# Stability Analysis for Viral Infection Model with Multitarget Cells, Beddington-DeAngelis Functional Response, and Humoral Immunity

Authors: Xinxin Tian; Jinliang Wang
Journal: Discrete Dynamics in Nature and Society
 (2015)

---

## Abstract

We formulate a (2n+2)-dimensional viral infection model with humoral immunity, n classes of uninfected target cells and  n classes of infected cells. The incidence rate of infection is given by nonlinear incidence rate, Beddington-DeAngelis functional response. The model admits discrete time delays describing the time needed for infection of uninfected target cells and virus replication. By constructing suitable Lyapunov functionals, we establish that the global dynamics are determined by two sharp threshold parameters: R
0 and R
1. Namely, a typical two-threshold scenario is shown. If R
0
≤
1, the infection-free equilibrium P
0 is globally asymptotically stable, and the viruses are cleared. If R
1
≤
1
<
R
0, the immune-free equilibrium P
1 is globally asymptotically stable, and the infection becomes chronic but with no persistent antibody immune response. If R
1
>
1, the endemic equilibrium P
2 is globally asymptotically stable, and the infection is chronic with persistent antibody immune response.


---

## Body

## 1. Introduction

In the study of mathematical models of infectious diseasesin vivo, it is an important problem to predict whether the infection disappears or the pathogens and the immune system persist. In general, the mathematical models of infectious diseasesin vivo have a general immune response. For instance, CTLs can kill infected cells or they can secrete soluble factors, which can inhibit viral replication. Another immune response to viral infection is antibody immune response (humoral immunity), which is widely developed to analyze the dynamics of infections agents such as HIV and malaria (see, e.g., [1, 2]).The basic viral infection model with antibody immunity response studied by Kajiwara and Sasaki [3] takes the following four-dimensional nonlinear differential equations:(1)
x
′
(
t
)
=
λ
-
d
x
(
t
)
-
β
x
(
t
)
v
(
t
)
,
y
′
(
t
)
=
β
x
(
t
)
v
(
t
)
-
p
y
(
t
)
,
v
′
(
t
)
=
k
y
(
t
)
-
μ
v
(
t
)
-
q
v
(
t
)
w
(
t
)
,
w
′
(
t
)
=
g
v
(
t
)
w
(
t
)
-
h
w
(
t
)
,where the uninfected CD
4
+ T cells, x
(
t
), are produced at a constant rate of λ, die at a rate of d, and become infected at a rate of β
x
v. The infected cells, y
(
t
), are produced at a rate of β
x
v. β and p are the infection rate and death rate of infected cells, respectively. Free virus particles, v
(
t
), are released from infected cells at a rate of k. μ is the death rate of virus. w
(
t
) denotes the density of the antibody immune response (B cells). g and μ are the birth rate and death rate of B cells. q is the B cells neutralization rate.In [3], Kajiwara and Sasaki studied the local stability of the equilibria of system (1). But local stability is not equivalent to global stability. The question of global stability in population models is a very interesting mathematical problem. Many authors have studied the global stability of virus dynamics models without delay using the second Lyapunov method. The Lyapunov function candidate for population biology models is the Volterra-type function which was applied in literatures [4–9] to prove global stability of the steady states of viral infections models with discrete intracellular delay and distributed delay. But they all ignore the antibody immune response. In [8], Wang and Zou investigated the dynamical behavior of in-host viral models with humoral immunity and intracellular delays. By means of constructing suitable Lyapunov functionals and Lasalle invariance principles, they established the global dynamics by two sharp thresholds.Most of the existing delayed HIV infection models are based on the assumption that the virus attacks one class of target cells,CD
4
+ T cells. In [10, 11], an HIV model with two target cells, CD
4
+ T cells and macrophages, has been proposed. Recently, a series of work [12–17] has been done for HIV models with two target cells with discrete-time and distributed delays. Lyapunov functionals are proved to be effective tool to establish the global asymptotic stability of their steady states [18–20].Letx
i
(
t
) and y
i
(
t
) be the populations of the uninfected target cells and infected cells of class i, respectively, and let v
(
t
) be the population of the virus particles. The basic virus infection model with multitarget cells proposed by Elaiw [15] takes the following form of ordinary differential equations: (2)
d
x
i
(
t
)
d
t
=
λ
i
-
d
i
x
i
(
t
)
-
β
i
x
i
(
t
)
v
(
t
)
,
d
y
i
(
t
)
d
t
=
β
i
x
i
(
t
)
v
(
t
)
-
p
i
y
i
(
t
)
,
d
v
(
t
)
d
t
=
∑
i
=
1
n
‍
k
i
y
i
(
t
)
-
c
v
(
t
)
,where, for i
=
1,2
,
…
,
n, λ
i, d
i, and β
i represent the rates of which new target cells are generated, the death rate constants, and the infection rate constants, respectively. The infected cells die with rate constants p
i. The virus particles are produced by the infected cells with rate constants k
i and are cleared with rate constant c.From system (2), the rate of infection of these viral dynamics models is assumed to be bilinear in the virus v and the uninfected CD
4
+ T cells x
i. However, the actual incidence rate is probably not linear over the entire range of v and x
i. Thus, it is reasonable to assume that the infection rate of viral infection model is given by the Beddington-DeAngelis functional response, β
i
x
i
v
/
(
1
+
a
i
x
i
+
b
i
v
), where a
i
,
b
i
≥
0 [7, 21]. Time delays are intrinsic to the viral infection and replication processes, since antigenic stimulation generating immune response may need a period of time. Let parameter τ account 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. Let m
i be a constant rate for infected but not yet virus-producing cells; the probability of surviving the time period from t
-
τ to t is e
-
m
i
τ.Combining model (1), model (2), and the preceding assumptions above, we arrive at the following viral infection model:(3)
x
i
′
(
t
)
=
λ
i
-
d
i
x
i
(
t
)
-
β
i
x
i
(
t
)
v
(
t
)
1
+
a
i
x
i
(
t
)
+
b
i
v
(
t
)
,
y
i
′
(
t
)
=
β
i
e
-
m
i
τ
x
i
(
t
-
τ
)
v
(
t
-
τ
)
1
+
a
i
x
i
(
t
-
τ
)
+
b
i
v
(
t
-
τ
)
-
p
i
y
i
(
t
)
,
v
′
(
t
)
=
∑
i
=
1
n
‍
k
i
y
i
(
t
)
-
μ
v
(
t
)
-
q
v
(
t
)
w
(
t
)
,
w
′
(
t
)
=
g
v
(
t
)
w
(
t
)
-
h
w
(
t
)
.Here the state variables x
i, y
i, v, and w and parameters λ
i
,
d
i
,
β
i
,
m
i
,
p
i
,
k
i
,
μ
,
q
,
g, and h have the same biological meanings as in the systems (1) and (2). a
i and b
i are constant.For system (3), we derive the basic reproductive number R
0 and antibody immune activation number R
1 and show that R
0 and R
1 completely determine the global dynamics. The global stability result for the equilibria is new for in-host models with Beddington-DeAngelis functional response, antibody immune response, and intracellular delays. Our proof utilizes a global Lyapunov functional that is motivated by the earlier works mentioned above. The global stability of equilibria rules out any possibility for Hopf bifurcations and existence of sustained oscillations.The paper is organized as follows. Section2 is devoted to showing the positivity and ultimate boundedness of the solutions for (1) under suitable initial conditions. Then, we introduce threshold parameter, the basic reproduction number for viral infection R
0, antibody immune response reproduction number R
1, and three possible equilibria for (3). In Section 3, we establish global asymptotic stability of infection-free equilibrium by constructing Lyapunov functional. For a special case i
=
1, we give the proof of global stability of equilibria for a 4-dimensional model in Section 4. We provide some numerical simulations which confirm our analysis in Section 5. In the last section, we offer a brief discussion.


## 2. Preliminaries

The initial conditions for system (3) take the following form:(4)
x
i
θ
=
ϕ
i
θ
,
y
i
θ
=
ψ
i
θ
,
v
θ
=
χ
θ
,
w
θ
=
φ
θ
,
ϕ
i
θ
,
ψ
i
θ
,
χ
θ
,
φ
θ
≥
0
,
θ
∈
-
τ
,
0
,
ϕ
i
0
,
ψ
i
0
,
χ
0
,
φ
0
>
0
,
i
=
1,2
,
3
,
…
,
n
,where (
ϕ
i
(
θ
)
,
ψ
i
(
θ
)
,
χ
(
θ
)
,
ψ
(
θ
)
)
∈
C
(
[
-
τ
,
0
]
,
R
+
2
n
+
2
).


### 2.1. Nonnegativity and Boundedness of Solutions

In the following, we establish the nonnegativity and boundedness of solutions of (3).Theorem 1.
Let(
x
i
(
t
)
,
y
i
(
t
)
,
v
(
t
)
,
w
(
t
)
) be any solution of system (3) satisfying the initial conditions (4); then x
i
(
t
)
,
y
i
(
t
)
,
v
(
t
), and w
(
t
) are all nonnegative for t
>
0 and are ultimately bounded.Proof.
First, fori
=
1,2
,
…
,
n, we prove that x
i
(
t
)
>
0 for all t
≥
0. Assume the contrary and let t
0
>
0 be such that x
i
(
t
0
)
<
0. Set t
01
=
inf
⁡
{
0
<
t
<
t
0
:
x
i
(
t
)
<
0
}. Then x
i
(
t
01
)
=
0, and from the first equation of system (3) we have x
i
′
(
t
01
)
=
λ
i
>
0. Hence x
(
t
)
<
0 for t
∈
(
t
01
-
ϵ
,
t
01
) and ϵ
>
0 sufficiently small. This contradicts x
(
t
)
≥
0 for t
∈
(
0
,
t
01
]. It follows that, for i
=
1,2
,
…
,
n, x
i
(
t
)
>
0 for t
≥
0.
Furthermore, if there existst
1
>
0, such that y
i
(
t
1
)
=
0, and y
i
(
t
)
>
0 for all t
∈
(
0
,
t
1
), then by the second equation of system (3), we can easily obtain y
i
′
(
t
)
≥
-
p
i
y
i
(
t
). And it follows that y
i
(
t
)
≥
e
-
p
i
t for all t
≥
0. This is a contradiction to the assumption of y
i
(
t
1
)
=
0. Hence, y
i
(
t
)
>
0, for all t
≥
0.
Finally, if there existst
2
>
0, such that v
(
t
2
)
=
0, and v
(
t
)
>
0, for t
∈
(
0
,
t
2
), then by the forth equation of system (3), we have v
′
(
t
)
≥
-
u
v
(
t
)
-
q
v
(
t
)
w
(
t
). And it follows that v
(
t
)
≥
e
-
(
u
+
q
w
(
t
)
) for all t
≥
0. This contradicts with the assumption of v
(
t
2
)
=
0. Hence, v
(
t
)
>
0, for all t
≥
0. Similarly, we have w
(
t
)
>
0, for t
∈
(
0
,
t
3
).
Next we show that the solution is ultimately bounded. It follows from the first equation of system (3) that (5)
x
i
′
(
t
)
≤
λ
i
-
d
i
x
i
(
t
)
.Thus limsup
⁡
t
→
∞
x
i
(
t
)
≤
λ
i
/
d
i and x
i
(
t
) is ultimately bounded. Let (6)
X
i
=
e
-
m
i
τ
x
i
t
-
τ
+
y
i
t
;then (7)
X
i
′
(
t
)
≤
λ
i
e
-
m
i
τ
-
σ
i
X
i
(
t
)
,where σ
i
=
min
⁡
{
d
i
,
p
i
}. It follows that limsup
⁡
t
→
∞
X
i
≤
M
1, where M
1
=
λ
i
e
-
m
i
τ
/
σ
i. This in turn implies, by the nonnegativity of x
i
(
t
) and y
i
(
t
), that limsup
⁡
t
→
∞
y
i
≤
M
1 and y
i
(
t
) is ultimately bounded. On the other hand, from the third equation of system (3) we have (8)
v
′
t
≤
∑
i
=
1
n
‍
k
i
M
1
-
u
v
-
q
v
w
;then limsup
⁡
t
→
∞
v
(
t
)
≤
M
2, where M
2
=
∑
i
=
1
n
‍
k
i
M
1
/
u and v
(
t
) is ultimately bounded. Let V
=
v
+
w; then (9)
V
′
≤
∑
i
=
1
n
‍
k
i
y
i
-
u
v
-
(
d
M
2
+
h
)
w
,where d
=
min
⁡
{
q
,
g
}; then let m
=
min
⁡
{
u
,
d
M
2
+
h
}; then (10)
V
′
≤
∑
i
=
1
n
‍
k
i
y
i
-
m
V
.It follows that limsup
⁡
t
→
∞
V
≤
M
3, where M
3
=
∑
i
=
1
n
‍
k
i
y
i
/
m. This in turn implies, by the nonnegativity of v
(
t
) and w
(
t
), that limsup
⁡
t
→
∞
w
(
t
)
≤
M
3 and w
(
t
) is ultimately bounded.


### 2.2. Reproduction Numbers and Steady States

The basic reproductive number of the virus for system (3) is (11)
R
0
=
∑
i
=
1
n
‍
R
i
,where R
i
=
λ
i
β
i
k
i
e
-
m
i
τ
/
d
i
p
i
μ
+
a
i
p
i
μ
λ
i. We will know that if R
0
≤
1, then infection-free equilibrium P
0
=
(
x
i
0
,
0,0
,
0
), x
i
0
=
λ
i
/
d
i, is the unique steady state. If R
0
>
1, then in addition to the uninfected steady state there exists an immune-free equilibrium P
1
=
(
x
i
1
,
y
i
1
,
v
1
,
0
), corresponding to the situation that infection becomes chronic but with no persistent antibody immune response, which is given by the following expressions: (12)
x
i
1
=
∑
i
=
1
n
‍
(
λ
i
b
i
k
i
-
λ
i
k
i
β
i
)
+
d
i
p
i
u
e
m
i
τ
+
λ
i
k
i
β
i
d
i
(
k
i
β
i
+
∑
i
=
1
n
‍
b
i
d
i
k
i
-
a
i
p
i
μ
e
m
i
τ
)
,
y
i
1
=
∑
i
=
1
n
‍
β
i
e
-
m
i
τ
k
i
λ
i
-
d
i
p
i
μ
-
a
i
p
i
μ
λ
i
∑
i
=
1
n
‍
p
i
k
i
d
i
b
i
+
β
i
k
i
p
i
-
p
i
2
a
i
μ
e
m
i
τ
(
R
i
-
1
)
,
v
1
=
∑
i
=
1
n
‍
k
i
μ
∑
i
=
1
n
‍
β
i
e
-
m
i
τ
k
i
λ
i
-
d
i
p
i
μ
-
a
i
p
i
μ
λ
i
∑
i
=
1
n
‍
p
i
k
i
d
i
b
i
+
β
i
k
i
p
i
-
p
i
2
a
i
μ
e
m
i
τ
(
R
i
-
1
)
.Now, we introduce an antibody immune response reproduction number: (13)
R
1
=
∑
i
=
1
n
‍
μ
(
∑
i
=
1
n
‍
p
i
k
i
d
i
b
i
+
β
i
k
i
p
i
-
p
i
2
a
i
μ
e
m
i
τ
)
g
k
i
(
∑
i
=
1
n
‍
β
i
e
-
m
i
τ
k
i
λ
i
-
d
i
p
i
μ
-
a
i
p
i
μ
λ
i
)
(
g
v
1
-
h
)
+
1
.Further, if (14)
R
0
>
1
+
∑
i
=
1
n
‍
μ
∑
i
=
1
n
‍
p
i
k
i
d
i
b
i
+
β
i
k
i
p
i
-
p
i
2
a
i
μ
e
m
i
τ
h
g
k
i
∑
i
=
1
n
‍
β
i
e
-
m
i
τ
k
i
λ
i
-
d
i
p
i
μ
-
a
i
p
i
μ
λ
i
,system (3) also has an endemic equilibrium P
2
=
(
x
i
∗
,
y
i
∗
,
v
∗
,
w
∗
), where (15)
x
i
∗
=
+
4
a
i
d
i
λ
i
+
4
λ
i
b
i
h
a
i
d
i
g
1
/
2
a
i
λ
i
-
d
i
-
b
i
d
i
h
g
-
β
i
h
g
Hi
+
a
i
λ
i
-
d
i
-
b
i
d
i
h
g
-
β
i
h
g
2
h
h
h
h
h
h
+
4
a
i
d
i
λ
i
+
4
λ
i
b
i
h
a
i
d
i
g
1
/
2
×
2
a
i
d
i
-
1
,
v
∗
=
h
g
,
y
i
∗
=
β
i
e
-
m
i
τ
x
i
∗
v
∗
p
i
(
1
+
a
i
x
i
∗
+
b
i
v
∗
)
,
w
∗
=
∑
i
=
1
n
‍
k
i
g
y
i
∗
q
h
-
μ
q
.It follows from (16)
R
1
=
R
0
-
∑
i
=
1
n
‍
μ
∑
i
=
1
n
‍
p
i
k
i
d
i
b
i
+
β
i
k
i
p
i
-
p
i
2
a
i
μ
e
m
i
τ
h
g
k
i
∑
i
=
1
n
‍
β
i
e
-
m
i
τ
k
i
λ
i
-
d
i
p
i
μ
-
a
i
p
i
μ
λ
ithat R
1
>
1 is equivalent to g
v
1
/
h
>
1. The latter g
v
1
/
h can be regarded as the immune activation effector. It is reasonable that immune response is activated in the case where R
1
>
1.


## 2.1. Nonnegativity and Boundedness of Solutions

In the following, we establish the nonnegativity and boundedness of solutions of (3).Theorem 1.
Let(
x
i
(
t
)
,
y
i
(
t
)
,
v
(
t
)
,
w
(
t
)
) be any solution of system (3) satisfying the initial conditions (4); then x
i
(
t
)
,
y
i
(
t
)
,
v
(
t
), and w
(
t
) are all nonnegative for t
>
0 and are ultimately bounded.Proof.
First, fori
=
1,2
,
…
,
n, we prove that x
i
(
t
)
>
0 for all t
≥
0. Assume the contrary and let t
0
>
0 be such that x
i
(
t
0
)
<
0. Set t
01
=
inf
⁡
{
0
<
t
<
t
0
:
x
i
(
t
)
<
0
}. Then x
i
(
t
01
)
=
0, and from the first equation of system (3) we have x
i
′
(
t
01
)
=
λ
i
>
0. Hence x
(
t
)
<
0 for t
∈
(
t
01
-
ϵ
,
t
01
) and ϵ
>
0 sufficiently small. This contradicts x
(
t
)
≥
0 for t
∈
(
0
,
t
01
]. It follows that, for i
=
1,2
,
…
,
n, x
i
(
t
)
>
0 for t
≥
0.
Furthermore, if there existst
1
>
0, such that y
i
(
t
1
)
=
0, and y
i
(
t
)
>
0 for all t
∈
(
0
,
t
1
), then by the second equation of system (3), we can easily obtain y
i
′
(
t
)
≥
-
p
i
y
i
(
t
). And it follows that y
i
(
t
)
≥
e
-
p
i
t for all t
≥
0. This is a contradiction to the assumption of y
i
(
t
1
)
=
0. Hence, y
i
(
t
)
>
0, for all t
≥
0.
Finally, if there existst
2
>
0, such that v
(
t
2
)
=
0, and v
(
t
)
>
0, for t
∈
(
0
,
t
2
), then by the forth equation of system (3), we have v
′
(
t
)
≥
-
u
v
(
t
)
-
q
v
(
t
)
w
(
t
). And it follows that v
(
t
)
≥
e
-
(
u
+
q
w
(
t
)
) for all t
≥
0. This contradicts with the assumption of v
(
t
2
)
=
0. Hence, v
(
t
)
>
0, for all t
≥
0. Similarly, we have w
(
t
)
>
0, for t
∈
(
0
,
t
3
).
Next we show that the solution is ultimately bounded. It follows from the first equation of system (3) that (5)
x
i
′
(
t
)
≤
λ
i
-
d
i
x
i
(
t
)
.Thus limsup
⁡
t
→
∞
x
i
(
t
)
≤
λ
i
/
d
i and x
i
(
t
) is ultimately bounded. Let (6)
X
i
=
e
-
m
i
τ
x
i
t
-
τ
+
y
i
t
;then (7)
X
i
′
(
t
)
≤
λ
i
e
-
m
i
τ
-
σ
i
X
i
(
t
)
,where σ
i
=
min
⁡
{
d
i
,
p
i
}. It follows that limsup
⁡
t
→
∞
X
i
≤
M
1, where M
1
=
λ
i
e
-
m
i
τ
/
σ
i. This in turn implies, by the nonnegativity of x
i
(
t
) and y
i
(
t
), that limsup
⁡
t
→
∞
y
i
≤
M
1 and y
i
(
t
) is ultimately bounded. On the other hand, from the third equation of system (3) we have (8)
v
′
t
≤
∑
i
=
1
n
‍
k
i
M
1
-
u
v
-
q
v
w
;then limsup
⁡
t
→
∞
v
(
t
)
≤
M
2, where M
2
=
∑
i
=
1
n
‍
k
i
M
1
/
u and v
(
t
) is ultimately bounded. Let V
=
v
+
w; then (9)
V
′
≤
∑
i
=
1
n
‍
k
i
y
i
-
u
v
-
(
d
M
2
+
h
)
w
,where d
=
min
⁡
{
q
,
g
}; then let m
=
min
⁡
{
u
,
d
M
2
+
h
}; then (10)
V
′
≤
∑
i
=
1
n
‍
k
i
y
i
-
m
V
.It follows that limsup
⁡
t
→
∞
V
≤
M
3, where M
3
=
∑
i
=
1
n
‍
k
i
y
i
/
m. This in turn implies, by the nonnegativity of v
(
t
) and w
(
t
), that limsup
⁡
t
→
∞
w
(
t
)
≤
M
3 and w
(
t
) is ultimately bounded.


## 2.2. Reproduction Numbers and Steady States

The basic reproductive number of the virus for system (3) is (11)
R
0
=
∑
i
=
1
n
‍
R
i
,where R
i
=
λ
i
β
i
k
i
e
-
m
i
τ
/
d
i
p
i
μ
+
a
i
p
i
μ
λ
i. We will know that if R
0
≤
1, then infection-free equilibrium P
0
=
(
x
i
0
,
0,0
,
0
), x
i
0
=
λ
i
/
d
i, is the unique steady state. If R
0
>
1, then in addition to the uninfected steady state there exists an immune-free equilibrium P
1
=
(
x
i
1
,
y
i
1
,
v
1
,
0
), corresponding to the situation that infection becomes chronic but with no persistent antibody immune response, which is given by the following expressions: (12)
x
i
1
=
∑
i
=
1
n
‍
(
λ
i
b
i
k
i
-
λ
i
k
i
β
i
)
+
d
i
p
i
u
e
m
i
τ
+
λ
i
k
i
β
i
d
i
(
k
i
β
i
+
∑
i
=
1
n
‍
b
i
d
i
k
i
-
a
i
p
i
μ
e
m
i
τ
)
,
y
i
1
=
∑
i
=
1
n
‍
β
i
e
-
m
i
τ
k
i
λ
i
-
d
i
p
i
μ
-
a
i
p
i
μ
λ
i
∑
i
=
1
n
‍
p
i
k
i
d
i
b
i
+
β
i
k
i
p
i
-
p
i
2
a
i
μ
e
m
i
τ
(
R
i
-
1
)
,
v
1
=
∑
i
=
1
n
‍
k
i
μ
∑
i
=
1
n
‍
β
i
e
-
m
i
τ
k
i
λ
i
-
d
i
p
i
μ
-
a
i
p
i
μ
λ
i
∑
i
=
1
n
‍
p
i
k
i
d
i
b
i
+
β
i
k
i
p
i
-
p
i
2
a
i
μ
e
m
i
τ
(
R
i
-
1
)
.Now, we introduce an antibody immune response reproduction number: (13)
R
1
=
∑
i
=
1
n
‍
μ
(
∑
i
=
1
n
‍
p
i
k
i
d
i
b
i
+
β
i
k
i
p
i
-
p
i
2
a
i
μ
e
m
i
τ
)
g
k
i
(
∑
i
=
1
n
‍
β
i
e
-
m
i
τ
k
i
λ
i
-
d
i
p
i
μ
-
a
i
p
i
μ
λ
i
)
(
g
v
1
-
h
)
+
1
.Further, if (14)
R
0
>
1
+
∑
i
=
1
n
‍
μ
∑
i
=
1
n
‍
p
i
k
i
d
i
b
i
+
β
i
k
i
p
i
-
p
i
2
a
i
μ
e
m
i
τ
h
g
k
i
∑
i
=
1
n
‍
β
i
e
-
m
i
τ
k
i
λ
i
-
d
i
p
i
μ
-
a
i
p
i
μ
λ
i
,system (3) also has an endemic equilibrium P
2
=
(
x
i
∗
,
y
i
∗
,
v
∗
,
w
∗
), where (15)
x
i
∗
=
+
4
a
i
d
i
λ
i
+
4
λ
i
b
i
h
a
i
d
i
g
1
/
2
a
i
λ
i
-
d
i
-
b
i
d
i
h
g
-
β
i
h
g
Hi
+
a
i
λ
i
-
d
i
-
b
i
d
i
h
g
-
β
i
h
g
2
h
h
h
h
h
h
+
4
a
i
d
i
λ
i
+
4
λ
i
b
i
h
a
i
d
i
g
1
/
2
×
2
a
i
d
i
-
1
,
v
∗
=
h
g
,
y
i
∗
=
β
i
e
-
m
i
τ
x
i
∗
v
∗
p
i
(
1
+
a
i
x
i
∗
+
b
i
v
∗
)
,
w
∗
=
∑
i
=
1
n
‍
k
i
g
y
i
∗
q
h
-
μ
q
.It follows from (16)
R
1
=
R
0
-
∑
i
=
1
n
‍
μ
∑
i
=
1
n
‍
p
i
k
i
d
i
b
i
+
β
i
k
i
p
i
-
p
i
2
a
i
μ
e
m
i
τ
h
g
k
i
∑
i
=
1
n
‍
β
i
e
-
m
i
τ
k
i
λ
i
-
d
i
p
i
μ
-
a
i
p
i
μ
λ
ithat R
1
>
1 is equivalent to g
v
1
/
h
>
1. The latter g
v
1
/
h can be regarded as the immune activation effector. It is reasonable that immune response is activated in the case where R
1
>
1.


## 3. Main Results

In this section, we consider the global asymptotic stability of the three equilibria. Throughout the paper, for easy notation, we adoptH
(
s
)
=
s
-
1
-
ln
⁡
s
≥
H
(
1
)
=
0 to simplify many of the expressions which follow.Theorem 2.
Consider system (3) and R
0 is defined by (11). The infection-free equilibrium P
0 of (3) is globally asymptotically stable if R
0
≤
1.Proof.
Define a Lyapunov functionalL
0 as follows: (17)
L
0
=
∑
i
=
1
n
‍
x
i
0
1
+
a
i
x
i
0
e
-
m
i
τ
H
x
i
x
i
0
+
y
i
+
p
i
k
i
v
+
p
i
q
k
I
g
w
+
β
i
e
-
m
i
τ
∫
t
-
τ
t
‍
x
i
(
θ
)
v
(
θ
)
1
+
a
i
x
i
(
θ
)
+
b
i
v
(
θ
)
d
θ
.Using λ
i
=
d
i
x
i
0 and calculating the time derivative of L
0 along (3) yield(18)
L
0
′
=
∑
i
=
1
n
‍
e
-
m
i
τ
1
+
a
i
x
i
0
1
-
x
i
0
x
i
(
t
)
x
i
′
(
t
)
+
y
i
′
(
t
)
hihh
+
β
i
e
-
m
i
τ
x
i
v
1
+
a
i
x
i
+
b
i
v
-
β
i
e
-
m
i
τ
x
i
(
t
-
τ
)
v
(
t
-
τ
)
1
+
a
i
x
i
(
t
-
τ
)
+
b
i
v
(
t
-
τ
)
e
-
m
i
τ
1
+
a
i
x
i
0
1
-
x
i
0
x
i
(
t
)
+
p
i
k
i
v
′
(
t
)
+
p
i
q
k
i
g
w
′
(
t
)
=
∑
i
=
1
n
‍
-
d
i
e
-
m
i
τ
x
i
(
1
+
a
i
x
i
0
)
x
i
-
x
i
0
2
hihh
-
β
i
e
-
m
i
τ
x
i
v
(
1
+
a
i
x
i
0
)
(
1
+
a
i
x
i
+
b
i
v
)
-
p
i
μ
k
i
v
-
p
i
q
h
k
i
g
w
hihh
+
β
i
e
-
m
i
τ
x
i
0
v
1
+
a
i
x
i
0
1
+
a
i
x
i
+
b
i
v
+
β
i
e
-
m
i
τ
x
i
v
1
+
a
i
x
i
+
b
i
v
=
∑
i
=
1
n
‍
-
d
i
e
-
m
i
τ
x
i
1
+
a
i
x
i
0
x
i
-
x
i
0
2
+
R
0
-
1
v
-
∑
i
=
1
n
‍
b
i
β
i
e
-
m
i
τ
x
i
0
(
1
+
a
i
x
i
0
)
(
1
+
a
i
x
i
+
b
i
v
)
v
2
-
p
i
q
h
k
i
g
w
.Thus, R
0
≤
1 ensures that L
0
′
<
0 for all x
i
>
0, y
i
≥
0, v
>
0, and w
>
0. And L
0
′
=
0 if and only if x
i
=
x
i
0, v
=
w
=
0 for R
0
≤
1. It is easy to show that P
0
=
(
x
i
0
,
0,0
,
0
) is the largest invariant set in {
(
x
i
,
y
i
,
v
,
w
)
∣
L
0
′
=
0
}. By LaSalle invariance principle, the equilibrium P
0
(
x
i
0
,
0,0
,
0
) is globally asymptotically stable. This completes the proof.Theorem 3.
Consider system (3) and R
0 and R
1 are defined by (11) and (13). The immune-free equilibrium P
1 of (3) is globally asymptotically stable if R
1
≤
1
<
R
0.Proof.
Define a Lyapunov functionalL
1 as follows:(19)
L
1
=
∑
i
=
1
n
‍
x
i
1
(
1
+
b
i
v
1
)
e
-
m
i
τ
1
+
a
i
x
i
1
+
b
i
v
1
H
x
i
x
i
1
+
y
i
H
y
i
y
i
1
hhhh
+
p
i
y
i
1
∫
t
-
τ
t
‍
H
1
+
a
i
x
i
1
+
b
i
v
1
x
i
1
v
1
x
i
(
θ
)
v
(
θ
)
1
+
a
i
x
i
(
θ
)
+
b
i
v
(
θ
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
×
1
+
a
i
x
i
1
+
b
i
v
1
x
i
1
v
1
d
θ
x
i
1
(
1
+
b
i
v
1
)
e
-
m
i
τ
1
+
a
i
x
i
1
+
b
i
v
1
H
x
i
x
i
1
+
p
i
v
1
k
i
H
v
v
1
+
p
i
q
k
i
g
w
.The derivative of L
1 is given by (20)
L
1
′
=
∑
i
=
1
n
‍
(
1
+
b
i
v
1
)
e
-
m
i
τ
1
+
a
i
x
i
1
+
b
i
v
1
1
-
x
i
1
x
i
x
i
′
+
1
-
y
i
1
y
i
y
i
′
hhhh
+
β
e
-
m
i
τ
x
i
v
1
+
a
i
x
i
+
b
i
v
-
β
i
e
-
m
i
τ
x
i
t
-
τ
v
t
-
τ
1
+
a
i
x
i
t
-
τ
+
b
i
v
t
-
τ
hhhh
+
p
i
y
i
1
ln
⁡
x
i
(
t
-
τ
)
v
(
t
-
τ
)
x
i
v
hhhh
×
1
+
a
i
x
i
+
b
i
v
1
+
a
i
x
i
(
t
-
τ
)
+
b
i
v
(
t
-
τ
)
1
-
x
i
1
x
i
(
1
+
b
i
v
1
)
e
-
m
i
τ
1
+
a
i
x
i
1
+
b
i
v
1
+
p
i
k
i
1
-
v
1
v
v
′
+
p
i
q
k
i
g
w
′
=
∑
i
=
1
n
‍
p
i
y
i
1
b
i
1
+
a
i
x
i
v
-
v
1
2
v
1
1
+
a
i
x
i
+
b
i
v
1
+
a
i
x
i
+
b
i
v
1
-
d
i
(
1
+
b
i
v
1
)
e
-
m
i
τ
x
i
(
1
+
a
i
x
i
1
+
b
i
v
1
)
x
i
-
x
i
1
2
hhhh
+
p
i
y
i
1
1
-
x
i
1
(
1
+
a
i
x
i
+
b
i
v
1
)
x
i
(
1
+
a
i
x
i
1
+
b
i
v
1
)
h
h
h
h
h
h
h
h
h
h
h
h
h
+
ln
⁡
x
i
1
(
1
+
a
i
x
i
+
b
i
v
1
)
x
i
(
1
+
a
i
x
i
1
+
b
i
v
1
)
hhhh
+
p
i
y
i
1
hhhh
×
1
-
y
i
1
1
+
a
i
x
i
1
+
b
i
v
1
x
i
t
-
τ
v
t
-
τ
y
i
x
i
1
v
1
1
+
a
i
x
i
t
-
τ
+
b
i
v
t
-
τ
h
h
h
h
h
h
h
+
ln
⁡
y
i
1
(
1
+
a
i
x
i
1
+
b
i
v
1
)
x
i
(
t
-
τ
)
v
(
t
-
τ
)
y
i
x
i
1
v
1
(
1
+
a
i
x
i
(
t
-
τ
)
+
b
i
v
(
t
-
τ
)
)
hhhh
+
p
i
y
i
1
1
-
v
1
y
i
y
i
1
v
+
ln
⁡
v
1
y
i
y
i
1
v
hhhh
+
p
i
y
i
1
1
-
1
+
a
i
x
i
+
b
i
v
1
+
a
i
x
i
+
b
i
v
1
+
ln
⁡
1
+
a
i
x
i
+
b
i
v
1
+
a
i
x
i
+
b
i
v
1
hhhh
-
p
i
y
i
1
b
i
1
+
a
i
x
i
v
-
v
1
2
v
1
1
+
a
i
x
i
+
b
i
v
1
+
a
i
x
i
+
b
i
v
1
+
p
i
q
(
g
v
1
-
h
)
k
i
g
w
.In the above calculations, we used the equations λ
i
=
d
i
x
i
1
+
β
i
x
i
1
v
1
/
(
1
+
a
i
x
i
1
+
b
i
v
1
), ∑
i
=
1
n
k
i
y
i
1
=
u
v
1
‍, and β
i
e
-
m
i
τ
x
i
1
v
1
/
(
1
+
a
i
x
i
1
+
b
i
v
1
)
=
p
i
y
i
1.
Further,R
1
≤
1
<
R
0 is equivalent to v
1
≤
h
/
g. Hence L
1
′ is always nonpositive under the condition R
1
≤
1
<
R
0, and L
1
′
=
0 if and only if x
i
=
x
i
1, y
i
/
y
i
1
=
v
/
v
1, and w
=
0 for R
1
<
1 or x
i
=
x
i
1, y
i
/
y
i
1
=
v
/
v
1 for R
1
=
1. Similarly, it follows from LaSalle invariance principle that the immune-free equilibrium P
1 is globally asymptotically stable. This completes the proof.Theorem 4.
Consider system (3) and R
1 is defined by (13). The endemic equilibrium P
2 of (3) is globally asymptotically stable if R
1
>
1.Proof.
Define a Lyapunov functionalL
2 as follows: (21)
L
2
=
∑
i
=
1
n
‍
x
i
∗
(
1
+
b
v
i
∗
)
e
-
m
i
τ
1
+
a
i
x
i
∗
+
b
i
v
∗
H
x
i
x
i
∗
+
y
i
∗
H
y
i
y
i
∗
hhhh
+
p
i
y
i
∗
∫
t
-
τ
t
‍
H
1
+
a
i
x
i
∗
+
b
i
v
∗
x
i
∗
v
∗
x
i
(
θ
)
v
(
θ
)
1
+
a
i
x
i
(
θ
)
+
b
i
v
(
θ
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
×
1
+
a
i
x
i
∗
+
b
i
v
∗
x
i
∗
v
∗
d
θ
+
p
i
v
∗
k
i
H
v
v
∗
+
p
i
q
w
∗
k
i
g
H
w
w
∗
.The derivative of L
2 is given by (22)
L
2
′
=
∑
i
=
1
n
‍
e
-
m
i
τ
(
1
+
b
i
v
∗
)
1
+
a
i
x
i
∗
+
b
i
v
∗
1
-
x
i
∗
x
i
x
i
′
+
1
-
y
i
∗
y
i
y
i
′
hhhh
+
β
i
e
-
m
i
τ
x
i
v
1
+
a
i
x
i
+
b
i
v
-
β
i
e
-
m
i
τ
x
i
t
-
τ
v
t
-
τ
1
+
a
i
x
i
t
-
τ
+
b
i
v
t
-
τ
hhhh
+
p
i
y
i
∗
ln
⁡
x
i
(
t
-
τ
)
v
(
t
-
τ
)
x
i
v
hhhh
×
1
+
a
i
x
i
+
b
i
v
1
+
a
i
x
i
t
-
τ
+
b
i
v
t
-
τ
1
-
x
i
∗
x
i
e
-
m
i
τ
(
1
+
b
i
v
∗
)
1
+
a
i
x
i
∗
+
b
i
v
∗
+
p
i
k
i
1
-
v
∗
v
v
′
+
p
i
q
k
i
g
1
-
w
∗
w
w
′
=
∑
i
=
1
n
‍
-
d
i
1
+
b
i
v
∗
e
-
m
i
τ
x
i
1
+
a
i
x
i
∗
+
b
i
v
∗
x
i
-
x
i
∗
2
+
∑
i
=
1
n
‍
p
i
y
i
∗
1
-
x
i
∗
(
1
+
a
i
x
i
+
b
i
v
∗
)
x
i
(
1
+
a
i
x
i
∗
+
b
i
v
∗
)
h
h
h
h
h
h
h
h
h
h
+
ln
⁡
x
i
∗
(
1
+
a
i
x
i
+
b
i
v
∗
)
x
i
(
1
+
a
i
x
i
∗
+
b
i
v
∗
)
+
∑
i
=
1
n
‍
p
i
y
i
∗
1
-
y
i
∗
(
1
+
a
i
x
i
∗
+
b
i
v
∗
)
x
i
(
t
-
τ
)
v
(
t
-
τ
)
y
i
x
i
∗
v
∗
(
1
+
a
i
x
i
(
t
-
τ
)
+
b
i
v
(
t
-
τ
)
)
h
h
h
h
h
h
h
h
+
ln
⁡
y
i
∗
(
1
+
a
i
x
i
∗
+
b
i
v
∗
)
x
i
(
t
-
τ
)
v
(
t
-
τ
)
y
i
x
i
∗
v
∗
(
1
+
a
i
x
i
(
t
-
τ
)
+
b
i
v
(
t
-
τ
)
)
-
∑
i
=
1
n
‍
p
y
i
∗
b
i
(
1
+
a
i
x
i
)
v
-
v
∗
2
v
∗
(
1
+
a
i
x
i
+
b
i
v
)
(
1
+
a
i
x
i
+
b
i
v
∗
)
+
∑
i
=
1
n
‍
p
i
y
i
∗
1
-
v
∗
y
i
y
i
∗
v
+
ln
⁡
v
∗
y
i
y
i
∗
v
+
∑
i
=
1
n
‍
p
i
y
i
∗
1
-
1
+
a
i
x
i
+
b
i
v
1
+
a
i
x
i
+
b
i
v
∗
+
ln
⁡
1
+
a
i
x
i
+
b
i
v
1
+
a
i
x
i
+
b
i
v
∗
.Therefore, L
2
′
≤
0 holds for all x
i
,
y
i
,
v
,
w
>
0. Thus, the endemic equilibrium P
2 is stable. And we have L
2
′
=
0 if and only if x
i
=
x
i
∗, y
i
=
y
i
∗, v
=
v
∗, and w
=
w
∗ hold. The largest compact invariant set in {
(
x
i
,
y
i
,
v
,
w
)
∣
L
2
′
=
0
} is the singleton {
P
2
}. Therefore, the endemic equilibrium P
2 is globally asymptotically stable by the LaSalle invariance principle when R
1
>
1. This completes the proof.


## 4. Special Case Wheni
=
1

Fori
=
1, system (3) becomes (23)
x
′
(
t
)
=
λ
-
d
x
(
t
)
-
β
x
(
t
)
v
(
t
)
1
+
a
x
(
t
)
+
b
v
(
t
)
,
y
′
(
t
)
=
β
e
-
m
τ
x
(
t
-
τ
)
v
(
t
-
τ
)
1
+
a
x
(
t
-
τ
)
+
b
v
(
t
-
τ
)
-
p
y
(
t
)
,
v
′
(
t
)
=
k
y
(
t
)
-
μ
v
(
t
)
-
q
v
(
t
)
w
(
t
)
,
w
′
(
t
)
=
g
v
(
t
)
w
(
t
)
-
h
w
(
t
)
.Here the state variables x, y, v, and w and parameters λ, d, β, a, b, m, p, k, μ, q, g, and h have the similar biological meanings as in the system (3).The basic reproductive number of the virus for system (23) is R
0
1
=
λ
β
k
e
-
m
τ
/
d
p
μ
+
a
p
μ
λ. We will know that if R
0
1
≤
1, then infection-free equilibrium P
0
=
(
x
0
,
0,0
,
0
), x
0
=
λ
/
d, is the unique steady state; if R
0
1
>
1, then in addition to the uninfected steady state there exists an immune-free equilibrium E
1
=
(
x
1
,
y
1
,
v
1
,
0
), corresponding to the situation that infection becomes chronic but with no persistent antibody immune response, which is given by the following expressions: (24)
x
1
=
λ
b
k
+
p
u
e
m
τ
k
β
+
b
d
k
-
a
p
μ
e
m
τ
,
y
1
=
d
p
μ
+
a
p
μ
λ
p
(
k
β
+
b
d
k
-
a
p
μ
e
m
τ
)
(
R
0
1
-
1
)
,
v
1
=
k
(
d
p
μ
+
a
p
μ
λ
)
p
μ
(
k
β
+
b
d
k
-
a
p
μ
e
m
τ
)
(
R
0
1
-
1
)
.Now, we introduce an antibody immune response reproduction number: (25)
R
1
1
=
p
μ
(
k
β
+
b
d
k
-
a
p
μ
e
m
τ
)
g
k
(
d
p
μ
+
a
p
μ
λ
)
(
g
v
1
-
h
)
+
1
.Further, if (26)
R
0
1
>
1
+
p
μ
(
k
β
+
b
d
k
-
a
p
μ
e
m
τ
)
h
g
k
(
d
p
μ
+
a
p
μ
λ
)
,system (3) also has an endemic equilibrium E
2
=
(
x
2
,
y
2
,
v
2
,
w
2
), where (27)
x
2
=
a
λ
-
d
-
b
d
h
g
-
β
h
g
2
+
4
a
d
λ
+
4
λ
b
h
a
d
g
a
λ
-
d
-
b
d
h
g
-
β
h
g
h
+
a
λ
-
d
-
b
d
h
g
-
β
h
g
2
+
4
a
d
λ
+
4
λ
b
h
a
d
g
×
2
a
d
-
1
,
v
2
=
h
g
,
y
2
=
β
e
-
m
τ
x
2
v
2
p
(
1
+
a
x
2
+
b
v
2
)
,
w
2
=
k
g
y
2
q
h
-
μ
q
.It follows from (28)
R
1
1
=
R
0
1
-
p
μ
(
k
β
+
b
d
k
-
a
p
μ
e
m
τ
)
h
g
k
(
d
p
μ
+
a
p
μ
λ
)that R
1
1
>
1 is equivalent to g
v
1
/
h
>
1. The latter g
v
1
/
h can be regarded as the immune activation effector. It is reasonable that immune response is activated in the case where R
1
1
>
1.The initial conditions for system (23) take the following form: (29)
x
θ
=
ϕ
1
θ
,
y
θ
=
ϕ
2
θ
,
v
θ
=
ϕ
3
θ
,
w
θ
=
ϕ
4
θ
,
ϕ
i
θ
≥
0
,
θ
∈
-
τ
,
0
,
ϕ
i
0
>
0
,
i
=
1,2
,
3,4
,where (
ϕ
1
(
θ
)
,
ϕ
2
(
θ
)
,
ϕ
3
(
θ
)
,
ϕ
4
(
θ
)
)
∈
C
(
[
-
τ
,
0
]
,
R
+
0
4
).Theorem 5.
Consider system (23). (i)
The infection-free equilibriumE
0 of (23) is globally asymptotically stable if R
0
1
≤
1.
(ii)
The immune-free equilibriumE
1 of (23) is globally asymptotically stable if R
1
1
≤
1
<
R
0
1.
(iii)
The endemic equilibriumE
2 of (23) is globally asymptotically stable if R
1
1
>
1.Proof.
Consider the following.
Proof of (i) of Theorem5. Define a Lyapunov functional L
0 as follows:(30)
L
0
=
x
0
1
+
a
x
0
e
-
m
τ
H
x
x
0
+
y
+
p
k
v
+
p
q
k
g
w
+
β
e
-
m
τ
∫
t
-
τ
t
‍
x
(
θ
)
v
(
θ
)
1
+
a
x
(
θ
)
+
b
v
(
θ
)
d
θ
.Using λ
=
d
x
0 and calculating the time derivative of U
0 along (23), we have(31)
U
0
′
=
e
-
m
τ
1
+
a
x
0
1
-
x
0
x
t
x
′
(
t
)
+
y
′
(
t
)
+
p
k
v
′
(
t
)
+
p
q
k
g
w
′
(
t
)
+
β
e
-
m
τ
x
v
1
+
a
x
+
b
v
-
β
e
-
m
τ
x
t
-
τ
v
t
-
τ
1
+
a
x
t
-
τ
+
b
v
t
-
τ
=
-
d
e
-
m
τ
x
(
1
+
a
x
0
)
x
-
x
0
2
-
β
e
-
m
τ
x
v
(
1
+
a
x
0
)
(
1
+
a
x
+
b
v
)
-
p
μ
k
v
-
p
q
h
k
g
w
+
β
e
-
m
τ
x
0
v
(
1
+
a
x
0
)
(
1
+
a
x
+
b
v
)
+
β
e
-
m
τ
x
v
1
+
a
x
+
b
v
=
-
d
e
-
m
τ
x
(
1
+
a
x
0
)
x
-
x
0
2
+
p
μ
v
(
1
+
a
x
)
k
(
1
+
a
x
+
b
v
)
(
R
0
-
1
)
-
p
μ
b
k
1
+
a
x
+
b
v
v
2
-
p
q
h
k
g
w
.Thus, R
0
1
≤
1 ensures that U
0
′
<
0 for all x
>
0, y
≥
0, v
>
0, and w
>
0. And U
0
′
=
0 if and only if x
=
x
0, v
=
w
=
0 for R
0
1
≤
1. It is easy to show that E
0
(
x
0
,
0,0
,
0
) is the largest invariant set in {
(
x
,
y
,
v
,
w
)
∣
U
0
′
=
0
}. By LaSalle invariance principle, the equilibrium E
0
(
x
0
,
0,0
,
0
) is globally asymptotically stable.
Proof of (ii) of Theorem5. Define a Lyapunov functional L
1 as follows: (32)
U
1
=
x
1
(
1
+
b
v
1
)
e
-
m
τ
1
+
a
x
1
+
b
v
1
H
x
x
1
+
y
1
H
y
y
1
+
p
v
1
k
H
v
v
1
+
p
q
k
g
w
+
p
y
1
∫
t
-
τ
t
‍
H
x
(
θ
)
v
(
θ
)
1
+
a
x
(
θ
)
+
b
v
(
θ
)
1
+
a
x
1
+
b
v
1
x
1
v
1
d
θ
.The derivative of U
1 is given by (33)
U
1
′
=
(
1
+
b
v
1
)
e
-
m
τ
1
+
a
x
1
+
b
v
1
1
-
x
1
x
x
′
+
1
-
y
1
y
y
′
+
p
k
1
-
v
1
v
v
′
+
p
q
k
g
w
′
+
β
e
-
m
τ
x
v
1
+
a
x
+
b
v
-
β
e
-
m
τ
x
(
t
-
τ
)
v
(
t
-
τ
)
1
+
a
x
(
t
-
τ
)
+
b
v
(
t
-
τ
)
+
p
y
1
ln
⁡
x
t
-
τ
v
t
-
τ
x
v
1
+
a
x
+
b
v
1
+
a
x
(
t
-
τ
)
+
b
v
(
t
-
τ
)
=
-
d
(
1
+
b
v
1
)
e
-
m
τ
x
(
1
+
a
x
1
+
b
v
1
)
x
-
x
1
2
+
p
q
(
g
v
1
-
h
)
k
g
w
+
p
y
1
1
-
x
1
1
+
a
x
+
b
v
1
x
1
+
a
x
1
+
b
v
1
+
ln
⁡
x
1
1
+
a
x
+
b
v
1
x
1
+
a
x
1
+
b
v
1
+
p
y
1
1
-
y
1
(
1
+
a
x
1
+
b
v
1
)
x
(
t
-
τ
)
v
(
t
-
τ
)
y
x
1
v
1
(
1
+
a
x
(
t
-
τ
)
+
b
v
(
t
-
τ
)
)
H
H
H
H
+
ln
⁡
y
1
(
1
+
a
x
1
+
b
v
1
)
x
(
t
-
τ
)
v
(
t
-
τ
)
y
x
1
v
1
(
1
+
a
x
(
t
-
τ
)
+
b
v
(
t
-
τ
)
)
+
p
y
1
1
-
v
1
y
y
1
v
+
ln
⁡
v
1
y
y
1
v
+
p
y
1
1
-
1
+
a
x
+
b
v
1
+
a
x
+
b
v
1
+
ln
⁡
1
+
a
x
+
b
v
1
+
a
x
+
b
v
1
-
p
y
1
b
(
1
+
a
x
)
v
-
v
1
2
v
1
(
1
+
a
x
+
b
v
)
(
1
+
a
x
+
b
v
1
)
.In the above calculations, we used the equations λ
=
d
x
1
+
β
x
1
v
1
/
(
1
+
a
x
1
+
b
v
1
), k
y
1
=
u
v
1, and β
e
-
m
τ
x
1
v
1
/
(
1
+
a
x
1
+
b
v
1
)
=
p
y
1.
Further,R
1
1
≤
1
<
R
0
1 is equivalent to v
1
≤
h
/
g. Hence U
1
′ is always nonpositive under the condition R
1
1
≤
1
<
R
0
1, and U
1
′
=
0 if and only if x
=
x
1, y
/
y
1
=
v
/
v
1, and w
=
0 for R
1
1
<
1 or x
=
x
1, y
/
y
1
=
v
/
v
1 for R
1
1
=
1. Similarly, it follows from LaSalle invariance principle that the immune-free equilibrium E
1 is globally asymptotically stable.Proof of (iii) of Theorem5. Define a Lyapunov functional U
2 as follows: (34)
U
2
=
x
2
(
1
+
b
v
2
)
e
-
m
τ
1
+
a
x
2
+
b
v
2
H
x
x
2
+
y
2
H
y
y
2
+
p
v
2
k
H
v
v
2
+
p
q
w
2
k
g
H
w
w
2
+
p
y
2
∫
t
-
τ
t
‍
H
x
(
θ
)
v
(
θ
)
1
+
a
x
(
θ
)
+
b
v
(
θ
)
1
+
a
x
2
+
b
v
2
x
2
v
2
d
θ
.The derivative of U
2 is given by (35)
U
2
′
=
e
-
m
τ
(
1
+
b
v
2
)
1
+
a
x
2
+
b
v
2
1
-
x
2
x
x
′
+
1
-
y
2
y
y
′
+
p
k
1
-
v
2
v
v
′
+
p
q
k
g
1
-
w
2
w
w
′
+
β
e
-
m
τ
x
v
1
+
a
x
+
b
v
-
β
e
-
m
τ
x
(
t
-
τ
)
v
(
t
-
τ
)
1
+
a
x
(
t
-
τ
)
+
b
v
(
t
-
τ
)
+
p
y
2
ln
⁡
x
(
t
-
τ
)
v
(
t
-
τ
)
x
v
1
+
a
x
+
b
v
1
+
a
x
(
t
-
τ
)
+
b
v
(
t
-
τ
)
=
-
d
(
1
+
b
v
2
)
e
-
m
τ
x
(
1
+
a
x
2
+
b
v
2
)
x
-
x
2
2
+
p
y
2
1
-
x
2
(
1
+
a
x
+
b
v
2
)
x
(
1
+
a
x
2
+
b
v
2
)
+
ln
⁡
x
2
(
1
+
a
x
+
b
v
2
)
x
(
1
+
a
x
2
+
b
v
2
)
+
p
y
2
1
-
y
2
(
1
+
a
x
2
+
b
v
2
)
x
(
t
-
τ
)
v
(
t
-
τ
)
y
x
2
v
2
(
1
+
a
x
(
t
-
τ
)
+
b
v
(
t
-
τ
)
)
H
H
H
H
+
ln
⁡
y
2
1
+
a
x
2
+
b
v
2
x
t
-
τ
v
t
-
τ
y
x
2
v
2
1
+
a
x
t
-
τ
+
b
v
t
-
τ
-
p
y
2
b
1
+
a
x
v
-
v
2
2
v
2
1
+
a
x
+
b
v
1
+
a
x
+
b
v
2
+
p
y
2
1
-
v
2
y
y
2
v
+
ln
⁡
v
2
y
y
2
v
+
p
y
2
1
-
1
+
a
x
+
b
v
1
+
a
x
+
b
v
2
+
ln
⁡
1
+
a
x
+
b
v
1
+
a
x
+
b
v
2
.Therefore, U
2
′
≤
0 holds for all x
,
y
,
v
,
w
>
0. Thus, the endemic equilibrium E
2 is stable. And we have U
2
′
=
0 if and only if x
=
x
2, y
=
y
2, v
=
v
2, and w
=
w
2 hold. The largest compact invariant set in {
(
x
,
y
,
v
,
w
)
∣
U
2
′
=
0
} is the singleton {
E
2
}. Therefore, the endemic equilibrium E
2 is globally asymptotically stable by the LaSalle invariance principle when R
1
1
>
1. This completes the proof.


## 5. Numerical Example

We now carry out some numerical simulations to support our theoretic results for the dynamics of the concentration of uninfected target cellsx
(
t
) [cells mL
-
1], productively infected cells y
(
t
) [cells mL
-
1], free virus in the serum v
(
t
) [virion cells mL
-
1] antibodies w
(
t
)
[
μ
g
]. We set m
=
0 and τ
=
0.5 for the sake of simplification. The values of the parameters λ, d, β, p, k, μ, q, g, and h are found in Pawelek et al. [22] and Wang et al. [23].Parameters in Figure1 are λ
=
46 cells mL
-
1, d
=
0.0046
day
-
1, β
=
4.8
×
1
0
-
7 mL virion-day
-
1, a
=
0.01, b
=
0.005, p
=
0.01
day
-
1, k
=
11.349
day
-
1, μ
=
0.25
day
-
1, q
=
0.03, g
=
0.0015, and h
=
2.9
day
-
1. In this case, we obtain R
0
1
=
0.215743 and R
1
1
=
0.127203. From (i) of Theorem 5, E
0
=
(
1
0
4
,
0,0
,
0
) is globally asymptotically stable.Figure 1
Trajectories ofx
(
t
), x
(
t
), x
(
t
), w
(
t
) for R
0
1
=
0.215743, R
1
1
=
0.127203, and E
0
=
(
1
0
4
,
0,0
,
0
) are globally asymptotically stable. x
(
t
), y
(
t
), v
(
t
), and w
(
t
) versus t are illustrated by (a), (b), (c), and (d).
(a)
(b)
(c)
(d)Parameters in Figure2 are λ
=
46 cells mL
-
1, d
=
0.0046
day
-
1, β
=
2.4
×
1
0
-
7 mL virion-day
-
1, a
=
0.01, b
=
0.005, p
=
0.01
day
-
1, k
=
11.349
day
-
1, μ
=
0.25
day
-
1, q
=
0.03, g
=
0.0015, and h
=
2.9
day
-
1. In this case, we obtain R
0
1
=
1.07872 and R
1
1
=
0.982187. From (ii) of Theorem 5, E
1
=
(
9924.5,34.7292,1576.57,0
) is globally asymptotically stable.Figure 2
Trajectories ofx
(
t
), y
(
t
), v
(
t
), w
(
t
) for R
0
1
=
1.07872, R
1
1
=
0.982187, and E
1
=
(
9924.5,34.7292,1576.57,0
) are globally asymptotically stable. x
(
t
), y
(
t
), v
(
t
), and w
(
t
) versus t are illustrated by (a), (b), (c), and (d).
(a)
(b)
(c)
(d)Parameters in Figure3 are λ
=
46 cells mL
-
1, d
=
0.0046
day
-
1, β
=
4.8
×
1
0
-
6 mL virion-day
-
1, a
=
0.01, b
=
0.005, p
=
0.01
day
-
1, k
=
11.349
day
-
1, μ
=
0.25
day
-
1, q
=
0.03, g
=
0.0015, and h
=
2.9
day
-
1. In this case, we obtain R
0
1
=
2.15743 and R
1
1
=
2.05092. From (iii) of Theorem 5, E
2
=
(
9818.03,83.7059,1933.33,8.0456
) is globally asymptotically stable.Figure 3
Trajectories ofx
(
t
), y
(
t
), v
(
t
), w
(
t
) for R
0
1
=
2.15743 and R
1
1
=
2.05092, and E
2
=
(
9818.03,83.7059,1933.33,8.0456
) are globally asymptotically stable. x
(
t
), y
(
t
), v
(
t
), and w
(
t
) versus t are illustrated by (a), (b), (c), and (d).
(a)
(b)
(c)
(d)


## 6. Conclusions and Discussion

From Theorem2 we can see that (36)
v
1
-
v
∗
=
∑
i
=
1
n
‍
k
i
μ
∑
i
=
1
n
‍
β
i
e
-
m
i
τ
k
i
λ
i
-
d
i
p
i
μ
-
a
i
p
i
μ
λ
i
∑
i
=
1
n
‍
p
i
k
i
d
i
b
i
+
β
i
k
i
p
i
-
a
i
p
i
2
μ
e
m
i
τ
×
(
R
1
-
1
)
>
0
,so the antibody immune response has a positive role in reduction of the virus. Huang et al. [7] investigated the global dynamics of basic in-host dynamics model with Beddington-DeAngelis infection rate. But their models did not conclude immune response and intracellular delay. On the other hand, we improve the results in [8] from bilinear incidence rate to Beddington-DeAngelis incidence rate. It is similar to the well-known Holling type II functional response but has an extra term b
v in the denominator which models mutual interference between viruses. When a
>
0, b
=
0, the Beddington-DeAngelis functional response is simplified to Holling type II functional response [21]. When a
=
0, b
>
0, it expresses a saturation response; our model will reduce to the model studied in [9].



---

*Source: 654507-2015-05-17.xml*