Acta Científica Venezolana, 55: 247-263, 2004

                                FOR HIV-1 INFECTION
                                                                   1                                    2
                                            Horacio Ortega and Miguel Martín-Landrove
                            Facultad de Medicina, Universidad Central de Venezuela and Cátedra de Física,
                                UNEXPO Antonio José de Sucre, VR La Yaguara, Caracas. A.P. 47636
                                            Los Chaguaramos, Caracas, 1041-A, Venezuela.
                            Departamento de Física, Centro de Resonancia Magnética, Facultad de Ciencias,
                                   Universidad Central de Venezuela, A.P. 47586, Caracas, 1041-A,
                     Venezuela and Instituto de Resonancia Magnética, La Florida/San Román, Caracas, Venezuela.

                                      Recibido: 26/11/02; Revisado: 29/05/03; Aceptado: 13/07/04

   ABSTRACT: A model of HIV-1 infection based on immunodominance concepts is presented. The model considers not only mutating virions
   and T-CD4 cells, but also viral reservoirs, such as macrophages and follicular dendritic cells. It also considers strong cytotoxic attack against
   reservoirs and extra cellular attack on virions, which are both coordinated by T-CD4 cells. As a first case, only one viral variant is dealt with,
   and approximations were used to obtain a manageable model. A stability criterion was found that marks the transition between progression
   and regression of the viral infection. This criterion was proven valid for the model without any approximation, i.e., two randomly mutant viral
   epitopes with two variants each. Data suggest that a) the role played by the reservoirs in the maintenance of the viral infection is a very
   important one, b) maintaining CTL attack on infected cells facilitates the control of HIV-1 infection, and c) therapy considerations should be
   addressed to these points. Keywords: reservoirs, virions, immune response, mutability, viral escape, HIV


   RESUMEN: Se presenta un modelo de infección por VIH-1 basado en el concepto de inmunodominancia. Además de viriones mutantes y
   células T-CD4, se considera también reservorios virales, entre ellos macrófagos y células dendríticas foliculares. También se considera
   ataque citotóxico contra los reservorios y ataque extracelular sobre los viriones, ambos coordinados por las células T-CD4. En primer lugar,
   se trabaja sólo con una variable viral, y se usan ciertas aproximaciones para obtener un modelo fácilmente manipulable, para el cual se
   encuentra un criterio de estabilidad que rige el paso de progresión a regresión de la infección. Este criterio mantiene su validez para el
   modelo sin uso de aproximaciones, esto es, dos epítopes que mutan al azar, cada uno de ellos con dos variantes. Los resultados sugieren:
   a) que el papel jugado por los reservorios en el mantenimiento de la infección es muy importante, b) que mantener el ataque de las células
   citotóxicas sobre los reservorios infectados facilita la labor del sistema inmune, y c) que la terapia debería orientarse preferiblemente hacia
   estos objetivos. Palabras clave: Reservorios, viriones, respuesta inmune, mutabilidad, escape viral, VIH

INTRODUCTION                                                                  persist in a non-functional state45. Recent reports state
                                                                              that: a) CTL attack to viral sources causes a radical
                                                                              decrease in viral charge in the primary stage of
  In 1995 Nowak et al.23 introduced a model of HIV-1                          infection25,26, b) activated CTL cells are short lived, their
infection based on the concept of immunodominance in
                                                                              presence in the blood stream needs continuous
order to explain variations of the human immune                               stimulation9,17,38, c) CTL cells are involved both in
response against virus which continually produce mutant                       cytopathic and non cytopathic control of virions5,19,44, d)
epitopes. HIV-1 infects T-CD4               cells causing damage              genetic manipulation and induction of apoptotic HIV-1
and shortening of their life cycle. Other cells as T-CD8 ,
                                                                       +      infected cells results in durable CTL responses14,35. A
macrophages, dendritic cells are also a target for the                        summary of CTL role in control of HIV-1 infection can be
virus. Several mechanisms have been proposed to justify                       found in42. A controversy currently is posed on the
                                  +                                           existence of humoral response and its effectiveness30,41,
the reduction of the T-CD4 cell life cycle due to the viral
                                                                              but this response exists and it is susceptible of being
infection, among them cytotoxicity, viral cytopathogenicity
                                                                              enhanced14,22,36. A long list of candidates for the viral
and apoptosis. Cytotoxicity occurs due to the attack
                                                                              replication site exists. Virions are produced in cellular
made by CTL to infected cells, the clonal and activation
                                                                              compartments in lymphoid tissue from the beginning of
processes of immunocompetent cells being mediated by
                                                                              the infection8,29. In advanced stages of AIDS, the viral
T-CD4 cells16. CTL can occasionally be present in high                                                                            +
                                                                              load is very high even when the T-CD4 cell population,
numbers, yet they are not associated to the control of
                                                                              which are the main accepted source of free virus, is
viral replication. If there is T-CD4 deficiency, CTL can
                                                                              depressed20,27. Moreover, there is good agreement
248                                                                                             Ortega and Martín-Landrove

between the number of cells infected by virus present in         several epitopes, or active regions, but the immune
lymphoid tissue and the quantity of free virus present in        system normally recognizes just one or two of them23.
blood11. Then, it is reasonable to assume the existence of       We suppose that virions exhibit only two epitopes, α
additional viral sources such as the follicular dendritic        with N variants 1, 2,....i,.....N , and β , with M
cells (FDPC) and peripheral blood mononuclear cells
(PBMC),       B-cells   among      these21.   Opportunistic      variants 1, 2,..... j,....M . A particular virion will be
infections , and immune stimulation, for instance by             designated by its density, vij , this is, we write vij for
vaccination40, seem to highlight the role of macrophages
as: a) site of attack by HIV-1 during the incubation period      vαi β j . Any cell set which is susceptible to be invaded by
of AIDS8, b) source of huge quantities of free virions, at a     the virus, and then to produce viral particles will be called
time when T-CD4          density is minimal, and c) not          a reservoir. An infected reservoir can exhibit viral
succumbing producers of both intracellular and extra             residues on its surface, and when they are recognized as
cellular HIV-1, although the detailed mechanisms that            alien by the immune system, specific cells deliver a lethal
govern this behavior are not clearly understood.                 hit against them. Densities for healthy reservoirs and for
Therefore, the existence of a cellular population with a         infected ones are denoted by m0 , and mij , respectively
long lifetime, which provides virions continuously to the
blood stream, seems to be stated beyond any doubt. The           (we also write mij for mαiβj ). T-CD4                cells acting mainly
use of viral inhibitor drugs causes a decrease of viral          as coordinators of the immune response are called z .
burden among others places, in lymphoid tissu2; this             Infected cells elicit a response of CTL cells addressed
therapy has lead to a decrease in deceases related to            against them. Cytotoxic cells directed against infected
AIDS. However, some troubles persist, because even
with the use of HAART (highly active antiretroviral              cells bearing variant i of epitope α are denoted by xi ,
therapy) some subsets of cells go on producing                   while those directed against cells bearing variant j of
virions3,34. Also, it has been reported that after               epitope β , by y j . Each equation for the present system
suppression of drug control in voluntaries, viral charge
                                                                 will be separately discussed. We begin by introducing the
and viral infection come back4, suggesting the presence
                                                                 following variables:
of sanctuaries where virions are kept safe. This fact
suggests that HAART treatment is a lifetime deal, which
produces no definitive cure.                                                      mi∗ = ∑ mij                     m∗ j = ∑ mij
  In the present paper we intend to describe the role of                                   j                               i
the reservoirs as a continuous source of HIV-1 infective
particles, and the effect of both cellular and humoral                         (uv )∗ = ∑ uij vij         ( γij vij )∗ = ∑ γij vij ,
immunity on the progression from HIV-1 infection to full                                 i, j                             i, j
AIDS. The main goal of our work is to find under what
conditions the immune system controls the viral                    The equation that states the dynamics of healthy
population in the proposed system, and how these                 reservoir cell density is:
conditions are modified by the presence of several
epitopes and mutations. To perform the stability analysis,
a simplified version of the original system of equations is            dm0
                                                                           = λ 0 − ( γ ij v ij ) * m              0   − ω0m      0
studied by standard means (linearization and evaluation                 dt
of eigenvalues by the Routh-Hurwitz procedure), and
later on the obtained stability criteria are used to verify        Reservoirs are produced in the hematopoietic organs at
numerically the behavior of the system without any               a rate λ0 . These reservoirs migrate to the blood stream
approximation, that is, situations where two epitopes can
be found, each one of them with two strains that can             (or to their final places), and they take virions, at a rate
mutate in a continuous way. For that case analytical              γij m0 vij , as a result of a contact interaction. γij
methods fail, but numerical integration of the system of         represents the probability for a healthy reservoir of being
equations shows that the stability criteria obtained in the      invaded once it contacts a virion. Reservoirs die after a
simplified case are still valid. Finally we point out that our                           −1
model can be extended easily to situations that deal with        mean time t0 = ( ω0 ) .
antiviral drugs7,24.                                               The equation for the infected reservoir cell density is:

THE MODEL                                                            d m ij
                                                                              = γ ij v ij m 0 − m ij   ( px   i   + q y j ) − ω 1 m ij
  In what follows we will use the notation utilized in23.
HIV-1 is a highly mutant virus; its mutation capability is                                                                             (2)
related to its reverse transcriptase replying-based
process. We consider the existence of free virions,                Infected reservoirs are produced as a result of the
                                                                 interaction of healthy reservoirs with viral particles, at a
denoted by vαβ . An antigen can simultaneously present
Model for HIV-1 infection                                                                                                                        249

rate γij m0 vij , and as they display viral epitopes, they are                     Virions are liberated by infected reservoirs, at a rate
                                                                                 gij mij , and by infected T-CD4 + cells, at a rate ξuij zvij .
attacked by cytotoxic cells, with density                          xi ( y j ),
                                                                                 The previous hypothesis is made considering that the
addressed against epitope αi ( β j ), at a rate pxi mij                                                          +
                                                                                 lifetime of infected T-CD4 cells is much shorter than
( qy j mij ). p and q are the dynamical coefficients ruling                      that of healthy ones13,31, and therefore at any moment
                                                                                 their average concentration is at quasi-static equilibrium
these interactions. This also means that CTL cells are
                                                                                 with the present density of healthy cells (a formal
specific, those addressed against epitope αi cannot                              deduction is shown in Appendix A). Virions are taken
exert any effect on epitope β j and vice versa. Infected                         from blood stream by reservoirs, at a rate γij m0 vij , and
reservoirs are supposed to die after a mean time                                 they are destroyed by an extra cellular response
             −1                                                                                          +
t1 = (ω1 ) , which is not affected essentially by the                            mediated by T-CD4 cells44. Such response includes
                                                                                 interaction with antibodies from B-cells22, targeting of the
                                                                                 120 gp viral envelope12, or targeting of its fragments15.
  The equations that describe the dynamics of cytotoxic
                                                                                 We describe this interaction by a function
cell densities are:
                                                                                  f ( B, xi , y j , z )vij , where B is the concentration of
   d xi                                                                          plasmatic cells. For simplicity we have assumed
        = η ci z m i* + ci z m i* xi − ω 2 xi                             (3)     f ( B, xi , y j , z ) = fij z , with fij a dynamic constant of
                                                                                 adjustment. Other apparently simpler alternatives lead to
  dy                                                                             an extra equation for B. In our model we do not consider
             = ηk j zm ∗      j   + k j zm ∗ j y     j   − ω2y        j   (4)                                +
                                                                                 any direct effect of T-CD4 cells over virions.
    dt                                                                             Equations (1) to (6) represent the complete system. We
                                                                                 discuss its stability by means of linearization and
  Cytotoxic cells are produced by activation from a pool                         evaluation of the resultant eigenvalues. Linearization
of precursor cells, at a rate ηci zmi• ( ηk j zm• j ), or by                     methods lead to equations with no analytical solution,
proliferation from cells previously activated, at a rate                         and due to this fact two very simple situations with
                                                                                 immunological relevance are considered first, in order to
ci zmi • xi ( k j zm• j y j ), both processes are regulated by                   obtain stability criteria which later on will be compared
the presence of ( z ) mediator cells, η is the ratio of the                      with the numerical solution without any approximations.
activation rate to proliferation rate, and ci ( k j ) is the                     We call these simple situations “cellular approximation”
                                                                                 and “humoral approximation”.
dynamical coefficient ruling the activation23. Note that
CTL cells can recognize antigens on the surface of
infected, antigen presenting cells (APC), but to develop
any cytopathic activity, they must also receive an                               CELLULAR APPROXIMATION, NO MUTATIONS
additional signal from T-CD4 cells. Activated cytotoxic
                                                              −1                   In this approximation we consider that CTL cells exert
cells eventually die after a mean time t2 = ( ω2 ) .
                                                                                 the main control on infected reservoirs, and that there
  The equation for the mediator T cell density is:                               exists no extra cellular attack on virions. We also
                                                                                 consider the existence of just one viral variant with
                  dz                                                             epitopes α and β , vαβ . We suppose a moderate viral
                     = λz − (u v )* z − ω z z                             (5)
                  dt                                                             invasion on reservoirs, γαβ vαβ → 0 in equation (1), and
 Mediator T cells are produced in the thymus at a rate                           on T-cells, uαβ vαβ →0 in equation (5), and we neglect the
λz , they are infected by any viral variant at a rate
                                                                                 humoral attack on virions,          ( ξu   αβ   − f αβ ) zvαβ → 0 in
(uv ) z , where uij is a dynamic interaction coefficient.                        equation (6). We obtain as null equilibrium point
Eventually they die after a mean time of t z = ( ωz ) .
                                                                          −1     (absence of virions and infected reservoirs); see the
                                                                                 appendix B for details:
Note that T-CD4 cells do not fight the virus directly, but
instead they act as mediators and activators of other                                         λ0       λ
cells.                                                                               m0 =        , z = z , vαβ = mαβ = xα = y β = 0 (7)
  The equation for the viral particle density is:                                             ω0      ωz

    d v ij
             = g ij m ij + ( ξ u ij − f ij ) z v ij − γ ij m 0 v ij       (6)    stable if:
250                                                                                               Ortega and Martín-Landrove

                          g   αβ
                                   < 1.                       (8)               λz                    λ g αβ γαβ                (11)
                                                                       hM ≡        ( f αβ − ξu αβ ) > 0
                           ω   1                                                ωz                   ω 0 ω1
  Condition (8) is easily fulfilled if the production of virions    and the non null equilibrium point is always unstable. We
in reservoirs is minute (then g αβ = 0 ), and this inequality       interpret inequality (11) by saying that the null equilibrium
means that the null point is stable in this approximation if        point is stable if the fraction of virions destroyed by
an infected reservoir creates less than one virion during           processes mediated by T-helper cells is higher than the
its lifetime, an interpretation already mentioned by some           fraction of virions produced by infected reservoirs in the
authors although they use a very different model39.                 same interval (see appendix D for details). Alternative
                                                                    interpretations (or no interpretation at all) of inequality
                                                                    (11) are also possible. We call hM “humoral
HUMORAL APPROXIMATION, NO MUTATIONS                                                                       λ0 g α β γ α β
                                                                    responsiveness”, and the factor                         as “viral
  Alternatively, it is possible to consider just the attack                                               ω 0 ω1
performed on virions by T-CD4+ humoral mediated                     growth”. If condition (11) is not fulfilled, then the viral
response. In this case, we neglect the decline of healthy           particle density increases drastically, as shown in
reservoirs due to viral invasion and also the clearing of           Figure 1. Linearization criterion fails both if
virions from blood stream for invasion to reservoirs,               γαβ = hM = 0 , and/or g αβ = hM = 0 (see below).
γ αβm0vαβ → 0 in equations (1) and (6), although this term
                                                                      We now consider several particular cases of interest:
is conserved in equation (2) due to that a)             γαβ m0vαβ   g αβ → 0 and/or γαβ → 0 , but hM ≠ 0 (reservoirs do not
represents the source of infected reservoirs in such                send virions to the blood, and/or there is no viral invasion
equation, and b) ω 1 m ij is small and comparable to                to them), then inequality (11) predicts that whenever a
                                                                    population of mediator cells exists, no matter how small it
γ α β m 0 v α β (in fact t1 ≡ lifetime of infected reservoirs is
                                                                    is, there will be control of the viral population. If uαβ = 0
rather long, therefore ω1 ≈ ( t1 )         is small).   We also     (there is not viral invasion to T cells, but virions continue
neglect the cytotoxic attack to infected reservoirs,                entering and reproducing into reservoirs), the null
      (              )
mαβ pxα + qy β → 0 in (2), and therefore equations                                               λz   λ g γ         
                                                                    population is stable if         > 0  αβ αβ      ,    i.e., it is
(3) and (4) become uncoupled from the system. As a                                               ωz ω0  ω1 f αβ
consequence, the following equilibrium points are
obtained (see appendix C for details):                              necessary that the equilibrium concentration of T-helper
                                                                    cells without infection is bigger than the medium rate of
                   λ0          λz                                   virions production, which is a more restrictive condition
            m0 =      ,   z=      ,   vαβ = mαβ = 0           (9)   than the previous one. If h M = 0 and γαβ ≠ 0 (there is
                   ω0          ωz
                                                                    no control on the viral population contribution coming
which is called the null equilibrium point, due to the              from T cells, although reservoirs continue producing
absence of infection. There exists also a non null                  virus), then any viral population is unstable. This situation
equilibrium point given by:                                         seems to be the current status of the therapy. If hM = 0
                                                                    and   γ αβ = 0,   perturbation procedures show that the
                                                                    equilibrium point is given by (9), stable Lyapunov (this
               λ           λ0 γαβ g αβ                              implies that minute virion densities can remain in the
          m0 = 0 , z =                        ,
              ω0       ω0 ω1 ( f αβ − ξu αβ )                                                                         dvαβ
                                                                    system). Finally if hM = 0 and g αβ = 0 , then           = 0 . In
                                                            (10)                                                       dt
                ω  λz ω0 ω1 ( f αβ − ξu αβ ) 
          vαβ = z                           − 1 ,                 this case, analytic approaches cannot provide an answer,
                u         ω z λ0 g αβ γαβ     
                                                                   but numerical integrations show that there is a finite
                 λ0 γαβ                                             equilibrium population given by m0 =             , vαβ = v0 ,
          mαβ =         vαβ                                                                                       ω0
                  ω1                                                        γ αβ                          λz
                                                                    m αβ =            v αβ , and z =                , where v0 is
                                                                           ω         
                                                                            0                        ω z + u αβ v 0
  Linearization procedure shows that the null equilibrium           the initial viral population. There exist reports of situations
point is stable if:                                                 of viral infection not progressing to full AIDS33.
Model for HIV-1 infection                                                                                                               251

  Figure 1 shows the equilibrium populations for an                      occurs is compatible with inequality (11). Figure 2 shows
system lacking of CTL cells and possessing just one viral                the temporal evolution for the same system under
variable, v11 , as a function of hM , the humoral                        conditions that lead to viral outgrowth (weak immune
                                                                         response). Notice that healthy reservoir density is
responsiveness. There is only viral persistence when the
                                                                         constant, that mediator T cells decrease, while viral load
humoral response is weak. It is clear that the point at                  and infected reservoirs densities increase continuously.
which the transition between viral growth and viral control

                                                        λ0γ 11 g11

Figure 1. Computer generated plot of equilibrium values of variables in our system in absence of cytolytic cells as a function of
hM =      ( f11 − ξu11 ) , humoral responsiveness. We consider the existence of just one epitope with one viral variable, which we call v11 .
So we use in this and in the following figures, α→1, and β → 1 . Values of parameters used in the calculations are:              λ0 = 0.25 ,

λ z = 4 . 00 , ω 2 = 140 , γ11 = 0 .35 , g11 = 0.65 , ω z = 7 . 0 0 ω0 = ω1 = 0 .28 , u11 = ξ = 1 . 80 , and we vary f 11 between 0

and 10. With this values the threshold for convergence,
                                                             λ 0 g αβ γ αβ    takes the value 0.73. Viral and infected reservoirs densities
                                                             ω 0 ω1
remain low only while immune response      hM is higher than the threshold. The step viral saturation → viral control occurs as predicted
by equation (11). For plotting, we used the 10000 for the number of steps, and we also truncated the integrator output for v11 and m11 . in
the divergent region.
252                                                                                                          Ortega and Martín-Landrove

Figure     2.   Temporal      evolution      of   a   system   with   weak   immune     response.      Parameters   used   in   the   calculation

were, f 1 1 = 3 . 50 , ξ = 2 . 5 0 ,   u11 = 3 .50 and all others as in Figure 1. With these values,         γ11 g11 ≅ 0 .73 and hM ≅ − 3 .00 .
                                                                                                       ω0 ω1

For this choice of parameters there exists just the null equilibrium point (unstable). As viral creation rate is higher than virions destruction

rate, viral and infected reservoirs densities grow, and slow T-helper cells decay occur.
Model for HIV-1 infection                                                                                                  253

THE FULL SYSTEM, NO MUTATIONS                                  two epitopes with two viral variants each one, b) there
                                                               are random mutations in any of the viral variants currently
                                                               present, c) each time a mutation takes place and a new
   When considering the complete system, that is to say,
                                                               viral variant of an epitope appears, a fraction α of the
with the presence of cytotoxic cells and without any
                                                               mutated viral variant is set as initial condition for that
approximation, it is found that there exists a null
                                                               variant, and a fraction (1- α ) of the not mutated viral
equilibrium population, given by equation (9), and
linearization procedures similar to those performed in         variant is set as initial condition for that epitope,
appendices B and C show that this point is stable if :         additional equations for the corresponding CTL and
                                                               infected reservoirs densities are added, and the
                                                               remaining of the system does not vary and d) the
        λz                      λ        g
           ( f α β − ξu α β ) > 0 γ α β ( α β − 1 )    (12)    mutation rate depends on γij , the viral infectivity to
        ωz                     ω0         ω1
                                                               reservoirs (our emphasis is on the role of reservoirs in
                                                               AIDS). An algorithm based on the Monte Carlo method
  We observe that a) inequality (12) is a combination of       was coupled to our numerical integrator for evaluating
inequalities (8) and (11) and b) it is easier for the immune   explicitly the effects of the virus mutability. We introduced
system to fulfill condition (12) than condition (11), due to   a temporal evolution for each viral variant, ruled by a
the existence of the negative term
                                           γαβ driving the                           (        )
                                                               quantity R= exp − sγ ij n∆t , where s is an adjustment
                                        ω0                     parameter which we call mutability, and which we could
system to the stable region. Figure 3 shows a numerical        arbitrarily vary, n is the number of elapsed time steps
integration for this system. Note that the transition          and ∆t , the time step interval. We compared R with Γ ,
instability → stability occurs at values of hM compatibles     the output of a random number generator between 0 and
with condition (12). Also, by comparing Figure 1 and           1, and every time that R < Γ a mutation takes place.
Figure 3, we can observe that the full system (with CTL        Notice that for short elapsed times, R ≈ 1 and mutations
cells) eradicates the infection at smaller values of hM        are unlikely, while for a sufficiently long elapsed time
than a system without CTL cells, (clearing of the infection    R → 0 and some of the possible mutations take place. In
occurs for lesser values of the immune response when           such a case, n is reset to zero for the particular variant.
                                                               Results obtained by this method show that a) the viral
cytolytic cells are present and moreover γαβ and gαβ
                                                               outgrowth is a consequence of the mutability, because
can take higher values than in absence of them without         for values of the parameters that would lead to viral
diverging).                                                    removal (any number of viral variants, without mutations)
  Treatment of the general equilibrium point, with non-        divergence occurs (with identical parameters and initial
null densities both of virions, vαβ and of infected            conditions), as shown in Figure 4, a plot of total viral

reservoirs,   mαβ    leads    to   eigenvalue    equations     density,   ∑v
                                                                           ij   ij   vs. time for some values of mutability, s.

analytically unmanageable, so that we limit ourselves in       b) It is sufficient that the immune system loses control
the remainder of this paper to a numerical treatment. In       upon a certain viral variant (through its respective hM ) to
any     instance    our    simulations     show     that       have viral outgrowth. c) For low values of mutability and
    λ                                                          the range the steps computed (up to 105), we found
hM = z ( fαβ − ξuαβ )   acts as a very important parameter     situations of non divergent dynamics. d) For moderate
    ωz                                                         values of s, there can be situations in which there is a
that controls the behavior of the system. Our simulations      long persistence in time of low or moderate values of the
(some of them running up to 106 steps) also suggest the        viral variables before the collapsing or the clearing of
possibility of non-progression to full AIDS for the infected   them. e) For moderate and high mutability the system
organism, a fact well documented33. We stress that our         becomes sensible to initial conditions, the higher the
system’s stability is related not only to the immune           mutability, the lower initial concentrations for viral or
control of viral output from T-CD4 cells, but also to the      infected reservoirs resulting in system collapse.
control of virions produced in reservoirs. This fact states      Nowak et al.23 classified the different behavior of
a departure from currently accepted evolution of HIV-1.        patients with conserved epitopes (always recognized by
                                                               immune system), as non-progressors, and others in
                                                               which immune response failed to recognize some viral
                                                               epitope (or shifted to another epitope), as fast
                                                               progressors. This fact was confirmed by our model (see
                                                               Figures 5-a and 5-b).
  HIV-1 is a highly mutant virus; its mutation capability is     Figure 5-a shows a case with fixed epitope α , and four
related to its reverse transcriptase replying-based            mutant variants in epitope β ( v11 , v12 , v13 , v14 ). This
process. In order to simulate the mutant system the
                                                               infection is easily controlled by the immune system.
following assumptions were taken: a) the system exhibit
                                                               Figure 5-b shows a case with two continually mutant
254                                                                                              Ortega and Martín-Landrove

epitopes, each one with two variants ( v11 , v12 , v21 , v22 ).    based in the lacking of CTL attack on them. Virions with
                                                                   conserved epitopes facilitate the immune system task
Now the immune system fails to control the infection and
                                                                   (see Figure 5-a). Mutability also results in a critical
viral outgrowth occurs.
                                                                   dependence on the initial concentrations of the temporal
                                                                   behavior of the system. Minute concentrations of virions
                                                                   do not imply compulsive progression to full AIDS, and
DISCUSSION                                                         numerical work suggests that it is possible to find
                                                                   situations of otherwise healthy HIV-1 infected subjects
                                                                   which remain in stable immune condition for long time.
  Inequalities (11) and (12) suggest that blocking the viral       Such situation could explain the report on non
invasion to reservoirs ( γ α β → 0 ), or the capability of         transmission of AIDS when the concentration of virions in
reservoirs of producing free virus ( g αβ → 0 ) is the             blood stream is low33.
                                                                     Opportunistic attack on the organism often shadows
easiest way of controlling viral growth. In such cases    any      the actual processes occurring during viral infection. It is
non-null population of T-CD4+ cells is enough for         the      generally expected that stable values for immune
fulfillment of this goal. The relative shift of           the      variables should lead to clearing opportunistic agents, but
convergence points given by equations (11),               (no      numerical work shows that there exists non diverging
                                                         ω1        points for the system (1) to (6), these values being
reservoirs), and (12) (with reservoirs) results to be          ,
                                                         gαβ       minute for z, T helper cells density, and m0 , healthy
this shift being large if there exists absence of viral            reservoirs density. If in the otherwise healthy system
                                                                   (HIV-1 infection is maintained under control) opportunistic
production by reservoirs, gαβ → 0 , or also if the lifetime        infections emerge, this breakdown and further damage of
for infected reservoirs is short, ω1 → ∞ , suggesting that         immune system being responsible of the posterior
                                                                   collapse of the whole organism.
therapeutic or CTL attack on them could result in better
expectances for HIV-1 control. Moreover if there is no
viral    production     inside    reservoirs,  gαβ → 0 ,
and    as    it   is    obtained     from    equation     (12)
λz                      λ
   ( f αβ − ξu αβ ) > − 0 γαβ , a condition that is fulfilled        We have made an approximate analytical approach,
ωz                     ω0                                          and an exact numerical one of the viral infection.
by every non-null equilibrium T-CD4+ cell population. This           They show that it is important to impair virus production
fact should be considered when checking the progression            inside the reservoirs and/or block the infection of these
of HIV-1 infected organisms to full AIDS.                          cells. If these requirements are fulfilled, a relatively
   Figure 1 and Figure 3 were plotted after numerical              moderate immune response is enough to control the viral
work. By comparing them, it is possible to confront                infection. The control on the T-CD4 cells alone drives the
asymptotic viral trends in absence/presence of cellular            system to the onset of AIDS. To annihilate the viral
attack on reservoirs, such attack resulting in the already         infection it is necessary to annihilate the infected
mentioned shift of the immune responsiveness threshold             reservoirs also. Attack to the infected reservoirs by
to lower values. Recent reports state the role of                  cytotoxic cells shows situations that are easier to handle
reservoirs in supporting viral production at late stages of        by the immune system, so it seems important to develop
HIV-1 infection, suggesting that the immune system fails           mechanisms for continually activating CTL.
to recognize infected reservoirs. Some mechanisms                    Viral mutability can lead to the appearance of more
responsible for this failure have been reported6,17,37,42.         efficient viral variants (higher values of γ ij or g ij ), to the
Lifetime of cytotoxic cells43, continuous production of
them, and attack CTL-CD4 mediated on infected cells1               shift of the cytotoxic response ( pi and/or q j tend to
should somehow be enhanced. We also mention the
                                                                   zero), or simply to viral accumulation and concerted
finding of non diverging dynamics (up to 105 computation
                                                                   attack on T cells. In any case, there is viral escape. It
steps) with the full system (one viral variable, no
                                                                   seems clear that any future therapy should include the
                                                                   control of any cell susceptible of being infected by the
   Mutation results in higher instability of the system,
                                                                   HIV-1, not only of T-helper cells, the only clinical
convergence points shifting toward higher values of the
                                                                   measure available at this moment. Also, therapy should
immune response. High mutability implies viral escape
                                                                   aim to the enhancement of the performance and lifetime
for situations that normally would produce immune
                                                                   of CTL. We recall that many advances have taken place
control if mutability, s, was low23, (see Figure 4).
                                                                   in the interpretation of the illness related to the HIV-1
   There is viral control only for low values of s. All
                                                                   infection, and it is possible to reflect these changes in the
simulations displayed were performed with identical
                                                                   therapy, so that reports of cures soon appear. Until this
parameter values, with the exception of the rate of
                                                                   happens, the avoidance of infection seems to be the best
mutation, s , so the advantage of new viral variables is
                                                                   prophylaxis against AIDS.
Model for HIV-1 infection                                                                                                               255

                                                                  λ0       g                       λ0γ 11 g11
                                                                     γ αβ ( αβ − 1)
                                                                  ω0       ω1                       ω0ω1

Figure 3. Equilibrium values of variables in our system with the presence of CTL cells, as a function of        hM =      ( f 11 − ξ u 11 ) .
Parameters used in the calculation are p1 = q1 = 2 .05 ,k1 = c1 = 0 .45 ; and    η = 0.02 , ω0 = ω1 = 0 .35 , and all others as in Figure 1.

Resulting densities of   xi , and y j are identical for this particular choice of parameters, and they overlap each other. Now there exists

viral saturation for small values of   hM , but the convergence point is shifted to lower values as predicted by equation (12), this shift is

caused by attack of CTL on virus producing cells. Note the existence of finite non diverging values for viral and infected reservoirs

densities. Cytotoxic cells exist just for appreciable density of reservoirs.
256                                                                                                               Ortega and Martín-Landrove

Figure 4. Time evolution of total viral density   ∑   ij
                                                           v ij   for three instances of the mutability s. For   s = 0.01 , low mutability, total virions

density fades quickly. For   s = 0.1 , moderate mutability, viral density oscillates before slow growth. For s = 1.0 , high mutability, total
virions density diverges. After mutations (marked by changes in the slope of the curves                s = 0.01 , and s = 0.1 ),there are transient
increases in the viral density caused by the current delay in CTL specific response. We used f11 = 5 .00 ,ξ = 1 .90 , u11 = 2 .25 ,

pi = q j = 2 .05 ,    ci = k j = 0.45 ,    η = 0 . 02 ,           and      the    remaining      parameters        as      in    Figure      3.     Now

λ0       g                               λ
    γ11 ( 11 − 1 ) ≅ 0 .21 < hM = 0.41 < 0 γ11 g 11 ≅ 0 .46 . T-helper cells concentrations (not shown) are high when viral burden is low
ω01      ω11                            ω0 ω1
and vice versa.
Model for HIV-1 infection                                                                                                                      257

Figure 5-a. Time evolution of the total viral density     ∑   ij
                                                                   v ij ≡ v , and T-CD4 cells, z, for a system with a conserved epitope (epitope 1).
There exist   v11 , v12 , v13 , v14   variants. As the attack of CTL     x1   against epitope 1 is always present, the immune system easily checks

the infection. We used for both 5 and 6 figures s=0.1 and for the remainder of parameters identical values as in Figure 4.
258                                                                                                             Ortega and Martín-Landrove

Figure 5-b. Time evolution of the total viral density        ∑   ij
                                                                      v ij ≡ v , and T-CD4 cells, z, for a system without conserved epitopes. There
exist   v11 , v12 , v 21 , v 22   variants. A possible evolution of the viral variants (among others) could be    v11 → v12 → v 22 . As the CTL
addressed against        v11 (the initial existing virion) are x1 , y1 , these mutations leave the immune system void of CTL addressed against
v 22 ( x 2 , y 2 ) , a delay being necessary for the creation of these cells, and viral escape occurs.

APPENDIX A                                                                        already explained in the text, and we include an attack on
                                                                                  infected T-CD4 + cells by CTL cells. Equation (6) can be
  The model can be expressed more generally by                                    written more generally as:
equations (1) to (6), adding the following equation:
          dzαβ                                                                            = gαβ mαβ + rαβ zαβ − fαβ zvαβ − γ αβ m0 vαβ − uαβ zαβ vαβ
                  = uzvαβ − ω4 zαβ − zαβ ( pxα + qy β )               (A-1)
            dt                                                                                                                              (A-2)
                                                                                    Where     rαβ   is the dynamical coefficient ruling the
where zαβ represents the T cell density infected by viral                         production of virions by infected T cells. The lifetime of an
variant αβ . Most of the terms in equation (A-1) are                              infected T cell is much shorter than that of a healthy
Model for HIV-1 infection                                                                                                                                                  259

cell38,39, for this reason we suppose that the population of                    The null point of equilibrium of this system is given by:
infected T cells is in quasi-static equilibrium with the
healthy ones. Additionally we suppose that an infected T                                   λ0                      λz
cell in its short time of life has few opportunities to                          m0 =           ,     z=                ,             vαβ = mαβ = xα = y β = 0
interact with a specific cytolytic cell, which is equivalent to                            ω0                      ωz
taking zαβ ( pxα + qyβ ) → 0 in (A-1). Then equation                                                                                                                  (B-7)
(A-1) becomes:
                                                                                Its linearization matrix is:
                                  = uzvαβ − ω 4 zαβ                  (A-3)                  −ω 0  0                    0                 0         0          0     
                            dt                                                              0    −ω1                   0                 0         0       γ αβ m0 
                                                                                                                                                                    
and in quasi-static equilibrium:                                                            0 ηcα z                −ω 2                  0         0          0 
                                                                                                                                                                    
                                                                                            0 ηkβ z                    0             −ω 2          0          0 
                                          u                                                 0     0                    0                 0      −ω z          0 
                                 zαβ =        zvαβ                   (A-5)
                                                                                                                                                                    
                                         ω4                                                                                                                 −γ αβ m0 
                                                                                            0
                                                                                                 gαβ                   0                 0         0                
  When substituting equation (A-5) into equation (A-2),
and defining       ξ ≡            − 1 , we obtain equation (6).
                           w4                                                  with        eigenvalues                  µ1 = −ω 0 ,                     µ 2 = µ3 = −ω 2 ,
                                                                               µ 4 = −ω z , and:
  Observe that viral invasion impairs infected T-CD4+
cells mediator role, so it is unnecessary to include an
equation for them in the model. There are alternative                             µ 2 + µ ( ω + γ αβ m 0 ) + γ αβ m 0 ( ω 1 − g αβ ) = 0
ways to arrive to equation (6), not shown here.

                                                                                 When applying the Routh-Hurwitz criterion to equation
                                                                               (B-9),    we    obtain      as condition  of   stability:
  The system, with the approximations given in the text                         m0γ αβ (ω1 − gαβ ) > 0 , or
                           dm0                                                                                                            <1                          (B-10)
                               = λ0 − ω 0m0                          (B-1)                                                       ω1

        d mαβ                                                                  APPENDIX C
                       = γ α β vα β m 0 − m α β      ( px   α   + qyβ     )
                                                                                 The system, with the approximations mentioned in the
        −ω 1m αβ                                                               text, is given by:
                                                                                                    dm         0
                                                                                                                    = λ                − ω          m                  (C-1)
   d xα                                                                                              dt
                                                                                                                                 0              0       0
        = η cα z m α β + cα z m α β xα − ω 2 xα
                                                                     (B-3)                 d m αβ
                                                                                                       = γ αβ m 0 v αβ − ω 1m αβ                                      (C-2)
   dy    β
              = η k β zm          αβ     + k β zm    αβ   yβ − ω      2   yβ
    dt                                                                                 d z
                                                                     (B-4)                 = λ             z       − u          α β   v   α β   z − ω         z   z    (C-3)
                                                                                       d t
                            = λz − ω z z                                          d vαβ
                                                                                                = g   αβ   m       αβ       −    (f       αβ    − ξ u αβ      )zv     αβ

              dv   αβ                                                           Its linearization matrix is:
                          = g αβ m αβ − γ            αβ   m 0 v αβ    (B-6)
260                                                                                                        Ortega and Martín-Landrove

    −ω 0          0                0                      0                   µ 3 + µ 2 [ ω 0 + ω z + uv αβ + z ( f − ξu )] +
   γ v           −ω1               0                   γ αβ m0     
    αβ αβ                                                                     µ [ ω 0 ( ω z + uv αβ ) + ( ω 0 + ω z )( f − ξu ) −
    0             0           −uαβ vαβ − ω z           − uαβ z                            λ0
                                                                              g αβ γ αβ      )]
    0
                 gαβ       ( ξ uαβ − fαβ )vαβ     ( ξ uαβ − fαβ )z 
                                                                                           ω0
                                                               (C-5)                                              λ0
                                                                                                       gαβ γ αβ      vαβ = 0                  (C-12)
  The system has two equilibrium points, which we call                                                            ω0
the null equilibrium point, and the general equilibrium
point, respectively.                                                      The independent term in equation (C-12) is negative,
  The eigenvalues µ , at the null equilibrium point, given              thus, according to the Routh-Hurwitz criterion one of
by:                                                                     system-associated eigenvalues always has a non-
                   λ0                                                   negative real part, and therefore the equilibrium point
          m0 =          ,    z=         , vαβ = mαβ = 0         (C-6)   given by equation (C-11) is unstable. This conclusion
                   ω0             ωz                                    does not hold if γ αβ = hM = 0 , and/or g αβ = hM = 0
                                                                        for in such case an eigenvalue is null and linearization
are solution of                                                         criterion fails.

                         µ + ω1                 m 0γ αβ     
 ( µ + ω 0 )( µ + ω z )                                       =0
                         g αβ              µ − ( ξ u − f )z 
                                                                       APPENDIX D
                                                                          In the following all the discussion is referred to the
we obtain    µ1 = −ω 0 , µ 2 = −ω z        , and                        system (C-1) to (C-4) and all our argumentation holds in
                                                                        a vicinity of its null equilibrium point. We notice that in
                                                                        equation (C-4) the term corresponding to the rate of viral
      µ 2 + µ [ ω 1 − z ( ξ uαβ − fαβ )] − g αβ γ αβ m0                 destruction mediated by T-helper cells is − f αβ zv αβ , but
      −ω 1 ( ξ uαβ − fαβ )z = 0                                         the term   ξu αβ zvαβ     is associated to production of virions
                                                                        at the same cells, so the net rate of virions variation
  When applying the Routh-Hurwitz criterion to (C-8) we
obtain that this point is stable if:                                                                              (
                                                                        mediated by T-CD4+ cells is f αβ − ξu αβ v αβ z . If this   )
                                                                        term is positive, there occurs viral destruction and the
                                            g αβ γ αβ                   opposite if it is negative. Now we define the
                 z ( f αβ − γ αβ ) > m 0                        (C-9)   instantaneous fraction of destroyed virions as
                                                                               dvαβ  = f − ξu
                                                                                      ( αβ     αβ ) z
                                                                         1                             ,                      (observe           that
  Or, if we use the equilibrium values given by (C-6),                  vαβ     dt  D

                 λz                 λ g γ                               although   vαβ      and d v α β approach separately to zero in a
                    ( f αβ − ξu ) > 0 αβ αβ                   (C-10)                              dt
                 ωz                ω0 ω1                                neighborhood of the equilibrium point, their ratio is a well
                                                                        defined   quantity).  The      equilibrium     value     for
  The general equilibrium point is given by:
                                                                        T-CD4+cells          is      λ 
                                                                                                  z = z ,             therefore       we     define

                 λ0         λ0 γαβ g αβ                                                               ωz 
        m0 =        ,z =                  ,                                      λz
                 ω0      ω0 ω1 ( f − ξu )                               hM ≡           ( f αβ − ξu αβ ) ,         and then we arrive to
                 ωz    λz ω0 ω1 ( f − ξu )            λ0 γαβ
        v αβ =                            − 1 ,m αβ =        vαβ               1      dv αβ  ,        this        is,   hM represents         the
                 u     ωz λ0 g αβ γαβ                  ω1             hM    ≈               
                                                                              v αβ    dt  D
                                                              (C-11)    fractional rate of virus population destroyed in
                                                                        equilibrium, by the immune system, by processes
(there is infection, and only positive equilibrium points
having biological meaning, we should restrain the                       mediated by T-helper cells. We call                         hM       “humoral
parameters values so that points obeying (C-11) fulfill this            responsiveness”. Now, the term corresponding to
requirement). The correspondent eigenvalue equation is:                 infected reservoirs creation in equation (C-2) is
Model for HIV-1 infection                                                                                                              261

γ αβ m0 vαβ       , and we define the rate of creation of                    hand side of (11) represents the fractional rate of virions
                                                                             produced in equilibrium by an infected reservoir in its
infected         reservoirs         as        dm α β                .      meantime, and we call “viral growth” to this factor. Then
                                                      = γαβ m 0 vαβ        this approximate system possesses a null and stable
                                               dt  C                       solution (no virus) if and only if the fraction of virions
Similarly, the term corresponding to virions creation                        destroyed by processes mediated by T-helper cells is
processes in equation (C-4) is given by                     g αβ m αβ        greater than the fraction of virions produced by infected
                                                                             reservoirs in their meantime, this is if the humoral
and     we       define    the      rate    of    virions    creation   as   responsiveness is higher than the viral growth.
 d vαβ                                 . Finally,    1
                                                            ≈ τ 1 , is the
        = g              αβ   m   αβ                ω1                     AKNOWLEDGEMENTS
  dt C
meantime of an infected cell. Then we obtain that:
                                                                               The authors wish to thank to Carla Ortega DiGregorio
                                                                             (Brunel University, Uxbridge, London) for grammatical
          λ0 γ α β g α β                    m0           m                   and editing revision of this paper.
                           = g αβ γαβ          ≈ g αβ γαβ 0
             ω 0 ω1                         ω1           ω1

                  1  d vαβ  1  d m αβ       
             =                         × τ1  ≈
                 m αφ  d t  C vαβ  d t       ,
              1 d vαβ              1 d m αβ      
                                            × τ1 
              vαβ d t              m αβ d t      
                             C                   C
where the factors          mαβ     and     vαβ   have been rearranged
and included in the parenthesis. We see that the right


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