A reaction–diffusion within-host HIV model with cell-to-cell transmission

Abstract

In this paper, a reaction–diffusion within-host HIV model is proposed. It incorporates cell mobility, spatial heterogeneity and cell-to-cell transmission, which depends on the diffusion ability of the infected cells. In the case of a bounded domain, the basic reproduction number \(R_0\) is established and shown as a threshold: the virus-free steady state is globally asymptotically stable if \(R_0<1\) and the virus is uniformly persistent if \(R_0>1\). The explicit formula for \(R_0\) and the global asymptotic stability of the constant positive steady state are obtained for the case of homogeneous space. In the case of an unbounded domain and \(R_0>1\), the existence of the traveling wave solutions is proved and the minimum wave speed \(c^*\) is obtained, providing the mobility of infected cells does not exceed that of the virus. These results are obtained by using Schauder fixed point theorem, limiting argument, LaSalle’s invariance principle and one-side Laplace transform. It is found that the asymptotic spreading speed may be larger than the minimum wave speed via numerical simulations. However, our simulations show that it is possible either to underestimate or overestimate the spread risk \(R_0\) if the spatial averaged system is used rather than one that is spatially explicit. The spread risk may also be overestimated if we ignore the mobility of the cells. It turns out that the minimum wave speed could be either underestimated or overestimated as long as the mobility of infected cells is ignored.

Full text

1 23
Journal of Mathematical Biology
ISSN 0303-6812
J. Math. Biol.
DOI 10.1007/s00285-017-1202-x
A reaction–diffusion within-host HIV
model with cell-to-cell transmission
Xinzhi Ren, Yanni Tian, Lili Liu &
Xianning Liu

1 23
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J. Math. Biol.
https://doi.org/10.1007/s00285-017-1202-x
Mathematical Biology
A reaction–diffusion within-host HIV model with
cell-to-cell transmission
Xinzhi Ren1·Yanni Tian1·Lili Liu2·Xianning Liu1
Received: 10 May 2017 / Revised: 26 December 2017
© Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract In this paper, a reaction–diffusion within-host HIV model is proposed. It
incorporates cell mobility, spatial heterogeneity and cell-to-cell transmission, which
depends on the diffusion ability of the infected cells. In the case of a bounded domain,
the basic reproduction number R0is established and shown as a threshold: the virus-
free steady state is globally asymptotically stable if R0<1 and the virus is uniformly
persistent if R0>1. The explicit formula for R0and the global asymptotic stability
of the constant positive steady state are obtained for the case of homogeneous space.
In the case of an unbounded domain and R0>1, the existence of the traveling
wave solutions is proved and the minimum wave speed c∗is obtained, providing the
mobility of infected cells does not exceed that of the virus. These results are obtained by
using Schauder fixed point theorem, limiting argument, LaSalle’s invariance principle
and one-side Laplace transform. It is found that the asymptotic spreading speed may
be larger than the minimum wave speed via numerical simulations. However, our
simulations show that it is possible either to underestimate or overestimate the spread
risk R0if the spatial averaged system is used rather than one that is spatially explicit.
The spread risk may also be overestimated if we ignore the mobility of the cells. It turns
out that the minimum wave speed could be either underestimated or overestimated as
long as the mobility of infected cells is ignored.
BXianning Liu
liuxn@swu.edu.cn
Lili Liu
liulili03@sxu.edu.cn
1Key Laboratory of Eco-environments in Three Gorges Reservoir Region (Ministry of Education),
School of Mathematics and Statistics, Southwest University, Chongqing 400715,
People’s Republic of China
2Complex Systems Research Center, Shanxi University, Taiyuan 030006,
People’s Republic of China
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Keywords HIV model ·Heterogeneity ·Mobility ·Basic reproduction number ·
Minimum wave speed
Mathematics Subject Classification 35K40 ·35K51 ·35B40 ·35C07 ·92D30
1 Introduction
The invading and establishing persistent infection processes of viruses in a host are
very complicated and include many factors such as the interactions between virus,
host cells, and immune system. Typically, for the infection of HIV, virions firstly must
cross some substantial physical barriers (e.g. the epithelia of genital or rectal mucosal
surfaces) to gain access to the target cells, where they establish a small founder pop-
ulation of infected cells. There have been two efficient mechanisms described for
this process: luminal entry into the epithelium via diffusive percolation for HIV and
transepithelial migration of HIV-carrying or HIV-infected cells (Anderson et al. 2010;
Carias et al. 2013). Subsequently, this founder population locally expands to dissem-
inate infection to the draining lymph node and then establishes a self-propagating
infection in secondary lymphoid organs via the bloodstream. However, it is not clear
how the virus disseminates to generate a systemic infection via the bloodstream (Fack-
ler et al. 2014; Haase 2010). In the dissemination of HIV, virions may enter the lymph
nodes from distal tissues via afferent lymph vessels, whereas T cells may immigrate
to these organs via high endothelial venules (HEVs) (Marchest and Gowans 1964;
Girard and Springer 1995). Moreover, virions and infected cells may be transported to
the bloodstream via efferent lymph vessels and the thoracic duct (Gowans and Knight
1964; Miller et al. 2005).
It is well known that the lymphoid tissues are among the primary sites of HIV
infection and replication, whose main target cells are CD4 T cells. It is believed
that HIV replicates and transmits mainly by cell-free viral particles. However, recent
experiments showed that the cell-to-cell viral transmission also contributes to the viral
persistence by the formation of the virological synapses during the stable contact
between uninfected cells and infected cells, and the cell-to-cell transmission con-
tributes to the same order as the free virus infection (Costiniuk and Jenabian 2014;
Huebner et al. 2009; Iwami et al. 2015; Komarova et al. 2013). Since CD4 T cells are
densely packed and may frequently interact in lymphoid tissues, it is reasonable to con-
jecture that the cell-to-cell transmission is particularly important in lymphoid tissues
and that the local propagation of HIV within lymphoid tissues may be fundamentally
different than in blood. Recently, Lorenzo-Redondo et al. (2016) showed that, after
treatment with antiretroviral drugs, the virus cannot be detected in the bloodstream,
whereas the virions are still able to reproduce and transport between tissues. It has
been shown that triggering pyroptosis requires cell-to-cell transmission, which is an
effective mechanism for CD4 T cell depletion in HIV infection (Doitsh et al. 2014;
Galloway et al. 2015). All these results highlight that HIV infection in tissues plays
an important role in viral persistence.
The lymphoid tissues consist of many lymph nodes with different sizes (Qatarneh
et al. 2006). Strain et al. (2002) summarized the two reasons for why localization
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is particularly relevant in the case of HIV infection: the virus is inherently unsta-
ble, and the target cells are densely packed in lymphoid tissues. In fact, the tissue
architecture and composition and biophysical parameters are different in different tis-
sues, which may influence the spread and replication of the virus (Fackler et al. 2014).
Lorenzo-Redondo et al. (2016) showed that HIV can continue to replicate in lymphoid
tissue sanctuary sites and then replenish the viral reservoir, which results in persis-
tent infection despite therapy. Therefore, the virus encounters remarkably different
physiological environments in the infected host, which may influence the dynamics
of HIV in vivo, especially in the early stage. For these reasons, to understand the viral
pathogenesis better, it is necessary to consider the spatial aspects of the tissues at the
early stage of infection.
Note that the lymph nodes with asymmetrical structures in terms of cell flux are
connected by lymph vessels (Ruddle 2014; Stacker et al. 2014), and T cells can travel
through the blood and lymph vessels (Kodera et al. 2008). Miller et al. (2002)showed
that the T cells within the intact lymph nodes can move with a mean velocity of
10.8±0.1µmmin
−1, and the tracks of T cells appear randomly over a long time
period by using two-photon laser microscopy. Beltman et al. (2007) concluded that
the migration behavior of T cells may depend on the environment of densely packed
lymphoid nodes. Nakaoka et al. (2016) showed that not all lymphoid tissues have both
efferent lymphatics and afferent lymphatics, but there are some other emigration or
immigration routes for these tissues such as M cells, HEVS and so on (Pabst 2012).
In fact, the moving behavior of lymphocytes is affected by many chemokines (such
as CCL19, CCL21, CXCL12, and CXCL13) and non-chemokine ligands (CXCR4-
tropic envelope glycoprotein gp120 of HIV-1IIIB) (von Andrian and Mempel 2003;
Green et al. 2009), whose distributions may vary in different tissues. Thus, depending
on the physiological environment and the tissue architectures, to study the influence
of the moving ability of lymphocytes on the dynamics of HIV infections in tissues,
we will consider heterogeneous mobility for cells.
Using humanized mouse models and using multiphoton intravital microscopy,
Murooka et al. (2012) was the first to visualize the behavior of HIV-infected human
T cells within a lymph node of a live animal. They found that most productively
infected cells migrate robustly at an average two-dimensional (2D)velocity of ∼7
µmmin
−1, which implies that HIV induces a partial reduction in the mobility of
infected T cells compared with uninfected cells. The reduction of the mobility of
the infected cells may be caused by Nef protein (Fackler et al. 2014). The virus also
appears to use the infected cells to travel through the body, and then infects other CD4
T cells.
Since the effective cell-to-cell transmission requires the formation of stable contacts
between target cells and infected cells, the mobility of infected cells may reduce
the stability of the contact and then destroy the cell-to-cell transmission. Sourisseau
et al. (2007) observed that the cell-to-cell transmission can be prevented by mildly
shaking the cell culture infected with virus. Therefore, we speculate that the effective
infection ability of cell-to-cell transmission is associated with the mobility of the
infected cells.
The movements of free HIV virions within and between tissues are restricted by
some factors: limited diffusions, biophysical barriers, immunological barriers and
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X. Ren et al.
so on (Sattentau 2008). There are many studies investigating the mobility of viri-
ons in tissues. Ewers et al. (2007) found that simian virus 40 receptor is more
than 2 orders of magnitude more mobile than the virion bound to it. Boukari
et al. (2009) showed that the moving behavior of HIV virus in human cervical
mucus is not a typical diffusion, but a combination of anomalous diffusion and
flow-like behavior. Virion mobility decreased 200-fold compared to that in aque-
ous solution. Strain et al. (2002) estimated the diffusion coefficient of HIV by
using the Stokes–Einstein relation, which was also applied to calculate the diffu-
sion constant of tumor virus (Salmeen et al. 1975). Moreover, Andreadis et al.
(2000) estimated the diffusivity of retroviruses Moloney murine leukemia virus by
using nonlinear regression analysis. The mentioned results imply that the diffusion
ability of virions varies from tissue to tissue or from environment to environ-
ment.
Because of the HIV infection propagation complexity, it is not easy to investi-
gate the whole process by experiment. However, mathematical models are effective
tools to understand the pathogenesis of HIV within the host. For decades, there have
been many investigations based on virus–host models to study the interaction between
virus and host (Perelson et al. 1996; Perelson and Nelson 1999; Tian and Liu 2014).
Many of these studies assume that the environment is well-mixed or homogeneous,
and ignore the mobility of the virions or cells. With homogeneous assumption, they
may underestimate or overestimate the key factors of the viral pathogenesis. Some
mathematical models have also been developed to investigate the influences of the
spatial aspect and mobility of the cells or virions on within-host dynamics. Wang and
Wang (2007) proposed a within-host HBV model in an infinite spatial domain and
established the existence of minimum wave speed when the diffusion ability of viri-
ons is sufficiently small. Wang et al. (2008) then proposed and analyzed a delayed
reaction–diffusion model in a bounded domain, where the time delay is associated
with the length of time interval between infection of a cell and production of new
virus particles for HBV. A general viral infection model inside a spherical organ was
studied by Dunia and Bonnecaze (2013), which includes the diffusion, penetration
and proliferation of immune cells. In order to study the effect of spatial heterogeneity
on viral spread dynamics, Wang et al. (2014) investigated a general virus infection
model by assuming that the parameters all depend on the space. All the mentioned
models can be used to describe a virus such as HBV where the host cells do not move.
However, the host cells of some viruses such as HIV can move as suggested in the
analyses. In order to study the repulsion effect of virion by infected cells, Lai and
Zou (2014) established a reaction–diffusion model and investigated its global dynam-
ics and spreading speed. Therein, the target cells and infected cells were assumed to
have the same mobility, and their mobilities were ignored when studying the traveling
wave solutions and spreading speed by simulation. Stancevic et al. (2013) proposed
a chemotaxis model to study the effects of inflammation during the early stage, and
focused on the Turing patterns. In order to study the influence of the lymphoid tis-
sues on HIV infection dynamics, Nakaoka et al. (2016) recently introduced a new
mathematical model that includes the lymphoid tissue volumes and the connectiv-
ity between different tissues. Recently, Pankavich and Parkinson (2016) studied a
reaction–diffusion model for HIV infection, by assuming that the target cells and
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infected cells have different mobility, and that the supply rate of the susceptible cells
depends on space. The global stability of the disease-free steady state was investigated,
but they did not achieve the dissipativity of the global solutions and the persistence of
the virus.
In the propagation of infection of HIV, viral propagation is a localized process
(Strain et al. 2002). Thus, we expect to predict overall HIV progression in terms of
its local dynamics. By developing a cellular automaton model of viral spread among
a small cluster of cells, Strain et al. (2002) showed that there exists a stable prop-
agation as a traveling wave. Recently, Bocharov et al. (2016) established a simple
delayed reaction–diffusion model to investigate the propagation of virus in tissues
and derived some spatiotemporal patterns of viral dynamics via studying the traveling
waves. Therefore, in the case of HIV infection, we are interested in the following
questions: whether there exist traveling wave solutions and how fast the virus spreads
within or between tissues, especially at the early stage of infection. Can the speed
of viral spread be decreased? It is well known that the spatial heterogeneity may
increase the transmission risk (Lou and Zhao 2011) in epidemic models, and the
mobilities of infected cells and virions may decrease the transmission risk (Lai and
Zou 2014). Murooka et al. (2012) found that the plasma viraemia of HIV is strongly
reduced when blocking the egress of migratory T cells from the lymph nodes into
efferent lymph at the early stage of infection, which implies that the spread rate of
HIV may depend on the mobility of infected cells. Because of the heterogeneity of
tissue environments, HIV may exist in some lymphoid tissues during drug therapy,
whereas other tissues or the bloodstream have undetectable levels. The mobility of
HIV virions or cells may also induce infection persistence when the virus exists in
some small lymphoid tissues. Thus, another interesting question arises: How does
the spatial heterogeneity of the environments within-host or the mobility of viri-
ons and cells influence the dynamics of HIV infection? It is well known that the
lymphatic system is distributed throughout the body, but it is difficult to investigate
this question in the body, especially at the early stage of infection. We shall explore
these questions by developing a mathematical model, which may provide some new
insights.
By these arguments, we know that the bloodstream is an important factor of HIV
infection. However, it is well known that blood circulates approximately once every
minute and there are only ≤2% of total lymphocytes that reside in the peripheral
blood, while it takes more than 6 h for T cells to cross the lymphatic system (Stekel
et al. 1997). Moreover, the cell-to-cell transmission mainly occurs in lymphoid tissues.
Based on this, we shall only be concerned with the dynamics of the virus within or
between tissues and ignore the factors of bloodstream.
Note that there are different mobilities between susceptible cells and infected cells
for HIV infection. Previous models did not consider the effects of cell-to-cell transmis-
sion and spatial heterogeneity at the same time, and theoretical proofs of the spreading
speed or traveling wave solutions are not obtained when the mobility of susceptible
cells and infected cells are present. In this paper, we will consider these factors by
investigating the following general reaction–diffusion HIV infection model with spa-
tial heterogeneity:
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⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
Tt=∇·(d1(x)∇T)+λ(x)−d(x)T−β1(x)TV −β2(x)TT∗,t>0,x∈Ω,
T∗
t=∇·(d2(x)∇T∗)+β1(x)TV +β2(x)TT∗−r(x)T∗,t>0,x∈Ω,
Vt=∇·(d3(x)∇V)+N(x)T∗−e(x)V,t>0,x∈Ω,
T(0,x)=T0(x)≥0,T∗(0,x)=T∗0(x)≥0,V(0,x)=V0(x)≥0,x∈Ω,
(1)
with
∂T(t,x)
∂ν =∂T∗(t,x)
∂ν =∂V(t,x)
∂ν =0,t>0,x∈∂Ω, (2)
where Ωis the spatial domain and νis the outward normal vector to ∂Ω. Since the
target tissues architecture and composition and the physiological environments within-
host are too complicated to describe, we take Ωas the target tissues and assume that
model parameters depend on space.
Here, T(t,x), T∗(t,x)and V(t,x)represent the concentrations of healthy cells
(CD4 T cells), infected cells and virions at time tand location x, respectively. Assume
that the initial data are continuous and bounded functions on Ω, and the mobility of
healthy cells, infected cells and virions take the standard Fick diffusion or Brownian
motion for simplification. The meanings and mean values of the parameters in (1) can
be found in Table 1. We assume that β2(x)is a continuous nonincreasing function of
d2(x), and β2(x)is small enough if d2(x)is sufficiently large. Since we mainly focus
on the early stage of infection, for simplification, the Neumann boundary conditions
are adopted in the case of bounded domain.
In this paper, we assume that all the location-dependent parameters are continuous,
strictly positive and uniformly bounded functions on Ω, and denote the maximum
value of a(x)over Ωas aand the minimum value of a(x)over Ωas a, where a=λ,
β1,β
2,d1,d2,d3,d,r,e,N.
The paper proceeds as follows. In the next section, we study the bounded domain
problem (1) with (2). Both the well-posedness of the solutions and the global dynamics
of the model are investigated. The traveling wave solution problem and the minimum
wave speed for the infinite domain are studied in Sect. 3. In Sect. 4, we study the
influences of spatial heterogeneity and diffusion ability on virion propagation via
numerical simulations. Some conclusions and discussions are presented in the last
section.
2 Dynamics in a bounded domain
In this section, we mainly focus on the conditions for viral persistence or extinction in
heterogeneous environments. We firstly show some results about the general system (1)
with (2) and then study the global dynamics of a special case where all the parameters
are homogenous expect for the diffusion coefficients.
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Tab l e 1 Model parameters and their mean values
Parameter Description (position x) Mean value Reference
λ(x)Recruitment rate of
CD4 cells
5.0×105cells/(day mL) Nakaoka et al. (2016)
d(x)Death rate of healthy
cells
0.01 day−1Nakaoka et al. (2016)
r(x)Death rate of infected
cells
0.5day
−1Nelson et al. (2000),
Perelson et al.
(1996)
e(x)Death rate of virions 3 day−1Nelson et al. (2000),
Perelson et al.
(1996)
N(x)Virus production rate 1000 virions/(cell day) Funk et al. (2005)
β1(x)Virus infection rate 1.2×10−10 mL/(virions cells day) Variable
β2(x)Cell-to-cell infection
rate
4.5×10−8mL /(cells cells day) Variable
d1(x)Diffusion rate of
healthy cells
0.09648 mm2day−1Miller et al. (2002)
d2(x)Diffusion rate of
infected cells
0.05 mm2day−1Variable
d3(x)Diffusion rate of
virions
0.17 mm2day−1Strain et al. (2002)
2.1 Analysis of the general system
In this subsection, we start with showing the well-posedness of system (1) with (2),
which includes the existence and uniqueness of the solutions.
Consider the following system:
ωt=∇·(d1(x)∇ω) +λ(x)−d(x)ω, t>0,x∈Ω,
∂ω(t,x)
∂ν =0,x∈∂Ω. (3)
By the similar arguments as those in Lou and Zhao (2011, Lemma 1), the following
lemma is valid.
Lemma 1 System (3)admits a unique positive steady state ω∗(x)which is globally
asymptotically stable in C(Ω,R). Furthermore, ω∗=λ/d if both λand d are positive
constants.
Let Y=C(Ω,R3)with the supremum norm ·
Y,Y+=C(Ω,R3
+). Then
(Y,Y+)is an ordered Banach space. Denote by Ti(t):C(Ω,R)→C(Ω, R)(i=
1,2,3)the C0semigroups associated with ∇·(di(x)∇)−μi(·)subjects to the Neumann
boundary condition, where μ1(x)=d(x), μ2(x)=r(x)and μ3(x)=e(x). That is
(Ti(t)φ)(x)=Ω
Gi(t,x,y)φ(y)dy,t>0,φ ∈C(Ω,R),
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where Gi(t,x,y)is the Green function associated with ∇·(di(x)∇)−μi(·)subjects
to the Neumann boundary condition. Then Ti(t)(i=1,2,3)is compact and strongly
positive for each t>0 by Smith (1995, Corollary 7.2.3). Obviously, there exists an
M>0 such that Ti(t)≤Meαitfor each t≥0, where αi<0 is the principal
eigenvalue of ∇·(di(x)∇)−μi(·)subjects to the Neumann boundary condition.
Define F=(F1,F2,F3)T:Y+→Yby
F1(ψ)(x)=λ(x)−β1(x)ψ1(x)ψ3(x)−β2(x)ψ1(x)ψ2(x),
F2(ψ)(x)=β1(x)ψ1(x)ψ3(x)+β2(x)ψ1(x)ψ2(x),
F3(ψ)(x)=N(x)ψ2(x),
where ψ=(ψ1,ψ
2,ψ
3)T∈Y+.Then system (1) with (2) can be rewritten as the
following integral equation:
u(t)=u∗(t)ψ +t
0
u∗(t−s)F(u(s))ds,(4)
where u(t)=(T(t), T∗(t), V(t))T,u∗(t)=diag(T1(t), T2(t), T3(t)).
It is easy to see that the subtangential conditions in Martin and Smith (1990, Corol-
lary 4) are satisfied. Thus, the following lemma is valid.
Lemma 2 For every initial data ψ∈Y+,system(1)with (2)has a unique noncon-
tinuable mild solution u(t,ψ) ∈Y+on the maximal interval of existence [0,τ
∞),
τ∞≤+∞. Moreover, this solution is a classical solution.
The well-posedness of the solutions and the existence of the global attractor of
system (1) with (2) can be established as follows.
Theorem 1 For every initial data ψ∈Y+,system(1)with (2)has a unique solution
u(t,x,ψ) ∈Y+on [0,+∞), and the solution semiflow Ψ(t)=u(t,·,ψ) :Y+→
Y+has a global attractor.
Proof The existence, uniqueness and positivity of the solution can be followed by
Lemma 2. Suppose τ∞<+∞, and then u(t,x,ψ)→+∞as t→+∞by Martin
and Smith (1990, Theorem 2). From the first equation of (1), we have
Tt≤∇·(d1(x)∇T)+λ−d T (x), t∈[0,τ
∞), x∈Ω. (5)
It follows from Lemma 1and the comparison principle that there exists a constant
M1>0 such that
T(t,x)≤M1,t∈[0,τ
∞), x∈Ω.
Then we can get
T∗
t≤∇·(d2(x)∇T∗)+β1M1V+β2M1T∗−r T ∗,t∈(0,τ
∞), x∈Ω,
Vt≤∇·(d3(x)∇V)+NT∗−eV,t∈(0,τ
∞), x∈Ω.
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Consider the following problem:
⎧
⎪
⎨
⎪
⎩
v1t=∇·(d2(x)∇v1)+β1M1v2+β2M1v1−rv1,t>0,x∈Ω,
v2t=∇·(d3(x)∇v2)+Nv1−ev2,t>0,x∈Ω,
∂v1(t,x)
∂ν =∂v2(t,x)
∂ν =0,x∈∂Ω.
(6)
It follows from the standard Krein–Rutman theorem that the eigenvalue problem asso-
ciated with system (6) has one principal eigenvalue λcorresponding to a strongly
positive eigenfunction φ=(φ1,φ
2). Thus, system (6) admits a solution σeλtφ(x)for
t≥0, where σsatisfying σφ =(v1(0,x), v2(0,x)≥(T∗(0,x), V(0,x)) for x∈Ω.
Then it follows from the comparison principle that
(T∗(t,x), V(t,x)≤σeλtφ(x), t∈[0,τ
∞), x∈Ω,
which implies that there exists a constant M2>0 such that
T∗(t,x)≤M2,V(t,x)≤M2,t∈[0,τ
∞), x∈Ω,
if τ∞<+∞,a contradiction. Therefore, the global existence can be derived.
We now show that the solution semiflow is point dissipative. By the comparison
principle, inequality (5) and Lemma 1, we obtain that there exist N1>0 and t1>0
such that
T(t,x)≤N1,t≥t1,x∈Ω.
Let P(t)=Ω(T(t,x)+T∗(t,x)+r(x)V(t,x)/N(x))dx.Then P(t)satisfies
Pt=Ω
(λ(x)−d(x)T(t,x)−r(x)T∗(t,x)−r(x)e(x)
N(x)V(t,x))dx
≤Ω
λ(x)dx−min
Ω{d(x), r(x), e(x)}P,t≥0.
Thus there exist N2>0 and t2>0 such that P(t)≤N2for all t≥t2.
It follows from Guenther and Lee (1996, Chapter 5) that G2(t,x,y)=
Σn≥1eτntϕn(x)ϕn(y), where τiis the eigenvalue of ∇·(d2(x)∇)−r(x)subjects
to the Neumann boundary condition corresponding to the eigenfunction ϕn(x), and
satisfies τ1>τ
2≥τ3≥···≥τn≥···. Then
G2(t,x,y)≤w1Σn≥1eτnt,t>0,
for some w1>0 since ϕnis uniformly bounded.
Denote by νn(n=1,2···)the eigenvalue of ∇·(d2∇)−rsubjects to the Neumann
boundary condition, and satisfy ν1=−r>ν
2≥ν3≥···≥νn≥···.It follows
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X. Ren et al.
from Wang (2010, Theorem 2.4.7) that νi≥τifor any i∈N+. Since νndecreases
like −n2, then for t>0,
G2(t,x,y)≤w1Σn≥1eνnt≤weν1t=we−rt ,
for some w>0.
Let t3=max{t1,t2}.By (4) and the comparison principle that for any t≥t3,
T∗(t,x)=T2(t)T∗(t3,x)+t
t3
T2(t−s)(β1(x)V(s,x)+β2(x)T∗(s,x))T(s,x)ds
≤Meα2(t−t3)T∗(t3,x)+t
t3Ω
G2((t−s), x,y)(β1(y)V(s,y)
+β2(y)T∗(s,y))T(s,y)dyds
≤Meα2(t−t3)T∗(t3,x)+t
t3
we−r(t−s)Ω
(β1V(s,y)
+β2T∗(s,y))N1dyds
≤Meα2(t−t3)T∗(t3,x)+wN1N2β2+β1N
rt
t3
e−r(t−s)ds
=Meα2(t−t3)T∗(t3,x)+wN1N2β2+β1N
r1−e−r(t−t3)
r
≤Meα2(t−t3)T∗(t3,x)+wN1N2β2r+β1N
r2.
Thus, lim supt→∞ T∗(t,x)wN1N2(β2r+β1N)/r2.By the similarly arguments,
there exists an N3>0 such that lim supt→∞ V(t,x)N3,which implies that the
system is point dissipative. It follows from Wu (1996, Theorem 2.2.6) that the solution
semiflow Ψ(t)is compact for any t>0. Therefore, Ψ(t)has a global attractor by
Hale (1988, Theorem 3.4.8). 
In the rest of this subsection, we first define the basic reproduction number of virus
and then show that it is a threshold for viral persistence or extinction.
It follows from Lemma 1that system (1) with (2) admits a unique virus-free steady
state E0=(T0(x), 0,0), where T0=ω∗(x).
Linearizing system (1)atE0, we can obtain the following equations:
⎧
⎨
⎩
P2t=∇·(d2(x)∇P2)+β1(x)T0P3+β2(x)T0P2−r(x)P2,t>0,x∈Ω,
P3t=∇·(d3(x)∇P3)+N(x)P2−e(x)P3,t>0,x∈Ω,
∂P2
∂ν =∂P3
∂ν =0,t>0,x∈∂Ω.
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Let (P2,P3)=eλt(ψ2(x), ψ3(x)). Then the system can be rewritten as:
⎧
⎨
⎩
λψ2=∇·(d2(x)∇ψ2)+β1(x)T0ψ3+β2(x)T0ψ2−r(x)ψ2,x∈Ω,
λψ3=∇·(d3(x)∇ψ3)+N(x)ψ2−e(x)ψ3,x∈Ω,
∂ψ2
∂ν =∂ψ3
∂ν =0,x∈∂Ω,
(7)
which is a cooperation system. The Krein–Rutman theorem implies that (7) admits
a unique principal eigenvalue λ0(T0)with a strongly positive eigenfunction (φ2(x),
φ3(x)).
Let Φ(t):C(Ω,R2)→C(Ω,R2)be the solution semigroup associated with the
following system:
⎧
⎨
⎩
P2t=∇·(d2(x)∇P2)−r(x)P2,t>0,x∈Ω,
P3t=∇·(d3(x)∇P3)+N(x)P2−e(x)P3,t>0,x∈Ω,
∂P2
∂ν =∂P3
∂ν =0,t>0,x∈∂Ω,
and define
F(x)=β2(x)T0β1(x)T0
00
.
Suppose that the distribution of initial infection is ψ=(ψ2(x), ψ3(x)). Then Φ(t)ψ
represents the distribution of those infective numbers as time evolves. Thus, the dis-
tribution of total new infective numbers is
L(ψ)(x)=∞
0
F(x)Φ(t)ψ dt.
It follows by the next generation operator that the basic reproduction number is given
by
R0=ρ(L),
where ρ(L)is the spectral radius of L.
By the similar arguments as those in Wang and Zhao (2012), the following lemma
holds.
Lemma 3 R0−1and λ0have the same sign. The steady state E 0is asymptotically
stable if R0<1, and it is unstable if R0>1.
Now we show that R0is a threshold for viral persistence or extinction.
Theorem 2 If R0<1, then the virus-free steady state E0is globally asymptotically
stable.
Proof By Lemma 3, there exists a δ>0 such that λ0(T0+δ) < 0.It follows from
the first equation of (1) that
Tt≤∇·(d1(x)∇T)+λ(x)−d(x)T,t>0,x∈Ω,
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which implies that there exists a t1>0 such that T(t,x)≤T0+δfor any t≥t1and
x∈Ω, with the aid of the comparison principle. Therefore,
T∗
t≤∇·(d2(x)∇T∗)+(T0+δ)(β1(x)V+β2(x)T∗)−r(x)T∗,t≥t1,x∈Ω,
Vt≤∇·(d3(x)∇V)+N(x)T∗−e(x)V,t≥t1,x∈Ω.
Suppose that α(φ2(x), φ3(x)) ≥(T∗(t1,x), V(t1,x)), where (φ2(x), φ3(x)) is the
eigenfunction corresponding to the principal eigenvalue λ0(T0+δ) < 0. Then the
comparison principle implies that
(T∗(t,x), V(t,x)) ≤α(φ2(x), φ3(x)eλ0(T0+δ)(t−t1),t≥t1.
Therefore, limt→∞(T∗(t,x), V(t,x)) =0,and T(t,x)is asymptotic to Eq. (3).
Thus, it follows by the theory of asymptotically autonomous semiflows (Thieme 1992,
Corollary 4.3) that limt→∞ T(t,x)=T0.The conclusion can be followed from these
results and Lemma 3.
Theorem 3 Suppose that R0>1, then there exists a η>0such that every solution
(T(t,x), T∗(t,x), V(t,x)) of system (1)and (2)with T (0,x)≡ 0,T∗(0,x)≡ 0and
V(0,x)≡ 0satisfies
lim inf
t→∞ T(t,x)≥η, lim inf
t→∞ T∗(t,x)≥η, lim inf
t→∞ V(t,x)≥η
uniformly for all x ∈Ω. Furthermore, system (1)with (2)admits at least one positive
steady state.
Proof Let X0={φ=(T,T∗,V)∈Y+:T∗(·)≡ 0 and V(·)≡ 0}.Obviously,
∂X0={(T,T∗,V)∈Y+:T∗(·)≡0orV(·)≡0}. The maximum principle and
Hopf boundary Lemma imply that X0is positively invariant for the solution semiflow
Ψ(t). Set M∂={φ∈∂X0:Ψ(t)∈∂X0,∀t≥0}and let ω(φ) be the omega limit
set of the forward orbit γ+(φ) ={Ψ(t)(φ) :t≥0}.
Claim 1. φ∈M∂ω(φ) =E0.
Since φ∈M∂, then T∗(t,x,φ) ≡0orV(t,x,φ) ≡0. Suppose that V(t,x,φ) ≡
0, and then T∗(t,x,φ) ≡0 by the third equation of (1). Therefore, T(t,x,φ) →T0
uniformly for x∈Ωas t→∞. If there exists a t0≥0 such that V(t0,x,φ) ≡ 0, then
V(t,x,φ) > 0 for all t>t0by the maximum principle and Hopf boundary Lemma,
which implies that T∗(t,x,φ) ≡0 for all t≥t0. It follows from the third equation of
(1) that V(t,x,φ) →0 uniformly for x∈Ωas t→∞. Then T(t,x,φ)is asymptotic
to Eq. (3). Thus, the theory of asymptotically autonomous semiflows (Thieme 1992,
Corollary 4.3) yields that T(t,x)→T0uniformly for x∈Ωas t→∞.
Since R0>1, then there exists a δ>0 such that λ0(T0−δ) > 0.
Claim 2. E0is a uniform weak repeller in the sense that lim supt→∞ Ψ(t)(φ) −
E0≥δfor all φ∈X0.
Suppose, by contradiction, there exists a φ0∈X0such that lim supt→∞ Ψ(t)(φ0)−
E0<δ.Then there exists a t1>0 such that
T(t,x,φ
0)>T0−δ, ∀t≥t1.
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Thus, we can get
T∗
t≥∇·(d2(x)∇T∗)+(T0−δ)(β1(x)V+β2(x)T∗)−r(x)T∗,t≥t1,x∈Ω,
Vt≥∇·(d3(x)∇V)+N(x)T∗−e(x)V,t≥t1,x∈Ω.
(8)
Denote by (φ2(x), φ3(x)) the eigenfunction corresponding to the principal eigenvalue
λ0(T0−δ) > 0. Suppose α>0 and satisfies α(φ2(x), φ3(x)) ≤(T∗(t1,x), V(t1,x)).
By the comparison principle, we obtain that
(T∗(t,x), V(t,x)) ≥α(φ2(x), φ3(x)eλ0(T0−δ)(t−t1),t≥t1,
which implies that limt→∞(T∗(t,x), V(t,x)) =(+∞,+∞), a contradiction.
Define a continuous function p:Y+→[0,∞] by
p(φ) =min{min
x∈Ω
φ2(x), min
x∈Ω
φ3(x)},φ∈Y+.
Obviously, p−1(0,∞)⊆X0,and phas the property that if either p(φ) =0 and
φ∈X0or p(φ) > 0, then p(Ψ (t)φ) > 0. Thus, pis a generalized distance function
for the semiflow Ψ(t):Y+→Y+(Smith and Zhao 2001). By the discussions
above, we can conclude that any forward orbit of Ψ(t)in M∂converges to E0, and
Ws(E0)∩X0=∅, where Ws(E0)is the stable subset of E0. Further, E0is an isolated
invariant set in Y+and no sets of {E0}formacyclein∂X0. It follows by Smith and Zhao
(2001, Theorem 3) that there exists a δ1>0 such that min{p(ψ) :ψ∈ω(φ)}>δ
1
for any φ∈X0,which induces that
lim inf
t→∞ T∗(t,x,φ) ≥δ1and lim inf
t→∞ V(t,x,φ) ≥δ1,∀φ∈X0.
By the arguments as those in Theorem 1, we find that there exist constants M>0
and t2>0 such that
T∗(t,x,φ) ≤Mand V(t,x,φ)≤M,t≥t2,∀x∈Ω.
Then T(t,x)satisfies:
Tt≥∇·(d1(x)∇T)+λ−(d+β1M+β2M)T,t≥t2,x∈Ω.
Thus, lim inf t→∞ T(t,x,φ) ≥δ2:= λ/(d+β1M+β2M), by the comparison
principle and Lemma 1.Letη:= min{δ1,δ
2}. The uniform persistence is obtained.
It follows from Magal and Zhao (2005, Theorem 4.7) that system (1) with (2)
admits at least one steady state in X0.We now show these steady states are positive
steady states. Assume (ψ1,ψ
2,ψ
3)is a steady state in X0. Then ψ2≡ 0 and ψ3≡ 0,
thus ψ2>0 and ψ3>0, by the maximum principle and Hopf boundary Lemma.
Note that the maximum principle implies that ψ1>0orψ1≡0. Suppose ψ1≡0.
Then ψ2≡0, by the second equation of the steady state system for system (1) and the
maximum principle, a contradiction. Thus, (ψ1,ψ
2,ψ
3)is a positive steady state. 
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2.2 Homogeneous space and heterogeneous diffusion ability
In this case, it is clear that system (1) with (2) has a virus-free steady state E0=
(T0,0,0), where T0=λ/d. We first show that the basic reproduction number is
R0=(β1T0N+β2T0e)/er, which is the same as the ODE system. The following
lemma can be directly followed by Wang and Zhao (2012, Theorem 3.4).
Lemma 4 Suppose that all the parameters in (1)with (2)are homogeneous except
for the diffusion coefficient di(i=1,2,3), then the basic reproduction number is
R0=β1T0N+β2T0e
er .
Obviously, system (1) with (2) has one positive steady state E1=(T1,T∗
1,V1)
if R0>1, where T1=re/(β1N+β2e), T∗
1=(λ −dT1)/r,V1=NT∗
1/e.In the
following, we show the global attractivity of the positive steady state.
Theorem 4 Suppose that all the parameters in (1)with (2)are homogeneous except
for the diffusion coefficient di(i=1,2,3). Then the following conclusions are valid.
1. If R0<1, then E0is globally asymptotically stable in the interior of Y+;
2. If R0>1, then E1is globally asymptotically stable in the interior of Y+.
Proof The assertion 1 is obvious by Theorem 2.
Suppose R0>1. Let Υ(t)=ΩW(t,x)dx, where
W=T−T1−T1ln T
T1+T∗−T∗
1−T∗
1ln T∗
T∗
1+β1
eT1V−V1−V1ln V
V1.
Then
Υ(t)=Ωd(T1−T)1−T1
T+f(t,x)dx−T1Ω
d1(x)|∇T|2
T2dx
−T∗
1Ω
d2(x)|∇T∗|2
T∗2dx−β1
eV1T1Ω
d3(x)|∇V|2
V2dx
≤0.
where
f(t,x)=β1T1V13−T1
T−T∗
1TV
T1V1T∗−T∗V1
T∗
1V+β2T1T∗
12−T1
T−T
T1≤0.
Obviously, Υ(t)=0 if and only if (T,T∗,V)=(T1,T∗
1,V1). Thus Υ(t)is a
Lyapunov functional, namely, Υ(t)≤0 for all t>0, and then the assertion 2 is valid
by LaSalle’s invariant principle (Henry 1981, Theorem 4.3.4). 
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Remark 1 Theorem 4implies that E1is the unique positive steady state of system (1)
with (2), and the global dynamics of the model in homogeneous space is the same as
those in the ODE model. In fact, if we consider a more general model with a virus
absorbing term and health cell function growth in homogeneous space, the eventual
properties of the model is the same as those in the ODE model in Pourbashash et al.
(2014).
3 Traveling wave solutions
We study the existence of the traveling wave solutions and the minimum wave speed
for system (1) in this section. The lymphoid tissue consists of 1000–1200 lymph nodes
and the size of lymph nodes is 2–38 mm (Qatarneh et al. 2006), which implies that
the sizes of HIV and CD4 T cells are much smaller than the lymphoid tissue. Thus,
for the sake of convenience, we assume that the domain is Rand the parameters are
all spatially homogeneous. By the analysis in Sect. 2.2, we only need to consider the
case of R0>1. The traveling wave solutions studied in this section are particular
solutions, which describe the spatial transition from one steady state to another.
Let (T(t,x), T∗(t,x), V(t,x)) =(T(ξ), T∗(ξ ), V(ξ)), where ξ=x+ct . Then
system (1) can be rewritten as:
⎧
⎪
⎨
⎪
⎩
d1Tξξ −cTξ+λ−dT −β1TV −β2TT∗=0,
d2T∗
ξξ −cT∗
ξ+β1TV +β2TT∗−rT∗=0,
d3Vξξ −cVξ+NT∗−eV =0.
(9)
For system (1), the traveling wave solutions satisfy system (9) and the boundary
conditions:
lim
ξ→−∞(T(ξ), T∗(ξ ), V(ξ)) =E0,lim
ξ→+∞(T(ξ), T∗(ξ ), V(ξ)) =E1.
Thus, the linearized system of (9)atE0=(λ/d,0,0)can be obtained as
⎧
⎪
⎨
⎪
⎩
d1Tξξ −cTξ−dT −β1T0V−β2T0T∗=0,
d2T∗
ξξ −cT∗
ξ+β1T0V+β2T0T∗−rT∗=0,
d3Vξξ −cVξ+NT∗−eV =0.
(10)
Let (T(ξ), T∗(ξ ), V(ξ)) =eλξ (ζ,η,ψ), and then system (10) is equivalent to
A(ζ,η,ψ)
T=cλ(ζ, η , ψ)T,(11)
where
A=⎛
⎝
d1λ2−d−β2T0−β1T0
0d2λ2−r+β2T0β1T0
0Nd
3λ2−e⎞
⎠.
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By simple calculations, we can obtain that the eigenvalues of Aare
λ1(λ) =d1λ2−d,λ
2(λ) =a+b+(a−b)2+4β1T0N
2,
λ3(λ) =a+b−(a−b)2+4β1T0N
2,
where a=d2λ2−r+β2T0,b=d3λ2−e.The minimum wave speed can be defined
as
c∗=inf
λ>0
λ2(λ)
λ:= inf
λ>0f(λ),
where
f(λ) =λ2(λ)
λ=
(d2+d3)λ2+β2T0−r−e+d2λ2−r+β2T0−d3λ2+e2+4β1T0N
2λ.
In the case of d2=d3,it is easy to show that
c∗=2d2c2
1+4β1T0N−(r+e−β2T0).
Set
A1=d2λ2−r+β2T0β1T0
Nd
3λ2−e.
Then |A1−cλ|=P1(λ)P2(λ) −β1T0N, where P1(λ) =d2λ2−cλ−r+β2T0,
P2(λ) =d3λ2−cλ−e.
Obviously, λ2(0)>0 since R0>1. Then λ2(λ) > λ2(0)/2>0forλ>0small
enough, which implies that limλ→0f(λ) =+∞.It is clear that limλ→+∞ f(λ) =
+∞.We now show that c∗is well defined.
Lemma 5 The following conclusions are all valid.
1. f (λ) > 0for all λ>0.
2. λ
2≥0,λ

2≥0for all λ>0.
3. f (λ) changes sign at most once for λ>0.
4. There exists a unique λ∗>0such that c∗=f(λ∗).
5. For each c >c∗, there exist λc>0,τ>1, and positive eigenvectors ωc=
(ω1c,ω
2c)T>0,ωτc=(ω1τc,ω
2τc)T>0such that
f(λc)=c,f(τ λc)<c,A1(λc)ωc=cλcωc,A1(τ λc)ωτc<cτλ
cωτc.
6. f (λ) > d2λfor any λ>0if d2≤d3.
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Proof (1)It is obvious that the conclusion holds when β2T0−r−e≥0.
Now we assume that β2T0−r−e<0, and then f(λ) > 0 for all λ≥λ−,where
λ−=√−(β2T0−r−e)/(d2+d3).
In the case of λ<λ
−,letg(λ) =(d2λ2−r+β2T0)(d3λ2−e). Then g(λ) =
2λ(2d2d3λ2+d3(−r+β2T0)−d2e)and g(0)=−(β2T0−r)e<β
1T0Nsince
R0>1. If d3(−r+β2T0)−d2e≥0,then gmax(λ) =g(λ−)=−(d3λ2
−−e)2≤0for
λ∈(0,λ
−],since g(λ) > 0 for all λ>0. If d3(−r+β2T0)−d2e<0,then gmax(λ) =
max(g(0), g(λ−)) < β1T0Nfor λ∈(0,λ
−], since g(λ) is decreasing in λ∈(0,λ
+)
and increasing in (λ+,∞), where λ+=√−(d3(−r+β2T0)−d2e)/(2d2d3). There-
fore, g(λ) < β1T0Nfor λ∈(0,λ
−], which implies that (a−b)2+4β1T0N−(a+
b)2>0, thus the conclusion is valid.
(2)By direct calculations, we can obtain that
λ
2=λ(d2+d3)√Δ+(a−b)(d2−d3)
√Δ≥0.
λ
2=Δ[(d2+d3)√Δ+(d2−d3)(a−b)]+2(d2−d3)2λ2[Δ−(a−b)2]
Δ3
2≥0,
where Δ=(a−b)2+4β1T0N.
(3)Since f(λ) =(λ
2λ−λ2)/λ2,and (λ
2λ−λ2)=λλ
2≥0,then the conclusion
is obvious.
(4)Obviously,
f(λ) =((d2+d3)λ2+r+e−β2T0)√Δ+(d2−d3)2λ4−c2
1−4β1T0N
2λ2√Δ,
where Δ=(a−b)2+4β1T0N,c1=r−e−β2T0.
Let g(λ) =((d2+d3)λ2+r+e−β2T0)√Δ+(d2−d3)2λ4−c2
1−4β1T0N.
Then
g(0)=(r+e−β2T0)c2
1+4β1T0N−c2
1−4β1T0N
=c2
1+4β1T0Nr+e−β2T0−c2
1+4β1T0N<0,
since R0>1.
It is easy to see that limλ→∞ g(λ) →+∞, thus there exists a unique λ∗>0
such that g(λ∗)=0 by the conclusion 3, which implies that f(λ∗)=0. Combining
these arguments, we can assert that f(λ) is decreasing in (0,λ
∗]and increasing in
[λ∗,+∞). Thus c∗=f(λ∗), where λ∗is the unique positive root of g(λ) =0.
(5)It is obvious that λ2and λ3are also the eigenvalues of A1, and λ2is the principle
eigenvalue of A1, thus the eigenvector corresponding to λ2is positive. By the definition
of f(λ), these conclusions can be directly derived from these results.
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(6)Let h(λ) =f(λ) −d2λfor any λ>0, and then h(λ) =h1(λ)/(2λ), where
h1(λ) =√Δ+(d3−d2)λ2+β2T0−r−e.Since R0>1 and
h
1(λ) =2(d2−d3)λ(a−b−√Δ)
√Δ≥0,
then h1(λ) > 0 for all λ>0, which implies that the conclusion is valid. 
In the following we show that the existence of traveling wave solutions, which
connect the two steady states E0and E1for c>c∗. In order to apply the Schauder fixed
point theorem, inspired by Weng and Zhao (2006), we first introduce the following
upper and lower solutions.
Let T=T0,T=max{0,T0−σeαξ},(T∗,V)T=ωceλcξ,(T∗,V)T=
max{ωceλcξ−qωτceτλ
cξ,0},where σ, α, q>0 can be defined later.
Lemma 6 The function (T∗,V)satisfies
0≥d2T∗ξξ −cT ∗ξ+β1T0V+β2T0T∗−r T ∗,
0≥d3Vξξ −cV ξ+N T ∗−e V ,
for any ξ∈R.
Proof Since (T∗,V)T=ωceλcξ, then
L(T∗,V)=d2T∗ξξ −cT ∗ξ+β1T0V+β2T0T∗−r T ∗
d3Vξξ −cV ξ+N T ∗−eV 
=eλcξ(A1−cλcI)ωc=0.
Thus, the conclusion holds. 
Lemma 7 Fo r 0<α<min{λc,c/d1}and σ>max{T0,(β
1T0ω2c+β2T0ω1c)/d}.
We have
d1Tξξ −cTξ+λ−dT −β1T V −β2T T ∗≥0,
for ξ= (ln T0/σ )/α := x0.
Proof It is obvious that the conclusion is valid for ξ>x0. Now, we assume that
ξ<x0<0, and then T=T0−σeαξ .Hence,
d1Tξξ −cTξ+λ−dT −β1T V −β2T T ∗
≥eαξ [σα(c−d1α) +dσ−β1T0ω2c−β2T0ω1c]≥0.

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Lemma 8 For q large enough and 1<τ<1+α/λc, we have
0≤d2T∗ξξ −cT∗ξ+β1T V +β2T T ∗−rT∗,for ξ=
ln ω1c
qω1τc
λc(τ −1):= x1,
0≤d3Vξξ −cVξ+NT∗−eV ,for ξ=
ln ω2c
qω2τc
λc(τ −1):= x2.
Proof We only consider the case x1≤x2,; the others can be obtained similarly. Let
q>max ⎧
⎨
⎩
ω2c
ω2τc
,ω2c
ω2τcT0
α−λc(τ−1)
α
,β1σω
2c+β2σω
1c
ω1τccτλ
c−d2(τ λc)2−β2T0+r−β1T0ω2τc⎫
⎬
⎭.
Then x2<(ln T0/σ )/α =x0.
The first inequality is obviously valid for ξ>x1, and the second is valid for ξ>x2.
In the case of x1≤ξ<x2<0,then T∗=0,V=ω2ceλcξ−qω2τceτλ
cξ>0,
T=T0−σeαξ >0.Since A1(τ λc)ωτc<cτλ
cωτcand ω1ceλcξ−qeτλ
cξω1τc≤0.
then
d3Vξξ −cVξ+NT∗−eV
=−Nω1ceλcξ+qeτλ
cξ−d3ω2τc(τλ
c)2+cτλ
cω2τc+eω2τc
>−Nω1ceλcξ+qeτλ
cξNω1τc≥0.
In the case of ξ<x1,then T∗=ω1ceλcξ−qω1τceτλ
cξ>0,V=ω2ceλcξ−
qω2τceτλ
cξ>0,T=T0−σeαξ >0.Thus
d2T∗ξξ −cT∗ξ+β1T V +β2T T ∗−rT∗
=qeτλ
cξcτλ
cω1τc−d2(τλ
c)2ω1τc−β2T0ω1τc+rω1τc−β1T0ω2τc
−β1σeαξ ω2ceλcξ−qω2τceτλ
cξ−β2σeαξ ω1ceλcξ−qω1τceτλ
cξ≥0,
since 1 <τ <1+α/λcand ξ<0. The inequality about Vcan be obtained
similarly. 
We now consider a truncated problem and then establish the existence of the semi-
traveling wave solutions that connect E0for system (9) by limiting argument.
Suppose c>c∗and l>maxi=0,1,2{−xi}. Consider the following truncated prob-
lem:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
d1Tξξ −cTξ+λ−dT −β1TV −β2TT∗=0,ξ∈(−l,l),
d2T∗
ξξ −cT∗
ξ+β1TV +β2TT∗−rT∗=0,ξ∈(−l,l),
d3Vξξ −cVξ+NT∗−eV =0,ξ∈(−l,l),
T(ξ) =T(ξ ), T∗(ξ) =T∗(ξ ), V(ξ) =V(ξ), ξ ∈{−l,l}.
(12)
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Let Il=[−l,l],X=C(Il)×C(Il)×C(Il), and
Λ={(T,T∗,V)∈X:T≤T≤T0,T∗≤T∗≤T∗,V≤V≤Vin Il}.
Obviously, Λis a closed convex set in X, which is equipped with the norm
(T,T∗,V)X=TC(Il)+T∗C(Il)+VC(Il).
It is obvious that T≥0,T∗≥0,V≥0 for any (T,T∗,V)∈Λ. Define
F:Λ→Xby
F(φ0,ϕ
0,ψ
0)=(φ,ϕ,ψ),
where (φ,ϕ,ψ)is the solution of the following problem:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
d1φξξ −cφξ+λ−dφ−β1φψ0−β2φϕ0=0,ξ∈(−l,l),
d2ϕξξ −cϕξ+β1φ0ψ0+β2φ0ϕ0−rϕ=0,ξ∈(−l,l),
d3ψξξ −cψξ+Nϕ0−eψ=0,ξ∈(−l,l),
φ(ξ) =T(ξ ), ϕ(ξ ) =T∗(ξ ), ψ (ξ) =V(ξ), ξ ∈{−l,l}.
(13)
Lemma 9 For any (φ0,ϕ
0,ψ
0)∈Λ, problem (13)has a unique solution. Moreover,
F(Λ) ⊂Λ.
Proof Since system (13) is not a coupled system, and the three equations in (13)are
all inhomogeneous linear equations, then the existence and uniqueness of the solutions
to system (13) can be followed by Hartman (1982, Theorem 3.1 of Chapter 12).
Denote by (φ,ϕ,ψ)the solution of system (13) corresponding to the initial data
(φ0,ϕ
0,ψ
0)∈Λ.
By the second equation of (13), we can get d2ϕξξ −cϕξ−rϕ=−β2φ0ϕ0−
β1φ0ψ0≤0forξ∈(−l,l), and ϕ(l)=0,ϕ(−l)=T∗(−l)>0 since l>
maxi=0,1,2{−xi}. Thus ϕ(ξ) > 0 for all ξ∈(−l,l)by the maximum principle. By
similar arguments, we can assert that φ(ξ) > 0,ψ(ξ)>0 for all ξ∈(−l,l).
Since (φ0,ϕ
0,ψ
0)∈Λ, it follows by the second equation of (13) that
d2ϕξξ −cϕξ+β1T0V+β2T0T∗−rϕ≥0,
d2ϕξξ −cϕξ+β1T V +β2T T ∗−rϕ≤0,
for all ξ∈(−l,l). Let w=T∗−ϕ, and then d2wξξ −cwξ−rw≤0 for all
ξ∈(−l,l)by Lemma 6. Note that w(±l)>0, it follows by the maximum principle
that w≥0 for all ξ∈Il, which implies that ϕ≤T∗for ξ∈Il.Letw=ϕ−T∗.
Then d2w
ξξ −cw
ξ−rw≤0 for all ξ∈(−l,x1)by Lemma 8,w(−l)=0,
w(x1)=ϕ(x1)>0. The maximum principle yields that w≥0 for all x∈[−l,x1].
Thus ϕ≥T∗for all ξ∈Il, since T∗=0 for all x>x1. Therefore, ϕ∈[T∗,T∗]for
all ξ∈Il.
By similar arguments as above and using the results in Lemmas 6,7and 8, we can
deduce that φ∈[T,T0],ψ ∈[V,V]for all x∈Il. This completes the proof. 
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Consider the following inhomogeneous linear equation:
wξξ +Awξ+f(ξ)w(ξ ) =h(ξ ). (14)
Lemma 10 (Fu 2014, Lemma 3.2) Let A be a positive constant and let f and h
be continuous functions on [a,b]. Suppose that w∈C[a,b]∩C2(a,b)satisfies
differential equation (14)in (a,b)and w(a)=w(b)=0.If
−C1≤f≤0and |h|≤C2on [a,b],
for some constants C1,C2,then there exists a positive constant C3,depending on
A,C1and the length of the interval [a,b],such that
wC([a,b])≤C2C3.
Lemma 11 (Fu 2014, Lemma 3.3) Let A,f and h be as those in Lemma 10. Sup-
pose that w∈C[a,b]∩C2(a,b)satisfies differential equation (14)in (a,b).If
wC([a,b])≤C0for some constant C0,then there exists a positive constant C4,
depending only on A,C0,C1,C2and the length of the interval [a,b]such that
wξC([a,b])≤C4.
Remark 2 It is clear that the above two lemmas are also valid for A<0if[a,b]=
[−l,l]and l>0.
Lemma 12 F is a continuous mapping.
Proof Suppose that (φ0,ϕ
0,ψ
0)∈Λ,(φ
0,ϕ

0,ψ
0)∈Λ, and let
(φ,ϕ,ψ)=F(φ0,ϕ
0,ψ
0), (φ,ϕ
,ψ)=Fφ
0,ϕ

0,ψ
0.
Let w=φ−φ. Then w(±l)=0 and satisfies:
wξξ −c
d1
wξ+f(ξ)w(ξ ) =h(ξ ), x∈(−l,l),
where
f=−1
d1
(d+β1ψ
0+β2ϕ
0), h=φ
d1β1ψ0−ψ
0+β2ϕ0−ϕ
0 .
Thus,
−C1≤f≤0 and |h|≤C2ψ0−ψ
0C(Il)+ϕ0−ϕ
0C(Il) on [−l,l],
where C1=(d+β1ω2ceλcl+β2ω1ceλcl)/d1,C2=T0(β1+β2)/d1.Then it follows
by Lemma 10 and Remark 2that there exists a C3, depending on C1,c,d1and l, such
that
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X. Ren et al.
wC(Il)=φ−φC(Il)≤C2C3[ψ0−ψ
0C(Il)+ϕ0−ϕ
0C(Il)].
By similar methods, we can derive that there exists a C4>0, depending on c,d2,
r,β
1,β
2,ω
1c,ω
2c,eλcland T0,such that
ϕ−ϕC(Il)≤C4[ψ0−ψ
0C(Il)+φ0−φ
0C(Il)+ϕ0−ϕ
0C(Il)].
There exists a C5>0, depending on c,d3,eand N, such that
ψ−ψC(Il)≤C5ϕ0−ϕ
0C(Il).
Combining these results, we can get
F(φ0,ϕ
0,ψ
0)−F(φ
0,ϕ

0,ψ
0)C(Il)
≤C6[ψ0−ψ
0C(Il)+φ0−φ
0C(Il)+ϕ0−ϕ
0C(Il)]
=C6(φ0,ϕ
0,ψ
0)−(φ
0,ϕ

0,ψ
0)C(Il),
where C6=C2C3+C4+C5, which implies the conclusion is valid. 
By the similar arguments as those in Fu and Tsai (2015, Lemma 2.7) and using
Lemma 11, we can obtain that Fis precompact. Then the Schauder fixed point theorem
can be used to establish the following lemma.
Lemma 13 System (12)has a solution (T,T∗,V)∈Λon Il.
Theorem 5 For any c >c∗,system(9)admits a solution (T(ξ ), T∗(ξ), V(ξ)) satis-
fying (T(−∞), T∗(−∞), V(−∞)) =(T0,0,0), and 0<T(ξ) < T0,T
∗(ξ) > 0,
V(ξ) > 0for any ξ∈R.Moreover, T ∗(ξ ) =O(eλcξ), V(ξ ) =O(eλcξ)as ξ→−∞,
and Tξ(−∞)=T∗
ξ(−∞)=Vξ(−∞)=0.
Proof Let {ln}n∈Nbe an increasing sequence, l1>maxi=0,1,2{−xi}, and ln→+∞
as n→+∞.Denote by (Tn,T∗
n,Vn)the solution of system (12) obtained in
Lemma 13 with l=ln. Obviously, {(Tn,T∗
n,Vn)}n≥Nare uniformly bounded in
[−lN,lN]for any integer N. Then, it follows from Lemma 11 that {(Tnξ,T∗
nξ,Vnξ)}n≥N
are uniformly bounded in [−lN,lN]. Hence, the equations of (12) imply that
{(Tnξξ,T∗
nξξ ,Vnξξ)}n≥Nare uniformly bounded in [−lN,lN]. We can also obtain that
{(Tnξξξ,T∗
nξξξ ,Vnξξξ)}n≥Nare uniformly bounded in [−lN,lN]. Therefore, the Arzela-
Ascoli’s theorem implies that there exists a subsequence {(Tnk,T∗
nk,Vnk)}such that
(Tnk,T∗
nk,Vnk)→(T,T∗,V)and
!Tnkξ,T∗
nkξ,Vnkξ"→!Tξ,T∗
ξ,Vξ",!Tnkξξ ,T∗
nkξξ ,Vnkξξ "→!Tξξ,T∗
ξξ,Vξξ"
uniformly in any compact interval of Ras k→+∞,forsomeT,T∗,Vin C2(R).
It is obvious that (T,T∗,V)is a solution of system (9) and satisfies T≤T≤T,
T∗≤T∗≤T∗,V≤V≤V. Thus, (T(−∞), T∗(−∞), V(−∞)) =(T0,0,0),
T∗(ξ) =O(eλcξ)and V(ξ ) =O(eλcξ)as ξ→−∞.
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Suppose that there exists a ξ1such that T(ξ1)=0, and then Tξ(ξ1)=0 and
Tξξ(ξ1)≥0 since T(ξ ) ≥0. But the first equation of (9)impliesthatd1Tξξ(ξ1)=
−λ<0, a contradiction. Therefore, T(ξ ) > 0 for all ξ∈R.
Suppose that there exists a ξ2such that V(ξ2)=0. Then Vξ(ξ2)=0 and
Vξξ(ξ2)≥0 since V(ξ ) ≥0. It follows by the third equation of (9) that d3Vξξ(ξ2)=
−NT∗(ξ2)≤0, which implies that T∗(ξ2)=0. Thus, T∗
ξ(ξ2)=0. Let u1=T,
u2=Tξ,u3=T∗,u4=T∗
ξ,u5=V,u6=Vξ. Then the traveling wave solution
problem (9) can be rewritten as
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
u
1=u2,
d1u
2=cu2−λ+du1+β1u1u5+β2u1u3,
u
3=u4,
d2u
4=cu4−β1u1u5−β2u1u3+ru3,
u
5=u6,
d3u
6=cu6−Nu3+eu5.
(15)
It is easy to see that Λ1={(u1,u2,0,0,0,0):u1∈R,u2∈R}is invariant for
system (15). Obviously, (u1(ξ2), u2(ξ2), u3(ξ2), u4(ξ2), u5(ξ2), u6(ξ2)) ∈Λ1, which
induces that T∗(ξ ) =V(ξ ) ≡0 for any ξ∈R, a contradiction. Hence, V(ξ) > 0for
all ξ∈R. By the same way, we can obtain that T∗(ξ ) > 0 for all ξ∈R.
Now, we show that T<T0. Suppose, by contradiction, there exists a ξ3>0 such
that T(ξ3)=T0. Then Tξ(ξ3)=0 and Tξξ(ξ3)≤0. Therefore, it follows from the
first equation in (9) that d1Tξξ(ξ3)=β1T0V(ξ3)+β2T0T∗(ξ3)>0, a contradiction.
Thus, T(ξ) < T0for all ξ∈R.
Integrating the first equation of (9)fromxto sfor fixed s,wehave
Tξ(x)=e−c
d1(s−x)Tξ(s)+1
d1s
x
e
c
d1(x−ξ)P(ξ )dξ,
where P(ξ) =λ−dT(ξ ) −β1T(ξ )V(ξ ) −β2T∗(ξ )T(ξ ). Thus,
lim sup
x→−∞ |Tξ(x)|≤max
ξ≤sP(ξ ) lim sup
x→−∞
1
d1s
x
e
c
d1(x−ξ)dξ≤1
cmax
ξ≤sP(ξ).
Hence, Tξ(−∞)=0 since P(−∞)=0. Similarly, we can derive that T∗
ξ(−∞)=
Vξ(−∞)=0. 
The existence of the semi-traveling wave solutions that connect E0is established
in Theorem 5. In the following, we show that the semi-traveling wave solutions are
indeed the traveling wave solutions connecting the two steady states by using LaSalle’s
invariance principle.
Denote by λ21 <0,λ
22 >0 the two roots of equation d2λ2−cλ−r=0,
and ρ2=d2(λ22 −λ21). Denote by λ31 <0,λ
32 >0 the two roots of equation
d3λ2−cλ−e=0,and ρ3=d3(λ32 −λ31).
Lemma 14 For any c >c∗, we can obtain that max{λ21,λ
31}<λ
c<min{λ22,λ
32}.
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Proof By the definitions of λcand ωcin Lemma 5, we can get
d2λ2
c−cλc−r=−β2T0ω1c+β1T0ω2c
ω1c
<0,
d3λ2
c−cλc−e=−Nω1c
ω2c
<0.
Thus, the conclusion is valid since d2>0 and d3>0.
Consider system (15), and let (u∗
1,u∗
3,u∗
5)=(T1,T∗
1,V1). Define
L1=cu1−d1u2+d1u2u∗
1
u1−cu∗
1u1
u∗
1
1
udu,
L2=cu3−d2u4+d2u4u∗
3
u3−cu∗
3u3
u∗
3
1
udu,
L3=cu5−d3u6+d3u6u∗
5
u5−cu∗
5u5
u∗
5
1
udu.
Let L=L1+L2+β1u∗
1L3/e. Then the derivative of Lalong the solution of system
(15) satisfies
L=du∗
1−u11−u∗
1
u1+β1u∗
1u∗
53−u∗
1
u1−u∗
3u1u5
u∗
1u∗
5u3−u3u∗
5
u∗
3u5
+β2u∗
1u∗
32−u∗
1
u1−u1
u∗
1−d1u2
2u∗
1
u2
1−d2u2
4u∗
3
u2
3−d3u2
6u∗
5β1u∗
1
u2
5e≤0.(16)
Lemma 15 For any c >c∗, denote by T,T∗,V the solution of (9)obtained in
Theorem 5corresponding to c.Assume that there exist a(c)>0,b(c)>0such that
T∗(ξ) < a(c), V(ξ ) < b(c)for any ξ∈R.Then there exist l1>0l2>0and l3>0
such that
|Tξ(s)|<l1T(s), |T∗
ξ(s)|<l2T∗(s), |Vξ(s)|<l3V(s),
for any s ≥0.
Proof It follows by the arguments as those in Theorem 5that there exists an L∗>0,
such that |Tξ(0)|<L∗T(0). Set
l1=max L∗,β1b(c)+β2a(c)
c,c+c2+4d1(β1b(c)+β2a(c))
2d1#.
Let Φ1(s)=Tξ(s)−l1T(s), and then Φ1(0)<0. We first show that Φ1(s)<0for
all s≥0. Suppose, by contradiction, there exists an s1>0 such that Φ1(s1)=0 and
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Φ1ξ(s1)≥0, which implies that Tξ(s1)=l1T(s1)and Tξξ(s1)≥l1Tξ(s1). It follows
by the first equation of (9) that
0=d1Tξξ(s1)−cTξ(s1)+λ−dT(s1)−β1T(s1)V(s1)−β2T(s1)T∗(s1)
>!d1l2
1−cl1−β1b(c)−β2a(c)"T(s1)≥0,
a contradiction. Thus, Tξ(s)<l1T(s)for all s≥0.
Let Φ2(s)=Tξ(s)+l1T(s), and then Φ2(0)>0. We now show that Φ2(s)>0
for all s≥0. Suppose, by contradiction, there exists an s2>0 such that Φ2(s2)=0
and Φ2ξ(s2)≤0. It can be divided into two cases: Φ2(s)≤0 for all s≥s2and there
exists an s3>s2such that Φ2(s3)=0 and Φ2ξ(s3)≥0.For the first case, it follows
by the first equation of (9) that for all s≥s2,
d1Tξξ(s)=cTξ(s)−λ+dT(s)+β1T(s)V(s)+β2T(s)T∗(s)
<(−cl1+β1b(c)+β2a(c))T(s)≤0.
Thus, Tξ(s)<Tξ(s2)=−l1T(s2)<0 for all s≥s2,which implies that T(s)→−∞
as s→+∞, a contradiction. For the second case, we have Tξ(s3)=−l1T(s3)<0
and Tξξ(s3)≥−l1Tξ(s3)>0.By the first equation of (9) that
0=d1Tξξ(s3)−cTξ(s3)+λ−dT(s3)−β1T(s3)V(s3)−β2T(s3)T∗(s3)
>(cl1−β1b(c)−β2a(c))T(s3)≥0,
a contradiction. Therefore, Tξ(s)>−l1T(s)for all s≥0.The other conclusions can
be obtained similarly. 
Lemma 16 Suppose that d2≤d3. For any c >c∗, denote by T ,T∗,V the solution
of (9)obtained in Theorem 5corresponding to c, then there exist a(c)>0,b(c)>0
such that T ∗(ξ ) < a(c), V(ξ ) < b(c)for all ξ∈R.
Proof Let Y=(cT −d1Tξ)/c,W=T∗,Z=(cT∗−d2T∗
ξ)/c,M=(cV −d3Vξ)/c.
By system (9), the solution satisfies the following equations:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
Tξ=c
d1(T−Y),
Yξ=1
c(λ −dT −β1TV −β2TW),
Wξ=c
d2(W−Z),
Zξ=1
c(β1TV +β2TW −rW),
Vξ=c
d3(V−M),
Mξ=1
c(NW −eV),
(17)
and limξ→−∞ Y(ξ ) =T0, limξ→−∞ Z(ξ ) =0 and limξ→−∞ M(ξ ) =0.
Consider the following system:
Wξ=c
d2
(W−Z), Zξ=−rW
c.(18)
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It follows from Ding and Huang (2016, Lemma 3.1) that (18) has a strictly monotone
decreasing solution (W1(ξ ), Z1(ξ )) defined for all ξ∈Rsatisfying that Z1(ξ) >
W1(ξ) > 0, (W1(ξ ), Z1(ξ )) →0asξ→+∞and (W1(ξ), Z1(ξ )) →+∞as
ξ→−∞.
Define σ:[0,+∞)→[0,+∞)as
σ(0)=0,σ(W)=Z1!W−1
1(W)"for W>0,
where W−1
1:(0,+∞)→Ris the inverse of W1. Then l∗:Z=σ(W)for W>0is
the orbit of {(W1(ξ), Z1(ξ )) :ξ∈R}. It is easy to see that σ(W)satisfies σ(W)>W
for W>0, σ(W)is monotone increasing, σ(W)→0asW→0 and σ(W)→+∞
as W→+∞.
If there exists a ξ0∈Rsuch that (W(ξ0), Z(ξ0)) ∈l∗, then it follows by the
comparison principle that Z(ξ ) ≥Z1(ξ ) =σ(W(ξ )) > W(ξ ) and W(ξ) ≤W1(ξ )
for all ξ≥ξ0, which implies that W(ξ ) is monotone decreasing in (ξ0,+∞). Thus,
W(ξ) ≤W(ξ0)for all ξ≥ξ0and W(ξ ) →0asξ→+∞, which yields that W(ξ ) ≤
a1(c):= ω1ceλcξ0for all ξ∈R. In the case that (W(ξ ), Z(ξ )) lies above l∗for all
ξ∈R,it is obvious that the above conclusion is valid and W(ξ) ≤a1(c):= ω1ceλcξ0
for any ξ0∈R.
Since V(−∞)=0 and T∗(ξ ) ≤ω1ceλcξ, it follows by the third equation of (9)
and Lemma 14 that for any ξ∈R:
V(ξ) =1
ρ3ξ
−∞
eλ31(ξ −t)NT∗(t)dt++∞
ξ
eλ32(ξ −t)NT∗(t)dt
≤Na1(c)
ρ3ξ
−∞
eλ31(ξ −t)dt++∞
ξ
eλ32(ξ −t)dt
=Na1(c)
e:= b1(c). (19)
By Lemma 15, we can derive that there exist l1>0, l2>0,and l3>0 such that
|Tξ(s)|<l1T(s)≤l1T0,|T∗
ξ(s)|<l2T∗(s)≤l2a1(c)and |Vξ(s)|<l3V(s)≤
l3b1(c)for any s≥0. By the previous arguments, we can get L(ξ ) →+∞as
ξ→+∞since W(ξ) →0asξ→+∞, which contradicts to Lξ≤0 for all ξ≥0.
Hence, the following claim holds.
Claim 1. Z(ξ )) < σ ( W(ξ)) for all ξ∈R.
We now show the following claim is valid.
Claim 2. |Y(ξ )|<H1:= 2T0+λd1/c2for all ξ∈R,which implies that −cH
1/d1≤
Tξ(x)≤c(T0+H1)/d1.
Since (T(−∞), Y(−∞)) =(T0,T0), then T(−∞)<H1and T(−∞)+
Y(−∞)<H1. Thus, there exists a ξ∗
0∈Rsuch that T(ξ ) +Y(ξ ) < H1for all
ξ≤ξ∗
0. Suppose that there exists a ξ>ξ
∗
0such that T(ξ ) +Y(ξ ) =H1and denote
the first time as ξ1. Then Tξ(ξ1)+Yξ(ξ1)≥0. However,
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Tξ(ξ1)+Yξ(ξ1)=c
d1
(T(ξ1)−Y(ξ1)) +1
c(λ −dT(ξ1)
−β1T(ξ1)V(ξ1)−β2T(ξ1)T∗(ξ1))
≤c
d1
(2T(ξ1)−H1)+1
cλ<0,
which is a contradiction. Therefore, T(ξ ) +Y(ξ ) < H1for all ξ∈R,and Y(ξ ) < H1
for all ξ∈Rsince T(ξ) > 0.
Obviously, there exists a ξ2∈Rsuch that T(ξ ) −Y(ξ ) < H1for all ξ≤ξ2.If
there exists a ξ3∈Rsuch that T(ξ3)−Y(ξ3)=H1, then
Tξ(ξ3)−Yξ(ξ3)=c
d1
(T(ξ3)−Y(ξ3)) −1
c(λ −dT(ξ3)
−β1T(ξ3)V(ξ3)−β2T(ξ3)T∗(ξ3))
≥cH
1
d1−1
cλ>0,
which implies that T(ξ ) −Y(ξ ) > H1for all ξ>ξ
3. Thus, T(ξ ) →+∞since
Tξ=c(T(ξ) −Y(ξ ))/d1, a contradiction. Therefore, T(ξ) −Y(ξ ) < H1for all
ξ∈R, and Y(ξ ) > −H1for all ξ∈R.
Claim 3. Z(ξ) ≤H2:= 2H1+r1(c)for all ξ∈R,where r1(c)>0 satisfies
σ−1(W)>λ/rfor all W≥r1(c).
The existence of r1(c)can be obtained by the definition of σ(W). Since Z(−∞)=0,
there exists a ξ4∈Rsuch that Z(ξ ) +Y(ξ) < H0:= H1+r1(c)for all ξ<ξ
4.
Suppose that there exists a ξ≥ξ4such that Z(ξ ) +Y(ξ) =H0, and denote the first
time as ξ5. Then Zξ(ξ5)+Yξ(ξ5)≥0. Moreover,
Zξ(ξ5)+Yξ(ξ5)=1
c(λ −dT(ξ5)−rW(ξ5)) < 0,
since W(ξ5)>λ/rby the fact that r1(c)<Z(ξ5)=H0−Y(ξ5)<σ(W(ξ5)), which
is a contradiction. Therefore, Z(ξ) < H0−Y(ξ ) ≤H2for all ξ∈R.
Claim 4.W(ξ ) < a(c):= 2H2for all ξ∈R.
Note that c>d2λcby the definition of λcand Lemma 5. It is easy to see that this
claim is valid for ξsmall enough. Suppose, by contradiction, there exists a t1∈R
such that W(t1)=2H2. From the third equation of (17) and Claim 3, we can get
Wξ(ξ) ≥cH
2/d2for all ξ≥t1, which implies that W(ξ ) →+∞as ξ→+∞.Thus
there exists a t2>t1such that rW(ξ ) > λ +cfor any ξ≥t2. Then
Yξ(ξ) +Zξ(ξ ) =λ−dT−rW
c≤−1,ξ>t2,
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which induces that limξ→+∞ Z(ξ ) =−∞by Claim 2. Hence, there exists a t3>t2
such that Z(ξ ) ≤−1/cfor any ξ≥t3.Thus,wehave
T∗(ξ) ≥e
c
d2(ξ−t3)T∗(t3)+1
c−1
c,
for any ξ≥t3,which contradicts T∗(ξ ) ≤ω1ceλcξfor any ξ∈R, since c/d2>λ
c.
Therefore, this claim is valid.
By the similar calculations as those in (19), we can obtain that V(ξ ) < b(c):=
Na(c)/e.
Theorem 6 Suppose that d2≤d3. For any c ≥c∗, there exists a positive traveling
wave solution (T(ξ), T∗(ξ ), V(ξ)) satisfying (9)and (T(−∞), T∗(−∞), V(−∞))
=(T0,0,0),(T(+∞), T∗(+∞), V(+∞)) =(T1,T∗
1,V1)and T (ξ ) < T0for any
ξ∈R.
Proof In the case of c>c∗. We only need to show that (T(+∞), T∗(+∞), V(+∞))
=(T1,T∗
1,V1), where (T,T∗,V)is the solution obtained in Theorem 5.Itfollows
from Lemmas 15 and 16 that there exist a(c)>0,b(c)>0, and l1>0, l2>0,
l3>0 such that T∗(ξ ) < a(c), V(ξ ) < b(c)for all ξ∈R, and
|Tξ(s)|<l1T(s), |T∗
ξ(s)|<l2T∗(s), |Vξ(s)|<l3V(s),
for any s≥0.
Let u1=T,u2=Tξ,u3=T∗,u4=T∗
ξ,u5=V,u6=Vξ,and
(u∗
1,u∗
3,u∗
5)=(T1,T∗
1,V1). Then system (9) can be rewritten as system (15). Define
D={(u1,u2,u3,u4,u5,u6):0<u1<T0,|u2|<l1u1,0<u3<a(c), |u4|<
l2u3,0<u5<b(c), |u6|<l3u5}.It is clear that Dis positively invariant.
Let L=L1+L2+β1u∗
1L3/e. Then Lis continuous and bounded below on D,
and the derivative of Lalong the solution of system (15) satisfies Lξ≤0, which
was shown by (16). Obviously, Lξ=0 if and only if (u1,u2,u3,u4,u5,u6)=
(u∗
1,0,u∗
3,0,u∗
5,0). Therefore, the conclusion can be derived by LaSalle’s invariance
principle.
Now, we consider the case of c=c∗.Let{ci}be a decreasing sequence sat-
isfying c∗<ci+1<ci<c∗+1 and ci→c∗as i→+∞.Let a∗=
8T0+4λd1/c∗2+2max
c∗≤c≤c∗+1r1(c),b∗=Na∗/e. Then for any ci,system
(9) has a positive solution Ψi=(Ti,T∗
i,Vi)satisfying 0 <Ti<T0,0<T∗
i≤
a(c)≤a∗,0<Vi≤b(c)≤b∗,(Ti(+∞), T∗
i(+∞), Vi(+∞)) =(T1,T∗
1,V1)and
(Ti(−∞), T∗
i(−∞), Vi(−∞)) =(T0,0,0),(Tiξ(−∞), T∗
iξ(−∞), Viξ(−∞)) =
(0,0,0), which implies that {Ψi}is uniformly bounded.
We now show {Ψiξ}is also uniformly bounded. Let s0be the local maximum or
minimum point of T∗
iξ, and then T∗
iξξ(s0)=0. It follows from the second equation
of (9) that
T∗
iξ(s0)=β1Ti(s0)Vi(s0)+β2Ti(s0)T∗
i(s0)−rT∗
i(s0)
ci
.
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Denote by a∗
1=(β1T0b∗+β2T0a∗)/ci<(β
1T0b∗+β2T0a∗)/c∗,a∗
2=−ra∗/ci>
−ra∗/c∗.Then a∗
2<T∗
iξ(s0)<a∗
1and a∗
2<T∗
iξ(−∞)=0<a∗
1. Suppose that
there exists an s1∈Rsuch that T∗
iξ(s1)>a∗
1, and then T∗
iξ(ξ) ≥a∗
1for all ξ≥s1,
which implies that T∗
i→+∞, a contradiction. Therefore, T∗
iξ(ξ) ≤a∗
1for all ξ∈R.
By the similar methods, we can also obtain that T∗
iξ(ξ) ≥a∗
2for all ξ∈R, which
yields that {T∗
iξ}is uniformly bounded. That {Tiξ}and {Viξ}are uniformly bounded
can be obtained similarly. Therefore, {Ψiξ}is uniformly bounded, and then {Ψiξξ}
and {Ψiξξξ}are also uniformly bounded since {Ψi}is the solution of system (9) with
c=ci. Arzela-Ascoli’s theorem implies that there exists a subsequence {cik}such that
(Tik,T∗
ik,Vik)→(T,T∗,V)and
!Tikξ,T∗
ikξ,Vikξ"→!Tξ,T∗
ξ,Vξ",!Tikξξ,T∗
ikξξ ,Vikξξ"→!Tξξ,T∗
ξξ,Vξξ"
uniformly in any compact interval of Ras ik→+∞,forsomeT,T∗,Vin C2(R).
And it is obvious that (T,T∗,V)is a nonnegative solution of system (9) and satisfies
0≤T≤T0,0≤T∗≤a∗,0≤V≤b∗.
By standard calculations, it is easy to obtain that there are no eigenvalues with zero
real parts for eigenvalue problem |A1−cλ|=0 with c>0. Thus (T0,0,0,0,0,0)is
a hyperbolic equilibrium of (15). We now show that (T,T∗,V)obtained previously
is a positive solution that connects E0=(T0,0,0)and E1=(T1,T∗
1,V1).For
simplification, we denote {cik}and (Tik,T∗
ik,Vik)by {ci}and (Ti,T∗
i,Vi), respectively.
Suppose that
sup
ξ∈R
(T0−Ti(ξ )) →0,sup
ξ∈R
T∗
i(ξ) →0,sup
ξ∈R
Vi(ξ) →0,as i→+∞.(20)
It follows from the Hartman–Grobman Theorem (Perko 1991; Zhang 2017) that
(Ti(ξ), T∗
i(ξ), Vi(ξ)) →(T0,0,0)as ξ→+∞for ilarge enough, which contradicts
(Ti(+∞), T∗
i(+∞), Vi(+∞)) =(T1,T∗
1,V1). Therefore, one of the three limits in
(20) does not hold. Hence, we can assume that for certain >0 small enough, up to
a subsequence if necessary, one of the following three inequations holds
sup
ξ∈R
(T0−Ti(ξ )) > , sup
ξ∈R
T∗
i(ξ) > , sup
ξ∈R
Vi(ξ) > .
Since (Ti(−∞), T∗
i(−∞), Vi(−∞)) =E0(T0,0,0), we can suppose by a translation
that
T0−≤Ti(ξ ) ≤T0,T∗
i(ξ) ≤, Vi(ξ) ≤, for ξ≤0
hold and one of
Ti(0)=T0−, T∗
i(0)=, V(0)=
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holds. Thus, the definition of (T,T∗,V)implies that
T0−≤T(ξ) ≤T0,T∗(ξ ) ≤, V(ξ ) ≤, for ξ≤0
and one of
T(0)=T0−, T∗(0)=, V(0)=
holds. And then by setting small enough, the Hartman–Grobman Theorem yields
that (T(−∞), T∗(−∞), V(−∞)) =E0(T0,0,0).
It follows by similar arguments as those in Theorem 5that T(ξ) > 0 for all ξ∈R.
We now show that T∗(ξ) > 0 and V(ξ ) > 0 for all ξ∈R.We first consider the case
that T(0)=T0−. Suppose that there exists an s2∈Rsuch that V(s2)=0. Then
T∗(ξ) =V(ξ ) ≡0 for all ξ∈Rby the similar arguments as those in Theorem 5.
Thus, T(ξ) satisfies
d1Tξξ −cTξ+λ−dT =0,ξ∈R,
T0−≤T(ξ) ≤T0for ξ≤0 and T(0)=T0−. It is clear that Tξ(0)≤0, and
Tξξ(0)=1
d1
(cTξ(0)−λ+dT(0)) ≤−d
d1
<0,
!e−c
d1ξTξ"ξ=1
d1
e−c
d1ξ(dT(ξ) −λ) ≤0,for ξ∈R,
by T≤T0.Thus, there exists an s3>0 small enough such that Tξ(s3)<0, and
Tξ(s)≤ec(s−s3)/d1Tξ(s3)for s>s3, which yields Tξ(s)→−∞as s→+∞.
Thus, T(ξ) < 0forξlarge enough, a contradiction. Therefore, V(ξ) > 0 for all
ξ∈R.Similarly, we can obtain that T∗(ξ) > 0 for all ξ∈R. Next, we consider the
case that T∗(0)=. Suppose that there exists an s4∈Rsuch that T∗(s4)=0. Then
T∗(ξ) =V(ξ ) ≡0 by the similar arguments as those in Theorem 5, which contradicts
T∗(0)=>0. Thus, T∗(ξ) > 0 for all ξ∈R. The positiveness of V(ξ) can be
obtained similarly. Finally, in the case where V(0)=, it is obvious that (T,T∗,V)
is a positive solution by the similar arguments as above. Therefore, we can conclude
that (T,T∗,V)is a positive solution.
By the positiveness of (T,T∗,V)and similar arguments as those in Theorem 5,it
is clear that T(ξ ) < T0. Since 0 <T<T0,0<T∗≤a∗,0<V≤b∗,we also
can obtain that (T(+∞), T∗(+∞), V(+∞)) =(T1,T∗
1,V1)by similar arguments
as those in c>c∗. This completes the proof. 
We now consider the nonexistence of the traveling wave solutions for 0 <c<c∗.
Obviously, the real eigenvalues of |A1−cλ|=0 must satisfy cλ=λ2or cλ=λ3.In
the case of 0 <c<c∗, it is easy to see that the real eigenvalues of |A1−cλ|=0must
satisfy cλ=λ3, and it has only one negative real eigenvalue, by the definition of c∗,
the arguments as those in Lemma 5and the properties of λ3/λ. Thus, |A1−cλ|=0
may have one or three positive real eigenvalues for 0 <c<c∗. In fact, it has only one
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positive real eigenvalue for the eigenvalue problem of this type in many studies, and
then the nonexistence of traveling wave solutions can be easily proved. However, we
find that there exist three positive real roots satisfying λ3/λ =cfor some 0 <c<c∗
in our system via numerical simulations. We need introduce the following lemma to
establish the nonexistence.
Lemma 17 For any 0<c<c∗,|A1−cλ|=0does not exist positive real root λ
such that P1(λ) < 0and P2(λ) < 0.
Proof Suppose, by contradiction, there exists a λ>0 such that |A1−cλ|λ=
P1(λ)P2(λ)−β1T0N=0, P1(λ)<0 and P2(λ)<0. Since c<c∗, then cλ<
λ2(λ)=(a+b+(a−b)2+4β1T0N)/2, where a=d1λ2+β2T0−r,b=
d3λ2−e. Thus, 0 >P1(λ)=a−cλ>(a−b−(a−b)2+4β1T0N)/2,
0>P2(λ)=b−cλ>(b−a−(a−b)2+4β1T0N)/2. Then
P1(λ)P2(λ)< a−b−(a−b)2+4β1T0N
2·b−a−(a−b)2+4β1T0N
2
=β1T0N,
which contradicts |A1−cλ|λ=0.
Theorem 7 Suppose that R0>1and 0<c<c∗, then there does not exist nonneg-
ative traveling wave solutions satisfying (9)and the conditions in Theorem 6.
Proof Suppose, by contradiction, there exists a nonnegative traveling wave solution
(T(ξ), T∗(ξ ), V(ξ)) satisfying the conditions in Theorem 6for some 0 <c<c∗.
It is easy to see that |A1−cλ|=0 has no pure imaginary root, thus there exists
aα>0 such that supξ≤0{T∗(ξ)e−αξ }<+∞,supξ≤0{V(ξ)e−αξ }<+∞,
supξ≤0{T∗
ξ(ξ)e−αξ }<+∞,supξ≤0{Vξ(ξ)e−αξ }<+∞,supξ≤0{T∗
ξξ(ξ )e−αξ }<
+∞,supξ≤0{Vξξ(ξ )e−αξ }<+∞ by using the Stable Manifold theorem (Perko
1991; Zhang and Wang 2014), we do not repeat them again.
It is easy to see that lims→−∞ Tξ(s)=0 by similar arguments as those in Theorem
5. Integrating the first equation of (9) from −∞ to ξ, we get
d1Tξ(ξ) −c(T(ξ ) −T0)=g1(ξ ),
where g1(ξ) =ξ
−∞[−d(T0−T(t)) +β1T(t)V(t)+β2T(t)T∗(t)]dt.Let$
T(ξ) =
T0−T(ξ), and then $
T(ξ) satisfies
−d1$
Tξ(ξ) +c$
T(ξ) =g1(ξ ).
Thus
$
T(ξ) =$
T(0)e
cξ
d1+1
d10
ξ
e
c
d1(ξ−t)g1(t)dt
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<$
T(0)e
cξ
d1+1
d10
ξ
e
c
d1(ξ−t)g2(t)dt,
where g2(ξ) =ξ
−∞[β1T(η)V(η) +β2T(η)T∗(η)]dη, since T(ξ ) < T0for all
ξ∈R.Since supξ≤0{T∗(ξ)e−αξ }<+∞ and supξ≤0{V(ξ )e−αξ }<+∞, then
g2(ξ) ≤Ceαξ for some C>0 and ξ≤0, and g2(ξ) =o(eα
2ξ)as ξ→−∞.
Thus, $
T(ξ) =o(eαξ)as ξ→−∞, where α=min{α/4,c/(2d1)}.Hence,
supξ≤0{(T0−T(ξ))e−αξ}<+∞.
Define the negative one-side Laplace transform (Zhang et al. 2016):
Jφ(λ) =0
−∞
e−λξ φ(ξ)dξ, λ ≥0,φ(ξ) ≥0.
Obviously, Jφ(λ) is defined in [0,λ
+
φ)such that λ+
φ≤+∞satisfying limλ→λ+
φJφ(λ) =
+∞ or λ+
φ=+∞. Thus λ+
T∗≥α,λ+
V≥α,λ+
T∗(T0−T)≥λ+
T∗+αand λ+
V(T0−T)≥
λ+
V+α.
Consider the following system:
d2T∗
ξξ −cT∗
ξ+β1TV +β2TT∗−rT∗=0,
d3Vξξ −cVξ+NT∗−eV =0.
By the negative one-side Laplace transform of the system, we can get
P1(λ)JT∗=−β1T0JV+h1,
P2(λ)JV=−NJ
T∗+h2,(21)
where h1=J(β1V+β2T∗)(T0−T)+(c−d2λ)T∗(0)−d2T∗
ξ(0)and h2=(c−d3λ)V(0)−
d3Vξ(0).
We now show that λ+
T∗=λ+
V<+∞.By the first equation of (21), we can obtain
that
Δ1(λ) := (d2λ2−cλ−r)JT∗+Jβ1TV+β2TT∗−(c−d2λ)T∗(0)+d2T∗
ξ(0)=0.
Suppose that λ+
T∗=+∞.Since JT∗>0 and Jβ1TV+β2TT∗>0forλ∈[0,λ
+
T∗),
then Δ1(+∞)=+∞, a contradiction. λ+
V<+∞can be obtained similarly. Suppose
λ+
T∗<λ
+
V<+∞. Then limλ→λ+
T∗JT∗=+∞and limλ→λ+
T∗JV<+∞, which
contradicts to the second equation of (21). Hence, λ+
T∗≥λ+
V. Similarly, we can get
λ+
T∗≤λ+
V. Therefore, λ+
T∗=λ+
V=λ+<+∞.
Suppose that P1(λ+)≥0. Then for λ=λ+,wehave
+∞ = (P1(λ)JT∗+β1T0JV)|λ+>(J(β1V+β2T∗)(T0−T)
+(c−d2λ)T∗(0)−d2T∗
ξ(0))|λ+,
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which contradicts the first equation of (21). Therefore P1(λ+)<0. By the similar
arguments as above, we can also obtain that P2(λ+)<0.
Multiplying the first equation by the second one of (21), we have
(P1(λ)P2(λ) −β1T0N)JT∗JV=−Nh1JT∗−β1T0h2JV+h1h2.
Therefore,
P1(λ+)P2(λ+)−β1T0N=lim
λ→λ+−Nh1JT∗−β1T0h2JV+h1h2
JT∗JV=0,
since hi(λ+)<+∞ (i=1,2). Thus, there exists a λ+>0 such that Pi(λ+)<
0(i=1,2)satisfying |A1−cλ+|=0, which contradicts Lemma 17.
4 Numerical simulations
In this section, we mainly focus on the influences of spatial heterogeneity and the
mobility of the cells or virions on the virus spread via numerical simulations. The
mean values of the parameters in model (1)aretakenasinTable1. We denote the
mean value of a(x)over Ωas %a, where a=λ, β1,β
2,d1,d2,d3,d,r,e,N.
Now, we give the detailed explanations of the parameter values. Note that the
estimated values of the parameters in model (1) are fixed values and independent of
the space in many studies or experiments, and these fixed values may reflect the mean
levels of these factors within-host. Thus, we shall adopt these values as the mean values
of the parameters in model (1). The parameters %
λ, %
d,%r,%eand %
Nare taken from the
literatures in Table 1. Through experimental-mathematical investigation, Iwami et al.
(2015) showed that the cell-to-cell transmission contributes over half of infection, and
Komarova et al. (2013) derived that the ratio of the free to cell-to-cell transmission
rate takes 1.0 on average under the assumption of quasi-steady state for free virus,
which implies that %
β1%
N/%eand %
β2should be assumed to be in the same order. Note
that the basic reproduction number of HIV is estimated to be between 8 and 10, thus
we assume that %
β1=1.2×10−10 mL virions−1cells−1day−1and %
β2=4.5×10−8
mL cells −1cells−1day−1, which implies that R0=8.5 for the homogeneous case
where all the parameters are taken mean values as in Table 1.
By the units of the parameters in Table 1, we find that the parameters %
β1,%
β2and
%
λare all volume based. In order to adapt them to a one-dimensional region and make
them constant, we assume that the region is a thin wire with a cross-sectional area 1
mm ×1 mm and Ω=(0,2)for simplification. The actual environments in vivo are
too complicated to find an explicit expression of the heterogeneity. Since lymphoid
tissue consists of lymphoid nodes, we let the environments be period and take the
amplitude as the heterogeneous parameter as those in (Allen et al. 2008; Lou and
Zhao 2011) in the following simulations, for the sake of convenience.
The diffusion coefficient of the uninfected cells can be followed by Miller et al.
(2002) that
%
d1=67 µm2min−1=0.09648 mm2day−1.
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X. Ren et al.
Fig. 1 Time variation of V(t,x) for different R0.aR0≈1.8638 >1. bR0≈0.8728 <1
Since the infected cells move slower than the uninfected cells do, then we assume that
%
d2=0.05 mm2day−1.
Noting that the diffusion coefficient of the virions is similar to those in (Andreadis
et al. 2000; Salmeen et al. 1975; Strain et al. 2002), we choose the diffusion coefficient
of virions to be that in Strain et al. (2002), that is,
%
d3=1.7×10−3cm2day−1=0.17 mm2day−1.
In the treatment of HIV, the reverse-transcriptase inhibitor and protease inhibitor
are usually used to prevent the synthesis of viral DNA from HIV RNA and to cause the
immature of virus produced by infected cells, respectively. Some studies (Agosto et al.
2015; Malbec et al. 2013) also showed that antiretroviral therapies and some broadly
neutralizing antibodies are likely effective to prevent cell-to-cell transmission. Thus
we can assume that the infection rates of free virus and infected cells, and the virus
production rate of infected cells would be decreased if treating the patients by these
measures. Now, let β1(x)=0.3×%
β1(1+0.5 cos(π x)),β2(x)=0.3×%
β2(1+
0.5 cos(π x)),N(x)=0.3%
N, and other parameters be fixed as their mean values in
Table 1. Then R0≈1.8638 and the virus is persistent (Fig. 1a with T0(x)=5.0×107,
T∗0(x)=0,V0(0)=25,V0(x)=0forx∈(0,2]). Let β2(x)=0.1×%
β2(1+
0.5 cos(π x)),V0(x)=200 ×(1+0.5 cos(π x)) for x∈(0,2)and other parameters
are the same as those in Fig. 1a. Then R0≈0.8728 and the virus is vanishing (Fig. 1b).
It is obvious that the transmission risk of HIV increases as R0increases. We now
consider the influence of the spatial heterogeneity on R0, which represents the trans-
mission risk. Let β2(x)=0.1×%
β2,N(x)=0.3%
N,β1vary and other parameters be
fixed as their mean values in Table 1. Then R0increases as the heterogeneous param-
eter kincreases (Fig. 2a) if β1(x)=0.3×%
β1(1.3+ksin(mπx)), which implies that
the infection risk is underestimated if we average the infection ability of free virus,
and the infection risk increases as the heterogeneity of the free virus infection rate
increases. However, in the case of β1(x)=0.3×%
β1(1.3+kcos(mπx)), the effect
of the heterogeneity on R0is complicated: in some cases R0decreases as kincreases
(m=3,k<0.7) and in other cases it is increasing as kincreases (Fig. 2b) (in fact
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A reaction-diffusion HIV model
0 0.2 0.4 0.6 0.8 1
0.9
0.92
0.94
0.96
0.98
1
1.02
1.04
k
R0
m=1
m=2
m=3
0 0.2 0.4 0.6 0.8 1
0.917
0.9175
0.918
0.9185
0.919
0.9195
0.92
0.9205
0.921
0.9215
k
R0
m=2
m=3
0 0.2 0.4 0.6 0.8 1
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
k
R0
m=1
m=11
m=21
0 0.2 0.4 0.6 0.8 1
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
k
R0
m=1
m=11
m=21
0 0.2 0.4 0.6 0.8 1
2.2239
2.2239
2.2239
2.224
2.224
2.224
2.224
2.224
2.224
2.224
2.224
k
R0
m=1
m=11
m=21
0 0.2 0.4 0.6 0.8 1
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
k
R0
m=1
m=6
(a) (b)
(c) (d)
(e) (f)
Fig. 2 The effect of the heterogeneous parameters on R0.aβ1(x)=0.3×%
β1(1.3+ksin(mπx)).bβ1(x)=
0.3×%
β1(1.3+kcos(mπx)).cN(x)=0.3×%
N(1.3+ksin(mπx)).dN(x)=0.3×%
N(1.3+kcos(mπx)).
eβ1=0.3×%
β1×(1.3+0.5cos(5πx)),d3=%
d3(1.3+kcos(mπx)),N(x)=0.3×%
N,β2=0.3%
β2×(1.3+
0.5cos(5πx)), and other parameters as their means values in Table 1.fβ1=0.3×%
β1(1.3+0.5sin(2πx)),
N(x)=0.3%
N,r(x)=0.5×(1.3+ksin(mπx),β2=0.1×%
β2(1.3+0.5sin(2πx)), and other parameters
as their means values in Table 1
R0increases as kincreases if m=1 by the similar simulation). All these simulations
also reveal that R0is a decreasing function with respect to m, which implies that the
fragmentation of the space is harmful to the virus. The influence of the heterogeneity
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X. Ren et al.
0 1 2 3 4 5
2.2238
2.224
2.2242
2.2244
2.2246
2.2248
2.225
2.2252
2.2254
k
R0
0 0.5 1 1.5 2
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
k
R0
a2=1
a2=21
a2=41
0 0.02 0.04 0.06 0.08 0.1
2.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6
2.65
2.7
k
R0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
a2=100
a2=10
a2=0
(a) (b)
(c) (d)
Fig. 3 The effect of the mobility of the cells or virus on R0or c∗.ad3=k%
d3.bd2=k%
d2.cd1=k%
d1.d
The effect of d2on the minimum wave speed c∗
of the cell-to-cell transmission rate β2on R0can be obtained similarly, and the effects
aresimilartothoseofβ1.
Let β1(x)=0.3×%
β1,β
2(x)=0.1×%
β2,N(x)vary, and other parameters be
fixed as their mean values in Table 1. Then R0decreases as kincreases (Fig. 2c)
if N(x)=0.3×%
N(1.3+ksin(mπx)), but increases as kincreases (Fig. 2d) if
N(x)=0.3×%
N(1.3+kcos(mπx)). The influence of the heterogeneity of other
parameters on transmission risk R0can be obtained similarly.
In the following, we mainly focus on the influence of the mobility of the cells or
virus on transmission risk R0and minimum wave speed c∗.
By the definitions of T0and R0, the basic reproduction number R0is independent of
the diffusion abilities of the cells and virions in the homogeneous case. It is independent
of the diffusion ability of uninfected cells if the recruitment rate and the death rate of
the uninfected cells are both homogeneous. So we should remove these assumptions
when investigating the effect of the mobility of the cells or virions. In the following, we
assume that the diffusion abilities are homogeneous for the virions and cells since the
influences of the heterogeneities of these parameters are small enough as in Fig. 2e. Let
d3=k%
d3and other parameters are the same as those in Fig. 2e. Then R0decreases as
kincreases (Fig. 3a) and R0→2.2239 as k→+∞.Fig. 3c shows that R0decreases
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A reaction-diffusion HIV model
−300 −200 −100 0 100 200 300
0
1
2
3
4
5
6
7
8
9x 109
Distance x
V(t,x)
t=150
t=200
t=250
0 50 100 150 200 250
−300
−200
−100
0
100
200
300
Time t
Distance x
a2=0
a2=0
a2=100
a2=100
(a) (b)
Fig. 4 The evolution of virus population. aEvolution of HIV virus. bThe boundaries of virus spread
as kincreases when d1=k%
d1,λ(x)=%
λ(1+0.9 cos(5πx)), d3=%
d3and other
parameters are the same as those in Fig. 2e.
For simplification, we now assume that β2(x)=β(x)/(a1+a2d2), where a1,
a2>0. Figure 3b implies that R0also decreases as kincreases if d2=k%
d2,a1=1,
β=0.3%
β2×(1.3+0.5 cos(5πx)),d3=%
d3and other parameters are the same as those
in Fig. 3a. Comparing Fig. 3a with b, we find that the transmission risk is overestimated
if ignoring the mobility of infected cells especially for a2large enough, and R0is more
sensitive to the mobility of infected cells than that of virions.
We now investigate the influence of d2on the minimum wave speed c∗, which
may be the asymptotic spreading speed of virus. Assume that a1=1,β =%
β2and
other parameters are taken to be their mean values (Table 1), Fig. 3d shows that the
dependence of c∗on d2for different a2. It shows that d2has different effects on c∗
especially for a2large enough such as a2=100. In the case that a2is not large enough,
the minimum wave speed c∗increases as d2increases. In the case that a2is large, the
dependence of c∗on d2is complicated.
Let T0(x)=5.0×107,T∗0(x)=0,V0(0)=10,V0(x)=0forx∈(−∞,0)∪
(0,+∞), a1=1,a2=0,β=%
β2, and other parameters be their mean values (Table
1). Then the evolution of HIV virus is the solution of (1), plotted in Fig. 4a. Note that
humps appear in the profile and the peak value is much larger than the steady state
value, which may be caused by the fact that the death rate of infected cells is larger than
that of the uninfected cells (Wang and Wang 2007). Assume that V∗=0.3, which is
the threshold value, above which virus population can be detected, then the boundaries
of virus spread are plotted in Fig. 4b. Inspired by Neubert and Parker (2004), we can
use the slope of boundaries in Fig. 4b to estimate the asymptotic spreading speed. Then
the asymptotic spreading speed is approximately equal to c=1.11 >c∗=0.9445. In
the case of a2=100 and other parameters are the same as those in Fig. 4a, we find that
the asymptotic spreading speed is approximately equal to c=0.73 >c∗=0.6666.
Note that these two cases all show that the asymptotic spreading speed is larger than
the minimum wave speed.
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Note that we do not consider the case where all the parameters vary with space in
the above simulations. What will happen if more parameters vary? We find that the
result is very complicated in this case. In the case that only r(x)is heterogeneous,
R0increases as the heterogeneous parameter kincreases in the similar way as above.
However, we can find the relation between R0and kis complicated if more parameters
vary, where the heterogeneity may decrease or increase R0depending on the actual
environment (Fig. 2f).
5 Conclusions and discussion
In this paper, we established a spatially explicit within-host HIV model to investigate
the influences of the mobility of cells or virions and spatial heterogeneity on HIV
pathogenesis. In order to obtain the persistence condition for virus in heterogeneous
space, we investigated the case of the bounded space with Neumann boundaries.
Firstly, the well-posedness of the solutions was proved, and then we introduced the
basic reproduction number R0, which was shown as a threshold: the virus vanishes
if R0<1 and persists if R0>1. In the case where the space is homogeneous, the
global behavior of the solutions was well established. We proved that the positive
constant steady state is globally asymptotically stable if R0>1. In order to study
the existence of traveling wave solutions and obtain the spreading speed of virus in
vivo, we proposed the traveling wave solution problem under an unbounded domain
Rwith spatially homogeneous parameters. Firstly, we derived the existence of the
semi-traveling wave solutions that connect the virus-free steady state when R0>1
and c>c∗. We then showed that the semi-traveling wave solutions also connect the
positive constant steady state when d2≤d3. The existence of critical traveling wave
solution was also obtained. By using the one-side Laplace transform, the nonexistence
of the traveling wave solutions was established when R0>1 and c<c∗, which
implies that the minimum wave speed c∗was well defined.
The heterogeneity is obvious in vivo environment, we then investigated the influ-
ence of spatial heterogeneity on the transmission risk R0via numerical simulations
since it is not easily obtained by mathematical analysis. We found that different hetero-
geneous parameters can cause different effects: Fig. 2a shows that the heterogeneity
of β1may increase R0, but the heterogeneity of N(x)may decrease R0as shown
in Fig. 2c. We also showed that the influence of the heterogeneity of the parameters
on R0depends on the actual environments (Fig. 2a, b). In the treatment of HIV, the
penetration abilities of drug at different anatomical sites are diversified (Svicher et al.
2014), which implies that the treatment may cause the fragmentation of the environ-
ments. The influence of the fragmentation on R0was also simulated, and we found
that the more the fragmentation of β1(x), β2(x)or N(x), the less transmission risk.
In the simulations, we only considered some simple cases, some new methods should
be established to explicitly explain the effect of the heterogeneity.
The influence of the diffusion rate of virions or cells was also studied. We derived
that with a higher diffusion rate of virions or cells, the transmission risk was smaller,
and the influence of the diffusion rate d2on the minimum wave speed c∗depends on the
formulation of the cell-to-cell transmission rate via numerical simulations. Figure 3b,
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c showed that the transmission risk is overestimated if ignoring the diffusion ability of
cells. Figure 3a, b showed that the transmission risk is more sensitive to the mobility
of infected cells than to that of virus.
For virus propagating, the spreading speed is a very important quantity, which can
be characterized as the minimum wave speed in many systems. In our system, the
simulations showed that the spreading speed is larger than the minimum wave speed.
This may be caused by the complexity of the system, which is neither a cooperative
system nor a competitive system. The explicit relationship between them and the
expression of the asymptotic spreading speed need to be investigated further. Although
the assumption that d2≤d3is reasonable for HIV propagation in studying the traveling
wave solution problem, it would be interesting to consider the case where d2>d3so
that the theory developed here can be extended to other systems.
We mainly focus on the effect of the diffusion ability of infected cells, which may
influence the cell-to-cell transmission ability. By the simulations, we can conclude that
decreasing the mobility of the T cells may increase the spreading risk, but decrease
the minimum wave speed, which may be the asymptotic spreading speed of the virus.
Thus, whether decreasing the infection risk or decreasing the spreading speed by
controlling the mobility of the T cells is an worthwhile studying optimizing problem.
Acknowledgements We are grateful to the editors and referees for their careful reading and valuable
comments which have led to a great improvement of this paper. We also thank Prof. Qihua Huang from
Southwest University for checking English in our last revision. This work is supported by the National
Natural Science Foundation of China (11671327,11271303, 11601293) and Chongqing Graduate Student
Research Innovation Project (CYB14053).
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