Download On the intra-host dynamics of HIV

Mathematical Biosciences 199 (2006) 1–25
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On the intra-host dynamics of HIV-1 infections
Nikolaos I. Stilianakis
a
a,b,*
, Dieter Schenzle
c
Department of Biometry and Epidemiology, Friedrich-Alexander-University of Erlangen-Nuremberg,
Waldstr. 6, 91054 Erlangen, Germany
b
Joint Research Centre, European Commission, T.P. 441, Via E. Fermi 1, 21020 Ispra (Va), Italy
c
Department of Medical Biometry, Eberhard-Karls-University of Tübingen, Westbahnhofstr. 55,
72070 Tübingen, Germany
Received 21 July 2004; received in revised form 23 May 2005; accepted 21 September 2005
Available online 15 December 2005
Abstract
An extension of a previously proposed theory for the pathogenesis of AIDS is presented and analyzed
using a mathematical modelling approach. This theory is based on the observation that human immunodeficiency virus type 1 (HIV-1) predominantly infects and replicates in CD4+-T cells, and that the infection
process within an infected individual is characterized by ongoing generation and selection of HIV variants
with increasing reproductive capacity. This evolutionary process is considered to be the reason for the gradual loss of immunocompetence and the final destruction of the immune system observed in most patients.
The extension presented here incorporates the effect of the permanently increasing susceptibility of CD4+-T
cell clones, as a result of the evolutionary process. The presented model reproduces and possibly explains a
wide variety of findings about the HIV infection process. Numerical results indicate that the effect of the
initial dose is minimal, and restricted to the primary phase of infection. According to the model predictions
the impact of the HIV evolutionary speed is crucial for the progression to disease. An important progression determinant is the initial infection rate, being a component of the viral reproductive capacity. An influential role in disease progression seems to be played by the initial CD4+-T cell count.
2005 Elsevier Inc. All rights reserved.
Keywords: HIV infection; Mathematical model; Disease progression; T-cell dynamics
*
Corresponding author. Address: Joint Research Centre, European Commission, T.P. 441, Via E. Fermi 1, 21020
Ispra (Va), Italy. Tel.: +39 0332 786427; fax: +39 0332 785236.
E-mail address: [email protected] (N.I. Stilianakis).
0025-5564/$ - see front matter 2005 Elsevier Inc. All rights reserved.
doi:10.1016/j.mbs.2005.09.003
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N.I. Stilianakis, D. Schenzle / Mathematical Biosciences 199 (2006) 1–25
1. Introduction
1.1. HIV infection and disease progression
Clinically, human immunodeficiency virus-1 (HIV) infection can be divided into three phases
[1,2]. During the initial phase, known as primary infection, the virus present in the host replicates
by infection of cells which express the CD4 receptor as well as the proper co-receptor, usually
CCR5 [3]. Yet, during this early stage of primary infection the CD4+-T lymphocytes represent
the largest group of target cells for HIV [4]. Within the first 3–6 weeks after infection about
50–75% of the patients develop an acute viral syndrom [5]. Symptomatic primary infection is characterized by a viremia with high numbers of infectious virus and infected cells in the peripheral
circulation. At the same time a significant reduction of the CD4+-T cells of about 30–40% is observed. Between one week and three months later an HIV specific immune response is mounted
and it timely coincides with a decrease of viremia and a recovery of the CD4+-T cells [6]. Virus
specific cytotoxic T lymphocytes (CTL) appear early and represent a critical immune component
for the control of primary infection [7,8]. The combined effects of CTL and other components of
the immune response as well as the limited number of target cells lead to a decrease of the viral
population in blood to a low level. However, it seems impossible for the immune system to generate permanent immunity.
The second phase of HIV infection is characterized by a long asymptomatic period between primary infection and the development of clinical immunodeficiency (acquired immunodeficiency syndrome, AIDS). There are two striking features of the asymptomatic phase: the permanent viral
replication in peripheral lymphatic tissue and the gradual loss of the CD4+-T cells. Although the
asymptomatic phase may represent a period of clinical latency there is permanent virus replication
during that time. Free virus can be detected in the circulation [9], but at a lower level compared to
the levels during primary infection or in patients who are in the third phase (AIDS) of the infection.
Infected cells are easily detectable in peripheral lymphoid tissue and their number directly correlates
with the virus level in plasma. This is an indication that lymphoid tissue represents a relevant source
for plasma virus [10,11]. The rate of decrease of CD4+-T cells seems to be determined by the level of
the permanent viral replication since patients with higher levels of plasma virus show a faster disease
progression [12]. The mechanism for the decrease of CD4+-T cells is not well understood [4].
The third and final phases of HIV infection is characterized by the development of clinical
immunodeficiency. Between one and two years before the emergence of AIDS there is often a faster decline of CD4+-T cells. Before this decline an increase of viral load is observed [13], whereas
viral replication is present not only in lymphoid tissue but also in many other organ systems
[14,15]. When the CD4+-T cell number drops below 200 mm3 opportunistic infections occur
and CD4+-T cells depletion continues leading to severe immunodeficiency and disease.
The period between primary infection and disease progression is highly variable between individuals and has a wide and skewed distribution function. The median estimate is 8–11 years, without treatment, for Europe and North America with similar numbers for Africa [16–19]. Some
relatively rare cases of HIV infections lead after only one year to AIDS, whereas a non-negligible
part remains asymptomatic even 15 years after primary infection [20]. Some estimates indicate
that a fraction of 8–15% of infected persons will be AIDS-free for more than 20 years without
any therapeutical intervention [21].
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During the last decade several theories have been presented in attempt to explain the pathogenic mechanisms responsible for the depletion of CD4+-T cells within an HIV infected individual. To determine those mechanisms is an essential step in understanding AIDS pathogenesis [22].
Many of these theories were accompanied with mathematical modelling. Deterministic as well as
stochastic approaches were used. Deterministic models focussed on the description and the potential explanation of the CD4+-T cell depletion and the corresponding dynamics of the virus. Many
of them were developed to specifically treat the issue of therapy regimes and to describe the associated dynamics. Stochastic models were used to explore the early stages of the infection. A concise review of these models with numerous references can be found in [23,24]. Almost all models
were developed to treat short term dynamics of a specific phase of the infection course or an intervention, making a significant contribution in uncovering interesting aspects of the HIV infection
dynamics. Little effort was made, however, to develop models that would describe the long term
dynamics including all three phases of HIV infection with and without therapeutic intervention.
In this work an effort to provide a general model for the long terms dynamics of HIV infection
and the progression to AIDS is made. The extended theory addresses mechanisms that govern disease progression of HIV infection.
1.2. A theory for HIV disease progression
HIV is a member of the family of lentiviruses. Therefore, viral replication is afflicted with many
errors. Three main aspects of the genetic variability of HIV can influence the pathogenesis: (i) the
development of tropism towards a more efficient replication in different cell types; (ii) antigenicity,
that leads to the development of escape mutants; and (iii) virulence. HIV variants with different
virulence can affect the pace and the severity of the disease. Antigenicity has received the bulk of
attention, with limited consideration of aspects like tropism and virulence sufficiently [25]. There
are clear indications that the evolution of more pathogenic variants has an impact on disease progression [26–30]. The extended theory focuses on the virulence and cell tropism aspects of HIV
infection.
HIV predominantly replicates in the CD4+-T lymphocytes HIV. A determinant of the reproduction rate is the rate at which HIV binds on these cells and enters the cells. For simplicity in
our model, this is termed as infection rate. The HIV population within an infected person is inhomogeneous. There is an ongoing emergence of new variants with different infection rates. Since
HIV particles compete for reproduction in the same cell populations obviously selection takes
place. This leads to a directed evolution of HIV towards higher cell infection rates. This evolutionary process is host-specific, if the interaction between HIV and CD4+-T cell involves surface structures that are differ between individuals. Such structures are present, for example, on the T-cell
receptor for MHC-II-restricted recognition. This receptor is integrated with the CD4 molecule,
the HIV receptor. These postulates combined with the assumption of an activated anti-HIV response at its maximal level, led to a simplified mathematical model that already reproduced some
of the main findings about the individual HIV infection courses [31,32]. The development of sensitive methods for measuring virus load in blood and in lymphoid tissue has led to the identification of valuable new data. These data support the predictions of the preliminary model. Based on
initial findings a more realistic modelling approach is presented here by incorporating additional
important features of the HIV immunopathology. With the extended model a more detailed
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analysis of the clinical and epidemiological data is possible. In this way the predictive power of the
model is improved.
This is shown with numerical results from the extended model with current best estimates of
crucial parameters. The qualitative features, and the predictive behavior of the extended model
is demonstrated. Implications from numerical solutions of the model are discussed.
2. Basic mechanisms
2.1. HIV variability and competitive selection
Most of the fundamental ideas of the proposed theory has been presented in detail in our previous work [31,32]. We epigrammatically review them here.
A striking feature of HIV infection is the high within-host genetic variability of HIV observed.
Hence, in many infected cells deleterious mutations should occur, most mutations are perhaps
neutral, but with some 1011 newly produced HIV particles per day in an infected individual,
HIV mutants with one or more of the following properties can be expected: an increased cell infection rate, an increased within-cell replication rate, a decreased cell removal rate or a decreased
virus removal rate. The population dynamics of virus-cell interaction is marked by competitive
exclusion between HIV mutants which compete for replication in the same susceptible cells, predominantly CD4+-T cells. If a virus particle enters a cell population with a fixed initial number
of susceptible cells, then it will generate another average number R0 of infectious virus particles.
Given two HIV variants, namely 1 and 2, and a persistent state with respect to virus 1 with basic
reproduction number R01, introduction of virus 2 with a reproduction number R02 leads to two
possible states (besides the unlikely equilibrium state with coexistence of both virus types, if
R01 = R02). If R02 < 1 < R01, then virus 2 will not replicate, whereas in case of R02 > R01 > 1 virus
1 becomes gradually replaced by virus 2.
During the transient phase, when both types of virus are still present, one may speak of an average effective virus reproduction number R0 which increases steadily with time from R01 to R02.
That competitive selection is involved in the case of HIV infection and can be demonstrated during treatment where fast selection of drug resistant HIV strains has been observed [33–35].
2.2. Host-specific evolution of HIV
HIV variation and selection result in a directed evolution of HIV towards higher reproductive
capacity, but according to the central hypothesis underlying the present work, this evolution can
be considered as highly host-specific. This means that upon transmission of HIV between individuals the evolutionary clock will be set back every time. Otherwise one would expect at the level of
the human population a noticeable evolution of HIV towards higher virulence. But this does not
seem to be the case.
Therefore, if HIV can mutate to improve its CD4+-T cell infection rate, and if HIV mutants
compete for reproduction within the same cell population, then a directed evolution takes place.
In the context of our model an increase of the cell infection rate will cause an irreversible increase
in the HIV basic reproduction number R0.
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A series of clinical and experimental results are consistent with the assumption, that HIV
increases its reproductive capacity during the course of an individual HIV infection, and that this
is associated with host-specific CD4 cell surface structures involved in the binding and entry of
HIV into CD4-T cells [26–30,36,37]. Current studies indicate that host-specific genetic factors
are involved and affect transmission and disease progression [36,38,39].
2.3. Deterministic description of HIV evolution
Evolution through generation and selection of random mutants is a process characterized by at
least two inherently stochastic elements. One is the time span until a new mutation arises. The
other is the magnitude of the resulting effect, in the present context the increase of the CD4-T cell
infection rate. Since, as reported above, an enormous number of HIV mutants arise each day in an
infected individual. And as the increase in the cell infection rate by any one mutant may be small,
it is reasonable to model the evolution of HIV with a deterministic approach.
To describe the effect of the increase of the within-host reproductive capacity of HIV we assume
that the infection rate is the driving force for the permanent increase and selection of HIV-mutants. We define a dynamical variable K(t), that upon infection has an initial value that can be
used to describe a subsequent steady increase of the HIV cell infection rate. Let j0 denote the
average cell infection rate of HIV upon primary infection, then the term K(t)j0 denotes the average cell infection rate of the mixture of HIV mutants currently present in an infected person. The
speed at which the factor K increases depends of course on the mutation rate of HIV in an infected
individual, which in turn depends on the number of newly infected cells per day. With these
assumptions one is led to consider the following modified simple model. Let X, Y and V be the
population of susceptible, infected CD4+-T cells and virus, respectively, and K the variable for
the increase of the infection rate as defined above.
dX
ð1Þ
¼ K lX j0 KVX ;
dt
dY
ð2Þ
¼ j0 KVX dY ;
dt
dV
¼ bY cV ;
ð3Þ
dt
dK
ð4Þ
¼ xK V ðK max KÞ;
dt
where X(0) = X0, Y(0) = Y0, V(0) = V0, K(0) = K0 and j0KVX denotes the number of newly infected CD4+-T cells per day. Kmax is a theoretical maximum value that can be reached for the
HIV evolutionary process within a host. The speed of the increase in K is determined by the model
parameter xK. Other parameters of the model are: the immigration rate of new susceptible cells K,
the natural death rate of susceptible cell l, the death rate of infected cells d, the production rate of
infectious viral particles b, and the viral clearance rate c.
The decisive difference to the classic virus-host model, is that the virus reproduction number R0
is a dynamic quantity
R0 ðtÞ ¼
bj0 KðtÞX 0
;
dc
ð5Þ
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which increases monotonically towards the value
R0 ¼
bj0 K max X 0
.
dc
ð6Þ
If the average cell infection rate increases slowly compared to the turnover times of cells and virus,
then the number of susceptible cells declines monotonically towards the value
X0
X ¼ ð7Þ
R0
and the number of virus particles and infectious cells increase monotonically towards the values
V ¼
blð1 1=R0 Þ
X0
dc
ð8Þ
and
c
Y ¼ V;
b
ð9Þ
respectively. Within the present modelling approach this result provides the basis for explaining
the progressive CD4+-T cell decline during HIV infection.
3. Model assumptions
3.1. CD4+-T lymphocytes as the main target cell population
Although HIV may infect different types of cells, CD4+-T cells are the main target cell population [3,4]. These cells permanently migrate through the body but at any time about 98% of
them are present in lymphoid tissue [40,41]. This means that only about 2% of the CD4+-T cell
population is in blood. Consequently, most cell infections take place in lymphoid tissue. This
raises the question whether measurements of viral load and infected cells during HIV-infection
are representative for the HIV activity [4]. Replication of HIV in macrophages may play an
important role, but their impact has been unclear. Moreover, the nature of other target cell
groups and their role is unknown [4]. Therefore, we assume in the approach presented here that
the effects of other cell groups besides the CD4+-T cells are negligible with respect to the
dynamics of infection, although such subpopulations may play an important role as reservoirs
for HIV.
3.2. CD4+-T cell count and turnover
The pool of CD4+-T cells consist of many cell clones. Each clone is characterized by a specific
T-cell receptor for interaction with antigen presenting cells. The current size of a specific CD4+-T
cell clone in an individual presumably depends on the current degree of antigenic stimulation. The
average number of CD4+-T cells in blood is 1000/mm3 [42]. The average blood amount in a 70 kg
person is about 5 l. This implies that the number of CD4+-T cells in the blood is 5 · 109 (2% of the
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total number [40,41]). About 1% of the weight of a 70 kg adult person is lymphoid tissue, which
means that an adult bears about 700 g lymphoid tissue. When the total number of CD4+-T cells is
2.5 · 1011 (50 times the amount in blood) the number of cells in lymphoid tissue is about
2.45 · 1011 (98% of the total number). Thus, the number of CD4+-T cells is 3.5 · 108/g. The value
2.5 · 1011 represents the normal value of the CD4+-T cells in a healthy individual. Assuming that
these cells have an average life span of about 50 days, which correspond to an average death rate
of 0.02/day, to maintain a steady state of 2.5 · 1011 a production of 5 · 109 cells per day is
required. This value is in line with data gained from short-term measurements of CD4+-T cell
numbers in patients after receiving antiviral therapy [33,34].
Since the number of mature CD4+-T cells is maintained by cell division, the number of new
cells per day depends on the number of existing cells and on the division rate per cell. If the division rate of the CD4+-T cells were constant, then declining numbers of CD4+-T cells would result
in a decrease of the daily production of new CD4+-T cells. In this case it would be very difficult to
explain the observed very high virus production during the final stage of disease. Therefore, it
seems more plausible that the CD4+-T cell clones are activated to compensate for a loss of cells
by increasing the cell division rate [43]. Thus, we assume that the number of newly produced
CD4+-T cells per day remains constant. This implies a tremendous increase in the CD4+-T cells
division rate throughout the course of an HIV infection [44].
3.3. CD4+-T cells with different HIV susceptibility
We assume that different CD4+-T cell clones possess different susceptibility to HIV infection.
This implies that different coexisting HIV variants are adapted to infection of cells from different
CD4+-T cell clones. There are indications that after primary infection the initial spread of HIV in
the CD4+-T cell pool colapses before the immune response is activated [45]. We suspect this happens due to the limited number of available susceptible CD4+-T cells. The originally presented
model [31,32] considering a single pool of CD4+-T cells is plausible with respect to the speed
at which the immune response is activated after primary infection. A slightly delayed response
is predicted to result in heavy exhaustion of CD4+-T cells which does not seem to be a realistic
feature.
Here we model the immune response slightly differently from the original model and incorporate the effect of the permanent increase of HIV cell susceptibility. This leads to more realistic
description of the disease progression mechanism.
To model the spread of HIV in hundreds or even thousands of specific CD4+-T cell clones
would result in a highly complex modelling approach. To avoid this problem the following approach is adopted. The total pool of CD4+-T cells is divided into two subpopulations of clones:
at any time t after primary infection X(t) denotes the current number of non-susceptible CD4+-T
cells, whereas S(t) denotes the complementary number of susceptible CD4+-T cells. These numbers are considered as dynamical variables. One major difference to the model suggested by Phillips [45] is that this model is of provisional nature, since it does not investigate what happens to
initially non-susceptible cells following primary infection.
The time dependency in the numbers X and S is here postulated as resulting from new HIV
variants with extended tropism for CD4+-T cells in previously non-susceptible cells. This is modeled in a similar way as the evolutionary increase of the CD4+-T cell infection rate in 3.2.
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N.I. Stilianakis, D. Schenzle / Mathematical Biosciences 199 (2006) 1–25
3.4. Production and elimination of productively infected CD4+-T cells
The number of productively infected CD4+-T cells is denoted as Y(t). These are cells producing
HIV-RNA and they arise mainly from susceptible CD4+-T cells according to the mass action law
involving the number of prevalent virus particles and an average CD4+-T cell infection rate. The
model does not further distinguish between direct cell-to-cell infection and infection by freely
moving HIV particles. Once a CD4+-T cell has become an infectious cell, it is assumed to produce
particles of one type of HIV variant. Thus, we exclude the possibility of superinfection of a cell
with different HIV variants. There is evidence that HIV seeks to prevent this [46].
Each infectious cell is assumed to die at an intrinsic rate lY. Infectious cells are assumed to be
removed, in addition, by the immune system at a rate dYZ(t), where Z(t) denotes the current antiHIV activity of the immune system.
3.5. Production and elimination of HIV
HIV particles are assumed to be produced during the lifetime of an infectious cell at a constant
rate b. This implies that during its life span an infected cell produces on average b/lY viral particles. Thus, the total number of particles produced per infectious cell is not a constant quantity
but is increased by weakening of the cellular immune response against infectious cells. The model
does not explicitly distinguish between intact and defective virus particles, but in reality the number of effectively infectious virus particles may be much smaller than the total number.
HIV particles are assumed to be removed at an intrinsic rate lV. In addition, HIV particles are
assumed to be removed by the immune system at a rate dVZ(t). Z(t) is the current effective antiHIV activity of the immune system.
3.6. Immune response against HIV
Due to the complexity of the immune response quantitative information about the dynamics of
interaction with viruses is very limited. This makes a detailed description of the immune response
against HIV difficult. The present approach limited by postulating that upon infection with HIV
the immune system activates an immune response which is independent from the amount of HIV
present in the body. As soon as the anti-HIV activity is initiated remains as long as possible at
high levels and gradually declines with the decline of the CD4+-T cells.
The cellular immune response is considered to be the dominant effective part of the immune
response at primary infection and during the full time course of the infection process [7]. Although
the existence of neutralizing antibodies has been shown, there is no clear evidence of an effective
contribution of the humoral immune response [47]. Modelling the immune response against HIV
is actually to model the effective immune response, that predominantly is more accurately the cellular immune response. We term this as anti-HIV activity in the model.
3.7. Initial condition upon primary infection
Upon primary infection the individuals CD4+-T cell population is supposed to be in a steady
state with a certain number of CD4+-T cell which divide into two fractions of non-susceptible and
susceptible cells with respect to HIV infection.
N.I. Stilianakis, D. Schenzle / Mathematical Biosciences 199 (2006) 1–25
9
Any individual is considered before infection with HIV to possess a number of CD4+ helper
lymphocytes and a number of CD8 cytotoxic cells which are ready to start proliferation upon
encounter with HIV derived antigens. Hence the anti-HIV activity is assigned some small initial
value Z(0) at time t = 0, which marks the begin of infection.
The initial infection dose is normally very small. We operate with V(0) = 1. Since the model is
insensitive with respect to the initial dose [32] a wide range of initial doses can be considered.
4. The extended modelling approach
4.1. The model
The original basic model of which the extension we introduce here has been presented and analyzed elsewhere [31,32] also with respect to the current estimates for certain parameters estimated
in clinical and experimental studies [33,34,48,49].
Based on the description of fundamental mechanisms and the assumptions described above and
considered in our approach the extended model includes the following dynamic variables:
X
S
Y
V
Z
P
K
total number of non-susceptible CD4+-T cells
total number of susceptible CD4+-T cells
total number of productively infected CD4+-T cells
total number of HIV particles
anti-HIV activity of the immune system
fraction of new CD4+-T cells entering the pool of susceptible CD4+-T cells
factor that describes the increase of the CD4+-T cell infection rate
The model consists of a system of non-linear differential equations and reads
dX
¼ að1 P Þ lX ;
dt
dS
S
¼ aP lS j0 KV
;
dt
ðP þ dÞ
dY
S
¼ j0 KV
ðlY þ dY ZÞY ;
dt
ðP þ dÞ
dV
¼ bY ðlV þ dV ZÞV ;
dt
dZ
¼ hgðV Þ þ q½f ðS þ X ÞZ max Z;
dt
dP
¼ xP V ðP max P Þ;
dt
dK
¼ xK V ðK max KÞ;
dt
ð10Þ
ð11Þ
ð12Þ
ð13Þ
ð14Þ
ð15Þ
ð16Þ
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N.I. Stilianakis, D. Schenzle / Mathematical Biosciences 199 (2006) 1–25
with
f ðN Þ ¼
1 þ bc
;
1 þ ðbN 0 =NÞc
ð17Þ
and g(V) = V/(a + V) where N = X + S the total number of non-infected CD4+-T cells and
N0 = X0 + S0, X(0) = X0, S(0) = S0, Y(0) = Y0, V(0) = V0, Z(0) = Z0, P(0) = P0, K(0) = K0 the
initial conditions.
The pool N of CD4+-T cells is divided into non-susceptible cells X and susceptible cells S, where
P denotes the proportion of new CD4+-T cells entering the pool of susceptible cells and 1 P the
proportion of new cells that still remain non-susceptible.
The terms a(1 P) and aP represent the immigration rate of new non-susceptible or susceptible
CD4+-T cells, respectively. These cells have a natural death rate l. The infection process between
cells and viruses is described by a mass action law term (j0KVS/(P + d)). The part S/(P + d) of this
term is of major importance because it describes the dynamic changes of the population of the susceptible cells expressed in the ratio between the susceptible cells population and the fraction of new
cells entering the pool of susceptible cells over time t. P is decisive for the model dynamics, because
depending on which fraction of cells are initially susceptible and how fast this fraction increases
over time it substantially determines the course of infection and progression to disease. It indicates
that many more cells can be attacked by the virus than can be combated by the immune system
cells. Infected cells are generated at the same rate at which susceptible cells are infected and die
at a rate lY. In addition they are removed due to the involvement of the anti-HIV activity at a rate
dYZ. Virus particles are produced from infected cells at a rate b and are cleared at a rate lV. Also
here an additional elimination factor, the anti-HIV activity is assumed (dVZ). The HIV specific
immune response is modelled, due to the limited knowledge about its dynamics, by a general equation, which has the intrinsic features of an immune response and, more importantly, it is coupled to
a time dependent decline of the CD4+-T cells. Thus, the HIV specific immune response is mounted
at a rate q and this happens independent of the number of the present particles. The dependence of
the immune response activation on the virus abundance is modelled by the function g(V), which
assumes a sigmoid dependence of the immune response on V. Thus, upon primary infection,
HIV will stimulate specific antibody producing and cytotoxic cells to proliferate at an intrinsic rate
q[f(S + X)Zmax] until the immune system practically limits itself independent of the number of HIV
particles and infected cells. The function f(N) is used to couple the activity of the immune system to
the amount of uninfected cells available and, hence, to take into account the fact that the activity of
the immune system becomes reduced if the number of CD4+-T cells is not sufficiently high. Also
f(S + X) is assumed to represent a sigmoid dependence on uninfected cells with a Hill coefficient
c. The fraction of new cells, that come from the pool of the susceptible cell, increases at a rate
xP, which corresponds to the generation and selection of HIV mutants. Finally, the HIV infection
rate increases at a rate xK per virus particle.
4.2. Qualitative features of the model
Through its extension the model becomes more complicated so that a full mathematical analysis is not possible. At the same time though, it becomes more realistic and applicable. Neverthe-
N.I. Stilianakis, D. Schenzle / Mathematical Biosciences 199 (2006) 1–25
11
less, even for the more complicated model it is still possible to define and calculate a virus reproduction number R0 :
R0 ¼
bj0 KS
.
ðlY þ dY ZÞðlV þ dV ZÞðP þ dÞ
ð18Þ
Table 1
Variables and parameter values used in model
Variables
N
X
S
Y
V
Z
P
K
Total number of non-infected CD4+-T cells
Number of non-susceptible CD4+-T cells
Number of susceptible CD4+-T cells
Number of productively infected CD4+-T cells
Number of infectious HIV particles
Anti-HIV activity
Proportion of new susceptible CD4+-T cells
Average infection rate of CD4+-T cells
Parameter
a
l
j0
lY
dY
b
lV
dV
h
q
xP
xK
+
CD4 -T cell production rate
Natural death rate of uninfected cells
Initial rate at which a HIV particle transforms a susceptible
CD4+-T cell to a productively infected cell
Death rate of productively infected cells
Maximum additional elimination rate of productively
infected cell through the anti-HIV activity
HIV production rate from infected cells
Clearance rate of infectious virus particles
Maximum additional elimination rate of virus particles
through the anti-HIV activity
HIV dependent immune activation rate
Autonomous immune activation rate
Rate of increase of the fraction of susceptible cells by
generation and selection of HIV mutants
Rate of the increase of reproduction per virus particle
Initial values
References
N0 = X0 + S0 = 2.5 · 1011
X0 = 0.7 · 2.5 · 1011
S0 = 0.3 · 2.5 · 1011
Y0 = 0
V0 = 1
Z0 = 106
P0 = 0.3
K0 = 1.0
[40,41]
[40,41]
[40,41]
Values
References
9
5 · 10 /day
[33,34]
0.02/day
[48]
1.0 · 1012/(particle Æ day)
0.6/day
0.6/day
[33,49]
150 particles/(cell Æ day)
6/day
5/day
[4]
[33,49]
106/day
0.1/day
1.4 · 1014/(particle Æ day)
1.1 · 1015/(particle Æ day)
Constants
Zmax
Pmax
Kmax
a
b
c
d
Values
Maximum anti-HIV activity
Maximum fraction of susceptible cells
Maximum infection rate of susceptible cells per infected cell
Constant
Constant
Constant
Constant
Derived quantities
R0
R0
1.0
1.0
20
103
0.2
2.0
102
Values
Basis reproduction number of HIV without anti-virus activity
Basis reproduction number of HIV with maximum anti-virus activity
10
2.75
12
N.I. Stilianakis, D. Schenzle / Mathematical Biosciences 199 (2006) 1–25
Eq. (18) has following meaning. If the variables S, Z, K would be kept fixed at the values S; Z; K; P
an HIV particle generates R0 secondary particles. At the start of an HIV infection at time t = 0 the
virus reproduction number therefore has the value:
R0 ¼
bj0 S 0
;
ðlY lV ÞðP 0 þ dÞ
ð19Þ
and if the anti-HIV activity has fixed maximum value Z = Zmax = 1 then for K = 1
R0 ¼
bj0 S 0
.
ðlY þ dY ÞðlV þ dV ÞðP 0 þ dÞ
ð20Þ
R0 is the initial reproduction number in the absence of any anti-HIV activity. The HIV reproduction number must be above one to establish persistent infection. More important is the reproduction number R 0 , in the presence of a fully activated anti-HIV activity and with maximum number
of susceptible cells S0. The value of R 0 decides whether an individual is able to eliminate an HIV
infection, which may rarely happen.
In the case R 0 > 1 an HIV infection will persist, as is the case for the vast majority of infected
persons. For parameter values which have been estimated in clinical studies or considered to
be reasonable guesses the values of the various dynamic variables then move rather smoothly
towards some final value.
At any time the system is in a sort of quasi-equilibrium which gradually shifts. This means, if all
evolutionary processes were stopped, all variables would remain constant afterwards.
4.3. Parameter values of the extended model
Each term presented in the model has a specific biological interpretation. However, for only
some of the corresponding parameters there are currently estimates from available clinical and
experimental data. Those estimates were used for the numerical results. The remaining parameter
values proposed below should be considered as reasonable choices (Table 1).
5. Model description of a typical HIV infection
The numerical results presented here were generated with the parameter values of Table 1.
These results describe the dynamics of the model for a typical HIV infection. In reality there is
a wide range of HIV infection courses and the typical course of infection corresponds to a small
fraction of patients. As the rounded numerical values for the model parameters in Table 1 indicate, this typical course has not been specifically fitted. Using variation of the parameter values
the whole spectrum of observed individual HIV infection courses could be reproduced.
5.1. Typical course of infection
Fig. 1(a) and (b) correspond to the typical infection courses know from the literature [1]. The
model can reproduce the characteristic phases of HIV infection, namely primary infection during
the first few weeks, the long asymptomatic period and the transition to AIDS.
N.I. Stilianakis, D. Schenzle / Mathematical Biosciences 199 (2006) 1–25
12
10
0.8
0.6
V
Y/(X+S+Y), (X+S+Y)/No, Z
1
11
10
0.4
0.2
0
(a)
13
10
0
2
4
6
8
Time (years)
10
10
12
0
2
4
(b)
6
8
Time (years)
10
12
12
10
10
V
10
8
10
0
(c)
0.2
0.4
0.6
0.8
1
(X+S+Y)/No
Fig. 1. (a) Typical clinical course of HIV infection predicted by the model with the parameter values from Table 1.
Dashed-line curve describes the anti-HIV activity Z; solid line represents the total CD4+-T cells (X + S + Y)/N0
standardized to the interval [0, 1]. The dashed–dotted line represents the fraction of infected cells Y/(X + S + Y). The
parallel to the x-axis indicates the value of 200/mm3 CD4+-T cells that marks progression to AIDS. (b) The dynamics of
the virus population and (c) trajectory between viral population and T cells that correspond to figures (a) and (b).
During primary infection there is a strong viremia, that causes the first decline of the CD4+-T
cells. Right after and partially simultaneously with the mounting of the anti-HIV activity virus
levels sharply decrease. The CD4+-T cell population then rebounds to a certain degree and the
anti-HIV activity reaches its maximum activity.
For about 10 years the virus load is slightly suppressed and slowly increases and the infection
shows the characteristics of a persistent infection in the presence of a protective anti-HIV activity.
Nevertheless the number of the CD4+-T cells declines more or less linearly. This is the most striking feature of HIV infection. Fig. 1(c) shows the trajectory of the viral population V and the
CD4+-T cells ((X + S + Y)/N0) and demonstrates the strong viremia during the fist weeks of
infection.
The effective immune response increases quickly and remains at high levels for several years.
After about 10 years the CD4+-T cell number drops below the level of 20% of the normal value
before infection, which is the definition to disease progression AIDS (Fig. 1(a)) [50]. This time
14
N.I. Stilianakis, D. Schenzle / Mathematical Biosciences 199 (2006) 1–25
12
10
0.8
10
0.6
10
V
Y/(X+S+Y), (X+S+Y)/No, Z
1
0.4
8
0.2
0
10
0
(a)
50
100
Time (days)
150
0
(b)
50
100
Time ( days)
150
Fig. 2. (a) Anti-HIV activity Z (dashed-line curve), total CD4+-T cells (solid line) and infected (dash–dot line) cells
during primary infection. All parameter values as in Table 1. (b) Viremia during primary infection. All parameter values
as in Table 1.
the immune system is no more able to control the infection. According to the model results the immune system holds only 50% of its potential, which is measured through the anti-HIV activity Z,
and it declines rapidly to 10% two years later. Thus, the immune system is also not able to efficiently
control other opportunistic infections. Due to the decline of the anti-HIV activity it is possible for
HIV to replicate and reach concentrations that are even higher than those of primary infection.
Because of the extreme pathological conditions that govern this stage of disease it might be that
the model loses its applicability for the very late part of the third phase of disease.
5.2. Primary infection
Numerical results of the model show that the dynamics of primary infection (Fig. 2(a) and (b))
is in line with the clinical observations [51,52].
After the start of infection the number of HIV particles grows exponentially and reaches after
14 days a maximum value of about 5 · 1011 particles. This virus load drops by 2.8-log units within
4 weeks. These values are close to those observed [7]. The HIV viremia causes a temporary reduction of the CD4+-T cells, which then recover and remain at a lower level than before infection.
The anti-HIV activity increases fast but it is not the only reason for the break down of the viremia.
Also the limitation of available target CD4+-T cells can be responsible [45]. This is part of the
intrinsic evolutionary dynamics of the model.
5.3. Evolutionary dynamics
The dynamics of the intra-host HIV evolution is represented through the variables P and K
(Figs. 3 and 4). Variable P determines which proportion of new CD4+-T cells enters the pool
of susceptible CD4+-T cells. In our standard numerical calculation it increases monotonically
over time and reaches a value of 0.96 after 12 years with the parameter values of the typical simulation. The dynamics of infection change drastically the higher the initial value for P is (Fig. 3).
For low values of P, indicating that the fraction of susceptible T cells is small at the beginning, the
N.I. Stilianakis, D. Schenzle / Mathematical Biosciences 199 (2006) 1–25
15
1
0.9
0.8
0.7
(X+S+Y)/No
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
Time (years)
8
10
12
Fig. 3. CD4+-T cell counts for two different initial values (P = 0.1 and P = 0.8) of the proportion of the new CD4+-T
cells coming into the pool of susceptible cells. For (P = 0.8) progression to disease is fast. All other parameter values as
in Table 1.
fraction of CD4+-T cells remains for a long time at high level, and after 12 years it is still at a level
of about 50% of its initial value. If the fraction of susceptible cells is large a fast progression to
disease is expected. This is interpreted to mean that through generation and selection of mutants
HIV increases its range of CD4+-T cell tropism over more and more CD4+-T cell clones, until
after 12 years almost all clones are equally susceptible to infection by HIV. This is a completely
new postulation of evolutionary dynamics which cannot be tested with available data yet. However, there is recent clinical work in support of this hypothesis [53]. It is known that HIV variants
with different phenotypes evolve in HIV infected persons [54,55], and that an antigen driven
expansion of T cell subpopulations takes part [56,57].
The variable K represents the factor, by which the infection rate of the CD4+-T cells increases
by generation and selection of HIV mutants. This factor increases slowly over years. Its growth
accelerates when the viral production during the third phase of the course of infection rises again
substantially. This effect corresponds to the observed intra-host evolution of HIV variants with
increasing replication capacity [26–30].
Fig. 4 shows how the HIV reproduction number increases in the population of susceptible
CD4+-T cells over time. The decline of the reproduction number during the first months can
be explained by the activity of the immune response. After this activation reaches its maximum
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N.I. Stilianakis, D. Schenzle / Mathematical Biosciences 199 (2006) 1–25
45
40
HIV reproduction number
35
30
25
20
15
10
5
0
0
2
4
6
8
10
12
Time (years)
Fig. 4. Changes in the value of the viral reproduction number during the course of infection.
level the reproduction number again increases quickly. At the end of the infection process an
exponential increase is given as disease progression is reached.
6. Model predictions about determinants of disease progression
6.1. Initial dose
Fig. 5 shows that the model results are independent from the initial dose. For Y(0) = 0 and
V(0) = 1 or Y(0) = 100 and V(0) = 106 the model predicts very small differences during primary
infection which disappear after short time. After 4 months the curves are almost identical.
6.2. HIV evolutionary speed
The parameters xP and xK, which determine the pace of the intra-host evolution of HIV are
essential determinants of the form and the duration of the infection. Fig. 6 shows three individual
courses with different values for xK. The CD4+-T cell reduction can look similar for several years
but the ultimate fate may be substantially different. A 5-fold reduction of the evolutionary speed
used in the typical simulation causes a prolongation of the life span of more than 2 years. Correspondingly, a 5-fold increase results in an acceleration towards disease progression.
N.I. Stilianakis, D. Schenzle / Mathematical Biosciences 199 (2006) 1–25
17
1
0.95
(X+S+Y)/No
0.9
0.85
0.8
0.75
0.7
0
20
40
60
80
100
120
140
160
180
Time (days)
Fig. 5. Total CD4+-T cell counts for Y0 = 0, V0 = 1 (solid line) and Y0 = 100, V0 = 106 (dashed line) during primary
infection.
Stronger effect can be observed in Fig. 7 for the parameter xP. Both figures indicate that combination of the two parameters xK and xP can generate an enormous repertoire of CD4+-T cell
curves. Figs. 6 and 7 implicate that the effect of the rate of the fraction of susceptible cells by generation and selection of HIV mutants is of major importance. This makes the combinatorial effect
of K and P crucial for the course of the infection. These results demonstrate the variability of the
CD4+-T cell counts during infection in several infected individuals and the flexibility of the model
to produce these results.
6.3. Initial infection rate
Infectious dose does not seem to play an important role, according to the model predictions.
However, the model points to the initial CD4+-T cell infection rate as a main determinant of the
infection and disease progression. This value depends on the state of the infected individual, since
a highly activated CD4+-T cell pool should favor CD4+-T cell infection. But the value can also depend on the composition of the infection inoculum of HIV variants. Fig. 8 shows changes due to
different R0 of which the different values were based on changes on the initial infection rate value.
The upper curve of Fig. 8 corresponds to an R0 = 10, which is the value for the simulation of
the typical case. The curve in the middle corresponds to a value of R0 = 15 and the lower curve to
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N.I. Stilianakis, D. Schenzle / Mathematical Biosciences 199 (2006) 1–25
1
0.9
0.8
0.7
(X+S+Y)/No
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
Time (years)
Fig. 6. Total CD4+-T cell counts for xK (middle), 5xK (lower) and xK/5 (upper).
a value R0 = 20. Depending on the value of R0 the effect on the CD4+-T cell decline and the
course of infection is substantial. R0 = 15 would lead to progression to AIDS after 8.5 years.
6.4. Initial CD4+-T-cell count
Based on the model predictions, the impact of the initial count of the CD4+-T cell pool on the
infection process is enormous. Fig. 9 demonstrates the difference between three different initial cell
numbers for a period of 12 years. For an initial value of 1200 cells per mm3 a rapid progression to
disease is predicted. A value of 800 cells per mm3 leads to a much smoother decline of the CD4+-T
cell count. Since the initial number of CD4+-T cells in healthy persons is highly variable this prediction would be of major clinical importance.
7. Discussion
An extended model of a previous approach [31,32] on a theory for the within-host dynamics of
HIV infection is presented, analyzed and numerically explored. The objective of the presented theory on AIDS pathogenesis is the assessment of the essential mechanisms underlying this process.
The model reproduces some of the most striking features of HIV infection. The model includes
N.I. Stilianakis, D. Schenzle / Mathematical Biosciences 199 (2006) 1–25
19
1
0.9
0.8
0.7
(X+S+Y)/No
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
Time (years)
Fig. 7. Total CD4+-T cells for xP (middle), 2xP (lower) and xP/2 (upper).
terms and parameters that enable description and prediction of biological mechanisms. The extended model allows a more realistic description of the HIV infection process and corrects some
artefacts of the original simpler model.
The model is based on the assumptions that HIV predominantly infects CD4+-T cells and that
the generation and selection of HIV mutants is towards higher reproduction capacity. The model
describes a persistent infection, which on a time scale of years, changes its intrinsic dynamics
through a mechanism that indirectly considers the effects of HIV variability. The model extension
is the incorporation of the tendency of the virus to increase its target cell pool, in other words, the
increasing cell tropism over time. The equation for the immune response represents only a general
description since immune dynamics are not well understood. The equation produces the dynamics
of the immune response by taking into account the initial dependence of the immune response on
antigen availability. Further autonomous immune activation is considered and is coupled to the
amount of T cells.
The effects of several factors on the dynamics of infection could be described. For instance, the
negligible impact of the initial dose, but the importance of the intra-host evolution of HIV. The
rate at which new CD4+-T cell clones are included in the target pool through tropism seems to be
of substantial influence for disease progression.
The model shows that the initial infection rate, which could also be represented with the initial
reproduction number as well as the base line count of CD4+-T cells are essential determinants of
20
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1
0.9
0.8
0.7
(X+S+Y)/No
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
Time (years)
Fig. 8. Total CD4+-T cell counts for R0 = 10, 15, 20 (from the top to the bottom).
the infection process. The latter prediction might seem a bit puzzling. It is based on the idea that
the greater the pool of susceptible cells the more likely is that the virus spreads faster because it
easier to find uninfected cells. This has been described for the initial phase of the HIV infection
period also elsewhere [45]. In addition, analysis of the HIV Swiss Cohort Study data showed
that higher baseline CD4+-T cell count correlates with faster subsequent decline of the count
(V. Müller, personal communication). Although the date of infection of many patients in this
study could be determined only approximately, the results support the model prediction. Further
clinical investigation is needed to test this aspect.
A major difference between the model presented here and other approaches is that the model
provides a description of all phases of HIV disease progression in observed time scales based
on the most important components of the infection process, including the anti-HIV specific immune response. Although the numerical solutions and the results presented here need further elaboration, it should be emphasized that most of the results are difficult to reproduce with the other
existing modelling approaches. One reason is that only a few models deal with the issue of the
mechanisms of disease progression. The vast majority of the models focus rather on treatment
strategies using approaches specifically developed for the short term dynamics of the therapeutic
interventions (see [23,24] for reviews).
The parameter values for the numerical results of the model were employed from estimates
from clinical and experimental studies. Some parameter values though were reasonable choices.
N.I. Stilianakis, D. Schenzle / Mathematical Biosciences 199 (2006) 1–25
21
1
0.9
0.8
0.7
(X+S+Y)/No
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
Time (years)
Fig. 9. Total CD4+-T cell counts after a reduction of the initial T cell number shown in Table 1 by 20% (upper curve) or
an increase of the same number by 20% (lower curve).
The numerical parameter values used are rounded values and no effort for a specific fit was made.
It should be possible to estimate the unknown parameters from combined data obtained under a
variety of different conditions and after various interventions. It would be of major help if clinicians could report more precise longitudinal data from observations on individual patients. One
of the most valuable uses of modelling is to indicate which parameters are most important in
understanding the infection process. It is essential that clinicians collect the data needed for the
identification of these parameters. Other effects that indicate the predictive power of the model
are outlined.
Recent studies [33] showed that the turnover rate of susceptible CD4+-T cells is relevant for
disease progression. The model could test the plausibility of different turnover rate estimates.
The model could explore the effects of the immune response on the disease progression. For instance, the strength of the immune response is associated with the speed of activating the immune
response. However, the model indicates that the speed of activation might be not so important
when considering the full course of the infection.
Latently infected cells, or other target cell subpopulations could be also incorporated and investigated with the model along with indirect cell destruction and co-infection processes. Differentiation between latently and actively infected cells could be of major interest since antigen-driven T
cell activation seems to play a major role in the HIV infection process. A model for the pathogenesis of AIDS that investigates the role of antigenic stimulation and the associated mechanisms in
22
N.I. Stilianakis, D. Schenzle / Mathematical Biosciences 199 (2006) 1–25
regulating the pathogenesis of HIV disease could successfully predict some of the key features of
disease progression [58]. Distinction in subpopulation of different types of CD4+-T cells would be
another possibility to make the model more realistic. However, the role of the each subgroup and
its contribution in the pathogenesis of AIDS has yet to be clarified. Thus, integration of such subgroups would increase both the complexity and the uncertainty of the model. Although superinfection is possible and has been observed it still seems to be a rare event [59]. This indicates the
natural immunity induced by primary infection can make it difficult for a superinfecting virus
to establish itself. We ignored superinfection at this stage since such cases are small portion of
the infected population.
The deterministic approach presented here is appropriate to analyze certain basic mechanisms
of the HIV infection process. However, there is also a need to understand the individual infection
courses of infected persons that incorporates of stochastic effects. Numeric simulation with the
generation of random numbers has been presented elsewhere [32]. This approach could be expanded to explore the whole spectrum of expected variability in parameter values. For instance,
based on such stochastic effects a plausible explanation for the variability of the incubation period
to AIDS can be proposed [32]. Other stochastic aspects like the prognostic value of the CD4+-T
cell count or that of the viral load or the correlation between CD4+-T cell count could also be
investigated. This and the development of optimal treatment strategies using appropriate modifications of the model to account for treatment interventions and associated effects such as drug
resistance development are subject of current work.
There is no doubt that the interaction between HIV and the human immune system is a highly
dynamic and multifactorial process. Therefore, it is essential to base therapeutic interventions on
a more solid theoretical ground than it has been the case until now. Treatment effects strongly
depend on the patients current health state which often is not particularly well defined. For most
treatment regimes it is unclear whether the patients will benefit in the long term. Therefore,
although the statistical analysis of clinical data is useful its value to understand mechanisms is
rather limited. The development of realistic mathematical models with prognostic power at the
individual level is needed. Besides better controlling infection treatment of side effects and drug
resistance may be better planed. Both aspects lead us to consider the optimization of treatment
strategies. Optimal treatment interruption and time-point of regime change may be more quantitatively considered. The fact that effective treatment of HIV infection is usually aggressive indicates that there is a need for well-planned therapeutic interventions.
Acknowledgment
We would like to thank Klaus Dietz for helpful discussions.
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