Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 578561, 25 pages
doi:10.1155/2012/578561
Research Article
Analysis of a HBV Model with Diffusion and
Time Delay
Noé Chan Chı́, Eric ÁvilaVales, and Gerardo Garcı́a Almeida
Facultad de Matemáticas, Universidad Autónoma de Yucatán, Anillo Periférico Norte,
Tablaje Catastral 13615, Colonia Chuburna Hidalgo Inn, 97203 Mérida, YUC, Mexico
Correspondence should be addressed to Eric ÁvilaVales, avila@uady.mx
Received 7 March 2012; Accepted 27 August 2012
Academic Editor: Alexander Timokha
Copyright q 2012 Noé Chan Chı́ et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
This paper discussed a hepatitis B virus infection with delay, spatial diffusion, and standard
incidence function. The local stability of equilibrium is obtained via characteristic equations. By
using comparison arguments, it is proved that if the basic reproduction number is less than unity,
the infection-free equilibrium is globally asymptotically stable. If the basic reproductive number is
greater than unity, by means of an iteration technique, sufficiently conditions are obtained for the
global asymptotic stability of the infected steady state. Numerical simulations are carried out to
illustrate our findings.
1. Introduction
Human infection with hepatitis B virus HBV is a major global health problem. Between 300
and 400 million people are chronically infected worldwide. The virus is contracted through
contact with blood or other fluids from the body, which could lead to develop viral persistence in the individual in the absence of strong antibody or some immune depression. Mathematical models have the potential to improve the understanding of the dynamics of this
disease; one of the earliest models is referred to as the basic virus infection model, introduced
by Nowak et al. 1. They proposed a basic mathematical model for uninfected susceptible
host cells hepatocytes, u, infected host cells, w, and free virus particles, v, as follows:
u̇t s − μut − βutvt,
ẇt βutvt − awt,
v̇t kwt − dvt,
1.1
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Journal of Applied Mathematics
where hepatocytes are produced at a rate s, uninfected cells die at rate μ and become and
infected at rate βutvt, infected hepatocytes are produced at rate βutvt and die at rate
awt. Free viruses are produced from infected cells at rate kwt and are removed at rate
dvt. It is assumed that all parameters are positive constants. Previous models assume that
the infectious process is instantaneous; that is, in the very moment that the virus enters an
uninfected cell, this one starts to produce virus particles; we know that this is not biologically
reasonable. Thus, models with delays have been considered; in 2, the authors studied the
following hepatitis B virus infection model with a time delay:
ẋt λ − dxt −
ẏt βxtvt
,
xt yt
βe−mτ xt − τvt − τ
− ayt,
xt − τ yt − τ
1.2
v̇t kyt − uvt.
The authors gave results about local and global stability of feasible equilibria.
For HBV infection, susceptible host cells and infected cells are hepatocytes and cannot
move under normal conditions, but viruses move freely in liver 3; therefore, the authors
introduce an HBV model with diffusion and delay. Xu and Ma 4 considered also a diffusion
model with delay but instead of bilinear response of the infection rate, they considered
saturation response.
In this work motivated by the work of Xu and Ma, we study the following model:
βux, tvx, t
∂u
L − dux, t −
,
∂t
ux, t wx, t
∂w βe−mτ ux, t − τvx, t − τ
− awx, t,
∂t
ux, t − τ wx, t − τ
1.3
∂v
DΔv kwx, t − pvx, t,
∂t
for t > 0, x ∈ Ω, with homogeneous Neumann boundary conditions
∂v
0,
∂η
1.4
and initial conditions
ux, θ φ1 x, θ ≥ 0,
vx, θ φ3 x, θ ≥ 0,
wx, θ φ2 x, θ ≥ 0,
θ ∈ −τ, 0, x ∈ Ω.
1.5
In the previous problem Ω is a bounded domain in Rn with smooth boundary ∂Ω, ∂/∂η
denotes the outward normal derivative on ∂Ω.
This paper is ordered as follows. In the next section we present a result about the
existence, uniqueness, and positivity. In Section 3 we discuss the local stability of each
Journal of Applied Mathematics
3
of the feasible equilibria of system 1.3, by analyzing the corresponding characteristic
equations. In Section 4, by using comparison arguments and an iterative technique, we
establish sufficient conditions for the global stability of the equilibria of system 1.3. In
Section 5 numerical simulations are carried out to illustrate our principal results and we
compare the effect of the diffusion and the delay on the system 1.3.
2. Preliminaries
Consider problem 1.3–1.5 and the following definitions.
Definition 2.1. A pair of functions U
u, w,
v , U
u, w,
v in C0, ∞ × Ω ∩ C1,2 0, ∞ ×
Ω are called coupled upper and lower solutions to system 1.3–1.5 if u
≤u
, w
≤ w,
v ≤ v
in Ω × −τ, ∞ and the following differential inequalities hold:
β
ux, t
vx, t
∂
u
≥ L − d
ux, t −
,
∂t
u
x, t wx,
t
βe−mτ u
x, t − τ
vx, t − τ
∂w
≥
− awx,
t,
∂t
u
x, t − τ wx,
t − τ
∂
v
≥ DΔ
v kwx,
t − pv x, t,
∂t
β
ux, t
vx, t
∂
u
≤ L − d
ux, t −
,
∂t
u
x, t wx,
t
2.1
βe−mτ u
x, t − τ
vx, t − τ
∂w
≤
− awx,
t,
∂t
u
x, t − τ wx,
t − τ
∂
v
≤ DΔ
v kwx,
t − pv x, t,
∂t
for x, t ∈ Ω × 0, ∞, and
∂w
∂w
≤0≤
,
∂η
∂η
∂
u
∂
u
≤0≤
,
∂η
∂η
∂
v
∂
v
≤0≤
∂η
∂η
x, t,
u
x, t ≤ φ1 x, t ≤ u
x, t ∈ ∂Ω × 0, ∞,
2.2
wx,
t ≤ φ2 x, t ≤ wx,
t,
v x, t ≤ φ3 x, t ≤ v x, t,
x, t ∈ Ω × −τ, 0.
The following lemma then follows from Theorem 3.4 developed by Redlinger 5.
and U
be a pair of coupled upper and lower solutions for problem 1.3–1.5
Lemma 2.2. Let U
and suppose that the initial functions φi (i 1, 2, 3) are Hölder continuous in −τ, 0 × Ω. Then
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problem 1.3–1.5 has exactly one regular solution Ux, t ux, t, wx, t, vx, t satisfying
≤U≤U
in Ω × −τ, ∞.
U
It is not hard to see that 0 0, 0, 0 and K K1 , K2 , K3 are a pair of coupled lowerupper solutions to problem 1.3–1.5, where
L
, sup φ1 ·, θCΩ,R ,
K1 max
d −τ≤θ≤0
kβe−mτ K1
K2 max
, sup φ2 ·, θ CΩ,R ,
ap
−τ≤θ≤0
k2 βe−mτ K1
K3 max
, sup φ3 ·, θ CΩ,R .
ap2
−τ≤θ≤0
2.3
Hence, 0 ≤ ux, t ≤ K1 , 0 ≤ wx, t ≤ K2 , 0 ≤ vx, t ≤ K3 for x, t ∈ Ω × −τ, ∞, and also, by
≡ 0 i 1, 2, 3, we have ux, t > 0, wx, t > 0, vx, t > 0
the maximum principle, if φi x, 0 /
for all t > 0, x ∈ Ω.
3. Local Stability
System 1.3 has the equilibrium E1 L/d, 0, 0. Let R0 βke−mτ /ap > 1 then system 1.3 has
a unique infected steady state E2 u∗ τ, w∗ τ, v∗ τ; the previous notation is because the
equilibrium involves τ and we use this as the parameter for the stability analysis, where
u∗ τ de−mτ
Le−mτ
,
aR0 − 1
w∗ τ Le−mτ R0 − 1
,
de−mτ aR0 − 1
Lke−mτ R0 − 1
.
v τ pde−mτ aR0 − 1
3.1
∗
Let 0 μ1 < μ2 < · · · be the eigenvalues of the operator −Δ on Ω with the homogeneous Neumann boundary conditions, and let Eμi be the eigenspace corresponding to μi
in C1 Ω.
3
Let X C1 Ω , let {φij ; j 1, 2, . . . , dim Eμi } be an orthonormal basis of Eμi , and
3
let Xij {cφij | c ∈ R }, then
X
∞
i0
Xi ,
Xi dim Epi j1
Xij .
3.2
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5
Let D diag0, 0, D, Z Zx, t Ux, t, Wx, t, V x, t, LZ DΔZ JE∗ Z JτE∗ Zt − τ,
where
JE ∗
⎛
⎞
βu v
βu
βw v
−
⎜−d −
⎟
⎜
u w2 u w2 u w ⎟
⎜
⎟,
⎝
0
−a
0 ⎠
0
k
−p
⎛
0
0
0
⎞
3.3
⎜ βe−mτ v w βe−mτ u v βe−mτ u ⎟
⎟
⎜
Jτ ⎜
−
⎟,
2
⎠
⎝ u w2
u
w
u w
0
0
0
and E∗ u, w, v represents any feasible steady state of the system 1.3. The linearization of
system 1.3 at E∗ is of the form Zt LZ. For each i ≥ 1, Xi is invariant under the operator
L, and λ is an eigenvalue of the matrix −μi D JE∗ JτE∗ for some i ≥ 1, then, there is an
eigenvector in Xi .
The characteristic equation on the equilibrium E1 is
λ d λ2 a1 λ a0 b0 τe−λτ 0,
3.4
where
a0 a p μI D ,
b0 τ −kβe−mτ .
a1 a p μi D,
3.5
The characteristic equation has the negative root λ −d. All other roots of 3.4 are given by
the transcendental equation
λ2 a1 λ a0 b0 τe−λτ 0.
3.6
fλ λ2 a1 λ a0 b0 τe−λτ ,
3.7
Let
if R0 > 1, note that for λ real and i 1 in this case μ1 0,
f0 a0 b0 ap − kβe−mτ < 0,
lim fλ ∞.
λ→∞
3.8
Hence, 3.7 has a positive root. Therefore, there is a characteristic root λ with positive real
part in the spectrum of L. Accordingly, if R0 > 1, the disease-free steady state E1 λ/d, 0, 0 is
unstable.
If R0 < 1, when τ 0 the coefficients of 3.7 are a1 and a0 b0 0, and under
the hypothesis R0 < 1 the coefficients are positive and according to the criterion of RouthHurwitz, the equilibrium E1 λ/d, 0, 0 is locally asymptotically stable.
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Journal of Applied Mathematics
For τ > 0 if iω ω > 0 is a solution of 3.6, separating in real and imaginary parts, we
obtain
−ω2 a0 −b0 τ cosωτ,
a1 ω b0 τ sinωτ.
3.9
Squaring and adding the above equations and taking z ω2 we obtain
z2 a21 − 2a0 z a20 − b0 τ2 0,
3.10
2
a21 − 2a0 a2 p μi D > 0,
a20 − b02 τ a p μi D − kβe−mτ a p μi D kβe−mτ > 0,
3.11
where
the last inequality is true because R0 < 1. Therefore there is no positive root z ω2 of 3.10.
In conclusion if R0 < 1 the equilibrium E1 λ/d, 0, 0 is locally asymptotically stable.
The characteristic equation of system 1.3 at the endemic equilibrium E2 u∗ , w∗ , v∗ is
of the form
λ3 a2 τλ2 a1 λ a0 τ b2 τλ2 b1 τλ b0 τ e−λτ 0,
3.12
where
a2 τ a d p μi D βv∗ w∗
,
u∗ w∗ 2
a1 a p μi D a p μi D d,
βw∗ v∗
,
a0 τ a p μi D d u∗ w∗ 2
b2 τ βe−mτ u∗ v∗
,
u∗ w∗ ∗
3.13
βe−mτ u∗ w∗
βe−mτ u∗
b1 τ d p μi D
−
k
,
u∗ w ∗
u∗ w∗ 2
βe−mτ u∗ v∗
βe−mτ u∗
−
dk
,
b0 τ d p μi D
u∗ w ∗
u∗ w∗ 2
when τ 0 becomes
λ3 a2 0 b2 0λ2 a1 b1 0λ a0 0 b0 0 0.
3.14
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Note that a2 0 b2 0 > 0; adding a0 0 b0 0 and replacing u∗ 0 and w∗ 0 we obtain
βw∗ 0v∗ 0
βu∗ 0v∗ 0
a0 0 b0 0 a p μi D
d
p
μ
D
> 0,
i
u∗ 0 w∗ 02
u∗ 0 w∗ 02
a2 0 b2 0a1 b1 0 − a0 0 b0 0
p μi D βv∗ 0
∗
∗
d a p μi D u 0 w 0
u∗ 0 w∗ 02
βv∗
∗
∗
× a d p μi D u 0 w 0
u∗ 0 w∗ 02
βv∗ 0
λβv∗ 0
∗
∗
a d u 0 w 0
> 0.
u∗ 0 w∗ 02
u∗ 0 w∗ 02
3.15
By the Routh-Hurwitz criteria, all roots have negative real parts if R0 > 1.
For the case τ > 0 we look for solutions λ iω ω > 0 for 3.12, separating real and
imaginary parts, it follows that
ω3 − a1 ω b2 τω2 − b0 τ sinωτ b1 τω cosωτ,
3.16
a2 τω − a0 τ − b2 τω − b0 τ cosωτ b1 τω sinωτ.
2
2
Squaring and adding the two equations, we derive that
3.17
ω6 C1 ω4 C2 ω2 C3 0,
where
C1 p μi D
2
d
βw∗ τv∗ τ
u∗ τ w∗ τ2
2
au∗ τ
∗
u τ w∗ τ
a
βe−mτ u∗ τv∗ τ
u∗ τ w∗ τ2
> 0,
2 w∗ τu∗ τ 2w∗ τ
C2 a2 d2 p μi D
u∗ τ w∗ τ2
βw∗ τ v∗ τ
βw∗ τv∗ τ
a2 βw∗ τv∗ τ
2d p μi D d > 0,
u∗ τ w∗ τ2
u∗ τ w∗ τ2
u∗ τ w∗ τ2
2
a p μi D βv∗ τde−mτ u∗ τ aw∗ τ
βw∗ τv∗ τ
∗
∗
C3 > 0,
d
2x
w
τ
τ
u∗ τ w∗ τ
u∗ τ w∗ τ3
3.18
implying that 3.17 has no positive roots z ω2 .
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Journal of Applied Mathematics
Theorem 3.1. If R0 < 1 the disease-free equilibrium is locally asymptotically stable; if R0 > 1 it is
unstable and the endemic equilibrium exists and it is locally asymptotically stable.
4. Global Stability
We will discuss in this section the global stability of the infected steady state and the diseasefree equilibrium. The technique of proof is to use comparison arguments and to successively
modify the coupled lower-upper solutions pairs.
Consider the following delay system:
u̇1 t a1 βe−mτ u2 t − τ
− au1 t,
a1 u1 t − τ
4.1
u̇2 t ku1 t − pu2 t,
with initial conditions
ui s φi s ≥ 0,
s ∈ −τ, 0, φi 0 > 0, φi ∈ C−τ, 0, R .
4.2
System 4.1 always have the trivial equilibrium A0 0, 0. If kβe−mτ > ap, then system 4.1
has a unique positive equilibrium A∗ u∗1 , u∗2 where
u∗1
a1 kβe−mτ − ap
,
ap
u∗2
a1 k kβe−mτ − ap
,
ap2
4.3
and according to 2, for system 4.1, one has the following.
Lemma 4.1. If kβe−mτ > ap, then the positive equilibrium A∗ u∗1 , u∗2 is globally stable.
If kβe−mτ < ap, then the equilibrium A0 0, 0 is globally stable.
Now we stablish and prove our result about global stability.
Theorem 4.2. Let ux, t, wx, t, vx, t be a solution to problem 1.3–1.5, let φi x, 0 /
≡ 0 i 1, 2, 3. If R0 > 1 and
H1 dpe−mτ > kβe−mτ − ap,
then
lim ux, t, wx, t, vx, t u∗ , w∗ , v∗ t→∞
uniformly for x ∈ Ω,
that is, the infected steady state E∗ is globally asymptotically stable.
4.4
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≡ 0 i 1, 2, 3.
Proof. Let ux, t, wx, t, vx, t be a solution to problem 1.3–1.5, let φi /
We have ux, t > 0, wx, t > 0, and vx, t > 0 for all x ∈ Ω, t > 0. Denote
u lim sup max ux, t,
t→∞
u lim inf min ux, t,
t→∞
x∈Ω
w lim sup max wx, t,
t→∞
t→∞
w lim inf min wx, t,
t→∞
x∈Ω
v lim sup max vx, t,
x∈Ω
x∈Ω
4.5
v lim inf min vx, t.
t→∞
x∈Ω
x∈Ω
First we look for upper solutions for the system 1.3. Let u1 x, t, w1 x, t, v1 x, t be a
solution for the following problem:
∂u1
L − du1 x, t,
∂t
t > 0, x ∈ Ω,
1
1
∂w1 βe−mτ u x, t − τv x, t − τ
− aw1 x, t,
∂t
u1 x, t − τ w1 x, t − τ
∂v1
DΔv1 x, t kw1 x, t − pv1 x, t,
∂t
∂u1 x, t ∂w1 x, t ∂v1 x, t
0,
∂t
∂t
∂t
t > 0, x ∈ Ω,
t > 0, x ∈ Ω,
4.6
t > 0, x ∈ ∂Ω,
u1 x, t ux, t,
v1 x, t vx, t,
w1 x, t wx, t,
t ∈ −τ, 0, x ∈ Ω.
We note that the solution of this system is an upper solution of system 1.3–1.5. For t > 0,
x ∈ Ω we have
0 ≤ ux, t ≤ u1 x, t,
0 ≤ wx, t ≤ w1 x, t,
0 ≤ vx, t ≤ v1 x, t.
4.7
From the first equation of 4.6
lim u1 x, t t→∞
L
M1u .
d
4.8
Hence, by comparison, for all > 0 sufficiently small, there exists t1 > 0 such that if t > t1
max u1 x, t ≤ M1u ,
x∈Ω
4.9
since is arbitrary and sufficiently small we can conclude that
u lim sup max ux, t ≤ M1u .
t→∞
x∈Ω
4.10
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Journal of Applied Mathematics
Now consider the problem related with the second and third equations of 4.6
1
∂ω2
∂t
1
βe−mτ M1u ω3 x, t − τ
1
M1u ω2 x, t − τ
1
∂ω3
∂t
1
1
− aω2 x, t,
1
1
DΔω3 x, t kω2 x, t − pω3 ,
1
∂ω2
∂η
1
∂ω3
∂η
1
0,
t > t1 , x ∈ Ω,
4.11
t > t1 , x ∈ ∂Ω,
1
ω2 x, t wx, t,
t > t1 , x ∈ Ω,
ω3 x, t vx, t,
t ∈ −τ, t1 , x ∈ Ω.
Consider the solution for
βe−mτ M1u u3 t − τ
u̇2 − au2 ,
M1u u2 t − τ
u̇3 ku2 − pu3 ,
u2 t max wx, t,
t > t1 ,
u3 t max vx, t,
x∈Ω
4.12
t > t1 ,
x∈Ω
t ∈ −τ, t1 .
Note that u1 t, u2 t is an upper solution for system 4.11, and using the assumption that
R0 > 1, by Lemma 4.1, it follows from 4.12 that
kβe−mτ − ap M1u ,
lim u2 t t→∞
ap
k kβe−mτ − ap M1u lim u3 t .
t→∞
ap2
4.13
Hence, for all > 0 sufficiently small, by comparison there exists a t2 > t1 such that if t > t2
max w1 x, t < M1w ,
x∈Ω
max v1 x, t < M1v ,
x∈Ω
4.14
where
kβe−mτ − ap M1u
,
ap
M1w
M1v
k kβe−mτ − ap M1u
.
ap2
4.15
Since > 0 is arbitrary and sufficiently small, we conclude that
w lim sup max wx, t ≤ M1w ,
t→∞
x∈Ω
v lim sup max vx, t ≤ M1v .
t→∞
x∈Ω
4.16
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11
Now for lower solutions, let u1 x, t, w1 x, t, v1 x, t be the solution for the following
problem:
βu1 x, tv1 x, t
∂u1
L − du1 x, t −
,
∂t
u1 x, t w1 x, t
t > t2 , x ∈ Ω,
∂w1 βe−mτ u1 x, tv1 x, t
− aw1 x, t,
∂t
u1 x, t w1 x, t
t > t2 , x ∈ Ω,
∂v1
DΔv1 x, t kw1 x, t − pv1 x, t,
∂t
t > t2 , x ∈ Ω,
∂u1 ∂w1 ∂v1
0,
∂η
∂η
∂η
u1 x, t 1
ux, t,
2
v1 x, t 1
vx, t,
2
4.17
t > t2 , x ∈ ∂Ω,
w1 x, t 1
wx, t,
2
t ∈ −τ, t2 , x ∈ Ω.
Note that the solution of 4.17 is a lower solution to 1.3–1.5. For all > 0 sufficiently
small, from the first equation of 4.17 and 4.16 it follows
∂u1
≥ L − du1 x, t − β M1v ,
∂t
t > t2 , x ∈ Ω.
4.18
By comparing the above equation with the following problem:
1
∂ω1
∂t
1
L − dω1 x, t − β M1v ,
∂ω1
0,
ωη
t > t2 , x ∈ ∂Ω,
t > t2 , x ∈ Ω,
1
ω1 x, t2 ux, t2 ,
2
4.19
x ∈ Ω,
we obtain
L − β M1v lim ω1 x, t ,
t→∞
d
4.20
so u1 x, t ≥ ω1 x, t, t > t2 , and x ∈ Ω. Hence, for all > 0 sufficiently small, there is a t3 > t2
such that if t > t3 ,
min u1 x, t ≥ N1u − ,
x∈Ω
4.21
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Journal of Applied Mathematics
where
N1u L − βM1v
.
d
4.22
Since > 0 is arbitrary sufficiently small, by comparison we conclude that
u lim inf min ux, t ≥ N1u t→∞
x∈Ω
L − βM1v
.
d
4.23
Now consider the following problem related with the second and third equations of 4.17:
∂ω2 βe−mτ N1u − ω3 x, t − τ
− aω2 x, t,
∂t
N1u − ω2 x, t − τ
∂ω3
DΔω3 x, t kω2 x, t − pω3 x, t,
∂t
∂ω2 ∂ω3
0,
∂η
∂η
ω2 x, t 1
wx, t,
2
ω3 x, t t > t3 , x ∈ Ω,
t > t3 , x ∈ Ω,
4.24
t > t3 , x ∈ ∂Ω,
1
vx, t,
2
t ∈ −τ, t3 , x ∈ Ω.
Now let us consider the solution for the problem
βe−mτ N1u − u3 t − τ
u̇2 t − au2 t,
N1u − u2 t − τ
u̇3 t ku2 t − pu3 t,
u2 t 1
min wx, t,
2 x∈Ω
u3 t t > t3 ,
t > t3 ,
1
min vx, t,
2 x∈Ω
4.25
t ∈ −τ, t3 , t > t3 ,
and according to Lemma 4.1
kβe−mτ − ap N1u − ,
lim u2 t t→∞
ap
k kβe−mτ − ap N1u − lim u3 t .
t→∞
ap2
4.26
Hence, for all > 0 sufficiently small, by comparison there exists a t4 > t3 such that if t > t4
min w1 x, t > N1w − ,
x∈Ω
min v1 x, t > N1v − ,
x∈Ω
4.27
Journal of Applied Mathematics
13
where
kβe−mτ − ap N1u
,
ap
N1w
N1v
k kβe−mτ − ap N1u
.
ap2
4.28
Since > 0 is arbitrary and sufficiently small, we conclude that
w lim inf min wx, t ≥ N1w ,
t→∞
x∈Ω
4.29
v lim inf min vx, t ≥ N1v .
t→∞
x∈Ω
Now we look for the closest upper and lower solutions. Let u2 , w2 , v2 be a solution for
the problem
βu2 x, tv1 x, t
∂u2
L − du2 x, t −
,
∂t
u1 w1 x, t
t > t4 , x ∈ Ω,
βe−mτ u2 x, t − τv2 x, t
∂w2
2
− aw2 x, t,
∂t
u x, t − τ w2 x, t − τ
∂v2
DΔvx, t kw2 x, t − pv2 x, t,
∂t
∂u2 ∂w2 ∂v2
0,
∂η
∂η
∂η
u2 x, t ux, t,
v2 x, t vx, t,
t > t4 , x ∈ Ω,
t > t4 , x ∈ Ω,
4.30
t > t4 , x ∈ ∂Ω,
w2 x, t wx, t,
t ∈ −τ, t4 , x ∈ Ω.
For all > 0 sufficiently small it follows form the first equation of 4.30 and the inequalities
4.27 and 4.14 that
βu2 x, t N1v − ∂u2
≤ L − du2 x, t −
,
∂t
M1u M1w t > t4 , x ∈ Ω.
4.31
2
Let ω1 x, t be the solution for the following problem:
2
∂ω1
∂t
2
L − dω1 x, t −
2
βω1 x, t N1v − ,
M1u M1w 2
∂ω1
0,
∂η
2
t > t4 , x ∈ ∂Ω,
ω1 x, t4 ux, t4 ,
x ∈ Ω,
t > t4 , x ∈ Ω,
4.32
14
Journal of Applied Mathematics
it follows that
2
lim ω x, t
t→∞ 1
L M1u M1w 2
u
.
d M1 M1w 2
β N1v − 4.33
2
By comparison we have that u2 ≤ ω1 , t > t4 , and x ∈ Ω. Hence, for all > 0 sufficiently
small, by comparison, there is a t5 > t4 such that if t > t5
max u2 x, t M2u ,
4.34
x∈Ω
where
M2u
L M1u M1w
u
.
d M1 M1w β N1v
4.35
Since 4.34 is valid for > 0 arbitrary and sufficiently small, by comparison we conclude that
u lim sup max ux, t ≤ M2u .
t→∞
4.36
x∈Ω
Now consider the following problem related with the second and third equations of 4.30:
2
∂ω2
∂t
2
βe−mτ M2u ω3 x, t
2
∂ω3
∂t
M2u
2
ω2 x, t
− τ
2
− aω2 x, t,
2
t > t5 , x ∈ Ω,
2
t > t5 , x ∈ Ω,
2
DΔω3 x, t kω2 x, t − pω3 x, t,
2
2
∂ω3
∂ω2
0,
∂η
∂η
2
ω2 x, t wx, t,
4.37
t > t5 , x ∈ ∂Ω,
2
ω3 x, t vx, t,
t ∈ −τ, t5 , x ∈ Ω.
Let u2 t, u3 t be the positive solution to the following problem:
u̇2 t βe−mτ M2u u3 t − τ
− au2 t,
M2u u2 t − τ
u̇3 t ku2 t − pu3 t,
u2 t max wx, t,
x∈Ω
t > t5 ,
4.38
t > t5 ,
u3 t max vx, t,
x∈Ω
t ∈ −τ, t5 .
Journal of Applied Mathematics
15
Then by Lemma 4.1 and the previous system we have
kβe−mτ − ap M2u ,
lim u2 t ap
k kβe−mτ M2u lim u3 t .
ap2
4.39
Hence for all > 0 sufficiently small, by comparison there is a t6 > t5 such that if t > t6
max wx, t < M2w ,
x∈Ω
max vx, t < M2v ,
x∈Ω
4.40
where
kβe−mτ − ap M2u
,
ap
k kβe−mτ M2u
v
M2 .
ap2
M2w 4.41
Since > 0 is arbitrary and sufficiently small, we conclude that
w lim sup max wx, t ≤ M2w ,
t→∞
x∈Ω
v lim sup max vx, t ≤ M2v .
t→∞
x∈Ω
4.42
Let u2 , w2 , v2 be a solution for the following problem:
∂u2
βu2 x, tv1 x, t
L − du2 x, t − 1
,
∂t
u x, t w1 x, t
t > t6 , x ∈ Ω,
∂w2 βe−mτ u2 x, t − τv2 x, t − τ
− aw2 x, t
∂t
u2 x, t − τ w2 x, t − τ
t > t6 , x ∈ Ω,
∂v2
DΔv2 x, t kw2 − pv2 x, t t > t6 , x ∈ Ω,
∂t
u2 x, t 1
ux, t,
2
w2 x, t 1
vx, t,
2
w2 x, t 4.43
1
wx, t,
2
t ∈ −τ, t6 , x ∈ Ω.
Then u2 , w2 , v2 and u2 , w2 , v2 are a pair of coupled lower and upper solutions to
system 1.3–1.5. Hence we have that for t ≥ t6 , x ∈ Ω
u2 x, t ≤ ux, t ≤ u2 x, t,
w2 x, t ≤ wx, t ≤ w2 x, t,
v2 x, t ≤ vx, t ≤ v2 x, t.
4.44
16
Journal of Applied Mathematics
For all > 0 sufficiently small, it follow from the first equation of 4.43, and the inequalities
βu2 x, t M1v ∂u2
≥ L − du2 x, t −
.
∂t
N1u − N1w − 2
4.45
2
By comparison we have that u2 x, t ≥ v1 x, t, t > t6 , and x ∈ Ω where v1 is the solution
to problem
2
∂v1
∂t
2
dv1 x, t
L−
2
βv1 x, t M1v ,
−
N1u − N1w − 2
∂v1
0,
∂η
4.46
t > t6 , x ∈ ∂Ω,
which has satisfies
L N1u − N1w − .
d N1u − N1w − β M1v 2
lim v1 x, t t→∞
4.47
Hence for all > 0 sufficiently small, by comparison, there is a t7 > t6 such that if t > t7
min u2 x, t ≥ N2u − ,
4.48
with
N2u
L N1u N1w
u
.
d N1 N1w βM1v
4.49
Since this holds true for arbitrary > 0 sufficiently small, by comparison we conclude that
u lim inf min ux, t ≥ N2u .
t→∞
4.50
x∈Ω
Now consider the following problem:
2
∂v2
∂t
2
βe−mτ N2u − v3 x, t − τ
N2u
2
∂v3
∂t
−
2
v2 x, t
2
2
2
DΔv3 x, t kv2 − pv3 x, t,
2
∂v2
∂η
2
− τ
2
− av2 x, t,
v2 x, t 1
wx, t,
2
2
∂v3
∂η
2
0,
v3 x, t t > t7 , x ∈ Ω,
t > t7 , x ∈ Ω,
t > t7 , x ∈ Ω,
1
vx, t,
2
t ∈ −τ, t7 , x ∈ Ω.
4.51
Journal of Applied Mathematics
17
Let u2 t, u3 t be the positive solution for the following problem:
v̇2 t βe−mτ N2u − u2 t − τ
− au2 t,
N2u − u2 t − τ
v̇3 t kv2 t − pv3 t,
v2 t 1
min,
2 x∈Ω
v3 t t > t7 ,
4.52
t > t7 ,
1
min vx, t,
2 x∈Ω
t ∈ −τ, t7 .
By Lemma 4.1 it follows
N2u − kβe−mτ − ap
,
ap
lim v2 t t→∞
lim v3 t t→∞
k N2u − kβe−mτ − ap
,
ap2
4.53
hence, for all > 0 sufficiently small, by comparison there exists a t8 > t7 such that if t > t8 ,
min w2 x, t > N2w − ,
x∈Ω
min v2 x, t > N2v − ,
x∈Ω
4.54
where
N2w
N2u kβe−mτ − ap
,
ap
N2v
kN2u kβe−mτ − ap
.
ap2
4.55
Since > 0 is arbitrary and sufficiently small, we conclude that
w lim inf min wx, t ≥ N2w ,
t→∞
x∈Ω
v lim inf min vx, t ≥ N2v ,
t→∞
x∈Ω
4.56
continuing this process, we derive six sequences Mnu , Mnw , Mnv , Nnu , Nnw , and Nnv n 1, 2, . . . such that, for n ≥ 2,
u
w
L Mn−1
Mn−1
u
,
w
v
d Mn−1 Mn−1
βNn−1
kβe−mτ − ap Mnu
w
,
Mn ap
k kβe−mτ − ap Mnu
Mnv ,
ap2
u
w
L Nn−1
Nn−1
u
Nn u
,
w
d Nn−1 Nn−1
βMnv
Mnu
18
Journal of Applied Mathematics
kβe−mτ − ap Nnu
,
ap
k kβe−mτ − ap Nnu
v
Nn .
ap2
Nnw 4.57
It is readily seen that
Nnu ≤ u ≤ u ≤ Mnu ,
Nnw ≤ w ≤ w ≤ Mnw ,
Nnv ≤ v ≤ v ≤ Mnv .
4.58
The sequences Mnu , Mnw , and Mnv are nonincreasing and the sequences Nnu , Nnw , and Nnv are
nondecreasing.
To prove the monotonicity of Nnu and Mnu , we follow the ideas of Uh Zapata et al. 6;
consider R0 βke−mτ /ap and the following expressions for N u , N w , N v , Mu , Mw , and Mv
u
w
L Mn−1
Mn−1
u
,
w
d Mn−1 Mn−1
βNn−1
u
w
L Nn−1
Nn−1
u
Nn u
,
w
d Nn−1 Nn−1
βMnv
Mnu
Mnw R0 − 1Mnu ,
Nnw
R0 −
1Nnu ,
Mnv Nnv
k
R0 − 1Mnu ,
p
4.59
k
R0 − 1Nnu .
p
We prove the result by induction so we first show that M2u − M1u ≤ 0,
M2u − M1u Le−mτ M1u
L L L
≤ − 0
u
u −
deM1 aR0 − 1N1 d d d
4.60
and N2u − N1u > 0,
N2u
−
N1u
LR0 − dR0 N1u − βh/p R0 − 1M2u
LR0 N1
u
− N1 N1
dR0 N1u βh/p R0 − 1M2u
dR0 N1u βh/p R0 − 1M2u
N1u
hβ
v
u
LR0 − LR0 βR0 M1 −
R0 − 1M2
p
dR0 N1u βh/p R0 − 1M2u
N1u
hβ
βh
u
u
R
−
1M
−
−
1M
R
R
0
0
0
2
1
p
dR0 N1u βh/p R0 − 1M2u p
βhR0 − 1N1u
R0 M1u − M2u
u
u
p dR0 N1 βh/p R0 − 1M2
>
βhR0 − 1N1u
R0 M1u − M2u > 0.
u
u
p dR0 N1 βh/p R0 − 1M2
4.61
Journal of Applied Mathematics
19
For the next step consider the function fx ax/bx c, with a, b, and c positive, which is
u
u
and Nnu ≥ Nn−1
,
monotone increasing. The induction hypothesis is Mnu ≤ Mn−1
u
Mn1
LR0 Mnu
LR0 Mnu
.
dR0 Mnu βNnu dR0 Mnu βh/p R0 − 1Nnu
4.62
The last function is increasing and by the induction hypothesis we have
u
Mn1
≤
u
u
LR0 Mn−1
LR0 Mn−1
≤
Mnu ,
u
u
dR0 Mn−1
βh/p R0 − 1Nnu dR0 Mn−1
βh/p R0 − 1Nnu
4.63
therefore the sequence Mnu is nonincreasing. For the sequence Nnu we use similar ideas and
the behaviour of the sequence Mnu just proved. One has
u
Nn1
LR0 Nnu
LR0 Nnu
.
u
v
u
LR0 Nn βMn1 LR0 Nnu βh/p R0 − 1Mn1
4.64
The last function is increasing and by the induction hypothesis we have
u
Nn1
≥
LR0 Nn−1
LR0 Nn−1
≥
Nnu ,
u
LR0 Nn−1 βh/p R0 − 1Mn1 LR0 Nn−1 βh/p R0 − 1Mnu
4.65
therefore the sequence Nnu is nondecreasing. The behaviour for the sequences Nnw , Nnv , Mnw ,
and Mnv follows from the nonincreasing sequence Mnu and the nondecreasing sequence Nnu .
Hence, the limit of each sequence in Nnu , Nnw , Nnv , Mnu , Mnw , and Mnv exists. Denote
x lim Mnu ,
x lim Nnu ,
y lim Mnw ,
y lim Nnw ,
z lim Mnv ,
z lim Nnv .
n→∞
n→∞
n→∞
n→∞
n→∞
4.66
n→∞
We therefore obtain from 4.57 and 4.66 that
dkβe−mτ kβ kβe−mτ − ap
0.
−
x−x
ap
ap2
4.67
20
Journal of Applied Mathematics
Noting that H1 holds and R0 > 1, it follows that
kβ kβe−mτ − ap
dkβe−mτ
>
,
ap
ap2
4.68
which together with the previous equation yields x x. We therefore derive from 4.57 y
4.66 that y y, z z. Noting that if R0 > 1, by Theorem 3.1, the virus-infected equilibrium
E2 is locally asymptotically stable, and if in addition H1 holds, we conclude that E2 is
globally asymptotically stable.
Now we prove the global stability for the disease-free equilibrium
Theorem 4.3. If R0 < 1 the disease-free equilibrium E1 L/d, 0, 0 of 1.3 is globally asymptotically
stable.
0, i 1, 2, 3. We
Proof. Let ux, t, wx, t, vx, t be a solution to problem with φi x, 0 /
have ux, t > 0, wx, t > 0, and vx, t > 0 for all x ∈ Ω. Let u1 x, t, w1 x, t, v1 x, t be
a solution to the following problem:
∂u1
L − du1 x, t,
∂t
∂w1 βe−mτ u1 x, t − τv1 x, t − τ
− aw1 x, t,
∂t
u1 x, t − τ w1 x, t − τ
4.69
∂v1
DΔv1 x, t kw1 x, t − pv1 x, t.
∂t
Therefore for t > 0, x ∈ Ω we have
0 ≤ ux, t ≤ u1 x, t,
0 ≤ wx, t ≤ w1 x, t,
0 ≤ vx, t ≤ v1 x, t.
4.70
We derive from the first equation that
lim u1 x, t t→∞
L
,
d
4.71
so we can conclude
lim sup max ux, t ≤
t→∞
L
.
d
4.72
Journal of Applied Mathematics
21
Approximation for u
×10
012
6
5
4
u 3
2
1
0
2000
Approximation for w
1000
800
w 600
400
200
100
t
0
−100
0
6
4
2
0
200
10
8
100
0
t
x
−100
0
2
4
6
x
8
10
Approximation for v
×106
6
5
4
v 3
2
1
0
200
0
100
0
t
−100
0
2
4
8
6
10
x
Figure 1: Simulations with parameters L 211 , d 3.78−2 , a 3.38d, p 0.67, β 1.45−6 , k 5.183 , m 0.2,
and D 0.5. R0 0.039426.
Hence, for > 0 sufficiently small, there exists a t1 such that u1 x, t ≤ L/d for all x ∈ Ω
and t ≥ t1 . Hence, wx, t, vx, t is a lower solution to the following problem:
1
∂ω2
∂t
βe−mτ L/d v1 x, t − τ
L/d 1
∂ω3
∂t
1
1
ω2 x, t
1
1
DΔω3 kω2 x, t − pω3 x, t,
1
1
∂ω3
∂ω2
0,
∂η
∂η
1
− τ
1
− aω2 x, t,
ω2 x, t wx, t,
1
t > t1 ; x ∈ ∂Ω,
ω3 x, t vx, t,
t ∈ −τ, t1 , x ∈ Ω.
4.73
22
Journal of Applied Mathematics
Approximation for u
Approximation for w
11
×10
15
×10012
6
5
4
u 3
2
1
0
200
100
t
10
w
5
0
0
−100
0
4
2
8
6
0
200
10
100
0
t
x
−100
0
2
4
6
8
10
x
Approximation for v
×1015
10
8
v
6
4
2
0
200
100
t
0
−100
0
2
4
6
8
10
x
Figure 2: Simulations with parameters L 211 , d 3.78−2 , a 3.38d, p 0.67, β 1.45−4 , k 5.18 × 103 ,
m 0.2, and D 0.1. R0 3.9426.
Consider u2 t, u3 t as the solution for
u̇2 βe−mτ L u3 t − τ
− au2 t,
L/d u2 t − τ
4.74
u̇3 ku2 t − pu3 t,
u2 t max wx, t,
x∈Ω
u3 t max vx, t,
x∈Ω
t ∈ −τ, t1 .
Then with R0 < 1 according to Lemma 4.1 we have that
lim u2 t 0,
t→∞
lim u3 t 0.
4.75
lim vt 0
4.76
t→∞
By comparison, it follows that
lim wt 0,
t→∞
t→∞
Journal of Applied Mathematics
23
×1016
2
8
Virus v(x, t)
Virus v(x, t)
×1015
10
6
4
2
0
300
00
200
Tim
100
e, t
0
−100
0
2
6
4
,x
o
i
t n
Loca
1.5
1
0.5
0
300
10
8
00
200
100
Tim
0
e, t
a D 0.01, τ 4
−100
0
2
4
10
8
6
,x
tion
Loca
b D 0.01, τ 0.9
×1016
2
15
Virus v(x, t)
Virus v(x, t)
×10
10
8
6
4
2
10
0
250 200
150
100 50
Time, t
5
0 −50
0
x
n,
1
0.5
0
300
o
ati
c
Lo
1.5
200
00
100
Time, t
c D 1, τ 4
0
−100 0
10
5
,x
tion
Loca
d D 1, τ 0.9
Figure 3: In this case the parameters are L 211 , d 3.78−2 , a 3.38d, p 0.67, β 1.45−4 , k 5.183 and
m 0.2.
uniformly for x ∈ Ω. Hence, for > 0 sufficiently small, by comparison there is a t2 ≥ t1 such
that if t ≥ t2 , wx, t < , vx, t < for all x ∈ Ω and t ≥ t2 .
As in the proof of Theorem 4.2 ux, t is an upper solution for the following problem:
1
1
1
∂ω1 L − dω1 − βω1 ,
1
∂ω1
0,
∂η
t ≥ t2 , x ∈ Ω,
1
ω1 x, t2 1
ux, t2 ,
2
4.77
x ∈ Ω,
from the above equation we have that
1
lim ω1 x, t t→∞
L
d β
4.78
24
Journal of Applied Mathematics
×1011
10
1.5
Virus v(x, t)
Virus v(x, t)
×1012
2
1
0.5
0
300
00
200
100
Tim
e, t
0
−100
0
2
4
8
6
8
6
4
2
0
300
10
100
e, t
,x
tion
Loca
200
0
Tim
a D 0.01, τ 0.9
0
2
10
,x
tion
Loca
×1011
10
2
1.5
Virus v(x, t)
Virus v(x, t)
−100
8
6
4
b D 0.01, τ 4
×1012
1
0.5
0
300
0
200
00
100
Tim
e, t
0
−100
0
2
4
8
6
10
8
6
4
2
0
300
00
200
Tim
e, t
,x
tion
Loca
100
c D 1, τ 0.9
0
−100
0
2
8
6
4
10
,x
tion
Loca
d D 1, τ 4
Figure 4: In this case the parameter are L 27 , d 3.78−2 , a 3.38d, p 0.67, β 1.45−4 , k 5.183 and
m 0.2.
uniformly for x ∈ Ω. Since this holds for arbitrary > 0 sufficiently small, by comparison we
conclude that
lim inf min ux, t ≥
t→∞
L
,
d
4.79
which together with 4.72 gives
lim ux, t t→∞
L
d
4.80
uniformly for x ∈ Ω. We already have by Theorem 3.1 that the disease-free equilibrium E1 is
locally asymptotically stable. And now we have proved that it is also globally asymptotically
stable.
Journal of Applied Mathematics
25
5. Numerical Simulations
In this section we illustrate some numerical solutions for systems 1.3. In the numerical
simulation display in Figure 1 we illustrate the stability for the disease-free equilibrium
according to Theorem 3.1. In this case the basic reproductive number is R0 0.039426. In
the graphics we see how the level of uninfected cells increases from the initial condition and
the number of infected cells and virus in the body goes to zero.
In Figure 2 consider the case R0 > 1 in this case we consider a bigger rate of infection
for the cells in the graphics we see how the number of infected cells and viruses increases
when the time passes, and when the number of susceptible cells decreases the number of
virus also decreases to the value v∗ .
Now in Figure 3 we just show the level of virus in different for different values of the
diffusion constant D and the delay τ. We see that a bigger delay increases the time needed
for the virus to reach the value v∗ , meanwhile a mayor value for the constant D just affects
the levels of the virus according to the space and does not affect significantly the time needed
for the virus to reach v∗ . In Figure 4 we consider a lower value for λ, which is the uninfected
cell production rate. In this case we see how the time to reach the value v∗ of the endemic
equilibrium is lower and again the diffusion rate has no significant effect on the time; its
effect is on the level of virus in the system. The delay is what really affect the time to reach
the value v∗ .
Acknowledgments
This paper was supported by UADY under Grant UADY-CA-34 and Mexican SNI under
Grants 15284 and 33365.
References
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vol. 93, no. 9, pp. 4398–4402, 1996.
2 X. Tian and R. Xu, “Asymptotic properties of a hepatitis B virus infection model with time delay,”
Discrete Dynamics in Nature and Society, vol. 2010, Article ID 182340, 21 pages, 2010.
3 K. Wang and W. Wang, “Propagation of HBV with spatial dependence,” Mathematical Biosciences, vol.
210, no. 1, pp. 78–95, 2007.
4 R. Xu and Z. Ma, “HBV model with diffusion and time delay,” Journal of Theoretical Biology, vol. 257,
pp. 499–509, 2009.
5 R. Redlinger, “Existence theorems for semilinear parabolic systems with functionals,” Nonlinear
Analysis. Theory, Methods & Applications, vol. 8, no. 6, pp. 667–682, 1984.
6 M. Uh Zapata, E. Avila Vales, and A. G. Estrella, “Asymptotic behavior of a competition-diffusion
system with variable coefficients and time delays,” Journal of Applied Mathematics, vol. 2008, Article ID
537284, 17 pages, 2008.
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International Journal of
Advances in
Combinatorics
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Mathematical Physics
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Volume 2014
Journal of
Complex Analysis
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Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
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Stochastic Analysis
Abstract and
Applied Analysis
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International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
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Applied Mathematics
Journal of
Function Spaces
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Volume 2014
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Volume 2014
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Volume 2014
Optimization
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014