗ + I ∗ ) − q I ∗ T ∗ − r T ( T + I ) T max − ( 1 − η ) β V + q I T ] + ( I − I ∗ ) [ ( 1 − η ) β V T I − r I T T max − r I I T max − ( 1 − η ) β V ∗ T ∗ I ∗ + r I T max ( T ∗ + I ∗ ) ] + ( 1 − η ) β + T ∗ V ∗ ( 1 − ε ) p I ∗ ( V − V ∗ V ) ( ( 1 − ε ) p I − ( 1 − ε ) p I ∗ V ∗ V )

= − s T T ∗ ( T − T ∗ ) 2 − r T T max ( T − T ∗ ) 2 − r T T max ( T − T ∗ ) ( I − I ∗ ) − ( 1 − η ) β ( T − T ∗ ) ( V − V ∗ ) + ( 1 − η ) β [ ( V T I − V ∗ T ∗ I ∗ ) ( I − I ∗ ) + T ∗ V ∗ ( 1 − ε ) p I ∗ V − V ∗ V ( ( 1 − ε ) p I − ( 1 − ε ) V ∗ p I ∗ V ) ] − q I ∗ T ∗ ( T − T ∗ ) + q I T ( T − T ∗ ) − r I T max ( T − T ∗ ) ( I − I ∗ ) − r I T max ( I − I ∗ ) 2

= − s T T ∗ ( T − T ∗ ) 2 − r T T max ( T − T ∗ ) 2 − r T + r I T max ( T − T ∗ ) ( I − I ∗ ) − r I T max ( I − I ∗ ) 2 + ( 1 − η ) β [ ( V T I − V ∗ T ∗ I ∗ ) ( I − I ∗ ) − ( T − T ∗ ) ( V − V ∗ ) + T ∗ V ∗ ( 1 − ε ) p I ∗ V − V ∗ V ( ( 1 − ε ) p I − 1 − ε V ∗ p I ∗ V ) ] − q 1 T T ∗ ( T 2 I ∗ + ( T ∗ ) 2 I − T ∗ T I − T ∗ T I ∗ )

= − s T T ∗ ( T − T ∗ ) 2 − 1 T max ( r T T + r I I − r T T ∗ − r I I ∗ ) ( T + I − T ∗ − I ∗ ) + ( 1 − η ) β T ∗ V ∗ ( V T V ∗ T ∗ − V T I ∗ I V ∗ T ∗ − V ∗ T ∗ I I ∗ V ∗ T ∗ + V ∗ T ∗ I ∗ V ∗ T ∗ I ∗ − T V T ∗ V ∗ + T V ∗ T ∗ V ∗ + T ∗ V T ∗ V ∗ − T ∗ V ∗ T ∗ V ∗ + I V I ∗ V − I ∗ V 2 I ∗ V V ∗ − V ∗ I I ∗ V + V ∗ I ∗ V V ∗ I ∗ V ) − q 1 T T ∗ ( ( T − T ∗ ) 2 I ∗ + ( T ∗ ) 2 ( I − I ∗ ) + T T ∗ ( I ∗ − I ) )

= − s T T ∗ ( T − T ∗ ) 2 − 1 T max ( r T T + r I I − r T T ∗ − r I I ∗ ) ( T + I − T ∗ − I ∗ ) + ( 1 − η ) β T ∗ V ∗ ( 1 + T T ∗ − V T I ∗ I V ∗ T ∗ − V ∗ I I ∗ V ) − q 1 T T ∗ ( ( T − T ∗ ) 2 I ∗ + T ∗ ( I − I ∗ ) ( T ∗ − T ) ) .

Note that

1 + T T ∗ − V T I ∗ I V ∗ T ∗ − V ∗ I I ∗ V = ( 3 − T ∗ T − V T I ∗ I V ∗ T ∗ − V ∗ I I ∗ V ) + ( T T ∗ + T ∗ T − 2 )

and

( T T ∗ + T ∗ T − 2 ) = ( T − T ∗ ) 2 T T ∗ .

Recall that:

s = ( 1 − η ) β T ∗ V ∗ + ( d T − r T + r T T ∗ + I ∗ T max ) T ∗ − q I ∗ .

furthermore,

T ∗ + I ∗ = T max ;

hence,

s = ( 1 − η ) β T ∗ V ∗ + d T T ∗ − q I ∗ .

By hypothesis, r T = r I this leads to:

d L d t = ( − d T T + q I ∗ T T ∗ − ( 1 − η ) β V ∗ T ) ( T − T ∗ ) 2 − r T T max ( T + I − T ∗ − I ∗ ) 2 + ( 1 − η ) β V ∗ T ∗ ( 3 − T ∗ T − V T I ∗ I V ∗ T ∗ − V ∗ I I ∗ V ) + ( 1 − η ) β V ∗ T ∗ ( T − T ∗ ) 2 T T ∗ − q 1 T T ∗ ( ( T − T ∗ ) 2 I ∗ + T ∗ ( I − I ∗ ) ( T ∗ − T ) ) = − d T T ( T − T ∗ ) 2 − r T T max ( T + I − T ∗ − I ∗ ) 2 − q 1 T ( I − I ∗ ) ( T ∗ − T ) + ( 1 − η ) β V ∗ T ∗ ( 3 − T ∗ T − V T I ∗ I V ∗ T ∗ − V ∗ I I ∗ V )

= − d T T ( T − T ∗ ) 2 − r T T max ( T + I − T ∗ − I ∗ ) 2 − q 1 T ( I − I ∗ ) ( T ∗ − T ) + ( 1 − η ) β V ∗ T ∗ ( 3 − ( T ∗ ) 2 I I ∗ V V ∗ + ( I ∗ V T ) 2 + ( I V ∗ ) 2 T T ∗ T T ∗ I I ∗ V V ∗ ) = − d T T ( T − T ∗ ) 2 − r T T max ( T + I − T ∗ − I ∗ ) 2 − q 1 T ( I − I ∗ ) ( T ∗ − T ) + 3 ( 1 − η ) β V ∗ T ∗ T T ∗ I I ∗ V V ∗ ( T T ∗ I I ∗ V V ∗ − 1 3 ( ( T ∗ ) 2 I I ∗ V V ∗ + ( I ∗ V T ) 2 + ( I V ∗ ) 2 T T ∗ ) ) .

Yet

1 3 ( ( T ∗ ) 2 I I ∗ V V ∗ + ( I ∗ V T ) 2 + ( I V ∗ ) 2 T T ∗ ) ≥ T T ∗ I I ∗ V V ∗

since the geometric mean is less than or equal to the arithmetic mean.

It should be noted that d L d t ≤ 0 and d L 2 d t = 0 holds if and only if ( T , I , V ) take the steady states values ( T * , I * , V * ) . Therefore, By the Lasalle invariance principle [

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.

The authors declare no conflicts of interest regarding the publication of this paper.

Nangue, A., Fokoue, C. and Poumeni, R. (2019) The Global Stability Analysis of a Mathematical Cellular Model of Hepatitis C Virus Infection with Non-Cytolytic Process. Journal of Applied Mathematics and Physics, 7, 1531-1546. https://doi.org/10.4236/jamp.2019.77104