There are too many mathematical models of HCV dynamics amongst those, the original model or model of Newmann [4] and its extended models as those in [5] [9] 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. [10]. 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

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

The model proposed by Dahari and coworkers [11] [13] expands on the standard HCV viral-dynamic model [4] of infection and clearance by incorporating density-dependent proliferation and death. Uninfected hepatocytes or noninfected hepatocytes, T, are infected at a rate β per free virus per hepatocyte. Infected cells, I, produce free virus at rate p per cell but also die with rate d I . 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 d T . Both infected and uninfected hepatocytes proliferate logistically with maximum rates r I and r T , respectively, as long as the total number of hepatocytes is less than T max . 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 η and the viral production rate by a fraction ε . It should be noted that η and ε are parameters which values are non-negative and less than one. A further comprehensive survey on the description of the model is given in [10] [14] [15]. Given the meanings of η and β , the term ( 1 − η ) β V T represents the mass action principle; β V T is the rate of infection of healthy hepatocytes T by interaction with virus V.

According to Reluga et al. [10] 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:

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 [10]. 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 [12] [16] [17] , 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 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. [18] in Section 2. In Section 3 we theoretically analyze with some assumptions the global stability of the model by constructing appropriate Lyapounov’s functions.

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. [18] The solutions of the Cauchy problem (4), (5), with positive initial data, exist globally in time in the future that is on [ t 0 , + ∞ [ .

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 [19] [20] to compute the basic reproduction number R 0 .

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. [18] T ∗ exists if and only if

s + q T max r I ( r I − δ ) > 0.

Lemma 2. [18] When it exists, T ∗ is defined by:

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 [18]. ,

We start this section by this lemma where the proof can be found in [18].

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 [18]. ,

Now let:

Δ 2 = | A 1 1 A 3 A 2 |

According to lemma 3 combined with the Routh-Hurwitz criterion [21] , we have the following results where the proofs can be found in [18].

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 .

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 [25] , the infection-free equilibrium is globally asymptotically stable if R 0 < 1 − q δ . We have seen previously that if R 0 > 1 , at least one of the eigenvalues of the Jacobian matrix evaluated at E 0 has a positive real part. Therefore, the infection-free equilibrium E 0 is unstable when R 0 > 1 . 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 [22] [23] [24] for the dynamic virus fondamental model.

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 [22] , the infected equilibrium point E * is globally asymptotically stable. This completes the proof of this theorem. ,