Mathematics in Applied Sciences and Engineering https://ojs.lib.uwo.ca/mase
Online First, pp.1-27 https://doi.org/10.5206/mase/14918

GLOBAL STABILITY OF A HCV DYNAMICS MODEL WITH CELLULAR

PROLIFERATION AND DELAY

ALEXIS NANGUE, ARMEL WILLY FOKAM TACTEU, AND AYOUBA GUEDLAI

Abstract. In this work, we propose and investigate a delay cell population model of hepatitis C virus

(HCV) infection with cellular proliferation, absorption effect, and a non-linear incidence function. First

of all, we prove the existence of the local solutions of the model, followed by the existence of the global

solutions and the positivity. Moreover, we determine the infection free equilibrium and the basic

reproduction rate R0, which is a threshold number in mathematical epidemiology. Then we prove the
existence and uniqueness of the infection persist equilibrium. We also proceed to study the local and

global stability of this equilibrium. We show that if R0 < 1, the infection free equilibrium is globally
asymptotically stable, which means that the disease will disappear and if R0 > 1, we have a unique
infection persist equilibrium that is globally asymptotically stable under some conditions. Finally, we

perform some numerical simulations to illustrate the obtained theoretical results.

1. Introduction

Hepatitis C is a disease caused by HCV, which is a virus that attacks liver cells, causing them to

become inflamed. The World Health Organization (WHO) estimates in [29] that globally, 71 million

people are chronic carriers of hepatitis C infection and that in 2016, around 399,000 people died from it,

most often as a result of cirrhosis or hepatocellular carcinoma (primary liver cancer). This organization

is leading actions to reduce the number of new cases of viral hepatitis by 90% and the number of deaths

associated with this disease by 65% by 2030. Due to the severity of hepatitis C infection, it is necessary

to develop the tools that help to understand this disease. It is for this reason that several mathematical

models have been developed to better understand the dynamics of the hepatitis C virus within the liver

itself [1, 2, 7, 18, 20, 19, 17]. Eric Avila Vales et al. in [2] studied an intra-host delay model, which is a

pioneer work which inspired the work in the present paper. Indeed, we note that in their model, the loss

of pathogens due to the absorption that we can find in [10, 30, 23] in uninfected cells is ignored. When

a pathogen enters an uninfected cell, the number of pathogens in the blood decreases by one. This is

called the absorption effect (see, for example, [4]). To place the model on a more solid biological basis,

we use the saturated infection rate (saturated infection rate found in [25, 30]) and cellular proliferation

effect. These three aspects added to the model studied in [2] make the model we are studying a more

realistic model in the biological sense. We note that in most intra-host models of virus dynamics, the

loss of pathogens due to the absorption into uninfected cells is ignored. In biology, it is natural that,

when pathogens are absorbed into susceptible cells, the numbers of pathogens are reduced in the blood

volume which : this phenomenon is called absorption effect. Hence, some researchers (see, for example,

[4, 16, 30] and the references therein) have included the absorption effect into their models. In several

of models with or without delay, the process of cellular infection by free virus particles is typically

modelled by the mass action principle, that is to say, the infection rate is assumed to occur at a rate

Received by the editors 30 April 2022; accepted 25 August 2022; published online September 10, 2022.

2020 Mathematics Subject Classification. 34A05, 34A06, 34A34, 34D20, 34D23, 37N25.

Key words and phrases. absorption effect, cellular proliferation, delay, global stability, Hepatitis C virus, wellposedness.

1

https://ojs.lib.uwo.ca/mase
https://dx.doi.org/https://doi.org/10.5206/mase/14918


2 A. NANGUE, A. W. FOKAM TACTEU, AND A. GUEDLAI

proportional to the product of the concentration of virus particles and uninfected target cells. This

principle is insufficient to describe the cellular infection process in detail, and some non-linear infection

rates have been proposed. Li and Ma [14], Song and Neumann [25] considered a virus dynamics model

with Monod functional response bxv
1+αv

. Regoes et al.[21] and Song and Neumann [26] considered a virus

dynamics model with the non-linear infection rates

bx
(v
k

)p
/
(

1 +
(v
k

)p)
and

bxv
q

(1 + αvp)
,

where p, q, k > 0 are constants, respectively. Recently, Huang et al. [12] considered a class of models

of viral infections with a non-linear infection rate and two discrete intracellular delays, and assumed

that the infection rate is given by a general non-linear function of uninfected target cells and free virus

particles F(x,v), which satisfies certain conditions.

The rest of this paper is organized as follows. Section 2 gives a description of the newly constructed

model. Section 3 deals with the existence, the positivity and boundedness of solutions of our model.

In Section 4, the threshold parameter R0(τ) of our DDE model (2.1) is derived and the existence
of the equilibria are discussed in relation to the value of R0(τ). Section 5 and section 6 show the
local and global stability of the infection free equilibrium and infection persist equilibrium respectively.

Additionally, some numerical results are displayed this supports the obtained theoretical results. Finally,

a brief discussion and some possible future ideas are presented in Section 7.

2. Model construction

In this section, motivated by what has been said previously, we propose an HCV infection model

with time delay, absorption effect and monod functional response, taking into account the proliferation

of both exposed cells and HCV infected hepatocytes :


dH(t)

dt
= λ + rHH(t)

(
1 −

H(t) + I(t)

k

)
−µH(t) −

βH(t)V (t)

1 + aV (t)

dI(t)

dt
=
βe−τmH(t− τ)V (t− τ)

1 + aV (t− τ)
+ rII(t)

(
1 −

H(t) + I(t)

k

)
−αI(t)

dV (t)

dt
= ηI(t) −γV (t) −

βH(t)V (t)

1 + aV (t)

(2.1)

with initial conditions

H(θ) = ϕ1(θ) ≥ 0, I(θ) = ϕ2(θ) ≥ 0, V (θ) = ϕ3(θ), −τ ≤ θ ≤ 0, (2.2)

where ϕ = (ϕ1,ϕ2,ϕ3) ∈C([−τ, 0],R3+) which is the Banach space of continuous functions
ϕ : [−τ, 0] −→R3+ =

{
(H,I,V ) ∈ R3 |H ≥ 0, I ≥ 0, V ≥ 0

}
with norm ‖ϕ‖ = sup

−τ≤θ≤0
{|ϕ1|, |ϕ2|, |ϕ3|}.

The model (2.1) is a modification of model (2) studied in [2] and later in [1]. The features of the

latter is as follows : H(t), I(t) and V (t) denote the concentration of uninfected hepatocytes (or target

cells), infected hepatocytes and free virus, respectively. All parameters are assumed to be positive

constants. Here, target cells are generated at a constant rate λ and die at a rate µ per uninfected

hepatocyte. These hepatocytes are infected at rate β per target cell per virion. Infected cells die at

rate α per cell by cytopathic effects. Because of the viral burden on the virus-infected cells, we assume

that µ ≥ α. In other words, we assume that the average life-time of infected cells
(

1
α

)
is shorter

than the average life-time of uninfected cells
(

1
α

)
[5, 24]. The proliferation of infected and uninfected

hepatocytes due to mitotic division obeys to a logistic growth. The mitotic proliferation of uninfected

hepatocytes is described by rHH(t) [1 − (H(t) + I(t))/k], and mitotic transmission occurs at a rate
rII(t) [1 − (H(t) + I(t))/k], which is the mitotic division of infected hepatocytes. It should be mentioned



GLOBAL STABILITY OF A HCV DYNAMICS MODEL WITH CELLULAR PROLIFERATION AND DELAY 3

that the model (2) in [2] has rI = rH. Uninfected and infected hepatocytes grow at the constant rate

rH and rI respectively , and k is the maximal number of total hepatocyte population proliferation.

Infected cells produce virions at an average rate µ per infected cell, and γ is the clearance rate of virus

particles. The population of virions decreases due to the infection at a rate βH(t)V (t)/[1 + aV (t)] :

this is absorption phenomenon. It should be noted that according to [20], to have a physiologically

realistic model, in an uninfected liver when k is reached, liver size should no longer increase i.e. λ ≤ µk.
We assume that the contacts between viruses and uninfected target cells are given by an infection rate

βH(t)V (t)/[1 + aV (t)], it is reasonable for us to assume that the infection has a maximal rate of β
a

.

The parameter τ accounts for the time between viral entry into a target cell and the production of new

virus particles. The recruitment of virus producing cells at time t is given by the number of cells that

were newly infected at time t − τ and are still alive at time t. Here, m is assumed to be a constant
death rate for infected but not yet virus-producing hepatocytes. Thus, the probability of surviving the

time period from t− τ to t is e−mτ .

3. Wellposedness

In this section, we show that our model (2.1) is mathematically and biologically well posed.

Theorem 3.1. All solutions of system (2.1) with initial conditions (2.2), where H(0) > 0, I(0) > 0,

V (0) > 0, are positive and under the initial conditions (2.2), the solution (H(t), I(t), V(t)) of model

(2.1) is existent and unique. Moreover, for any positive solution (H(t), I(t), V(t)) of system (2.1)

we have : lim sup
t→+∞

H(t) ≤ H0 =
[
(rH −µ) +

(
(rH −µ)2 + 4rHλk

)1/2] k
2rH

, the existence of constants

MI > 0 and MV > 0 such that I(t) < MI, V (t) < MV .

Proof. Firstly, we deal with the fact that R3+ is positively invariant with respect to the dde model

system (2.1). We prove the positivity by contradiction. Suppose H(t) is not always positive. Then, let

t0 > 0 be the first time such that H(t0) = 0. From the first equation of system (2.1),
dH(t0)
dt

= λ > 0.

By our hypothesis this means that H(t) < 0 for t ∈ (t − ε,t0), where ε is an arbitrary small positive
constant. Implying that exist t′0 < t0 such that H(t

′
0) = 0 : this is a contradiction because we take t0

as the first value which H(t0) = 0 . It follows that T(t) is always positive. We now show that I(t) > 0

for all t > 0. By considering the second equation of system (2.1), one has :

I(t) = I(0) exp

(
−αt +

∫ t
0
rI

(
1 −

H(u) + I(u)

k

)
du

)
+ exp

[
−αt +

∫ t
0
rI

(
1 −

H(u) + I(u)

k

)
du

]
×
∫ t

0

[
βe−τmH(u− τ)V (u− τ)

1 + aV (u− τ)
exp

(
−αu +

∫u
0
rI

(
1 −

H(θ) + I(θ)

k

)
dθ

)]
du.

(3.1)

u − τ ∈ [−τ, 0] since u ∈ [0,τ] and furthermore by assumption for t ∈ [−τ, 0] we have : H(t) >
0, I(t) > 0, V (t) > 0, H(0) > 0, I(0) > 0 and V (0) > 0. We deduce that∫ t

0

[
βe−τmH(u− τ)V (u− τ)

1 + aV (u− τ)
exp

(
−αu +

∫ u
0

rI

(
1 −

H(θ) + I(θ)

k

)
dθ

)]
du

is positive for all t ∈ [0,τ] and hence I(t) > 0 for all t ∈ [0,τ]. Let us show by recurrence on n that
I(t) > 0 for all t ∈ [nτ, (n + 1)τ]. Let Pn, the proposition : ∀n ∈ N I(t) > 0 ∀t ∈ [nτ, (n + 1)τ].

a) For n = 0, P0 is verified.

b) Suppose that for n ∈ N, I(t) > 0 for all t ∈ [nτ, (n + 1)τ] and show that I(t) > 0 for all
t ∈ [(n+1)τ, (n+2)τ]. According to (3.1), for u ∈ [(n+1)τ, (n+2)τ], we have u−τ ∈ [nτ, (n+1)τ].



4 A. NANGUE, A. W. FOKAM TACTEU, AND A. GUEDLAI

By recurrence assumption, the term∫ t
0

[
βe−τmH(u− τ)V (u− τ)

1 + aV (u− τ)
exp

(
−αu +

∫ u
0

rI

(
1 −

H(θ) + I(θ)

k

)
dθ

)]
du

is positive. Therefore I is positive on [(n + 1)τ, (n + 2)τ] and consequently Pn+1 is true. Hence

I(t) > 0 for all t > 0.

We finally show that V (t) > 0 for all t > 0. Since I(t) > 0 for all t > 0 and η > 0, we have :

dV (t)

dt
≥
(
−γ −

βH(t)

1 + aV (t)

)
V (t).

Integrating the previous expression on [0, t], we obtain :

V (t) ≥ V0 exp
(∫ t

0

(
−γ −

βH(u)

1 + aV (u)

)
du

)
,

which ensures that V (t) > 0 for all t > 0. Thus deduce that R3+ is positively invariant with respect to

model (2.1).

Secondly, the existence and uniqueness of the solution (H(t),I(t),V (t)) can be easily proved by using

the following theorems(Theorem 2.1 and Theorem 2.2 Page 19 in [13]).

Finally, let us show that the solution (H(t),I(t),V (t)) is uniformly bounded. For any positive

solution of system (2.1), we have

lim sup
t→+∞

H(t) ≤ H0 =

[
(rH −µ) +

(
(rH −µ)2 + 4

rHλ

k

)1/2]
k

2rH

since the first equation of (2.1) yields

dH(t)

dt
≤ λ− (µ−rH)H(t) −

rH
k
H2(t).

Then there is a t1 > 0 such that for any sufficiently small ε > 0 one has H(t) < H0 + ε for t > t1. Now

let for t ≥ 0, define U(t) as below

U(t) = H(t) + I(t) + β

∫ t
t−τ

e−m(t−s)
H(s)V (s)

1 + aV (s)
ds. (3.2)

Taking the derivation of the previous expression along the solution, collecting and simplifying some

terms, we obtain, for t ≥ 0,

dU(t)

dt
= λ + ((H(t) + I(t)) (rH + rI))

(
1 −

H(t) + I(t)

k

)
−µH(t) −αI(t)

−(rHI(t) + rIH(t))
(

1 −
H(t) + I(t)

k

)
−mβ

∫ t
t−τ

e−m(t−s)
H(s)V (s)

1 + aV (s)
ds

= λ +

(
rH + rI

4

)
k −

(
rH + rI

k

)(
H(t) + I(t) −

k

2

)2
−µH(t) −αI(t)

−(rHI(t) + rIH(t))
(

1 −
H(t) + I(t)

k

)
−mβ

∫ t
t−τ

e−m(t−s)
H(s)V (s)

1 + aV (s)
ds,

≤ λ +
(
rH + rI

4

)
k −µH(t) −αI(t) −mβ

∫ t
t−τ

e−m(t−s)
H(s)V (s)

1 + aV (s)
ds

≤ λ +
(
rH + rI

4

)
k − bU(t),



GLOBAL STABILITY OF A HCV DYNAMICS MODEL WITH CELLULAR PROLIFERATION AND DELAY 5

where b = min{µ,α,m}. It follows that

lim sup
t→+∞

U(t) ≤
4λ + (rH + rI)k

4b
,

that is, there exist t2 > 0 and M1 > 0 such that U(t) < M1 for t > t2. Then I(t) has an upper bound

MI. It follows from the third equation of system (2.1), that, for t ≥ 0

dV (t)

dt
≤ ηI(t) −γV (t),

from which we have that

lim sup
t−→+∞

V (t) ≤
η

γ

4λ + (rH + rI)k

4b
.

Then there exists MV > 0 such that V (t) < MV for t > t2. This completes the proof of Theorem 3.1

. �

Remark 3.1. From Theorem 3.1, one has that the solution of initial value problem (2.1), (2.2) enters

the region

Γ =
{

(H,I,V ) ∈ R3+|0 ≤ H(t) ≤ H0, 0 ≤ I(t) ≤ MI, 0 ≤ V (t) ≤ MV
}
.

Hence Γ, of biological interest, positively-invariant under the flow induced by the problem (2.1), (2.2).

Now, we determine the equilibria of model (2.1).

4. Equilibria and basic reproduction number

4.1. Infection free equilibrium and basic reproduction number. Model (2.1) always has the

uninfected equilibrium E0 = (H0, 0, 0).

By using similar techniques in [9] and [27], we obtain the basic reproduction number (spectral radius

of next generation matrix) for model (2.1) as

R0(τ) =
1

α

[
rI

(
1 −

H0
k

)
+
ηβe−τmH0
γ + βH0

]
.

Here, R0(τ) is the average number infected cells produced by one infected hepatocyte after introducing
infected hepatocyte into fully susceptible hepatocyte population, that plays a crucial role in the dy-

namics. Generally, the basic reproduction number R0(τ) helps us to decide whether viruses clean out
with time or not. E0 = (H0, 0, 0) is the trivial equilibrium of model (2.1).

To find the other equilibrium of (2.1), we solve the following algebraic system :


λ + rHH

(
1 −

H + I

k

)
−µH −

βHV

1 + aV
= 0, (4.1)

βe−τmHV

1 + aV
+ rII

(
1 −

H + I

k

)
−αI = 0, (4.2)

ηI −γV −
βHV

1 + aV
= 0. (4.3)

4.2. Infection persist equilibrium.



6 A. NANGUE, A. W. FOKAM TACTEU, AND A. GUEDLAI

4.2.1. Existence of an infection persist equilibrium.

Proposition 4.1. The system (2.1) possesses an infection persist equilibrium denoted as E1 = (H1,I1,V1)

if R0(τ) > 1.

Proof. From (4.3), one has

I1 =
γ

η
V1 +

βH1V1
η(1 + aV1)

. (4.4)

Reporting (4.4) into (4.1) yields the following second degree algebraic equation in H1 :

−
(
rH
k

+
rHβV1

kη(1 + aV1)

)
H21 +

(
rH −µ−

rH
k

γ

η
V1 −

βV1
1 + aV1

)
H1 + λ = 0. (4.5)

The discriminant of the algebraic equation (4.5) is given by :

∆ =

(
rH −µ−

rH
k

γ

η
V1 −

βV1
1 + aV1

)2
+ 4λ

(
rH
k

+
rHβV1

kη(1 + aV1)

)
> 0. (4.6)

Thus, equation (4.5) has a unique positive root known as :

H1 =

(
rH −µ−

rH
k

γ
η
V1 −

βV1
1 + aV1

)
+
√

∆

2

(
rH
k

+
rHβV1

kη(1 + aV1)

) .
We can once again denote H1 = f(V1). Substituting (4.4) into (4.2), one gets:

βe−τmH1V1
1 + aV1

+ rI

(
γ

η
V1 +

βH1V1
η(1 + aV1)

)(
1 −

H1 +
γ
η
V1 +

βH1V1
η(1+aV1)

k

)
−α

(
γ

η
V1 +

βH1V1
η(1 + aV1)

)
= 0,

which is equivalent to

βe−τmH1
1 + aV1

+ rI

(
γ

η
+

βH1
η(1 + aV1)

)(
1 −

H1 +
γ
η
V1 +

βH1V1
η(1+aV1)

k

)
−α

(
γ

η
+

βH1
η(1 + aV1)

)
= 0,

since V1 > 0. Taking into account the fact that H1 = f(V1) one has

βe−τmf(V1)

1 + aV1
+ rI

(
γ

η
+

βf(V1)

η(1 + aV1)

)1 − f(V1) + γηV1 + βf(V1)V1η(1+aV1)
k


−α(γ

η
+

βf(V1)

η(1 + aV1)

)
= 0.

On the other hand we also have

−
(
rH
k

+
rHβV1

kη(1 + aV1)

)
f(V1)

2
+

(
rH −µ−

rH
k

γ

η
V1 −

βV1
1 + aV1

)
f(V1) + λ = 0. (4.7)

Let

F(V1) =
βe−τmf(V1)

1 + aV1
+ rI

(
γ
η

+
βf(V1)
η(1+aV1)

)1 − f(V1) + γηV1 + βf(V1)V1η(1+aV1)
k




−α
(
γ
η

+
βf(V1)
η(1+aV1)

)
.

(4.8)

Obviously, F is continuous on [0; +∞[. We have, for V1 = 0,

F(0) = βe−τmf(0) + rI

(
γ

η
+
βf(0)

η

)(
1 −

f(0)

k

)
−α

(
γ

η
+
βf(0)

η

)
,



GLOBAL STABILITY OF A HCV DYNAMICS MODEL WITH CELLULAR PROLIFERATION AND DELAY 7

with

f(0) =

(rH −µ) +
√

(rH −µ)
2

+ 4λ
rH
k

2
(rH
k

) = H0.
We finally obtain

F(0) = α

(
γ

η
+
βH0
η

)[
ηβe−τmH0
α(γ + βH0)

+
rI
α

(
1 −

H0
k

)
− 1

]
which is equivalent to :

F(0) = α

(
γ

η
+
βH0
η

)
(R0 − 1) . (4.9)

R0(τ) > 1, implies F(0) > 0. From 0 < H1 = f(V1) ≤ C, with C > 0 according to Theorem 3.1, we

have lim
V1→+∞

βe−τmf(V1)

1 + aV1
= 0 since lim

V1→+∞

βe−τmC

1 + aV1
= 0 and, consequently

lim
V1→+∞

−α
(
γ

η
+

βf(V1)

η(1 + aV1)

)
= −

αγ

η

and

lim
V1→+∞

rI

(
γ

η
+

βf(V1)

η(1 + aV1)

)1 − f(V1) + γηV1 + βf(V1)V1η(1+aV1)
k


 = −∞.

Therefore

lim
V1→+∞

F(V1) = −∞.

The intermediate value theorem ensures the existence of V1 > 0 such that F(V1) = 0. The existence of

V1 also ensures the existence of H1 and I1. Therefore the infection persist equilibrium E1 = (H1,I1,V1)

exists. �

Now let’s take a look at uniqueness.

4.2.2. Uniqueness of the infection persist equilibrium.

Proposition 4.2. Let Λ = [4λ + (rH + rI)k]/4b. If e
−τm > α/η and Λ/k ≤ 1/2 then the infection

persist equilibrium E1 = (H1,I1,V1) is unique when it exists.

Proof. From equation (4.8), one has

F ′(V1) =

(
β

1 + aV1

(
e−τm −

α

η

)
+

rIβ

η(1 + aV1)


1 − f(V1) + γηV1 + βf(V1)V1η(1+aV1)

k


 + A

)
f′(V1)

+
aβf(V1)

(1 + aV1)2

(
α

η
−e−τm

)
−
rI
k

(
γ

η
+

βf(V1)

η(1 + aV1)

)(
γ

η
+

βf(V1)

η(1 + aV1)2

)

−
rIaβf(V1)

η(1 + aV1)2


1 − f(V1) + γηV1 + βf(V1)V1η(1+aV1)2

k


 ,

where

A = −
rI
k

(
γ

η
+

βf(V1)

η(1 + aV1)

)(
1 +

βV1
η(1 + aV1)

)
.

The expression of A can be rewritten in the following form :

A =
rIβ

η(1 + aV1)


1 − f(V1) + γηV1 + βf(V1)V1η(1+aV1)

k


− rIγ

ηk
−

rIβ

η(1 + aV1)
.



8 A. NANGUE, A. W. FOKAM TACTEU, AND A. GUEDLAI

Thus

F ′(V1) =

[
β

1 + aV1

(
e−τm − α

η

)
+ rIβ

η(1+aV1)


1 − 2f(V1) + 2γη V1 + 2βf(V1)V1η(1+aV1)

k




−
rIγ

ηk

]
f′(V1) −

rIaβf(V1)
η(1+aV1)2


1 − f(V1) + γηV1 + βf(V1)V1η(1+aV1)

k




+
aβf(V1)

(1 + aV1)2

(
α
η
−e−τm

)
− rI

k

(
γ

η
+

βf(V1)
η(1+aV1)

)(
γ

η
+

βf(V1)
η(1+aV1)2

)
.

Dividing equation (4.7) by f(V1), we obtain

−
(
rH
k

+
rHβV1

kη(1 + aV1)

)
f(V1) +

(
rH −µ−

rH
k

γ

η
V1 −

βV1
1 + aV1

)
+

λ

f(V1)
= 0. (4.10)

Using the implicit differentiation we get from 4.10 :

f′(V1) =

(
rHγ

ηk
+

β

(1 + aV1)2
+

rHβf(V1)

kη(1 + aV1)2

)(
rH
k

+
λ

f(V1)2
+

rHβV1
kη(1 + aV1)

)−1
.

We deduce from the latter that

F ′(V1) = −
β

(1 + aV1)2

(
e−τm − α

η

)(
rHγ
ηk

+ β
(1+aV1)2

+
rHβf(V1)
kη(1+aV1)2

)(
rH
k

+ λ
f(V1)2

+ rHβV1
kη(1+aV1)

)−1
− rIβ

η(1+aV1)


1 − 2f(V1) + 2γη V1 + 2βf(V1)V1η(1+aV1)

k




×
(
rHγ
ηk

+ β
(1+aV1)2

+
rHβf(V1)
kη(1+aV1)2

)(
rH
k

+ λ
f(V1)2

+ rHβV1
kη(1+aV1)

)−1

+
rIγ

kη

(
rHγ
ηk

+ β
(1+aV1)2

+
rHβf(V1)
kη(1+aV1)2

)(
rH
k

+ λ
f(V1)2

+ rHβV1
kη(1+aV1)

)−1
−
rIaβf(V1)

η(1 + aV1)


1 − f(V1) + γηV1 + βf(V1)V1η(1+aV1)

k


 + aβf(V1)

(1 + aV1)2

(
α
η
−e−τm

)
−rI

k

(
γ

η
+

βf(V1)
η(1+aV1)

)(
γ

η
+

βf(V1)
η(1+aV1)2

)
.

The terms
aβf(V1)

(1 + aV1)2

(
α

η
−e−τm

)
,

−rIβ
η(1 + aV1)


1 − 2f(V1) + 2γη V1 + 2βf(V1)V1η(1+aV1)

k


(rHγ

ηk
+

β

(1 + aV1)2
+

rHβf(V1)

kη(1 + aV1)2

)
×

(
rH
k

+
λ

f(V1)2
+

rHβV1
kη(1 + aV1)

)−1
and

−
β

(1 + aV1)2

(
e−τm −

α

η

)(
rHγ

ηk
+

β

(1 + aV1)2
+

rHβf(V1)

kη(1 + aV1)2

)(
rH
k

+
λ

f(V1)2
+

rHβV1
kη(1 + aV1)

)−1



GLOBAL STABILITY OF A HCV DYNAMICS MODEL WITH CELLULAR PROLIFERATION AND DELAY 9

are negative since
α

η
−e−τm ≤ 0 and Λ

k
≤ 1

2
. Once more we deduce that

F ′(V1) ≤
rIγ

kη

(
rHγ

kη
+

rHβf(V1)

(1 + aV1)2kη
+

rHβf(V1)

(1 + aV1)kη
+

rHβ
2f2(V1)

(1 + aV1)3γkη

)(rH
k

)−1

−
rIaβf(V1)

η(1 + aV1)2


1 − f(V1) + γηV1 + βf(V1)V1η(1+aV1)

k


− rI

k

(
γ

η
+

βf(V1)

η(1 + aV1)

)
×

(
γ

η
+

βf(V1)

η(1 + aV1)2

)
,

since (
rHγ

ηk
+

β

(1 + aV1)2
+

rHβf(V1)

kη(1 + aV1)2

)(
rH
k

+
λ

f(V1)2
+

rHβV1
kη(1 + aV1)

)−1
≤

(
rHγ

kη
+

rHβf(V1)

(1 + aV1)2kη
+

rHβf(V1)

(1 + aV1)kη
+

rHβ
2f2(V1)

(1 + aV1)3γkη

)(rH
k

)−1
.

Thus

F ′(V1) ≤
rI
k

(
γ

η
+

βf(V1)

η(1 + aV1)

)(
γ

η
+

βf(V1)

η(1 + aV1)2

)
−
rI
k

(
γ

η
+

βf(V1)

η(1 + aV1)

)(
γ

η
+

βf(V1)

η(1 + aV1)2

)

−
rIaβf(V1)

η(1 + aV1)2


1 − f(V1) + γηV1 + βf(V1)V1η(1+aV1)

k


 ,

F ′(V1) ≤ −
rIaβf(V1)

η(1 + aV1)2


1 − f(V1) + γηV1 + βf(V1)V1η(1+aV1)

k


 .

Therefore

F ′(V1) < 0.

The fact that F is strictly decreasing function allows us to state that V1 is unique. Uniqueness of V1
implies those of H1 and I1. Therefore one can concludes that E1 = (H1,I1,V1) is unique. �

5. Asymptotic stability analysis of the infection free equilibrium

The aim of this section is to study the local and global stability of the infection free equilibrium.

5.1. Local stability analysis of E0. The following result gives conditions for equilibrium E0 to be

locally asymptotically stable.

Proposition 5.1. If R0(τ) < 1, then the infection free equilibrium E0 of system (2.1) is locally asymp-
totically stable.

Proof. The characteristic equation associated to the Jacobian matrix at the infection free equilibrium,

E0 = (H0, 0, 0) is given by the following determinant :

PJ (E0)(X) =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

(
rH −µ− 2rHk H0

)
−X −

rH
k
H0 −βH0

0
(
rI − rIk H0 −α

)
−X βe−τ(m+X)H0

0 η (−γ −βH0) −X

∣∣∣∣∣∣∣∣∣∣∣∣∣∣
= 0. (5.1)



10 A. NANGUE, A. W. FOKAM TACTEU, AND A. GUEDLAI

Computation of the determinant (5.1) yields :

PJ (E0)(X) =
(
X −

(
rH −µ− 2

rH
k
H0

))(
X2 +

(rI
k
H0 + γ + α−rI + βH0

)
X
)

+
(
−rIγ −rIβH0 + αγ + αβH0 +

rI
k
H0γ +

rI
k
H20β −ηβe

(m+X)τH0

)
= 0.

Since

rH

(
1 −

H0
k

)
= µ−

λ

H0
,

the first factor of the characteristic equation PJ (E0)(X) = 0 is

X =
(
rH −µ− 2

rH
k
H0

)
−µ = −

(
λ

H0
+
rHH0
k

)
,

which have a negative eigenvalue. The other two eigenvalues satisfy the following transcendental poly-

nomial

X2 + a2X + a3 + b3(τ)e
−Xτ = 0, (5.2)

where

a2 =
rI
rH

λ

H0
−
rI
rH

µ + λ + α + βH0,

a3 = −rIγ
(

1 −
H0
k

)
−rIβH0

(
1 −

H0
k

)
+ α (γ + βH0)

and

b3(τ) = −ηβe−mτH0.
When τ = 0, (5.2) yields

X2 + a2X + a3 + b3(0) = 0. (5.3)

Note that a2 > 0 as − rIrH µ + α > 0 since rI ≤ rH and µ ≤ α, and

a3 + b3(0) = −α(γ + βH0)
[
rI
α

(
1 −

H0
k

)
+

ηβH0
γ + βH0

− 1
]

= −α(γ + βH0) [R0(0) − 1] .

If R0(0) < 1 then a3 +b3(0) > 0. It follows that for τ = 0, according to Routh-Hurwitz criteria [8, 3],
infection free equilibrium E0 = (H0, 0, 0) is locally asymptotically stable.

Now, let us consider the distribution of the roots of (5.2) when τ > 0.

Assume that, X = ωi, (ω > 0) is a solution of (5.2). The substitution of X = ωi, (ω > 0) into (5.2)

yields :

−ω2 + iωa2 + a3 + b3(τ) cos ωτ − ib3(τ) sin ωτ = 0,
then separating in real and imaginary parts the previous equation, we obtain{

a3 −ω2 = −b3(τ) cos ωτ
a2ω = b3(τ) sin ωτ.

Squaring and adding the last two equations, we get{
(a3 −ω2)2 = b23(τ) cos2 ωτ,
a22ω

2 = b23(τ) sin
2 ωτ,

and simplifications yields

ω4 + (a22 − 2a3)ω
2 + (a23 − b

2
3(τ)) = 0.

Furthermore, if

ω2 = Z; A = a22 − 2a3; B(τ) = a
2
3 − b

2
3(τ),



GLOBAL STABILITY OF A HCV DYNAMICS MODEL WITH CELLULAR PROLIFERATION AND DELAY 11

we obtain equation

F(Z) = Z2 + AZ + B(τ) = 0, (5.4)

where

A = (γ + βH0)
2

+

(
rIλ

rHH0
−
rIµ

rH
+ α

)2
> 0

and

B(τ) = a23 − b
2
3(τ) = (a3 − b3(τ)) (a3 + b3(τ)) .

We know that

a3 + b3(τ) = −α(γ + βH0)(R0(τ) − 1).

Thus,

B(τ) = α(γ + βH0)
2(1 −R0(τ))

(
rIλ
rHH0

− rIµ
rH

+ α + µβe
−τmH0

γ+βH0

)
.

Since

−
rIH

rH
+ α > 0,

if R0(τ) < 1, then B(τ) > 0. Now as A > 0, B(τ) > 0 and ω > 0, then F(Z) > 0 for any Z > 0 which
contradicts F(Z) = 0. This show that characteristic equation (5.4) does not have pure imaginary roots

when R0(τ) < 1.
Now, let us show that equation (5.2) has all its roots with real negative part when R0(τ) < 1. Let

P(X) = X2 + a2X + a3 et q(X) = b3(τ).

and

a = −rI
(

1 −
H0
k

)
+ α et p = γ + βH0,

thus :

a2 = a + p et a3 = ap,

and

P(X) = X + (a + p)X + ap.

Let us verify if the four conditions of Theorem 1 p.187 [6] are satisfied.

1) Using the Routh-Hurwitz criteria [8, 3], if P(X) = 0, it follows that if a + p = a2 > 0 and

ap = a3(γ + βH0)

(
−rI

(
1 −

H0
k

)
+ α

)
> 0,

then the real part of X 2: Re(X) < 0. Here the first condition holds.
2) For 0 ≤ y < +∞, we have:

p(−iy) = −y2 − i(a + p)y + ap
p(−iy) = −y2 + i(a + p)y + ap

= p(iy)

q(−iy) = ηβH0e−mτ

= q(iy).

And the second condition is satisfied.



12 A. NANGUE, A. W. FOKAM TACTEU, AND A. GUEDLAI

3) For 0 ≤ y < +∞

p(iy) = −y2 + (a + p)iy + ap
|p(iy)|2 = (ap−y)2 + y(a + p)2

= (a2 + y2)(p2 + y2),

thus |p(iy)| ≥ ap. Note that for R0(τ) < 1,

ap−ηβH0emτ = (γ + βH0)
(
−rI

(
1 −

H0
k

)
+ α−

ηβH0e
−mτ

γ + βH0

)
=

(γ + βH0)

α
(1 −R0(τ)) > 0.

Thus

ap > ηβH0e
−mτ.

Therefore

ap > |q(iy)|,

and it follows that

|p(iy)| > |q(iy)|.

4) The last condition holds since:

|q(X)|
|p(X)|

=
ηβH0e

−mτ

X2 + (a + p)X + ap

and

lim
|X|→+∞

∣∣∣∣q(X)p(X)
∣∣∣∣ = 0.

Finally, if R0(τ) < 1, the infection free equilibrium E0 of system (2.1) is locally asymptotically stable.
This completes the proof. �

5.2. Global stability analysis of E0. For biological models and virus dynamics models in particular,

it is interesting to study the stability of positive equilibria. All hepatocytes populations must persist.

It is also necessary for all hepatocyte population to be present initially. Therefore, a genuine concept

of global stability for positive equilibrium points in biological models is that every model solution

that starts in the positive orthant R3+ must remain there for all finite values of t and converge to the

equilibrium when t tends to ∞.
In this section, applying Lyapunov functionals as in Vargas-De-Leon [28], we consider the global

stability of the infection free equilibrium E0.

Theorem 5.2. Assume that the condition rH = (γ + βH0)rIe
τm/γ holds. If R0(τ) < 1, then the

infection free equilibrium E0 = (H0, 0, 0) of system (2.1) is globally asymptotically stable in R
3
+.

Proof. Define the Lyapunov functional

U(t) = emτrI

∫ H(t)
H0

η −H0
η

dη + emτrHI(t) +
rHβH0V (t)

γ + βH0

+rHβ

∫ τ
0

H(t−ω)V (t−ω)
1 + cV (t−ω)

dω. (5.5)



GLOBAL STABILITY OF A HCV DYNAMICS MODEL WITH CELLULAR PROLIFERATION AND DELAY 13

U is defined and continuous for any positive solution (H(t),I(t),V (t)) of system (2.1). Let us calculate

the derivative of U(t) along a positive solution of (2.1). We have:

dU(t)

dt
= emτrI

(H −H0)
H

Ḣ(t) + rHe
τmİ(t) +

rIβH0
γ + βH0

V̇ (t)

+βrH
d

dt

∫ τ
0

H(t−ω)V (t−ω)
1 + aV (t−ω)

dω.

Since u = t−ω and

βrH
d

dt

∫ τ
0

H(t−ω)V (t−ω)
1 + aV (t−ω)

dω = −βrH
d

du

∫ t−τ
t

H(u)V (u)

1 + aV (u)
du,

it follows that

dU(t)

dt
= emτrI

(H −H0)
H

(
λ + rH

(
1 −

H + I

k

)
−µH −

βHV

1 + aV

)
+
rHβH(t− τ)V (t− τ)

1 + aV (t− τ)
+ rHe

mτrII

(
1 −

H + I

k

)
−rHαemτI

+
rHβH0ηI

γ + βH0
−
rHβH0γV

γ + βH0
−

rHβ
2H0HV

(γ + βH0)(1 + aV )
−
rHβH(t− τ)V (t− τ)

1 + aV (t− τ)

+
rHβHV

1 + aV
.

Using the fact that

rH −µ =
rHH0
k
−

λ

H0
,

we get

dU(t)

dt
= emτrI

(H −H0)
H

(
λ +

rH
k
H0H −

λ

H0
H −

rH
k
IH −

βHV

1 + aV

)
+emτIrHrI

(
1 −

H0
k

)
−emτIrHrI

(
H −H0

k

)
−rHemτrI

I2

k
−rHγαemτI

+
rHβH0ηI

γ + βH0
−
rHβH0γV

γ + βH0
−

rHβ
2H0HV

(γ + βH0)(1 + aV )
+
rHβHV

1 + aV
;

= −
rHrI
H0

emτ ((H −H0) + I)
2

+ emτrHαI(R0(τ) − 1)

−λemτ
rI(H −H0)2

HH0
+ βH0V

(
rIe

mτ

1 + aV
−

rHγ

γ + βH0

)
+
βHV

1 + aV

(
−rIemτ +

rHγ

γ + βH0

)
.

Therefore,

dU(t)
dt
≤ emτrI

(H−H0)2
HH0

− rH
k
rIe

mτ ((H −H0) + I)
2

+ emτrHαI(R0(τ) − 1) (5.6)

since
rHγ

γ + βH0
= rIe

mτ.

Thus
dU(t)
dt
≤ 0 since R0(τ) < 1.Furthermore,

dU(t)
dt

= 0 if and only if H(t) = H0, I(t) = 0 and V (t) = 0.

Therefore, the largest compact invariant set in
{

(H(t),I(t),V (t)) /
dU(t)
dt

= 0
}

when R0(τ) ≤ 1 is
E0 = (H0, 0, 0), where E0 is the infection free equilibrium. This shows that lim

t−→∞
(H(t),I(t),V (t)) =

(H0, 0, 0). By the Lyapunov-LaSalle invariance theorem for delay differential systems (Theorem 2.5.3

in [13]), this implies that E0 is globally asymptotically stable in the interior of R
3
+. �



14 A. NANGUE, A. W. FOKAM TACTEU, AND A. GUEDLAI

5.3. Numerical results. In this section, we present some numerical simulations which validate our

theoretical results. To explore system (2.1) and illustrate the stability of infection free equilibrium

solution, we consider the set of following parameters value :

τ = 5; rH = 0, 05; rI = 0.0428; m = 0, 021; k = 1200 ;

a = 0, 001; β = 9, 2419.10−7; µ = 0, 02; γ = 0, 02;

α = 0, 021; λ = 20; η = 0, 2 .

(5.7)

We obtain Figure 1.

0 100 200 300 400 500 600 700 800 900

t

0

200

400

600

800

1000

1200

(a)

0
80

100

200

60 1200

V
(t

)
300

Phase Space

I(t)

400

40 1000

H(t)

500

20 800

0 600

(b)

Figure 1. Dynamical behaviour of system (2.1) with the set of parameter values (5.7)

We have R0(5) = 0.5513 < 1 and the infection free equilibrium E0 = (1140.8; 0; 0) is globally
asymptotically stable. The graph in (a) shows the time series of the solutions with constant initial

conditions. The graph in (b) shows the trajectory in the phase diagram of system 2.1, which illustrates

the stability of the uninfected E0 with the history functions ϕ1(θ) = 800, ϕ2(θ) = 80, ϕ3(θ) = 300

(first trajectory); ϕ1(θ) = 750, ϕ2(θ) = 75, ϕ3(θ) = 250 (second trajectory); ϕ1(θ) = 700, ϕ2(θ) = 70,

ϕ3(θ) = 200 (third trajectory).

6. Asymptotic stability analysis of the infection persist equilibrium

The aim of this section is to study the local and global stability of the infection persist equilibrium

for system (2.1).

6.1. Local stability analysis of E1. We now study the local stability behaviour of the infection persist

equilibrium E1 when R0(τ) > 1 for system (2.1). Thus, linearizing system (2.1) at the infection persist
equilibrium E1 = (H1,I1,V1), we obtain that the associated transcendental characteristic equation is

given by



GLOBAL STABILITY OF A HCV DYNAMICS MODEL WITH CELLULAR PROLIFERATION AND DELAY 15

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

− λ
H1
− rH

k
H1 −X −rHk H1 −

βH1
(1+aV1)2

βe−(m+X)τ

1+aV1
− rI

k
I1 −βe

−τmH1V1
(1+aV1)I1

− rI
k
I1 −X βe

−(m+X)τH1
(1+aV1)2

− βV1
1+aV1

η −γ − βH1
(1+aV1)2

−X

∣∣∣∣∣∣∣∣∣∣∣∣∣∣
= 0.

When we take into account the identities

rH −µ = −
λ

H1
+
rH
k
H1 +

rH
k
I1 +

βV1
1 + aV1

,

rI −α = −
βe−τmH1V1
(1 + aV1)I1

+
rI
k
H1 +

rI
k
I1,

the characteristic equation reduces to

X3 + a2(τ)X
2 + a1(τ)X +

[
b1(τ)X + b2(τ)

]
e−Xτ + a0(τ) = 0. (6.1)

where

a2(τ) =
λ

H1
+
rH
k
H1 +

βe−mτH1V1
(1 + aV1)I1

+
rI
k
I1 + γ +

βH1
(1 + aV1)2

,

a1(τ) =
βe−τmH1V1
(1 + aV1)I1

γ +
rI
k
γI1 +

rIβH1I1
(1 + aV1)2

+
β2e−τmH21V1
(1 + aV1)3I1

+

(
λ

H1
+
rH
k
H1

)
×

(
βe−τmH1V1
(1 + aV1)I1

+
rH
k
I1 +

βH1
(1 + aV1)2

+ γ

)
−
rH
k
H1I1 −

β2H1V1
(1 + aV1)3

,

a0(τ) =

(
λ

H1
+
rH
k
H1

)(
βe−mτH1V1
(1 + aV1)I1

γ +
rI
k
γI1 +

rIβH1I1
k(1 + aV1)2

+
β2e−τmH21V1
(1 + aV1)3I1

)
+
rH
k
H1

(
−
rI
k
γI1 −

rIβH1I1
k(1 + aV1)2

)
+

βH1
(1 + aV1)2

×

(
−
rIηI1
k
−

rIβV1I1
k(1 + aV1)

−
β2e−τmH1V

2
1

(1 + aV1)2I1

)
,

b1(τ) = −
ηβe−(m+X)τH1

(1 + aV1)2
+
rH
k
H1

βe−(m+X)τV1
(1 + aV1)

,

b2(τ) = −
(
λ

H1
+
rH
k
H1 +

rH
k
I1

)
ηβe−(m+X)τH1

(1 + aV1)2
+
rHβe

−(m+X)τH1V1
k(1 + aV1)

γ

−
βH1

(1 + aV1)2
ηβe−(m+X)τV1

(1 + aV1)
.

Define

P(λ,τ) = X3 + a2(τ)X
2 + a1(τ)X + a0(τ)

and

Q(λ,τ) = b1(τ)X + b2(τ).



16 A. NANGUE, A. W. FOKAM TACTEU, AND A. GUEDLAI

When τ = 0, we have from (6.1) that

P(λ, 0) + Q(λ, 0) = X3 + a2(0)X
2 + (a1(0) + b1(0))X + b2(0) + a0(0) = 0. (6.2)

By the Routh-Hurwitz criterion the conditions for the real part of X to be negative are a2(0) > 0,

a0(0) + b2(0) > 0 and Q = a2(0)(a1(0) + b1(0)) − (a0(0) + b1(0)) > 0. In our case a2(0) > 0. In other
hand, we have

a1(0) + b1(0) =
βH1V1

(1 + aV1)I1
γ

aV1
1 + aV1

+
rI
k
γI1 +

rIβH1I1
(1 + aV1)2

+
rH
k
H21

β

(1 + aV1)2

+

(
λ

H1
+
rH
k
H1

)(
βH1V1

(1 + aV1)I1
+ γ

)
+

λ

H1

rI
k
I1 +

rHH1I1
k

(rH −rI)

−rHH1
(

1 −
H1 + I1

k

)
β

(1 + aV1)2
+

βH1
(1 + aV1)2

µ

=
aβH1V

2
1 γ

(1 + aV1)2I1
+

βH1
(1 + aV1)2

(
µ + rII1 −rH

(
1 −

2H1 + I1
k

))

+

(
λ

H1
+
rH
k
H1

)(
βH1V1

(1 + aV1)I1
+ γ

)
+

λ

H1

rI
k
I1

+
rHH1I1

k
(rH −rI) +

rIγI1
k

,

since

ηβH1
(1 + aV1)3I1

=
βH1V1γ

(1 + aV1)2I1
+

β2H21V1
(1 + aV1)3I1

and

−
β2H1V1

(1 + aV1)3
= −

βλ

(1 + aV1)2
−rHH1

(
1 −

H1 + I1
k

)
β

(1 + aV1)2
+

βH1
(1 + aV1)2

µ.

It follows that if

µ + rII1 −rH
(

1 −
2H1 + I1

k

)
> 0, (6.3)

then

a1(0) + b1(0) > 0. (6.4)



GLOBAL STABILITY OF A HCV DYNAMICS MODEL WITH CELLULAR PROLIFERATION AND DELAY 17

Furthermore, using (4.3) we obtain

a0(0) + b2(0) =

(
λ

H1
+
rH
k
H1

)(
1 −

1

1 + aV1

)
βH1V1

(1 + aV1)I1
γ

+
λ

H1

(
rI
k
γI1 +

rIβH1I1
k(1 + aV1)2

)
+

rHβH1V1
k(1 + aV1)

γ +
ηβ2H1V1

(1 + aV1)3

−
rIβH1I1

k(1 + aV1)2

(
γ

I1
V1 +

βH1V1
1 + aV1I1

)
−
rIβ

2H1V1I1
k(1 + aV1)3

−
β3H21V

2
1

(1 + aV1)4I1
,

=

(
λ

H1
+
rH
k
H1

)(
1 −

1

1 + aV1

)
βH1V1

(1 + aV1)I1
γ

+
λ

H1

(
rI
k
γI1 +

rIβH1I1
k(1 + aV1)2

)
+

βH1V1γ

k(1 + aV1)
×

(
rH −

rI
1 + aV1

)
−

rIβ
2H21V1

k(1 + aV1)3
−
rIβ

2H1V1I1
k(1 + aV1)3

−
β3H21V

2
1

(1 + aV1)4I1

+
β2H1V1

(1 + aV1)3

(
γ

I1
V1 +

βH1V1
(1 + aV1)I1

)
=

(
λ

H1
+
rH
k
H1

)(
1 −

1

1 + aV1

)
βH1V1

(1 + aV1)I1
γ

+
λ

H1

(
rI
k
γI1 +

rIβH1I1
k(1 + aV1)2

)
+

βH1V1
k(1 + aV1)

γ×

(
rH −

rI
1 + aV1

)
−

rIβ
2H21V1

k(1 + aV1)3
−
rIβ

2H1V1I1
k(1 + aV1)3

+
β2H1V

2
1

(1 + aV1)3I1
γ,

=

(
λ

H1
+
rH
k
H1

)
aβH1V

2
1

(1 + aV1)2I1
γ +

βH1V1
k(1 + aV1)

γ

(
rH −

rI
1 + aV1

)

+
λ

H1

(
rI
k
γI1 +

rIβH1I1
k(1 + aV1)2

)
+
rIβ

2H1V1
(1 + aV1)3

(
1 −

H1 + I1
k

)

+
β2H1V1

(1 + aV1)3

(
V1
I1
−rI

)
.

Finally, if

rI <
V1
I1
,

then

a0(0) + b2(0) > 0.

Thus, if τ = 0, by the Routh-Hurwitz criterion, we have the following theorem :

Theorem 6.1. Assume R0(0) > 1, if Q > 0 and rI < V1I1 then the unique infection persist equilibrium
E1 = (H1,I1,V1) is locally asymptotically stable as τ = 0.



18 A. NANGUE, A. W. FOKAM TACTEU, AND A. GUEDLAI

Now we are going to check if it is possible to have a complex root with a positive real part for τ > 0,

assuming

H(1) : a0 + b0 > 0, a2(a1 + b1) − (a0 + b0) > 0.
Note that X = 0 is not root of the equation because a0 + b2 > 0. Suppose then that X = iω, (ω > 0),

is a root of equation given by:

P(X,τ) + Q(X,τ)e−Xτ = 0 (6.5)

with

P(X,τ) = X3 + a2(τ)X
2 + a1(τ) + a0(τ),

Q(X,τ) = b1(τ)X + b2(τ)

As X = iω is a root of equation (6.5), we have

P(iω,τ) + Q(iω,τ)e−iωτ = 0

and

−iω3 −a2(τ)ω2 + ia1(τ)ω + a0(τ) + (b1(τ)iω + b2(τ)) e−iωτ = 0.
And it follows that

−iω3 −a2(τ)ω2 + ia1(τ)ω + a0(τ) + (ib1(τ)ω + b2(τ)) (cos ωτ − i sin ωτ) = 0

We obtain : −iω3 −a2(τ)ω2 + ia1(τ)ω + a0(τ) + ib1(τ)ω cos ωτ + b2(τ) cos ωτ + b1(τ)ω cos ωτ
− ib2(τ) sin ωτ = 0. Consequently, we have the following relation :{

a0(τ) −a2(τ)ω2 = −b2(τ) cos ωτ − b1(τ)ω sin ωτ (6.6)
a1(τ)ω −ω3 = −b1(τ)ω cos ωτ + b2(τ) sin ωτ. (6.7)

Multiplying (6.6) by −b1(τ)ω and (6.7) by b2(τ), we get :
(−a0(τ) + a2(τ)ω2)b1(τ)ω + (a1(τ)ω −ω3)b2(τ)

b22(τ) + b
2
1(τ)ω

2
= sin ωτ.

Therefore

sin ωτ =
(a2(τ)b1(τ) − b2(τ))ω3 + (a1(τ)b2(τ) −a0(τ)b1(τ))ω

b22(τ) + b
2
1(τ)ω

2
.

Similarly, multiplying (6.6) by b2(τ) and (6.7) by b1(τ)ω, we obtain:

cos ωτ =
b1(τ)ω

4 + [a2(τ)b2(τ) −a1(τ)b1(τ)]ω2 −a0(τ)b2(τ)
b22(τ) + b

2
1(τ)ω

2
.

Moreover for X = iω, we get :

P(iω,τ) = −iω3 −a2(τ)ω2 + ia1(τ)ω + a0(τ),
and

Q(iω,τ) = ib1(τ)ω + b2(τ).

Therefore one has :

P(iω,τ)

Q(iω,τ)
=

i(a2(τ)b1(τ) − b2(τ))ω3 + (a1(τ)b2(τ) −a0(τ)b1(τ)ω)
b22(τ) + b

2
1(τ)ω

2

−

(
b1(τ)ω

4 +
(
a2(τ)b2(τ) −a1(τ)b1(τ)

)
ω2 −a0(τ)b2(τ)

)
b22(τ) + b

2
1(τ)ω

2
.

It follows that :

sin(ωτ) = Im
(
Q(iω,τ)

P(iω,τ)

)



GLOBAL STABILITY OF A HCV DYNAMICS MODEL WITH CELLULAR PROLIFERATION AND DELAY 19

and

cos(ωτ) = −Re
(
Q(iω,τ)

P(iω,τ)

)
.

Thus from the fact that sin2(ωτ) + cos2(ωτ) = 1 we have

|Q(iω,τ)|2

|P(iω,τ)|2
= 1 and |Q(iω,τ)|2 = |Q(iω,τ)|2.

We conclude that ω is a positive root of the equation |P(iω,τ)|2 −|Q(iω,τ)|2 = 0. Furthermore

|P(X,τ)|2 = ω6 + a22(τ) − 2a1(τ)ω
4 + a21(τ) − 2a0(τ)a2(τ)ω

2 + a20(τ)

and

|Q(X,τ)|2 = b22(τ) + b
2
1(τ)ω

2.

We get

ω6 −Aω4 + Bω2 + C = 0 (6.8)

where
A = a2(τ) − 2a1(τ),
B = a21(τ) − 2a0(τ)a2(τ) − b21(τ),
C = a20(τ) − b22(τ).

Let z = ω2 , we obtain

z3 + Az2 + Bz + C = 0 (6.9)

Suppose that (6.9) has at least one positive root. Let z0 be the smallest value of its roots. Then (6.8)

has the root ω0 =
√
z0 and from (6.7) we get the value of τ associated with ω0 such that X = ωi is a

pure imaginary root of (6.5). This value of τ is given by:

τ0 =
1

ω0
arccos

[
b2(a2ω

2
0 −a0) + b1ω0(ω30 −a1ω0)

b22 + b
2
1ω

2
0

]
.

Ultimately we can summarize what precedes in the following result. Thus according to Theorem 2.4

p.48 [22], we have :

Theorem 6.2. Suppose (H(1)) is verified,

(1) If C ≥ 0 and Λ = A2 − 3B < 0, then the roots of (6.5) have a negative real part for all τ ≥ 0,
therefore the infection persist equilibrium E1(H1,I1,V1) is locally asymptotically stable.

(2) If C < 0 or C ≥ 0, z1 > 0 and z31 + Az21 + Bz1 + C ≤ 0, then all the roots of the equation
(6.5) have a negative real part when τ ∈ [0,τ0], and therefore the infection persist equilibrium
E1(H1,I1,V1) is locally asymptotic stable in [0,τ0].

6.2. Global stability analysis of E1.

Theorem 6.3. Denote

ε = µ−rH +
rH
k

(H1 + I1), Λ =
4λ + (rH + rI)k

4b
, et Ω =

ηΛ

γ
.

Assume

rH = rIe
mτ, ε > 0 and k >

β3H1V
2
1 Λ(1 + aΩ)

4εη2(1 + aV1)4
+

aβΛV1
ηI1(1 + aV1)

,

then the unique infection persist equilibriumE1 = (H1,I1,V1) for system (2.1) is globally asymptotically

stable for any τ ≥ 0.



20 A. NANGUE, A. W. FOKAM TACTEU, AND A. GUEDLAI

Proof. Define a Lyapunov functional for E1

L(t) = L̃(t) +
βH1V1
1 + aV1

L+(t), (6.10)

where

L̃(t) = rIe
mτ

∫ H
H1

η −H1
η

dη + rHe
mτ

∫ I
I1

η − I1
η

dη + rI
emτβH1V1
ηI1(1 + aV1)

∫ V
V1

(
1 −

V1(1 + η)

η(1 + aV1)

)
dη

and

L+(t) = rH

∫ τ
0

(
H(t− τ)V (t− τ)(1 + aV1)
H1V1(1 + aV (t− τ))

− 1 − ln
H(t−ω)V (t− τ)(1 + aV )
H1V1(1 + aV (t− τ))

)
dω.

We have :

dL+(t)

dt
=

d

dt
rH

∫ τ
0

(
H(t−ω)V (t−ω)(1 + aV1)

H1V1(1 + aV (t−ω))
− 1 − ln

H(t−ω)V (t−ω)(1 + aV )
H1V1(1 + aV (t−ω))

)
dω.

Denote

u = t−ω,

we obtain :

dL+(t)

dt
= −rH

du

dt

d

du

∫ t−τ
t

(
H(u)V (u)(1 + aV1)

H1V1(1 + aV (u))
− 1 − ln

H(u)V (u)(1 + aV )

H1V1(1 + aV (u))

)
dω

= −rH

[
H(u)V (u)(1 + aV1)

H1V1(1 + aV (u))
− 1 − ln

H(u)V (u)(1 + aV )

H1V1(1 + aV (u))

]t−τ
ω=t

,

= −
rHH(t− τ)V (t− τ)(1 + aV1)

H1V1(1 + aV (t− τ))
+ rH ln

H(t− τ)V (t− τ)(1 + aV )
H1V1(1 + aV (t− τ))

+
rHHV (1 + aV1)

H1V1(1 + aV )
−rH ln

HV (1 + aV1)

H1V1(1 + aV )
,

= −
rHH(t− τ)V (t− τ)(1 + aV1)

H1V1(1 + aV (t− τ))
+ rH ln

I1H(t− τ)V (t− τ)(1 + aV )
IH1V1(1 + aV (t− τ))

,

+
rHHV (1 + aV1)

H1V1(1 + aV )
+ rH ln

H1
H

+ rH ln
IV1(1 + aV )

I1V (1 + aV1)
.

Besides, a direct calculation yields :

dL̃(t)

dt
= rIe

mτ (H −H1)
H

Ḣ + rHe
mτ (I − I1)

I
İ + rIe

mτ βH1V1
ηI1(1 + aV1)

(
1 −

V1(1 + aV )

V (1 + aV1)

)
V̇ ,

= (H −H1)rIemτ
(
λ

H
−
rH
k

(H + I) −
βV

1 + aV
+ rH −µ

)
+rHe

mτ (I − I1)
(
βe−τH(t− τ)V (t− τ)
I(1 + aV (t− τ))

−
rI
k

(H + I) + rI −α
)

+rIe
mτ βH1V1

ηI1(1 + aV1)

(
1 −

V1(1 + aV )

V (1 + aV1)

)(
ηI −γV −

βHV

1 + aV

)
.

Hence using (4.1), (4.2) and (4.3) we get :



GLOBAL STABILITY OF A HCV DYNAMICS MODEL WITH CELLULAR PROLIFERATION AND DELAY 21

dL̃(t)

dt
= rIe

mτ (H −H1)

(
λ

H
−
rH
k

(H + I) −
βV

1 + aV1
−

λ

H1
+
rH
k

(H1 + I1)

+
βV1

1 + aV1

)
+ rHe

mτ (I − I1)

(
βe−mτH(t− τ)V (t− τ)

I(1 + aV (t− τ))
−
rI
k

(H + I)

+
rI
k

(H1 + I1) −
βe−mτH1(t− τ)V1(t− τ)

I1(1 + aV1)

)
+
rIe

mτβH1V1
ηI1(1 + aV1)

×

(
1 −

V1(1 + aV )

V (1 + aV1)

)(
(ηIV1 − I1V )

V1
+

βV H1
(1 + aV1)

−
βV H

(1 + aV )

)
.

Cancelling identical terms with opposite signs and collecting terms yields consecutively

dL̃(t)

dt
= rIe

mτ (H −H1)

(
−λ(H −H1)2

HH1
−
rH
k

[(H −H1) + (I − I1)]

−β
(

V

1 + aV
−

V1
1 + aV

))
+ rH(I − I1)

(
βH(t− τ)V (t− τ)
I(1 + aV (t− τ))

−
βH1V1

I1(1 + aV1)
−
rI
k
emτ ((H −H1) + (I − I1))

)
+
r1e

mτβH1V1
ηI1(1 + aV1)

×

(
1 −

V1(1 + aV )

V (1 + aV1)

)(
η(IV1 − I1V )

V1
+

βV H1
(1 + aV1)

−
βV H

(1 + aV )

)
= −λrIemτ

(H −H1)2

HH1
+
(
−
rH
k
rIe

mτ (H −H1)2

−
rH
k
rIe

mτ (H −H1)(I − I1) −
rIrH
k

emτ (H −H1)(I − I1)

−
rIrH
k

emτ (I − I1)2
)

+
βHIVI
1 + aV1

(
−emτ

rHHV (1 + aV1)

H1V1(1 + aV )
+
rHH(t− τ)V (t− τ)(1 + aV1)

H1V1(1 + aV (t− τ))

)

+
βH1V1
1 + aV1

(
−rHemτIV1(1 + aV )

I1V (1 + aV )
−
rHI1H(t− τ)V (t− τ)(1 + aV1)

H1IV1(1 + aV (t− τ))

)

+
βH1V1
1 + aV1

(
rIe

mτ H

H1
+ rIe

mτ V (1 + aV1)

V1(1 + aV )
−rIemτ

V

V1
+ rIe

mτ (1 + aV )

1 + aV1

)

−
rIe

mτβ2H1V1
η(1 + aV1)3

(V −V1)(H −H1) +
rIe

mτaβ2H1HV1(V −V1)2

ηI1(1 + aV1)3(1 + aV )
.



22 A. NANGUE, A. W. FOKAM TACTEU, AND A. GUEDLAI

= −λrIemτ
(H −H1)2

HH1
−
rHrI
k

emτ

(
(H −H1) + (I − I1)

)2
+
βH1V1
1 + aV1

(
−emτ

rIHV (1 + aV1)

H1V1(1 + aV )
+
rHH(t− τ)V (t− τ)(1 + aV1)

H1V1(1 + aV (t− τ))

)
+
βH1V1
1 + aV1

(
emτrIV (1 + aV1)

V1(1 + aV )
−emτrI

V

V1
+
emτrI(1 + aV )

1 + aV1
−emτrI

)
+
βH1V1
1 + aV1

(
3rIe

mτ −rIemτ
H1
H
−
emτrIV1(1 + aV )I

I1V (1 + aV1)

−
rHI1H(t− τ)V (t− τ)(1 + aV1)

H1IV1(1 + aV (t− τ))

)
−
rIe

mτβ2H1V1
η(1 + aV1)3

(V −V1)(H −H1)

+
rIe

mτaβ2H1HV1(V −V1)2

ηI1(1 + aV1)3(1 + aV )
+
rIe

mτβH1V1
1 + aV1

(
H1
H

+
H

H1
− 2

)
.

Additionally,
H1
H

+
H

H1
− 2 =

(H −H1)2

HH1
,

thus,

dL̃(t)

dt
= rIe

mτ

(
−λ +

βH1V1
1 + aV1

)
(H −H1)2

HH1
−
rHrI
k

emτ

(
(H −H1) + (I − I1)

)2

+
βH1V1
1 + aV1

(
−
emτrIHV (1 + aV1)

H1V1(1 + aV )
+
rHH(t− τ)V (t− τ)(1 + aV1)

H1V1(1 + aV (t− τ))

)

+
βH1V1
1 + aV1

(
−
rIe

mτV (1 + aV1)I

I1V1(1 + aV )
−
rHI1H(t− τ)V (t− τ)(1 + aV1)I

H1IV1(1 + aV (t− τ))

+ 3rIe
mτ −rIemτ

H

H1

)
+ rIe

mτ βH1V1
1 + aV1

(
1 −

V1(1 + aV )

V (1 + aV1)

)(
V (1 + aV1)

V1(1 + aV )
−
V

V1

)

−
r1e

mτβ2H1V1
η(1 + aV1)3

(V −V1)(H −H1) +
r1e

mτaβ2H1HV1(V −V1)2

ηI1(1 + aV1)3(1 + aV )
.

The fact that

λ−
βH1V1
1 + aV1

= (µ−rH)H1 +
rHH1
k

(H1 + I1)

yields

dL̃(t)

dt
= −rIemτ

ε

H
(H −H1)2 −

rHrI
k

emτ ((H −H1) + (I − I1))
2

+
βH1V1
1 + aV1

(
−emτ

r1HV (1 + aV1)

H1V1(1 + aV )
+
rHH(t− τ)V (t− τ)(1 + aV1)

H1V1(1 + aV (t− τ))

)



GLOBAL STABILITY OF A HCV DYNAMICS MODEL WITH CELLULAR PROLIFERATION AND DELAY 23

+
βH1V1
1 + aV1

(
3emτrI −emτrI

H1
H
−
emτr1IV1(1 + aV )

I1V (1 + aV1)

−
rHI1H(t− τ)V (t− τ)(1 + aV1)

H1IV1(1 + aV (t− τ))

)
−

krIe
mτβH1

(1 + aV )(1 + aV1)2
(V −V1)2

−rI
emτβ2H1V1
η(1 + aV1)3

(V −V1)(H −H1) +
rIe

mτaβ2H1HV1(V −V1)2

ηI1(1 + aV1)3(1 + aV )
.

It follows that

dL̃(t)

dt
≤ −rIemτ

ε

H

(
(H −H1) +

β2HH1V1
2η(1 + aV1)3

(V −V1)
)2

+
rIe

mτβH1
(1 + aV )(1 + aV1)2

(
−k +

β3H1V
2
1 Λ(1 + aΩ)

4εη2(1 + aV1)4
+

aβΛV1
ηI1(1 + aV1)

)
(V −V1)2

+
βH1V1
1 + aV1

(
−emτ

rIHV (1 + aV1)

H1V1(1 + aV )
+
rHH(t− τ)V (t− τ)(1 + aV1)

H1IV1(1 + aV (t− τ))

)

+
βH1V1
1 + aV1

(
3emτrI −emτrI

H1
H
−
emτr1IV1(1 + aV )

I1V (1 + aV1)

−
rHI1H(t− τ)V (t− τ)(1 + aV1)

H1IV1(1 + aV (t− τ))

)
−
rIrH
k

emτ
(

(H −H1) + (I − I1)
)2
.

According to (6.10), we have:

dL

dt
=
dL̃

dt
+
βH1V1
1 + aV1

dL+
dt

.

Thus,

dL(t)

dt
≤−

rHrI
k

emτ ((H −H1) + (I − I1))
2 −−

rIrH
k

emτ
(

(H −H1) + (I − I1)
)2

−rIemτ
ε

H

(
(H −H1) +

β2HH1V1
2η(1 + aV1)3

(V −V1)
)2

+
rIe

mτβH1
(1 + aV )(1 + aV1)2

×

(
−k +

β3H1V
2
1 Λ(1 + aΩ)

4εη2(1 + aV1)4
+

aβΛV1
ηI1(1 + aV1)

)
(V −V1)2

−
rIe

mτβH1V1
(1 + aV1)

(
H1
H
− 1 − ln

H1
H

)
−
rIe

mτβH1V1
(1 + aV1)

(
IV1(1 + aV )

I1V (1 + aV1)
− 1

− ln
IV1(1 + aV )

I1V (1 + aV1)

)
−rIemτ

βH1V1
1 + aV1

(
H(t− τ)V (t− τ)(1 + aV1)
H1IV1(1 + aV (t− τ))

− 1

− ln
H(t− τ)V (t− τ)(1 + aV )
H1IV1(1 + aV (t− τ))

)
.



24 A. NANGUE, A. W. FOKAM TACTEU, AND A. GUEDLAI

The function x 7→ x− 1 − ln x is positive on ]0; +∞[, therefore, since

−k +
β3H1V

2
1 Λ(1 + aΩ)

4εη2(1 + aV1)4
+

aβΛV1
ηI1(1 + aV1)

≤ 0 and ε > 0

we deduce that
dL(t)

dt
≤ 0.

Furthermore,
dL(t)
dt

= 0 if and only if H(t) = H(t − τ) = H1, V (t) = V (t − τ) = V1 and I(t) = I1.
Therefore, the largest compact invariant set M is the singleton {E1}, where E1 is the infection persist
equilibrium. This shows that lim

t−→∞
(H,I,V ) = (H1,I1,V1). By the Lyapunov-LaSalle invariance theo-

rem for delay differential systems (Theorem 2.5.3 in [13]), this implies that E1 is globally asymptotically

stable in the interior of R3+. �

6.3. Numerical results. In this section, we present some numerical simulations which validate our

theoretical results. To explore system (2.1) and illustrate the stability of infection persist equilibrium

solution, we consider the set of following parameter value :

τ = 10; β = 0.0027; rH = 0.01; m = 0.02; rI = 0.0061; λ = 5;

µ = 0.02; a = 0.001; η = 0.5 ; γ = 2.1; α = 0.05 ; ε = 0.0116 > 0;

Λ = 550; E0(379.7959; 0; 0); E1 = (156.9218; 38.0708; 7.5522)

k = 1200 >
β3H1V

2
1 Λ(1+aΩ)

4εη2(1+aV1)4
+ aβΛV1

ηI1(1+aV1)
= 8.0643.

(6.11)

For these parameter values, calculation by the formula of basic reproduction number leads to R0 =
2.0729 > 1; and thus, the infection persist equilibrium E1 = (156.9218; 38.0708; 7.5522) is globally

asymptotically stable. This conclusion is demonstrated in Figure 2). The graph in (a) shows the time

series of the solutions with constant initial conditions. The graph in (b) shows the trajectory in the

phase diagram of system 2.1, which illustrates the stability of the infected E1 with the history functions

ϕ1(θ) = 120, ϕ2(θ) = 80, ϕ3(θ) = 20 (first trajectory); ϕ1(θ) = 210, ϕ2(θ) = 120, ϕ3(θ) = 25 (second

trajectory); ϕ1(θ) = 330, ϕ2(θ) = 200, ϕ3(θ) = 30 (third trajectory).

7. Conclusion

In order to better understand the dynamics of HCV viral infection, this paper presents a mathematical

study on the global dynamics of improved intra-host HCV models based on models in [2] and [1]. In this

work we have established results about the local and global stability of equilibria known as infection

free equilibrium and infection persist equilibrium. We can conclude that the stability of infection free

equilibrium is completely determined by the value of the basic reproductive number R0(τ). If R0(τ) < 1,
then the infection free equilibrium will be asymptomatically stable and unstable if R0(τ) > 1. For the
infection persist equilibrium, we established conditions to ensure the local stability. We have needed

another conditions to ensure the global stability for this equilibrium. Most of the results obtained in

the present work generalize, in the same framework, those obtained by Eric Avila Vales et al. in [1] in

the sense that in [1] the proliferation rates of infected and uninfected hepatocytes are identical and the



GLOBAL STABILITY OF A HCV DYNAMICS MODEL WITH CELLULAR PROLIFERATION AND DELAY 25

0 100 200 300 400 500 600 700 800 900 1000

t

0

50

100

150

200

250

300

350

(a)

0
300

20

400

40

V
(t

)

200

Phase space

60

I(t) H(t)

80

200100

0 0

(b)

Figure 2. Dynamical behaviour of system (2.1) with the set of parameter values (6.11)

absorption phenomenon was absent.

This work can be developed in many ways as follows:

(i) By taking into consideration the effect of several time discrete delays that may occur during

the infection process, diffusion of cells and virions or other biological processes. For example,

the following model can be study:


∂H(x,t)

∂t
= DH∆H(x,t) + λ + rHH(x,t)

(
1 −

H(x,t) + I(x,t)

k

)
−µH(x,t) −

βH(x,t)V (x,t)

1 + aV (x,t)
,

∂I(x,t)

∂t
= DI∆I(x,t) +

βe−τ1m1H(x,t− τ1)V (x,t− τ1)
1 + aV (x,t− τ1)

+rII(x,t)

(
1 −

H(x,t) + I(x,t)

k

)
−αI(x,t)v,

∂V (x,t)

∂t
= DV ∆V (x,t) + ηe

−τ2m2I(x,t− τ2) −γV (x,t) −
βH(x,t)V (x,t)

1 + aV (x,t)
,

where the parameter τ1 accounts for the time between viral entry into a target cell and the

production of new virus particles, after which the cells produce virus at per capita rate ηe−τ2m2

with a delay of τ2. The constant m1 is assumed to be the death rate for cells that are infected

but not yet producing virus, so that e−τ1m1 is the probability of surviving from time t− τ2 to
time t. Likewise the constant m2 is assumed to be the birth rate of the virions so that e

−τ2m2

is the probability of producing a fraction of η virions from time t− τ2 to time t. And DH, DI
and DV give the rates at which the target cells, the infected cells and the virus particles diffuse

respectively;

(ii) By fitting the model with real data and finding a better estimation for the values of parameters.

(iii) Using a more general incidence rate as a function with certain desired properties, as considered

in [11] or considering distributed delays in the equations for the infected cells and the virus

particles, as is discussed in [15].

(iv) By investigating HCV co-infections with other viruses.



26 A. NANGUE, A. W. FOKAM TACTEU, AND A. GUEDLAI

Disclosure statement

No potential conflict of interest was reported by the authors.

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Corresponding author, Higher Teachers’ Training College , University of Maroua, Cameroon, P.O.Box

55 Maroua

Current address: same

Email address: alexnanga02@gmail.com

Higher Teachers’ Training College , University of Maroua, Cameroon, P.O.Box 55 Maroua

Current address: University of Maroua, Faculty of Sciences, Department of Mathematics and Computer Science,

Cameroon, P.O. Box 814 Maroua

Email address: tacteufokam@gmail.com

Higher Teachers’ Training College , University of Maroua, Cameroon, P.O.Box 55 Maroua

Current address: University of Maroua, Faculty of Sciences, Department of Mathematics and Computer Science,

Cameroon, P.O. Box 814 Maroua

Email address: guedlaiayoubagg@gmail.com


	1. Introduction
	2. Model construction
	3. Wellposedness
	4. Equilibria and basic reproduction number
	4.1. Infection free equilibrium and basic reproduction number
	4.2. Infection persist equilibrium

	5. Asymptotic stability analysis of the infection free equilibrium
	5.1. Local stability analysis of E0
	5.2. Global stability analysis of E0
	5.3. Numerical results

	6. Asymptotic stability analysis of the infection persist equilibrium
	6.1. Local stability analysis of E1
	6.2. Global stability analysis of E1
	6.3. Numerical results

	7. Conclusion
	Disclosure statement
	References