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Modelling Vaccine Effects on HIV-1 Viral Dynamics
JOHN GUARDIOLA and ANTONIA VECCHIO*
Istituto di Genetica e Biofisica ‘A. Buzzati Traverso’
and
Istituto per le Applicazioni del Calcolo ‘Mauro Picone’*
Consiglio Nazionale delle Ricerche
via P. Castellino, 111, I-80131 Naples
ITALY
http://www.iigb.na.cnr.it
http://www.iam.na.cnr.it
Abstract: - Mathematical models of viral dynamics are useful for describing viral progression upon
infection and viral decline after drug treatment. HIV-1 models including intracellular delays and
pharmacological delays appear to be a more accurate representation of the biology of the virus and to
provide better estimates of the values of the kinetic parameters of infection. In the present paper, we
extend integro-differential equation models, previously developed for simulating the effect of antiretroviral inhibitors on virus progression, to the prediction of the impact of a hypothetical vaccine on
HIV-1 propagation. The vaccine simulated here, either as preventive or therapeutic, is envisaged as
stimulating an anti-HIV specific cytolytic T cell response of variable strength, able to kill chronically
or latently infected cells. An “immunological” delay, accounting for the time required for the immune
system to respond to the challenge, is also incorporated into the model.
Key-words:- Mathematical models, integro-differential equations, HIV-1 vaccine, virus progression,
time delays, CTL, eradication
1
INTRODUCTION
Emergent diseases caused by viruses constitute a
relevant treat to human health due to the presently
limited panoply of drugs available and to the lack
of tested efficacious vaccines.
Mathematical models of viral dynamics
have proved useful in describing viral progression
upon infection and viral decline after drug
treatment. In the case of the human
immunodeficiency virus-1 (HIV-1), such models
were constructed to provide the means to predict
important kinetic constants of virus replication and
decay in vivo, in the presence and in the absence of
anti-retroviral inhibitors[8,2]. Similarly, kinetic
parameters of hepatitis C virus (HCV) [6] or
hepatitis B virus (HBV) [3] infections obtained by
the solution of various models were found to be in
good agreement with those estimated by direct
determination of plasma virus levels following
perturbation of the chronically infected (quasi-)
steady state.
Treatment of these viruses with anti-viral
drugs or combinations of them in highly active
anti-retroviral therapy (HAART) causes rapid
reduction in plasma virus load, viral decline
occuring in several phases determined by the virus
properties and by the pharmacodynamical
characteristics of the drug used [3]. Current models,
incorporating intracellular time-delays for virus
replication
in
the
infected
cells
and
“pharmacological” delays related to drug efficacy,
satisfactorily explain some of these phases [2,3],
even though they may require ad hoc assumptions
to facilitate a mathematical analysis and to enable
to derive simple solutions [3]. Both intracellular
delays and pharmacological delays were found
useful to refine the predictions of a model and to
improve the determination of some kinetic
parameters of the infection.
Time-delays are introduced into these
models to account for the intracellular phase of the
virus life-cycle, whereby it is assumed that virus
production may lag by a delay behind the
infection of the cell. This implies that recruitment
of virus-producing cells at time t is not given by the
density of newly infected cells, but by the density
of cells that were newly infected at time t-. Such a
1
delay can be considered fixed [2,3] or distributed
according to a probability density function [5].
Analogously, pharmacological delays have
been incorporated into models describing the effect
of drugs onto the virus replication to reflect the lag
in the pharmacological effect due to the time
required for drug absorption, distribution and
penetration into the target organ or cell type. The
assumption taken for sake of simplification is that
drug effectiveness undergoes a step-wise increase
from 0 to 100% after some fixed pharmacological
delay  p which may be characteristic of different
drugs. Model formulations may depend on the
mechanism of action of the drug considered. For
example, models in which substances inhibiting the
production of virus by the infected cells, such as
reverse transcriptase inhibitors [2,3] differ from
models in which the action of the drug results in
the production of non-infectious virus particles
(see, for example, ref. 7). However, basic models of
viral dynamics generally contain three unknown
functions: the population of cells susceptible to
infection, or target cells, S(t), the population of
infected cells that produce virus, I(t) and the virus
present in the plasma, V(t). A constant influx  and
a constant relative death rate  is generally
assumed for target S(t) cells, although this may be
an oversimplification. Furthermore, to facilitate the
mathematical analysis and to enable to derive
simple analytical solutions when modelling the
effect of inhibitory drugs, the population of
uninfected cells, infected virus-producing cells and
free virus are considered at a steady-state level [3].
HIV
constitutes
a challenge
for
mathematical modelling due to the complexity of
the viral life cycle, in particular considering the
interaction of the virus with the immune system [4],
the emergence of drug resistant mutants [8], the
importance of virus compartmentalization in
different organs [6] or the occurrence of different
phases of virus progression on significantly
different time-scales [3]. Most models, however,
albeit useful for parameter estimation and for
supporting patient monitoring, may not accurately
reflect such a complexity. In the present paper,
adopting available models as the starting point, we
explore the effect of a hypothetical anti-HIV
vaccine. Anti-HIV vaccines can be basically
envisaged in two forms: a vaccine capable of
inducing neutralizing anti-HIV antibodies, acting
to reduce virus load and a vaccine inducing antiHIV specific cytolytic T cells capable of killing
HIV-infected cells, controlling the number of HIVproducing cells (see ref. 1 and ref.s therein).
Studies of the immune response in clinically
asymptomatic seropositive patients supports in fact
the contention that humoral and cellular immunity
may be useful for controlling HIV infection. An
ideal anti-HIV vaccine should thus contain both
activities. Presently, we modify existing models
featuring an intracellular delay by incorporating the
effect of killer vaccines based on cytolytic T cells
and having different efficacies. A “vaccine” or
“immunological” delay determined by the time
required for the vaccine to elicit a protective
response from the immune system is also taken into
consideration.
2 Problem Formulation
Several mathematical models have been proposed
to describe viral dynamics of HIV-1 infection
[2,3,5,7]. All models derive from a simplified
phenomenological model of virus infection such as
that reported in Fig. 1. Susceptible cells (S) are
provided at a constant rate  and removed by
clearance at a constant relative rate . Infected cells
(I) are derived upon infection of plasma viruses (V)
of uninfected S cells at a rate of infection k and are
in turn removed by clearance at a rate . Infected
cells produce viral particles at a rate p, thus filling
the viral plasma compartment from which viruses
are removed at a clearance rate c. The effect of
inhibitors used in highly active anti-retroviral
therapy is exerted at different levels, depending on
the mode of action of the drug considered. For
example, reverse transcriptase (RT) inhibitors act at
the level of infection of S cells by the virus, thus
essentially reducing the value of k, whereas
protease inhibitors interfere with the ability of
viruses produced by I cells to cause de novo
infections, thus lowering the value of p. In
principle, for 100% effective drugs, k and p must
reach a 0 value.
Fig. 1 Phenomenological model of HIV-1 virus
infection
2
A classical mathematical formulation of the
basic model, using differential equations, is that
reported by Herz et al. [3]:
dS / dt     S (t )  kS (t )V (t ),
dI / dt  k (t ) S (t )V (t )   I (t ),
dV / dt  p (t ) I (t )  cV (t ),
where k and p are explicitly considered (in their
case, but not in other models) variable with time t
to account for the effect of drugs added at later
times. In the variant used by Perelson et al. [7], the
model incorporates the fact that in the presence of
protease inhibitors plasma viruses belong to two
separate pools, the infectious virus particles, VI, and
the non-infectious virus particles VNI (Fig. 2), the
variations of which are described by two distinct
equations (see also below). S is instead considered
constant. A parameter (where  = 1 for a drug
100% effective) is introduced to account for the
action of the protease inhibitor.
proposed by Herz et al. [3]. In the former case, after
primary infection the I cell is considered to
progress through various stages until it reaches a
stage of maturation in which it produces virus. The
transition rate coefficient from one stage to the
next, where cell death and virus release occur at the
last stage, is considered constant and the delay is
the time required to complete this multistep
process. In the other case, it is taken that virusproducing cells at time t are those infected at time t
– , where  is the delay. Furthermore, a constant
~
death rate  different from the  rate of virus
producing cells, is assumed for cells infected, but
~
not yet producing virus and a probability e   of
surviving from time t –  to time t is incorporated
into the equation [3]. Mittler et al. [5] have
modified the model proposed by Perelson et al. [7]
to estimate the role of a variable intracellular delay.
Such a distributed delay is accounted for by a
density function f (x ) , assuming that viruses
which infect cells at time t were produced by cells
infected t-x time units before. In this model a
pharmacological delay is also introduced using the
Heavyside function h(t), where h(t) = 0 when t < p
and h(t) = 1 when t ≥ p and p is the
pharmacological delay. The equations of this model
(derived from those of Mittler et al., 1998 using the
notations of Fig. 1 and 2) are the following:

dI / dt   kS ( x)V I (t  x)e  mx dx  I (t ),
0
dV I / dt  (1  h(t ))(1   ) pI (t )  cVI (t ),
Fig. 2 Phenomenological model of the effect of
protease inhibitors on HIV-1 virus
infection
2.1
Intracellular delays
Models of HIV-1 infection based on intracellular
delays are more accurately descriptive of the
phenomenological aspects of virus life cycle and
predictive of the values of kinetic parameters than
models without delays. The intracellular delay
refers to the lag which observed between the time
at which the virus infects the cell and the time at
which the infected cell becomes productive. A
fixed delay is considered both in the model
proposed by Grossman et al. [2] and in that
(1)
dV NI / dt  h(t )pI (t )  cV NI (t ).
The exponential factor e-mx in the first equation of
problem (1) accounts for the loss of infected cells
between the time of the initial infection and the
release of the first virions [5]. The solution is then
obtained by converting the
integro-differential
equations (1) into an equivalent set of ordinary
differential equations [5] and
by solving it
numerically.
2.2 Modelling vaccine effects on viral
progression
Anti-retroviral RT inhibitors affect de novo
infection of target cells by plasma viruses, but do
not block production of virus particles by
previouslly infected cells. Nonetheless, HAART
3
yields an esponential fall of virus titers in treated
patients followed by a slower decline in residual
viremia. The biphasic behaviour of this kinetics is
related to the finding that most infected cells are
short lived, a fact, which in combination with the
inhibition of de novo infection due to the drug,
explains the initial very rapid reduction of viral
burden. The subsequent lower rate of decline
observed in Phase II of the treatment and after
HAART corresponds to longer half-lives of cells
chronically and latently infected before the start of
the therapy and reflects the ability of HIV-1 to
attack multiple cellular compartments during its life
cycle [2]. Thus, although HAART is very efficient
in reducing the level of viremia during Phase I of
the treatment, viruses cannot be eradicated, leading
to the probability of rebound and leaving room for
the accumulation of resistant mutants. As
mentioned, models in which protease inhibitors are
considered imply production of VI and VNI particles
in various ratios, depending on the effectiveness of
the drug used. De novo infection is not inhibited
and in the expected decline is strongly dependent
on the estimate of virus turnover rates.
Furthermore, as compared to inhibitors having a
different mode of action, the initial decline of
plasma virus is slower because infectious viruses
that were produced before the start of the therapy
continue to infect cells.
Eradication of virus infection by HAART
is thus hardly expected, either in real-life situation
or in ideal model conditions, thus justifying the
need for an efficacious HIV-1 vaccine able to
remove all kinds of virus producing cells. Such a
vaccine should in principle induce a cytolytic T cell
(CTL) response targeted against cells infected by
HIV-1 and expressing virus-specific antigens. The
feasibility of this approach is supported by the
evidence that anti-HIV CTL are recruited in the
course of HIV-1 infection and that, in animal
systems, depletion of CD8+ T cells causes a
dramatic rise in plasma viremia (ref. 10 and ref.s
therein). We have developed and analyzed a set of
models, incorporating distributed intracellular
delays, immunological delays and distributed
survival probability, aimed at predicting the
possibility of virus eradication by hypothetical
preventive, as well as therapeutic anti-HIV-1
vaccines for which different levels of efficacy are
supposed. In the present paper, we discuss a
preliminary model obtained by the adaptation of
the model proposed by Mittler et al. [5] and thus
based on a system of Volterra integro-differential
equations. Elsewhere, we shall present more
refined models based on delay integro-differential
equations approaches and on visual modelling.
3
Problem Solution
Our mathematical model describing the effect of a
hypothetical vaccine on HIV-1 progression is based
on the phenomenological model of infection
summarized in Fig. 1, except that no drug
inhibition is considered, but killing by cytolytic T
cells of I cells is assumed. The mathematical
models also refers to the in vivo kinetic pattern of
progression of HIV-1 infection schematized in Fig.
3.
Fig. 3 Schematic time-course of HIV-1 infection
in long-term progressor patients
As it now well known, the initial phase of
HIV-1 infection is characterized by a very rapid
initial increase of plasma virus titer, followed by a
clinically asymptomatic lag period which may last
several years before another burst of viremia sets in
because of a change in the properties and
replication rate of the virus. The model, therefore,
aims at reproducing the long term kinetics of
infection and the resulting multiphasic behaviour.
Furthermore, the hypothetical vaccine that we
simulate can be either a preventive vaccine, the
inoculation of which took place before infection or
a therapeutic vaccine, inoculated immediately after
the infection, during the lag period or just at the
onset of the final phase of the infection. The
following integro-differential equations are thus
proposed:
4
t
dI / dt  kS (t )  f (t  x)V ( x)dx  (   (t )) I (t ),
0
dV / dt  p (t ) I (t )  cV (t ),
( 2)
t
dS / dt    S (t )  kS (t )  f (t  x)V ( x)dx,
0
t  0, tf , I (0)  I 0 , V (0)  V0 , S (0)  S 0 .
Where

x n 1
b
f ( x) 
e
,
(n  1)!b n
x
(3)
 p t r
p(t )   1
,
 p2 t   r
p1  p 2 ,
  0 t   i
 (t )   1
,
 2 t   i
0   2  1.
The parameters n and b describe the shape and the
scale of the density function f, respectively;  r  0
is the viremia resurgence delay and  i  0 is the
immunological delay, i.e. the time required for the
immunological system to develop a reaction in
response to vaccination. Note that in this model the
delay, in analogy with Mittler et al. [5], is described
by the probability density function f(x) assuming
that viruses infecting susceptible cells were
produced by cells infected x time units before. The
function p(t) representing the rate of infection is
defined as a step function with a discontinuity
point in t   r where  r is the time at which the
asymptomatic lag ends and a drastic resurgence of
viremia is observed. The function  (t ) ,
representing vaccine efficacy, is also a step
function which assumes null value in absence of
vaccine ( t   i ), i.e. before the time required for
the immunological sysytem to develop a reaction in
response to vaccination. For a fully effective
vaccine we have  2  1 . When  i  0 , the
vaccine can be considered a preventive vaccine.
Assuming different values of  i  0 allows instead
to describe a therapeutic vaccine gaining efficacy at
different time-points during the process. It must be
kept in mind, however, that this constitutes an
oversimplification as priming of cytolytic CD8(+)
T cells by a vaccine depends on the help provided
by CD4(+) T cells, which, in this case, are also the
target of HIV-1: the ability of the immune system
to respond to the antigenic challenge may be thus
impaired in parallel with the decrease of CD4(+)
cell counts typically observed in HIV-1 infected
patients. This models also differs from the model of
Mittler et al. [5] for the time scale considered
(which includes all phases of a long term
progression process) and because S is variable.
Of course, the analytical (true) solution of
(2) cannot be found and a numerical (approximate)
solution must be computed in order to know the
behaviour of the unknown functions I(t),S(t),V(t).
Before proceeding to this, we note that (as shown
in [5]), thanks to the form of f(x), given in (3), the
system of three integro-differential equations (2)
can be transformed into an equivalent system of
n+3 ordinary differential equations, n appearing in
(3). What is the advantage of this transformation?
Even if a variety of methods there exists for
computing the numerical solution of a system of
integro-differential equations (see for example [9]
and the refs therein), user friendly numerical codes
for solving ordinary differential equations are much
more developed than those for solving integrodifferential equations. Therefore, if n is not too
large, it may be convenient, instead of solving a
system of 3 integro-differential equations, to solve
an higher dimension system of ordinary differential
equations. The transformed problem is the
following system of nonlinear first order ordinary
differential equations
1
dE1 / dt  V (t )  E1 (t ),
b
1
dE j / dt  E j 1 (t )  E j (t ) , j  2,..., n
b
dI / dt  kS (t ) E n (t )  ( (t )   ) I (t ),
(4)


dV / dt  p (t ) I (t )  cV (t ),
dS / dt    S (t )  kS (t ) E n (t ),
E j (0)  0, j  1,..., n,
I (0)  I 0 , V (0)  V0 , S (0)  S 0 ,
where E j (t ) are auxiliary functions with no
biological
meaning
introduced
by
the
transformation [5] and the meaning of the other
symbols are already specified in (2). Problem (4)
now appears as a classical Cauchy problem of the
type
5
t  0, tf ,
dY / dt  F (t , Y (t )),
Y (0)  Y0 ,
kinetic behaviour predicted (Fig. 4) does indeed
parallel the real-life dynamics of infection shown in
Fig. 3.
(5)
where Y and F are the following vectors belonging
to  n 3
 E1 
 1 / b(V  E1 ) 
E 
 1 / b( E  E ) 
1
2
 2


  






 (7 )
1
/
b
(
E

E
)
E
n

1
n
Y n
(6), F 
 


I 
kSEn  (   ) I 
V 
 pI  cV

 


  S  kSEn 
 S 
Due to the discontinuity of p(t) and  (t ) , the
second member (7) of the Cauchy problem (5) is
not continuous w.r.t. the independent variable t and
thus the existence and the uniquiness of the
solution of (5) is not assured. In order to overcome
this drawback, we split system (5) into three
different problems whose second member is
continuous w.r.t t. For sake of simplicity let us
assume  i   r , then (5) is equivalent to
dYA / dt  FA (t , YA (t )),
t  0, i ,
YA (0)  Y0 ,
(8)
dYB / dt  FB (t , YB (t )),
t   i , r  ,
YB ( i )  YA ( i )
(9)
dYC / dt  FC (t , YC (t )),
t   r , tf  ,
YC ( r )  YB ( r )
Here, YA , YB , YC
and FA , FB , FC
p(t )  p1
 (t)  1
Furthermore, assuming (t) = 1, i.e. the
hypothesis of maximal effectiveness of the
treatment, we found that a preventive vaccine
(delivered at time i = 0) may be in fact able to
block the infection (not shown) and a therapeutic
vaccine (delivered after infection with a time lag
such that 0 < i ≤ r) may control the viral surge of
the last phase of infection normally leading to overt
disease and death (Fig. 4). The latter conclusion
holds true only if the ability of a patients to mount
a full immune response is maintained during the
asymptomatic lag period. In addition, vaccines
exhibiting less that 100% efficacy may fail to
eradicate the infection.
(10)
have the same meaning of (6)
have the same form of (7) with
the following values of p(t) and  (t ) respectively
A)
Fig.4 Effect of a hypothetical vaccine on viral
dynamics using model (2). Curve a (solid line)
represents viral progression in the absence of
vaccine, while curve b (dotted line) represents the
effect of a 100% efficacious vaccine assuming i =
50 and r = 80 time units.
, B)
p(t )  p1
 (t)   2
C)
p(t )  p 2
 (t)   2
.
Each of problems (8-10)
is then solved
numerically by using a classical fourth order
Runge-Kutta methods.
Numerical solutions show that, using a wide range
of values for the various parameters considered, the
4.
Conclusion
The extent to which virus propagation from
infected cells to uninfected, susceptible cells is
halted by anti-retroviral therapies based on the use
of pharmacological drugs or may be arrested by a,
presently unavailable, vaccine is obviously of great
importance, both from the clinical point of view
and for the understanding of the details of HIV-1
natural history. Mathematical models, relying on a
vast body of experimental observations, have been
developed which permit the description of the
events occurring during the asymptomatic phase of
6
the viral infection and of the short-term kynetic
perturbations of the resulting quasi-steady state
caused by the use of anti-retroviral drugs.
The possibility that HAART may not
achieve eradication of viral infection [2] makes it
mandatory to continue ongoing efforts to develop a
vaccine. Mathematical models simulating the effect
of vaccination on HIV infected patients and
considering a longer time scale of the process
suffer from the lack of experimental data available.
Our current understanding of the viral life cycle
and immunobiology, in particular, and knowledge
of the immune system, in general, may nonetheless
allow to estimate their predicting value to some
extent.
We discuss here a model which brings into
consideration features of the biology of the system
such as the delay which is observed between the
time a cell is infected by the virus and the time it
starts to produce infective particles (intracellular
delay) and the time necessary to the immune
system to mount an immunological reaction
(immunological delay). The model reproduces the
kinetics of viral production observed in vivo and
allows to simulate preventive as well as therapeutic
vaccination schemes. However, other important
aspects of HIV-1 biology need to be kept in mind
in order to construct a more realistic representation.
In fact, as mentioned earlier, the fact that the virus
is able to exploit different cell compartments as
well as the fact the CD4(+) T cells are an important
intersection between viral life cycle and the
mechanisms of immune response need to be
accounted for. In addition, the key role of genetic
variability, whereby escape mutants (that is virus
carrying altered antigenic properties and thus
escaping immune surveillance) may be selected for,
must be certainly taken into consideration.
This work was supported by grants from
the AIDS Programme, Ministry of Health-ISS,
from MURST Biotechnology Programme L. 95/95
and EU Quality of Life – Health Research
Programme contract QLRT- 2001-00620.
Immunology Today, 1998, Vol. 19, pp. 528-532.
[3] Herz, A.V.M., Bonhoeffer, S., Anderson, R.M.,
May, R.M., Nowak, M.A., Viral dynamics in vivo:
limitations on the estimate of intracellular delay
and virus decay, Proc. Natl. Acad. Sci. USA, Vol.
93, 1996, pp. 7247-7251.
[4] Korthals Altes, H., Wodarz, D., Jansen, V.A.A.,
The dual role of CD4 T helper cells in the infection
dynamics of HIV and their importance for
vaccination, J. Theor. Biol., 2002, Vol. 214, pp.
633-646.
[5] Mittler, J.E., Sulzer, B., Neumann, A.U.,
Perelson, A.S., Influence of delayed viral
production on viral dynamycs in HIV-1 infected
patients, Mathematical Biosciences, 1998, Vol.
152, pp. 143-163.
[6] Muller, V., Marée, A.F.M., De Boer, R.J.,
Release of virus from lymphoid tissue affects
human immunodeficiency virus type 1 and hepatitis
C virus kinetics in the blood, Journal of Virology,
2001, Vol. 75, pp. 2597-2603.
[7] Perelson, A.S., Neumann, A.U., Markowitz, M.,
Leonard, J.M., Ho, D.D., HIV-1 dynamics in vivo:
virion clearance rate, infected cell life-span, and
viral generation time, Science, 1996, Vol. 271, pp.
1582-1586.
[8] Stilianakis, N. I., Boucher, C.A.B., De Jong,
M.D., Van Leeuwen, Schuurman, R., De Boer,
R.J., Clinical data sets of human immunodeficiency
virus type 1 reverse transcriptase-resistant mutants
explained by a mathematical model, Journal of
Virology, 1997, Vol. 71, pp. 161-168.
[9]Vecchio,
A.,
Stability
of
backward
differentiation formula for Volterra integrodifferential equations, J. Comp. Appl. Math., 2000,
Vol.115, pp.565-576.
[10] X, J., Bauer, D.E., Tuttleton, S.E., Lewin, S.,
Gettie, A., Blanchard, J., Irwin, C.E., Safrit, J.T.,
Mittler, J., Weinberger, L., Kostrikis, L.G., Zhang,
L., Perelson, A.S., Ho, D.D., Dramatic rise in
plasma viremia after CD8(+) T cell depletion in
simian immunodeficiency virus-infected macaques,
J. Exp. Med., 1999, Vol. 189, pp. 991-998.
References:
[1] DeBerardinis, P., Sartorius, R., Fanutti, C.,
Perham, R.N., Del Pozzo, G., Guardiola, J., Phage
display of peptide epitopes from HIV-1 elicits
strong cytolytic responses, Nature Biotechnol.,
2000, Vol. 18, pp. 873-876.
[2] Grossman, Z., Feinberg, M., Kuznetsov, V.,
Dimitrov, D., Paul, W., HIV infection: how
effective is drug combination treatment?
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