A mathematical model of human T-cell lymphotropic virus type 1 in vivo with cell-to-cell infection and mitosis is
formulated and studied. The basic reproductive number R0 is derived. It is proved that the dynamics of the model can
be determined completely by the magnitude of R0. The infection-free equilibrium is globally asymptotically stable
(unstable) if R0<1(R0>1). There exists a chronic infection equilibrium and it is globally asymptotically stable if R0>1.
1. Introduction
Human T-cell lymphotropic virus type 1 (HTLV-1) is a pathogenic retrovirus and persists indefinitely in the infected hosts [1–4]. There are approximately 10–20 million infected people worldwide [5, 6]. HTLV-1 is associated causatively with a large number of pathologies. A slowly progressive neurologic disease HTLV-1 associated myelopathy/tropical spastic paraparesis (HAM/TSP) [7] and adult T-cell leukemia (ATL) are two most common forms of the disease [8]. The majority of HTLV-1 infected individuals remain lifelong asymptomatic carriers (ACs). The remaining 0.25–3% of infected individuals develop into HAM/TSP [9]. The virus can be transmitted from mother to child, through sexual contact, and by needle sharing and contaminated blood products [5, 9].
In HTLV-1 infection, the initial infection is subclinical. The virus preferentially integrates into the genome of host T lymphocytes. Since the virions are almost undetectable from extracellular matrix, the viral burden is quantified as the proportion of peripheral blood mononuclear cells that carry an integrated HTLV-1 provirus. About 90–95% of the proviral load in chronic HTLV-1 infection is carried by CD4+ T cells and 5–10% by CD8+ T cells [10–13].
To persist within the host, HTLV-1 requires two routes: (i) infectious spread to uninfected cells via cell-to-cell contact known as the virological synapse, cellular conduits, extracellular viral assemblies, and transinfection via dendritic cells [9] and (ii) clonal expansion, which would actively promote mitotic proliferation of infected cells, and pass on the provirus to daughter cells. It is assumed that infection of an individual with HTLV-1 occurs in two stages; the virus is thought to initially spread from T cells to T cells, primarily CD4+ T cells, and later to persist by clonal expansion of infected cells [12].
It has been observed that HTLV-1 infection has a lower rate of proviral genetic variation than HIV infection, which suggests that the vertical transmission through mitotic division rather than horizontal transmission through cell-to-cell contact plays an important role [14, 15]. HTLV-1 succeeds in causing a persistent infection with a high proviral load and remains approximately stable in one individual over years. In order to identify the underlying mechanism of HTLV-1 persistence in vivo and the key factors determining the HTLV-1 provirus load and the disease risk, Asquith and Bangham [1] have used a combination of mathematical and experimental techniques to propose a model of HTLV-1 persistence. Mitosis is the main route of viral replication, and the expression of HTLV-1 proteins, particularly Tax, is required to promote the selective expansion of cells that harbour a provirus [5, 16–18], though the majority of infected cells are not expressing viral protein. Although the Tax expressing is silenced in the majority of surviving cells and a small proportion (0.03%–3%) of infected cells can express Tax, the cells with Tax expression proliferate more rapidly than silently infected and uninfected cells, leading to the selective expansion of infected cells and an increase in proviral load [1]. The small proportion of infected cells that express viral proteins play a crucial role, and the very high provirus load in HTLV-1 infection is maintained by proliferation of infected T cells, induced by the Tax protein of HTLV-1 [19].
It has been observed that the CD4+ T cells population from HAM/TSP patients express higher levels of tax mRNA than CD4+ T cells from ACs. Tax expression at any given proviral load is significantly higher in the HAM/TSP patients than that in the ACs [20]; thus a high rate of viral protein expression is associated with a large increase in the prevalence of HAM/TSP, and Tax expression is a significant predictor of the disease [1].
Most of the existing models have considered the persistence and pathogenesis for HTLV-1 infection of CD4+ T cells. Mathematical models that take into account both infectious and mitotic routes have also been developed to describe the interaction in vivo among HTLV-1 [14, 20–22]. Motivated by the new hypothesis of HTLV-1 infection by Asquith and Bangham, we construct a model with three compartments, healthy CD4+ T cells x, resting infected CD4+ T cells u, and Tax-expressing infected CD4+ T cells y, to investigate the dynamics of the HTLV-1 infection. The model is formulated and the required conditions are given in Section 2. The stability of equilibria is presented in Section 3. The simulations are done in Section 4. The concluding remarks are given in Section 5.
2. Model Formulation
In this section, we construct a mathematical model including the spontaneous HTLV-1 antigen Tax expression, cell-to-cell contact, and mitotic infectious routes to describe the viral dynamics. Let x(t) be the number of healthy CD4+ T cells at time t, let u(t) be the number of the resting infected CD4+ T cells at time t, and let y(t) be the number of Tax-expressing infected CD4+ T cells at time t. We consider only HAM/TSP among nonmalignant HTLV-1 infection diseases; the dynamics of ATL and other aggressive malignancies may be very different. Although mitosis occurs in all CD4+ T cells as a natural process, normal homeostatic proliferation occurs at a very slower rate than that of selective mitotic division in Tax-expressing infected cells. We ignore the effects of passive homeostatic proliferation of the healthy and resting infected CD4+ T cells to simplify the model.
Healthy CD4+ T cells are produced in bone marrow at a constant rate λ [23, 24]; we assume that the new cells generated in the bone marrow are uninfected. The infected CD4+ T cells can make the healthy CD4+ T cells get infected through cell-to-cell contact. The infectious incidence is described by a bilinear term βxy, where β is the transmission coefficient among CD4+ T cells [25]. The newly infected cells experience an irreparable destruction by the strong adaptive immune responses. As a result, a small fraction σβxy, σ∈(0,1), survives after the immune attack and becomes the resting infected cells [14, 22]. Every day, a small proportion τ of resting infected cells express Tax with τ∈(0.3%,3%) [26]. The mitotic transmission of HTLV-1 involving selective clonal expansion of these Tax-expressing CD4+ T cells occurs at a rate s. The newly infected cells from mitosis to the resting infected cells compartment are ɛsy, ɛ∈(0,1), with (1-ɛ)sy staying in the Tax-expressing infected CD4+ T-cell compartment. The transfers among those three compartments are shown in Figure 1.
The schematic diagram of the HTLV-1 infection in vivo.
From the mechanism of the HTLV-1 infection and the schematic diagram we can have the following model consisting of three differential equations;
(1)dxdt=λ-βxy-μ1x,dudt=σβxy+ɛsy-τu-μ2u,dydt=τu+(1-ɛ)sy-μ3y.
In model (1), μ1, μ2, and μ3 are the removal rate of healthy CD4+ T cells, resting infected CD4+ T cells, and Tax-expressing infected CD4+ T cells, respectively. From epidemiological background, it is natural to assume that the initial values of these variables and parameters are nonnegative.
We define the basic reproductive number of model (1) by the next generation matrix approach given in [27]. Let
(2)F=[0σβx+ɛs0s-ɛs],V=[τ+μ20-τμ3].
The calculation shows that the spectral radius (the basic reproductive number) of FV-1 is
(3)R0=ρ(FV-1)=σβτλ(τ+μ2)μ1μ3+τɛs(τ+μ2)μ3+(1-ɛ)sμ3.
The basic reproductive number, R0, gives the average number of the secondary infections caused by a single Tax-expressing infected CD4+ T cell during its whole infectious period. The secondary infection caused by a single Tax-expressing infected CD4+ T cell through horizontal transmission is σβ·(λ/μ1)·(τ/(τ+μ2))·(1/μ3); the secondary infection caused by a single Tax-expressing infected CD4+ T cell through mitotic transmission is ɛs·(τ/(τ+μ2))·(1/μ3)+(1-ɛ)s·(1/μ3).
Throughout the paper, we use the assumption
(A1)s<(τ+μ2)μ3τ+μ2(1-ɛ).
The inequality (A1) is equivalent to that τɛs/(τ+μ2)μ3+(1-ɛ)s/μ3<1, which requires that the average number of the secondary infections by a single Tax-expressing infected CD4+ T cell through mitosis should not be larger than one. If the inequality in (A1) does not hold, then the number of the infected cells may increase to infinity. The biological interpretation of (A1) is to keep the solutions of the model bounded. From condition (A1), we have s<μ3/(1-ɛ); that is, μ3>(1-ɛ)s. We can get the following nonnegative and bounded conclusions on the solutions of model (1).
Theorem 1.
The solutions (x(t),u(t),y(t)) of model (1) with the nonnegative initial conditions are nonnegative and bounded for all t>0 if (A1) holds.
Proof.
It is easy to have
(4)dx(t)dt|x=0=λ>0,du(t)dt|u=0=σβxy+ɛsy≥0,dy(t)dt|y=0=τu≥0.
From Lemma 2 in [28], we know that any solutions of model (1) with nonnegative initial conditions will be nonnegative for all t>0.
It follows from the first equation of model (1) that
(5)dxdt=λ-βxy-μ1x≤λ-μ1x,
which leads to limt→+∞supx≤λ/μ1. Let L=x+u+((τ+μ2)/τ)y; from model (1) we can obtain
(6)dLdt=dxdt+dudt+τ+μ2τdydt=λ+(σ-1)βxy-μ1x-Gy≤λ-μ1x-Gy,
where G=((τ+μ2)/τ)(μ3-(1-ɛ)s)-ɛs>0 since (A1) holds. The inequality in (6) implies that L=x+u+((τ+μ2)/τ)y will decrease along the solutions curve of model (1) if μ1x+Gy>λ. Geometrically, all solution trajectories of model (1) will go through the plane x+u+((τ+μ2)/τ)y=L from outside to inside if μ1x+Gy>λ.
Let L0 be the maximal value of the function x+((τ+μ2)/τ)y on the bounded domain
(7)G0={(x,y)∣x≥0,y≥0,μ1x+Gy≤λ},
and let M0 be the maximal value of the function σβxy+ɛsy on the bounded domain
(8)G1={(x,y)∣x≥0,y≥0,x+τ+μ2τy≤L0}.
When x+((τ+μ2)/τ)y≤L0 holds, the second equation of model (1) yields
(9)dudt=σβxy+ɛsy-τu-μ2u≤M0-(τ+μ2)u.
From the comparison principle and (9), it follows that there exists a positive um=M0/(τ+μ2), such that du/dt≤0 when u>um and x+((τ+μ2)/τ)y≤L0.
For any given initial values x(0)=x0≥0, u(0)=u0≥0, and y(0)=y0≥0, there exists a plane P, given by the equation
(10)P:x+u+τ+μ2τy=L0+um+x0+u0+τ+μ2τy0,
such that the point (x0,u0,y0) locates inside the domain with the boundaries x=0, u=0, y=0, u=um+x0+u0+((τ+μ2)/τ)y0, and P. It is not difficult to verify that those two planes u=um+x0+u0+((τ+μ2)/τ)y0 and P have the intersection line x+((τ+μ2)/τ)y=L0. The equations in (6) and (9) imply that
(11)dudt≤0,ifu=um+x0+u0+τ+μ2τy0≥um,mmmmmmmmmmmmmmmix+τ+μ2τy≤L0,dLdt≤0ifx+τ+μ2τy≥L0.
Those inequalities imply that the domain with the boundaries x=0, u=0, y=0, u=um+x0+u0+((τ+μ2)/τ)y0, and P is positively invariant for solutions of model (1). That is, any solution of model (1) with nonnegative initial value is bounded.
With a similar argument as used in the proof of Theorem 1, we know that the domain
(12)Γ={(x,u,y)∣0≤x≤λμ1,0≤u≤um,y≥0,x+u+τ+μ2τy≤L0+umλμ1}
is positively invariant with respect to model (1). In fact, the solutions of model (1) located on the boundary planes of Γ, x=λ/μ1, or u=um, or x+u+((τ+μ2)/τ)y=L0+um, will enter Γ0, where Γ0 is the interior of Γ. From (5), (6), and (9) we can prove that all the solutions of model (1) with positive initial values will enter Γ when the time is large enough. We will investigate the dynamic behavior of model (1) on Γ in the rest of the paper.
The straightforward calculation shows that model (1) has two equilibria: the infection-free equilibrium P0=(x0,0,0), located on the boundary of Γ, where x0=λ/μ1, and the chronic infection equilibrium P1=(x1,u1,y1), where
(13)x1=λβy1+μ1,u1=(μ3-(1-ɛ)s)y1τ,y1=μ1β(R0-1)(τ+μ2)μ3(τ+μ2)μ3-τs-μ2(1-ɛ)s.x1, u1, and y1 are positive if and only if R0>1 and (A1) holds. We have the following conclusion on the existence of the equilibrium of model (1).
Theorem 2.
If R0≤1, then P0=(λ/μ1,0,0) is the only equilibrium of model (1). If R0>1 and (A1) holds, then P1=(x1,u1,y1) is the unique chronic infection equilibrium.
3. Stability Analysis of Equilibria3.1. Stability of Infection-Free Equilibrium
Intuitively, if R0<1, then a Tax-expressing infected CD4+ T cell will produce less than one secondary infection on average in its lifetime. This fact may lead to the extinction of the infection. We will try to prove the global stability of the infection-free equilibrium when R0<1.
Theorem 3.
If R0<1, then the infection-free equilibrium P0 of model (1) is stable, and it is unstable if R0>1.
Proof.
We use the linearized system of model (1) to discuss the stability of P0. The characteristic equation of the matrix of the linearized system of model (1) at the infection-free equilibrium P0 is
(14)(ρ+μ1)(ρ2+b0ρ+c0)=0,
where b0=μ3(1-R0+(σβτx0+τɛs)/(τ+μ2)μ3)+τ+μ2, c0=(1-R0)(τ+μ2)μ3. From the Routh-Hurwitz criterion, it is easy to know that all the roots of (14) have negative real parts if R0<1, and (14) has at least one root with positive real part if R0>1. This completes the proof.
Theorem 4.
If R0<1, then the infection-free equilibrium P0 of model (1) is globally asymptotically stable in Γ.
Proof.
We consider a Lyapunov function L=τu+(τ+μ2)y. Calculating the derivative of L along the solutions of model (1) gives
(15)dLdt|(1)=τdudt+(τ+μ2)dydt=y(τσβx+τɛs+(τ+μ2)(1-ɛ)s-(τ+μ2)μ3)≤y(τσβλμ1+τɛs+(τ+μ2)(1-ɛ)s-(τ+μ2)μ3)=yμ3(τ+μ2)(R0-1).
Therefore, R0<1 implies that (dL/dt)|(1)≤0 for all t>0, and (dL/dt)|(1)=0 only if y=0. From the inequality in (15) we can have that limt→∞y(t)=0, limt→∞u(t)=0. By using the limiting theory for ordinary differential equations we can have limt→∞x(t)=λ/μ1. That is, the infection-free equilibrium P0 attracts all solutions of model (1) with initial values in Γ. The global stability conclusion of Theorem 4 is proved.
3.2. Stability of the Chronic Infection EquilibriumTheorem 5.
Assume that (A1) holds; if R0>1, then the unique chronic infection equilibrium P1 of model (1) is stable.
Proof.
The characteristic equation of the matrix of the linearized system of model (1) at the chronic infection equilibrium P1 is
(16)ρ3+b1ρ2+c1ρ+d1=0,
where
(17)b1=μ3-(1-ɛ)s+τ+μ2+βy1+μ1,c1=(τ+μ2)(μ3-(1-ɛ)s)+(βy1+μ1)(μ3-(1-ɛ)s)+(βy1+μ1)(τ+μ2)-(σβτx1+ɛτs),d1=(βy1+μ1)(τ+μ2)(μ3-(1-ɛ)s)-ɛτsβy1-σβτμ1x1-μ1ɛτs.
Since βy1+μ1=λ/x1, σβτx1+ɛτs=(τ+μ2)(μ3-(1-ɛ)s), we have c1=(λ2/x1)b0>0, d1=μ1μ3(τ+μ2)(R0-1). The straightforward calculation yields b1c1-d1>0. According to the Routh-Hurwitz criterion, we can see that all the roots of (16) have negative real parts if R0>1. This completes the proof of Theorem 5.
The following two lemmas, which can be found in [29], are used for the study of the uniform persistence of model (1). We show that the disease persists when R0>1; that is, the infected proportion of the CD4+ T cells persists above a certain positive level for sufficiently large t.
Let f:X→X be a continuous map and X0⊂X an open set. Define ∂X0=X/X0 and M∂:={x∈∂X0∣fn(x)∈∂X0,n≥0}.
Lemma 6 (see [<xref ref-type="bibr" rid="B30">29</xref>]).
If f:X→X is compact and point dissipative, then there is a connected global attractor A that attracts each bounded set in X.
Lemma 7 (see [<xref ref-type="bibr" rid="B30">29</xref>]).
Let f:X→X be a continuous map and X0⊂X an open set. Assume that
f(X0)→X0 and f has a global attractor A;
the maximal compact invariant set A∂=A∩M∂ of f in ∂X0, possibly empty, admits a Morse decomposition {M1,…,MK} with the following properties:
Mi is isolated in X;
Ws(Mi)∩X0=ϕ for each 1≤i≤k.
Then there exists ρ>0 such that, for any compact internally chain transitive set L with L⊄Mi for all 1≤i≤k, we have infx∈Ld(x,∂X0)>ρ.
We deal with the uniform persistence of model (1) now. Let X={(x,u,y)∣x≥0,u≥0,y≥0}, X0={(x,u,y)∣x≥0,u>0,y>0}; define ∂X0=X/X0, and M∂={(x(0),u(0),y(0))∈∂X0∣Φt(x(0),u(0),y(0))∈∂X0,t≥0}, where Φt:X→X is the semiflow defined by model (1).
Proposition 8.
One has M∂={(x,0,0)∣x≥0}.
Proof.
We first show that M∂⊂{(x,0,0)∣x≥0}; that is, if (x(0),u(0),y(0))∈M∂, then u(0)=y(0)=0. Due to the definition of M∂, we can get Φt(x(0),u(0),y(0))∈∂X0 for all t≥0, especially, Φ0(x(0),u(0),y(0))=(x(0),u(0),y(0))∈∂X0. If M∂⊂{(x,0,0)∣x≥0} does not hold, then at least one of u(0), y(0) is greater than zero. Without loss of generality, we assume that u(0)>0. When u(0)>0 we can prove that u(t) and y(t) are all greater than zero for t∈[0,1]. In fact, from the second equation of model (1) we have
(18)dudt=σβxy+ɛsy-τu-μ2u≥-(τ+μ2)u,t∈[0,1].
It follows that
(19)u(t)≥u(0)exp[-(τ+μ2)]≜M1>0.
From the third equation of model (1) we have
(20)dydt=τu+(1-ɛ)sy-μ3y≥τM1+(1-ɛ)sy-μ3y;
then, for t∈[0,1], we can have
(21)y(t)≥τM1μ3-(1-ɛ)s[1-exp[-(μ3-(1-ɛ)s)t]]+y(0)exp[-(μ3-(1-ɛ)s)t]≥τM1μ3-(1-ɛ)s[1-exp[-(μ3-(1-ɛ)s)]]>0.
The inequalities u(t)≥M1>0 and y(t)≥(τM1/(μ3-(1-ɛ)s))[1-exp[-(μ3-(1-ɛ)s)]]>0 for t∈[0,1] imply that (x(t),u(t),y(t))∈X0 for t∈[0,1]. From the definition of M∂ and (x(t),u(t),y(t))∈X0 for t∈[0,1] we know that Φ0(x(0),u(0),y(0))∉∂X0 if u(0)>0. This contradiction implies that (x(0),u(0),y(0))∈M∂ only if u(0)=y(0)=0; that is, M∂⊂{(x,0,0)∣x≥0}.
On the other hand, for any initial values (x(0),0,0)∈{(x,0,0)∣x≥0}, we have du/dt=0, dy/dt=0, and u(t)=y(t)=0 for t≥0, {(x,0,0)∣x≥0}⊂M∂. The proposition is proved.
From Proposition 8, we can get the conclusion that M∂ is the maximal invariant set in ∂X0. Next we show that the solutions with the initial values in X0 cannot go to the boundary.
Proposition 9.
Assume that (A1) holds. If R0>1, then there exists a δ>0 such that the solution of model (1) with initial value (x(t0),u(t0),y(t0))∈X0 satisfies limt→+∞supmax{u(t),y(t)}>δ.
Proof.
If the conclusion in Proposition 9 does not hold, then, for any δ>0, there exists a T such that u(t)≤δ and y(t)≤δ for all t>T. Consider the following equation:
(22)dx^dt=λ-βδx^-μ1x^.
The solution of (22) with the any initial value x(t0)>0 is
(23)x^(t)=λβδ+μ1[1-exp[(βδ+μ1)(t0-t)]]+x(t0)exp[(βδ+μ1)(t0-t)],
and limt→+∞x^(t)=λ/(βδ+μ1). For ε1>0, there exists a T1>T, such that x^(t)>λ/(βδ+μ1)-ε1 holds when t≥T1. x^1(δ)=λ/(βδ+μ1) is an equilibrium of (22). The fact that limδ→0x^1(δ)=λ/μ1=x0 implies that x^1(δ)≥x0-ε1 when δ is small enough. By the comparison principle, we can have x(t)≥x^(t) and x(t)≥x0-2ε1, for t>T1.
Consider the following linear system:
(24)du^dt=σβ(x0-2ε1)y^+ɛsy^-τu^-μ2u^,dy^dt=τu^+(1-ɛ)sy^-μ3y^.
The characteristic equation is
(25)ρ2+b2ρ+c2=0,
where b2=τ+μ2+μ3-(1-ɛ)s>0, c2=(1-R0)(τ+μ2)μ3+2τσβε1. From the expression of c2 we see that c2<0 if R0>1 and ε1 is small enough. Let ρ1 and ρ2 be the two roots of ρ2+b2ρ+c2=0 and ρ1>0>ρ2. The solution of model (24) with the initial value (u^(0),y^(0))>0 satisfies
(26)(u^(t),y^(t))T=d1ξ1exp(ρ1t)+d2ξ2exp(ρ2t),
where ξ1 and ξ2 are the eigenvectors corresponding to ρ1 and ρ2, respectively. d1 and d2 are two constants depending on (u^(0),y^(0)). The solution expression of model (24) indicates that max{u^(t),y^(t)}→∞ as t→∞. For the same initial values, the comparison principle implies that u(t)>u^(t) and y(t)>y^(t), where u(t) and y(t) are the solutions of model (1). Subsequently, we have u(t)→∞ or y(t)→∞ as t→∞. The contradiction shows that Proposition 9 holds true.
By using Propositions 8 and 9, we can get the uniform persistence of model (1).
Theorem 10.
Assume that (A1) holds. If R0>1, then model (1) is uniformly persistent with respect to (X0,∂X0); that is, there exists a positive number η such that min{limt→∞infx(t),limt→∞infu(t),limt→∞infy(t)}≥η.
Proof.
X and X0 are positively invariant for model (1). Φt is point dissipative and compact. By Lemma 6 we know that there is a connected global attractor A for Φt that attracts each bounded set in X.
From the discussion of Proposition 8, we know that M∂ is the maximal compact invariant set in ∂X0. Since we choose the Morse decomposition of M∂ as {P0} and ∪x∈M∂ω(x)={P0}, the set {P0} is isolated. Proposition 9 shows that the solutions of model (1) with initial values in X0 cannot go to the boundary, which implies that Ws(P0)∩X0=ϕ. It follows from Lemma 7 that model (1) is uniformly persistent with respect to (X0,∂X0).
The following lemmas in [30–32] are used to study the global stability of the chronic infection equilibrium P1. We will show that all the solutions of model (1) in Γ0 converge to P1 if R0>1.
Let x→f(x)∈Rn be a C1 function for x in an open set D⊂Rn. Consider the system of differential equations
(27)dxdt=f(x).
Let x(t,x0) be the solution of model (27) satisfying x(0,x0)=x0.
A set K is said to be absorbing in D for model (27) if x(t,K1)⊂K for each compact K1⊂D and sufficiently large t. We make the following two basic assumptions.
There exists a compact absorbing set K⊂D.
System (27) has a unique equilibrium x¯ in D.
System (27) is said to have the Poincaré-Bendixson Property if any nonempty compact omega limit set that contains no equilibrium is a closed orbit [31]. It is known that a three-dimensional competitive system has the Poincaré-Bendixson property in a convex region.
Lemma 11 (see [<xref ref-type="bibr" rid="B31">30</xref>]).
Let D∈Rn be convex. The autonomous system dx/dt=f(x), x∈D, is cooperative in D if there exists a diagonal matrix P=diag(α1,…,αn) (αi=-1 or 1, i=1,2,…,n), such that P(∂fi/∂xj)(x)P≥0, for i≠j, x∈D; that is, all off-diagonal entries of P(∂f/∂x)(x)P are nonnegative. It is competitive in D if there exists a diagonal matrix P=diag(α1,…,αn) (αi=-1 or 1, i=1,2,…,n), such that P(∂fi/∂xj)(x)P≤0, for i≠j, x∈D; that is, all off-diagonal entries of P(∂f/∂x)(x)P are nonpositive.
Lemma 12 (see [<xref ref-type="bibr" rid="B35">32</xref>]).
Assume that n=3 and D is convex; suppose that model (27) is competitive in D; then it satisfies the Poincaré-Bendixson property [32].
Lemma 13 (see [<xref ref-type="bibr" rid="B34">31</xref>]).
Assume that the following conditions hold.
Assumptions (H1) and (H2) hold;
model (27) satisfies the Poincaré-Bendixson property;
for each periodic solution x=p(t) with p(0)∈D, model (27) is asymptotically stable;
(-1)ndet((∂f/∂x)(x¯))>0.
Then the unique equilibrium x¯ is globally asymptotically stable in D.
Next, we show that model (1) is a competitive system which implies that model (1) has the Poincaré-Bendixson property.
Theorem 14.
Model (1) is competitive in Γ.
Proof.
The Jacobian matrix of model (1) is
(28)J(x,u,y)=[-βy-μ10-βxσβy-τ-μ2σβx+ɛs0τ(1-ɛ)s-μ3].
Choose P=diag(1,-1,1); we can obtain
(29)PJP=[-βy-μ10-βx-σβy-τ-μ2-σβx-ɛs0-τ(1-ɛ)s-μ3].
All off-diagonal entries of PJP are nonpositive. It follows from Lemma 11 that model (1) is competitive in the convex region Γ.
Now, we are ready to prove the global stability of the unique chronic infection equilibrium P1 of model (1).
Theorem 15.
Assume that (A1) holds. If R0>1, then the unique chronic infection equilibrium P1 of model (1) is globally asymptotically stable in Γ0.
Proof.
From Theorem 10 and Lemma 6, we know that Φt is compact and point dissipative, and there is a global attractor A for Φt. Subsequently, model (1) satisfies (H1). From Theorem 2, model (1) satisfies (H2). By Theorem 14 and Lemma 12, model (1) has the Poincaré-Bendixson property. Thus conditions (1) and (5) of Lemma 13 hold.
The second compound system of the linearized system along a periodic solution (x(t),u(t),y(t)) of model (1) is
(30)dXdt=-(βy+μ1+τ+μ2)X+(σβx+ɛs)Y+βxZ,dYdt=τX-(βy+μ1-(1-ɛ)s+μ3)Y,dZdt=σβyY-(τ+μ2-(1-ɛ)s+μ3)Z.
In order to verify that model (30) is asymptotically stable, we define a Lyapunov function
(31)V(X,Y,Z;x,u,y)=sup{|X|,uy(|Y|+βλσβλ+ɛsμ1|Z|)}.
From the uniform persistence, we know that the orbit O of the periodic solution (x(t),u(t),y(t)) has a positive distance from the boundary of Γ. There exists a constant c>0 such that
(32)V(X,Y,Z;x,u,y)≥csup{|X|,|Y|,|Z|}.
For all (X,Y,Z)∈R3 and (x,u,y)∈O, we have the following estimates on the right derivatives along the solutions (X(t),Y(t),Z(t)) of model (30):
(33)D+(|X|)=X|X|(-(βy+μ1+τ+μ2)X+(σβx+ɛs)Y+βxZ)≤-(βy+μ1+τ+μ2)|X|+(σβx+ɛs)|Y|+βx|Z|=-(βy+μ1+τ+μ2)|X|+(σβx+ɛs)yu·uy(|Y|+βxσβx+ɛs|Z|)=-(βy+μ1+τ+μ2)|X|+(σβx+ɛs)yu·uy(|Y|+βλσβλ+ɛsμ1|Z|);D+(|Y|)=Y|Y|(τX-(βy+μ1-(1-ɛ)s+μ3)Y)≤τ|X|-(βy+μ1-(1-ɛ)s+μ3)|Y|;D+(|Z|)=Z|Z|(σβyY-(τ+μ2-(1-ɛ)s+μ3)Z)≤σβy|Y|-(τ+μ2-(1-ɛ)s+μ3)|Z|.
From (33) we have
(34)D+(uy(|Y|+βλσβλ+ɛsμ1|Z|))=u′y-uy′y2(|Y|+βλσβλ+ɛsμ1|Z|)+uy(D+|Y|+βλσβλ+ɛsμ1D+|Z|)≤(u′u-y′y)·uy(|Y|+βλσβλ+ɛsμ1|Z|)+uy(τ|X|-(βy+μ1-(1-ɛ)s+μ3)|Y|)+uy·βλσβλ+ɛsμ1×(σβy|Y|-(τ+μ2-(1-ɛ)s+μ3)|Z|)=(u′u-y′y)·uy(|Y|+βλσβλ+ɛsμ1|Z|)+uy·βλσβλ+ɛsμ1|Z|(-τ-μ2+(1-ɛ)s-μ3)+uyτ|X|+uy|Y|(σβλσβλ+ɛsμ1-μ1+(1-ɛ)s-μ3-βy+σβλσβλ+ɛsμ1βy)≤(u′u-y′y)·uy(|Y|+βλσβλ+ɛsμ1|Z|)+uyτ|X|+uy|Y|(-μ1+(1-ɛ)s-μ3)+uy·βλσβλ+ɛsμ1|Z|(-τ-μ2+(1-ɛ)s-μ3)≤uyτ|X|+(u′u-y′y+(1-ɛ)s-μ3-min{μ1,τ+μ2}y′y)×uy(|Y|+βλσβλ+ɛsμ1|Z|).
The inequalities in (33) and (34) lead to
(35)D+V(t)≤max{g1(t),g2(t)}V(t),
where
(36)g1(t)=-(βy+μ1+τ+μ2)+(σβx+ɛs)yu,g2(t)=uyτ+u′u-y′y+(1-ɛ)s-μ3-min{μ1,τ+μ2}.
After rewriting the last two equations of model (1), we find that
(37)(σβx+ɛs)·yu=u′u+τ+μ2,y′y=uyτ+(1-ɛ)s-μ3.
From (36) and (37), we obtain
(38)max{g1(t),g2(t)}≤u′u-min{μ1,τ+μ2},∫0ωmax{g1(t),g2(t)}dt≤∫0ω(u′u-min{μ1,τ+μ2})dt=lnu(t)|0ω-ωmin{μ1,τ+μ2}=-ωmin{μ1,τ+μ2}.
The inequalities in (35) and (38) imply that V(t)→0 as t→∞, which leads to (X(t),Y(t),Z(t))→0 as t→∞ because of (32). As a result, the second compound system (30) is asymptotically stable. This verifies condition (6) of Lemma 13.
Let J(P1) be the Jacobian matrix of model (1) at P1. Then we have
(39)(-1)3det(∂f∂x(P1))=-[-βy1-μ10-βx1σβy1-τ-μ2σβx1+ɛs0-τ(1-ɛ)s-μ3]=(βy1+μ1)((μ3-(1-ɛ)s)(τ+μ2)+ɛsτ)+σβτμ1x1>0.
Condition (9) of Lemma 13 holds. The chronic infection equilibrium P1 of model (1) is globally asymptotically stable in Γ0 since all conditions of Lemma 13 are satisfied.
4. Numerical Simulation
Numerical simulations are done to demonstrate the results in Section 3. The sensitive analysis is given to show the effects of the model parameters on the solutions.
In numerical simulations, the time scale is a day. The rate of healthy CD4+ helper T cells produced in the bone marrow, λ, is 15–25cells/mm3/day. The coefficient of infectious transmissibility, β, is 0.0005–0.003mm3/cell/day. The proportion of infected cells expressing Tax, τ, is (0.003–0.03)/day. The removal rates of healthy CD4+ T cells, resting infected CD4+ T cells, and Tax-expressing infected CD4+ T cells, μ1, μ2, and μ3, are taken to be the value 0.01–0.05/day. The death rate of the Tax-expressing infected CD4+ T cells is considerably shorter than the natural lifespan of CD4+ T cells [33].
In Figure 2, we use the following set of parameters: λ=20, β=0.001, μ1=μ2=1/30, s=0.05, σ=0.01, ɛ=0.9, τ=0.03, μ3=0.05, and R0=0.5832<1. All solutions converge to the infection-free equilibrium P0.
Global stability of the infection-free equilibrium P0 when R0<1.
In Figure 3, we use the following set of parameters: λ=20, β=0.001, μ1=1/30, μ2=0.02, s=0.1, σ=0.1, ɛ=0.8, τ=0.03, μ3=0.09, and R0=1.1556>1. All solutions converge to the chronic infection equilibrium P1.
Global stability of the infection-free equilibrium P1 when R0>1.
A sensitivity analysis quantifies how changes in the values of the input parameters alter the value of the outcome variable [34]. The sensitivity analysis is performed to explore the behavior of model (1) by calculating the partial rank correlation coefficients (PRCC) for each input parameter, which are sampled by the Latin hypercube sample (LHS) and R0 (Table 1). Figure 4 shows that a significantly strong positive correlation exists between parameters β and R0 (PRCC = 0.8597; P value = 0 < 0.01). The second sensitive parameter to R0 is τ (PRCC = 0.3421; P value = 0 < 0.01). The result indicates that the cell-to-cell contact transmission and Tax expression contribute a lot to the viral infection.
PRCC results and P value.
Parameters
β
τ
s
μ1
μ2
μ3
PRCC
0.8597
0.3421
0.0214
-0.0575
-0.0813
-0.0610
P value
0.0000
0.0000
0.4994
0.0696
0.0103
0.0545
Sensitivity analysis of R0 with the parameters came from LHS sampling.
The sensitivity analysis result shows that β and τ are two significant parameters for the infection. We illustrate the impact of β and τ on the magnitude of the chronic infection equilibrium P1 by numerical simulations. The curves in Figures 5(a) and 5(b) show the dependence of u1 and y1 on the parameters β and τ, respectively. The surfaces in Figures 5(c) and 5(d) give the values of u1 or y1 as the functions of β and τ, respectively. Those curves and surfaces in Figure 5 indicate that u1 and y1 will increase with β and τ. For any given τ, u1 or y1 increases very fast for small β and quite slow for large β.
The impact of β and τ on the magnitude of the chronic infection equilibrium P1.
5. Concluding Remarks
We have formulated and studied a mathematical model of HTLV-1 in vivo including the spontaneous HTLV-1 antigen Tax expression, cell-to-cell contact, and mitotic infectious route to the viral dynamics. The persistence of the model is discussed. Sufficient conditions are established for the global asymptotic stability of the infection-free equilibrium and chronic infection equilibrium. The sensitivity analysis by PRCC with the LHS sample is presented to show the impact of the parameters on the model dynamics.
As we know, infected cells from HAM/TSP patients have a significantly higher probability of expressing Tax protein than infected cells from ACs. When an infected individual has settled at a chronic infection state, the proportion of Tax-expressing cells in infected cells is y¯/(u¯+y¯), where u¯=((μ3-(1-ɛ)s)/τ)y¯. Hence (∂/∂τ)(y¯/(u¯+y¯))=(μ3-(1-ɛ)s)/(μ3-(1-ɛ)s+τ)2>0. That is, a faster rate of spontaneous expression of the Tax results in a higher proportion of y in infected CD4+ T cells which influence the risk of HAM/TSP.
It follows from our sensitivity analysis that β and τ are significantly sensitive to the reproduction number R0. In particular, increasing the rate of Tax expression results in a reduction of the proportion of proviral cell at the equilibrium state. This conclusion implies that Tax expression should be controlled in the therapeutic intervention in order to reduce the risk of HAM/TSP.
Our conclusions are based on a simple model; with the recent progress in HTLV-1 pathogenesis and new findings in immune reactions against HTLV-1 infection and Tax expression, more factors should be investigated in improved models.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by a Grant (no. 104519-010) from the International Development Research Center, Ottawa, Canada, and the Natural Science Foundation of China (no. 11171267, no. 11301314).
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