Global Dynamics of an HTLV-1 Model with Cell-to-Cell Infection and Mitosis

and Applied Analysis 3 a single Tax-expressing infected CD4 T cell through mitotic transmission is εs ⋅ (τ/(τ + μ 2 )) ⋅ (1/μ 3 ) + (1 − ε)s ⋅ (1/μ 3 ). Throughout the paper, we use the assumption

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 4 + T cells and 5-10% by 8 + T cells [10][11][12][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 4 + 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

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 () be the number of healthy 4 + T cells at time , let () be the number of the resting infected 4 + T cells at time , and let () be the number of Taxexpressing infected 4 + T cells at time .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 4 + 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 4 + T cells to simplify the model.
Healthy 4 + T cells are produced in bone marrow at a constant rate  [23,24]; we assume that the new cells generated in the bone marrow are uninfected.The infected 4 + T cells can make the healthy 4 + T cells get infected through cell-to-cell contact.The infectious incidence is described by a bilinear term , where  is the transmission coefficient among 4 + T cells [25].The newly infected cells experience an irreparable destruction by the strong adaptive immune responses.As a result, a small fraction ,  ∈ (0, 1), survives after the immune attack and becomes the resting infected cells [14,22].Every day, a small proportion  of resting infected cells express Tax with  ∈ (0.3%, 3%) [26].The mitotic transmission of HTLV-1 involving selective clonal expansion of these Tax-expressing 4 + T cells occurs at a rate .The newly infected cells from mitosis to the resting infected cells compartment are ,  ∈ (0, 1), with (1−) staying in the Tax-expressing infected 4 + T-cell compartment.The transfers among those three compartments are shown in Figure 1.
From the mechanism of the HTLV-1 infection and the schematic diagram we can have the following model consisting of three differential equations; In model (1),  1 ,  2 , and  3 are the removal rate of healthy 4 + T cells, resting infected 4 + T cells, and Taxexpressing infected 4 + 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 The calculation shows that the spectral radius (the basic reproductive number) of  −1 is The basic reproductive number,  0 , gives the average number of the secondary infections caused by a single Tax-expressing infected 4 + T cell during its whole infectious period.
From Lemma 2 in [28], we know that any solutions of model (1) with nonnegative initial conditions will be nonnegative for all  > 0.
With a similar argument as used in the proof of Theorem 1, we know that the domain is positively invariant with respect to model (1).In fact, the solutions of model ( 1) located on the boundary planes of Γ,  = / 1 , or  =   , or  +  + (( +  2 )/) =  0 +   , will enter Γ 0 , where Γ 0 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.

Stability Analysis of Equilibria
3.1.Stability of Infection-Free Equilibrium.Intuitively, if  0 < 1, then a Tax-expressing infected 4 + 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  0 < 1.
Proof.We use the linearized system of model (1) to discuss the stability of  0 .The characteristic equation of the matrix of the linearized system of model ( 1) at the infection-free equilibrium  0 is where From the Routh-Hurwitz criterion, it is easy to know that all the roots of ( 14) have negative real parts if  0 < 1, and ( 14) has at least one root with positive real part if  0 > 1.This completes the proof.

Stability of the Chronic Infection Equilibrium
Theorem 5. Assume that (A1) holds; if  0 > 1, then the unique chronic infection equilibrium  1 of model (1) is stable.
Proof.The characteristic equation of the matrix of the linearized system of model ( 1) at the chronic infection equilibrium  1 is where Since According to the Routh-Hurwitz criterion, we can see that all the roots of ( 16) have negative real parts if  0 > 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  0 > 1; that is, the infected proportion of the 4 + T cells persists above a certain positive level for sufficiently large .
From Proposition 8, we can get the conclusion that   is the maximal invariant set in  0 .Next we show that the solutions with the initial values in  0 cannot go to the boundary.Proposition 9. Assume that (A1) holds.If  0 > 1, then there exists a  > 0 such that the solution of model (1) with initial value (( 0 ), ( 0 ), ( 0 )) ∈  0 satisfies lim  → +∞ sup max{(), ()} > .
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  0 > 1, then model (1) is uniformly persistent with respect to ( 0 ,  0 ); that is, there exists a positive number  such that Proof. and  0 are positively invariant for model (1).Φ  is point dissipative and compact.By Lemma 6 we know that there is a connected global attractor  for Φ  that attracts each bounded set in .
From the discussion of Proposition 8, we know that   is the maximal compact invariant set in  0 .Since we choose the Morse decomposition of   as { 0 } and ∪ ∈  () = { 0 }, the set { 0 } is isolated.Proposition 9 shows that the solutions of model ( 1) with initial values in  0 cannot go to the boundary, which implies that   ( 0 ) ∩  0 = .It follows from Lemma 7 that model ( 1) is uniformly persistent with respect to ( 0 ,  0 ).
The following lemmas in [30][31][32] are used to study the global stability of the chronic infection equilibrium  1 .We will show that all the solutions of model (1) Let  → () ∈   be a  1 function for  in an open set  ⊂   .Consider the system of differential equations Let (,  0 ) be the solution of model ( 27) satisfying (0,  0 ) =  0 .A set  is said to be absorbing in  for model (27) if (,  1 ) ⊂  for each compact  1 ⊂  and sufficiently large .We make the following two basic assumptions.
( 1 ) There exists a compact absorbing set  ⊂ .27) has a unique equilibrium  in .
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.
Then the unique equilibrium  is globally asymptotically stable in .
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 Choose  = diag(1, −1, 1); we can obtain All off-diagonal entries of  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  1 of model (1).
The second compound system of the linearized system along a periodic solution ((), (), ()) of model ( 1) is In order to verify that model ( 30) is asymptotically stable, we define a Lyapunov function From the uniform persistence, we know that the orbit O of the periodic solution ((), (), ()) has a positive distance from the boundary of Γ.There exists a constant  > 0 such that  (, , ; , , ) ≥  sup {|| , || , ||} .
Let ( 1 ) be the Jacobian matrix of model ( 1) at  1 .Then we have Condition (9) of Lemma 13 holds.The chronic infection equilibrium  1 of model ( 1) is globally asymptotically stable in Γ 0 since all conditions of Lemma 13 are satisfied.

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 4 + helper T cells produced in the bone marrow, , is 15-25 cells/mm 3 /day.The coefficient of infectious transmissibility, , is 0.0005-0.003mm 3 /cell/day.The proportion of infected cells expressing Tax, , is (0.003-0.03)/day.The removal rates of healthy 4 + T cells, resting infected 4 + T cells, and Tax-expressing infected 4 + T cells,  1 ,  2 , and  3 , are taken to be the value 0.01-0.05/day.The death rate of the Tax-expressing infected 4 + T cells is considerably shorter than the natural lifespan of 4 + T cells [33].
In Figure 3, we use the following set of parameters:  = 20,  = 0.001,  1 = 1/30,  2 = 0.02,  = 0.1,  = 0.1,  = 0.8,  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  0 (Table 1).Figure 4 shows that a significantly strong positive correlation exists between parameters  and  0 (PRCC = 0.8597;  value = 0 < 0.01).The second sensitive parameter to  0 is  (PRCC = 0.3421;  value = 0 < 0.01).The result indicates that the cell-to-cell contact transmission and Tax expression contribute a lot to the viral infection.
The sensitivity analysis result shows that  and  are two significant parameters for the infection.We illustrate the impact of  and  on the magnitude of the chronic infection equilibrium  1 by numerical simulations.The curves in Figures 5(a 5(c) and 5(d) give the values of  1 or  1 as the functions of  and , respectively.Those curves and surfaces in Figure 5 indicate that  1 and  1 will increase with  and .For any given ,  1 or  1 increases very fast for small  and quite slow for large .

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 /( + ), where  = (( 3 − (1 − ))/).Hence (/)(/( + )) = ( 3 − (1 − ))/( 3 − (1 − ) + ) 2 > 0. That is, a faster rate of spontaneous expression of the Tax results in a higher proportion of  in infected 4 + T cells which influence the risk of HAM/TSP.
It follows from our sensitivity analysis that  and  are significantly sensitive to the reproduction number  0 .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.

Figure 1 :
Figure 1: The schematic diagram of the HTLV-1 infection in vivo.

Figure 5 :
Figure 5: The impact of  and  on the magnitude of the chronic infection equilibrium  1 .

Table 1 :
PRCC results and  value.