Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol166.Issue1.Jul2000:

Mathematical Biosciences 166 (2000) 69±84
www.elsevier.com/locate/mbs

A competition model for viral inhibition of host cell
proliferation
Sarah Holte a,*, Michael Emerman b
a

b

Division of Public Health Science, Fred Hutchinson Cancer Research Center, 1100 Fairview Ave North MW-500,
Seattle, WA 98109, USA
Division of Human Biology, Fred Hutchinson Cancer Research Center, 1100 Fairview Ave North MW-500, Seattle,
WA 98109, USA
Received 11 January 2000; received in revised form 27 April 2000; accepted 5 May 2000

Abstract
Some viruses encode proteins that promote cell proliferation while others, such as the human immunode®ciency virus (HIV), encode proteins that prevent cell division. It has been hypothesized that the selective advantage determining which strategy evolves depends on the ability of the virus to induce a cellular
environment which maximizes both virus production and cell life span. In HIV, the protein that causes cell
cycle arrest is Vpr. In this paper, we develop a mathematical model, based on di€erence equations, to study
the competition between two genotypes of HIV ± one that encodes this protein (Vpr+) and one that does

not (Vpr)). In particular, we are interested in parameters that could be di€erent between the in vitro
condition, where the Vpr) genotype dominates, and the in vivo condition, where the Vpr+ genotype
dominates. Our model indicates that the infected cell death-rate, the viral half-life, and the dynamics of the
target cell population all e€ect the competition dynamics between the Vpr+ and Vpr) viral genotypes.
Perturbing any of these parameters from the in vitro estimates while holding the others ®xed has no a€ect
on the competition outcome, i.e., the Vpr) genotype dominates. Perturbing the infected cell death-rate and
the target cell source causes a switch in competitive outcome, although not necessarily at values, which
represent the in vivo condition. Adding a perturbation in the viral half-life from in vitro to in vivo condition
results in a switch of the competitive advantage from the Vpr) genotype to the Vpr+ genotype with parameters for all three mechanisms set to estimated in vivo values. Ó 2000 Published by Elsevier Science
Inc. All rights reserved.
Keywords: HIV; Competition dynamics; Vpr protein

*

Corresponding author. Tel.: +1-206 667 6975; fax: +1-206 667 4812.
E-mail address: sarah@hivnet.fhcrc.org (S. Holte).

0025-5564/00/$ - see front matter Ó 2000 Published by Elsevier Science Inc. All rights reserved.
PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 2 0 - 1

70

S. Holte, M. Emerman / Mathematical Biosciences 166 (2000) 69±84

1. Introduction
Viruses are obligate parasites with diverse ways of interacting with their hosts. Some viruses
initiate lytic infections, which result in high levels of virus replication followed by clearance of the
virus. On the other hand, other viruses establish life-long persistent infections in which the virus
has established a quasi-homeostasis with the organism. In general, the mechanisms underlying
persistent infections are poorly understood and require a complex relationship between levels of
replication that are bene®cial to the virus, but not so detrimental to the host that viral transmission is compromised [1]. One way in which persistent infections are maintained is by making
the viral life cycle dependent on pathways that induce cellular proliferation. For example, human
T cell leukemia virus (HTLV), the causative agent of adult T cell leukemia is a retrovirus that
incorporates its genetic material into the host cell genome and promotes cell growth. Other viruses, such as the human papilloma viruses, also encode proteins that promote cell proliferation
and are propagated within the host essentially by replication of the infected cell. Not surprisingly,
these viruses can be oncogenic. Other viruses use additional means of driving infected cells into
replication [2].
The human immunode®ciency virus (HIV), the causative agent of acquired immune de®ciency
syndrome (AIDS), also establishes a persistent infection and incorporates its genetic material into
the host cell chromosome. However, HIV infected cells have a short half-life after infection [3].

Moreover, not only does HIV not promote cell proliferation, it also encodes a protein, called Vpr,
that prevents cells from progressing through the cell cycle and dividing [4]. Expression of Vpr is
able to delay or arrest cells in the phase of the cell cycle just before mitosis occurs (called G2
phase) [4].
In cell culture, mutants of HIV bearing inactivating mutations in the vpr gene often outgrow the
wild-type population. In the absence of a functional Vpr, these mutants of HIV do not cause
infected cells to delay or arrest in the G2 phase of the cell cycle. In contrast, in vivo, nearly all HIV
isolates encode a functional vpr gene, and genotypes which are initially defective in vpr (Vpr))
revert to wild-type (Vpr+) over time [5]. We previously o€ered an explanation for why cell cycle
arrest by HIV is selected for in vivo, while it is selected against in vitro [5]. It was shown that
during the G2 phase of the cell cycle, the viral production rate was signi®cantly increased. Thus, it
was hypothesized that arrest in the G2 phase (which is induced by Vpr) would o€er a selective
advantage in vivo when death of the infected cells is accelerated [5]. A simple mathematical model
was presented which indicated that under the in¯uence of the infected cell death-rates observed in
vivo, HIV which encodes a functional Vpr protein (Vpr+) would achieve the selective advantage.
This simple model, and conjecture about the role of infected cell death-rate in competition outcome between viruses which do or do not arrest the cell cycle, motivated us to develop a more
detailed mathematical model to assess the a€ect of infected cell death-rate on the competition
dynamics between viral genotypes which di€er in their a€ect on the progression of the cell cycle.
In this work, we describe that model and propose two other factors as possible mechanisms,
which determine a selective advantage for cell cycle arrest: (1) The dynamics of the uninfected

target cell population, and (2) clearance of free virus, which is also accelerated in vivo. Our model
uses di€erence equations to describe the temporal dynamics of the two interacting HIV genotypes.
While di€erence equations are more often used to describe dynamics of synchronized populations
with non-overlapping generations, they have also been used in a variety of settings with mixed

S. Holte, M. Emerman / Mathematical Biosciences 166 (2000) 69±84

71

generations. Examples include models for HIV pathogenesis after treatment with anti-retroviral
drugs [6] and mathematical descriptions of epidemics [7,8]. We chose to use di€erence equations
due to the discrete nature of the data we plan to collect, and the ease with which stochastic
variability can be incorporated. Our model contains compartments representing free virus that is
either Vpr+ or Vpr), infected cells that are producing virus that is either Vpr+ or Vpr), and
uninfected target cells. By varying parameters in the model which correspond to biological parameters that di€er between in vivo and in vitro conditions, we are able to simulate a variety of
conditions, and theoretically predict how the two genotypes interact in the setting of co-infection
by the two genotypes. In this work, we focus on infected cell death-rates, the dynamics of the
uninfected source population, and the half-life of infectious viral RNA. Other parameters in the
model could also a€ect the competition outcome between the two genotypes of HIV, however, we
chose to focus on these three parameters since they are known to di€er between the in vivo and in

vitro conditions. They are also parameters that can be perturbed in an experimental setting.
Stability of a zero equilibrium for one of the population compartments in a model is the
mathematical equivalent of extinction, e.g., when the zero equilibria for the population of Vpr+
infected cells is stable, the Vpr+ genotype becomes extinct and the Vpr) genotype is the competitive winner. Thus, we focus on bifurcations in stability of zero equilibria for the compartments
representing cells that are infected with viruses that are either Vpr+ or Vpr). An understanding of
these bifurcations, i.e., changes in stability, helps us to predict when viral genotypes that inhibit
host cell proliferation attain the competitive advantage and which biological factors may a€ect the
competitive outcome. Such model-based predictions can be used to guide laboratory experiments
to further understand the interaction of these two populations.

2. The di€erence equation model
In this section, we develop a mathematical model to study the competition dynamics of the viral
genotypes of HIV that either do or do not cause cell cycle arrest (Vpr+ and Vpr), respectively).
We modeled mixed populations of virus and infected cells containing both Vpr+ and Vpr) genotypes of HIV-1. When cells become infected by HIV, the infection status is characterized as
acutely infected (those subject to extensive cytopathic e€ects of infection, high levels of proviral
DNA, and superinfection of cells). Infected cells are initially in the acute state. Cells which survive
acute infection become chronically infected (characterized by immunity to superinfection, no signs
of cytopathic e€ect, and small number of provirus per cell [9]). We di€erentiate between acutely
and chronically infected cell compartments, with ¯ow from acutely to chronically infected populations. We chose to include compartments, for both chronically and acutely infected cells, since
it seems likely that when the infected cell death-rate is accelerated (in vivo condition), the dynamics of the viral competition is dominated by the acutely infected cells. In the in vitro condition,

where the infected cell death-rate is extended, the chronically infected cells may play the dominant
role in determining the competitive outcome. A compartment for uninfected cells is the ®nal
compartment included in this model.
We use the following notation for compartments and parameters in the model. Upper case
notation denotes compartments while lower case notation denotes parameters. Model compartments are described below, and model parameters are listed in Table 1.

72

S. Holte, M. Emerman / Mathematical Biosciences 166 (2000) 69±84

Table 1
In vitro model parameters
c
n‡
a
n‡
c
nÿ
a
nÿ

c
c
r‡
rÿ
r
d‡
a
d‡
c
dÿ
a
dÿ
c
du
d
i
s

Infectious viral clearance rate
Infectious viral production rate by cells acutely

infected with the Vpr+ genotype
Infectious viral production rate by cells
chronically infected with the Vpr+ genotype
Infectious viral production rate by cells acutely
infected with the Vpr) genotype
Infectious viral production rate by cells
chronically infected with the Vpr) genotype
Rate at which acutely infected cells become
chronically infected
Birth-rate for Vpr+ infected cells
Birth-rate for Vpr) infected cells
Birth-rate for uninfected cells
Death-rate for Vpr+ acutely infected cells
Death-rate for Vpr+ chronically infected cells
Death-rate for Vpr) acutely infected cells
Death-rate for Vpr) chronically infected cells
Death-rate for uninfected cells
Density dependent overall cell death-rate
Probability of infection
Constant rate of target cell replacement

0.12 per virion per houra
0.028 virions per hourb
0.0028 virions per cell per hourb
0.014 virions per cell per hourb
0.0014 virions per cell per hourb
0.01 cells per hourc
0.019 per cell per hourd
0.035 per cell per hourd
0.035 per cell per hour d
0.05 per cell per houre
0.025 per cell per houre
0.05 per cell per houre
0.01 per cell per houre
0.01 per cell per houre
10ÿ11f
2  10ÿ6 per virion per target cell per hourg
0 cells per hourh

a

Corresponds to viral half-life in vitro of 6 h [10].
See text.
c
Based on data in [11].
d
Corresponds to doubling time of 36 h for Vpr+ infected cells and 20 h for Vpr) and uninfected cells [5].
e
Corresponds to life expectancies (inverse of the death-rate) of 20 h for both Vpr+ and Vpr) acutely infected cells,
100 h for Vpr) chronically infected and uninfected cells, (Figs. 1 and 2 [11]) and 40 h for Vpr+ chronically infected cells
[12].
f
Corresponds to maximal carrying capacity of 2:4  109 . Set to allow larger carrying capacity than expected in vivo to
account for divisions of culture that occurs in experimental setting.
g
Estimate of probability of infection in 1 h could vary from 0.05 to less that 5  10ÿ7 with viral clearance c in the
ranges we consider in this paper. Fig. 5 [13]. We assumed infectivity to be the same for both Vpr+ and Vpr) genotypes
[14].
h
In vitro condition, no additional target cells added after start of experiment.

b

Model compartments
Vt ‡
Vt ÿ
A‡
t
Aÿ
t
Ct‡
Ctÿ
Tt
Xt

infectious Vpr+ viral population at time t
infectious Vpr) viral population at time t
cells acutely infected with the Vpr+ genotype at time t
cells acutely infected with the Vpr) genotype at time t
cells chronically infected with the Vpr+ genotype at time t
cells chronically infected with the Vpr) genotype at time t
uninfected target cells at time t
total cell population (uninfected and infected) at time t

S. Holte, M. Emerman / Mathematical Biosciences 166 (2000) 69±84

73

The model equations are as follows:
‡
‡ ‡
‡
ˆ eÿc Vt ‡ ‡ …n‡
Vt‡1
a At ‡ nc Ct †;
…r
A‡
t‡1 ˆ e

‡ ÿd‡ ÿcÿdX †
t
a

‡
Ct‡1
ˆ e…r

‡ ÿd‡ ÿdX †
t
c

…2:1†
‡

ÿiVt
†Tt ;
A‡
t ‡ …1 ÿ e

Ct‡ ‡ …1 ÿ eÿc †A‡
t ;

ÿ
ÿ ÿ
ÿ
ˆ eÿc Vt ÿ ‡ …nÿ
Vt‡1
a At ‡ nc Ct †;
…r
Aÿ
t‡1 ˆ e

ÿ ÿdÿ ÿcÿdX †
t
a

ÿ
Ct‡1
ˆ e…r

ÿ ÿdÿ ÿdX †
t
c

Tt‡1 ˆ e…rÿdu ÿdXt ÿi

ÿ

ÿiVt
†Tt ;
Aÿ
t ‡ …1 ÿ e

Vt‡ ÿiÿ Vtÿ †

Tt ‡ s:

…2:3†
…2:4†

Ctÿ ‡ …1 ÿ eÿc †Aÿ
t ;

‡

…2:2†

…2:5†
…2:6†
…2:7†

In developing the di€erence equation model we chose the time unit to be 1 h. In Eqs. (2.1) and
(2.4) the ®rst terms represent the clearance of free virus. We assume that virus decays exponentially with decay parameter c so that eÿc represents the proportion of viral RNA remaining after
1 h. The second terms in Eqs. (2.1) and (2.4) represent the number of new virions produced in the
unit of time from t to t ‡ 1. Note that we distinguish between the number of virions produced by
each infected cell depending on whether or not the cell is infected with the Vpr+ or Vpr)
genotype, and whether the cell is acutely or chronically infected.
Eqs. (2.2), (2.3), (2.5), and (2.6) describe the dynamics of the four types of infected cells, acutely
or chronically infected, and infected with the Vpr+ or Vpr) genotype. Note that we include the
density dependent death factor, dXt , in the equations for all cell populations. This term limits the
size of the total population of cells to some `carrying capacity' and is used frequently to more
realistically describe population sizes since continued exponential growth is not possible [7]. In
laboratory experiments, this issue is addressed by dividing culture every few days so that carrying
capacity is not reached, although we include it here as a necessity for conducting equilibrium
analysis. The ®rst term in each of the equations represents the density in those compartments at
time t ‡ 1 based on the size of those compartments at time t, and the combined birth and deathrates [e.g., …r‡ ÿ d‡
a ÿ dXt † for acutely infected Vpr+ cells]. For acutely infected cells we also allow
for ¯ow to the chronically infected compartment at rate c. The second terms in the equations for
acutely infected cells account for new infections in one unit of time. To derive these terms, we
compute the probability of a cell‡ not becoming infected by the Vpr+ genotype in 1 h using the
‡
binomial distribution as …1 ÿ i†Vt  eÿiVt so that the number of infections by the Vpr+ genotype
‡
between t and t ‡ 1 is approximately …1 ÿ eÿiVt †Tt . A similar derivation is applied for infections by
the Vpr) genotype. For chronically infected cells, the second term represents the contribution to
that compartment from the acutely infected cell populations.
The calculations to determine the amount of infectious viral RNA produced by acutely and
ÿ
ÿ
‡
chronically infected cells and cells infected with the Vpr+ or Vpr) genotype …n‡
c ; na ; nc ; and na †
utilize the following assumptions: (1) The average amount of viral RNA produced by all acutely
infected cell types (cells infected with Vpr+ or Vpr) and cells in G1 or G2 phase of cell cycle) is

74

S. Holte, M. Emerman / Mathematical Biosciences 166 (2000) 69±84

50 virions per hour, based on continuous release, which is demonstrated in [14]; (2) the amount of
viral RNA produced in the G2 phase is four times the amount produced in the G1 phase per viral
DNA [5] (with twice as much viral DNA in G2 as in G1); (3) the amount of viral RNA produced
acutely infected cells is 10 times the amount produced by chronically infected cells [11]; and (4)
both genotypes remain in the G1 phase of the cell cycle for 18 h, the Vpr) genotype remains in the
G2 phase for 2 h, and the Vpr+ genotype remains in the G2 phase for 18 h [5]. We also assumed
that 1/1000 of the virions produced are actually infectious. For example, n‡
a is calculated as follows: If n1 is the rate of viral production in the G1 phase of the cell cycle and n2 is the rate of viral
production in the G2 phase of the cell cycle, then n1 ˆ average viral production rate  …combined
time in G1 and G2)/(time in G1 ‡ 2  4  time in G2) ˆ …50  36†=f1000  …18 ‡ 2  4  18†g
and n2 ˆ 4  n1 . Then the average rate of viral production for cells infected with the Vpr+
genotype is (n1  time in G1 ‡ n2  time in G2†=…time in G1 ‡ time in G2† ˆ …0:011
18  0:044  18†=…18 ‡ 18† ˆ 0:028.
Eq. (2.7) describes the dynamics of the uninfected cells. The expression ÿi‡ Vt ‡ ÿ iÿ Vt ÿ † in the
®rst term represents the loss of cells from the population of uninfected cells to the various infected
cell compartments. The remaining terms in the exponential, r ÿ du ÿ dXt , represent the combined
birth and death-rate for uninfected cells, as in Eqs. (2.3)±(2.6) for infected cell compartments.
Finally, the term s in Eq. (2.7) is the parameter we will vary to simulate the e€ects of various
sources of uninfected cells into the system.
As mentioned in Section 1, we investigate via the mathematical model three factors that could
di€er between in vivo and in vitro conditions, and their in¯uence on genotype selection:
· the infected cell death-rate,
· dynamics of the uninfected target cell population, and
· the clearance-rate of viral RNA.
Recall that the idea behind the relevance of the infected cell death-rate is that it is higher in
vivo than in vitro. Therefore, since viral production is signi®cantly higher in the G2 phase of the
cell cycle, the ability to arrest the cell cycle in G2 (due to Vpr) would give the virus a selective
advantage in vivo since the accelerated death-rate limits the number of cell divisions. By varying
ÿ
‡
ÿ
d‡
a ; da ; dc and dc we can use the mathematical model to make predictions about the e€ect that
the rate of infected cell death has on the viral dynamics. We will be most interested in variations
ÿ
of d‡
c and dc , since these rates are most dramatically altered with accelerated in vivo infected cell
death, although our approach is to add a single additional increment to all four death-rates, in
order to simulate additional cell killing in vivo, most likely due to the immune response.
ÿ
‡
ÿ
Mathematically, we will add an increment, , to each of d‡
a ; da ; dc , and dc in Eqs. (2.2), (2.3),
(2.5) and (2.6).
The second variable we consider is the dynamics of the target cell population. In this work, we
consider only two possibilities, which could be easily replicated in laboratory experiments: no
additional source of target cells after initiation of the experiment or a constant replenishment of
additional target cells throughout the experiment at speci®ed time intervals. Other possibilities for
the dynamics of the target cell population may include activation or deactivation of target cells
which might be dependent on the amount of virus present or other immunological factors, although we do not consider those mechanisms in this work. The parameter s in the model
quanti®es a constant source of target cells. By setting s to zero we simulate mathematically the
e€ect of no source of target cells, and by varying s we can assess the e€ect of constant target cell

S. Holte, M. Emerman / Mathematical Biosciences 166 (2000) 69±84

75

replacement, which is the mechanism most likely to be carried out in in vitro experiments. If other
mechanisms for target cell replacement are proposed, then they could be incorporated into the
model through more detailed modeling of the uninfected target cell compartment.
The third factor we consider is the half-life of infectious virus. This has been estimated as 6 h in
vitro [10] and 20 min in vivo [3]. We assess the e€ects of these two values for viral clearance
in combination with the parameters described in the previous two paragraphs. The parameters c
in the model corresponds to the clearance-rate for infectious viral RNA.

3. Results
3.1. Temporal results
To begin, we simulated the system dynamics by iterating the di€erence equations 600 times, to
assess the cellular trajectories for 25 days (600 h) after initiation of the experiment. At time zero,
we assumed that the density of uninfected cells is 106 per ml, with 5000 cells infected with HIV-1
that is Vpr+ and an equal amount infected with HIV-1 that is Vpr). There is an initial spike in the
percentage of cells infected with HIV-1 that is Vpr+, but after about 10 days that genotype is
essentially extinct (Fig. 1).
Next, we perturbed each of the three parameters of interest in this work separately while
holding the other two ®xed: the infected cell death-rate, the uninfected cell source rate, and the
viral death-rate. Perturbing each of these parameters separately did not result in any change in
the competition outcome. However, for cells chronically infected with the Vpr) genotype, if the
death-rate is perturbed to be less than the birth-rate, s ˆ 0, then both populations die out.

Fig. 1. Simulated competition between Vpr) and Vpr+ genotypes under standard in vitro conditions: no source of
uninfected target cells, chronically infected Vpr) cell life expectancy 100 h, viral half-life 6 h. Trajectories in bold
represent Vpr+ genotype. Cells infected with the Vpr) genotype attain the competitive advantage approximately 15
days after start of the experiment.

76

S. Holte, M. Emerman / Mathematical Biosciences 166 (2000) 69±84

Fig. 2. Simulated competition between Vpr) and Vpr+ genotypes under perturbed conditions. (a) represents trajectories
when uninfected cell source is manipulated so that 100 000 uninfected cells are added to the system every hour, Vpr)
infected cell life expectancy is 100 h and viral half-life is 6 h. (c) represents trajectories when 100 000 uninfected cells are
added to the system every hour and Vpr) infected cell life expectancy is 25 h and viral half-life is 6 h. (e) represents
trajectories when 100 000 uninfected cells are added to the system every 24 h and the Vpr) infected cell life expectancy is
25 h and viral half-life is 6 h, and (g) represents trajectories when 100 000 uninfected cells are added to the system every
hour, Vpr) infected cell life expectancy is 25 h, and viral half-life is 20 min. Trajectories in bold represent Vpr+ genotype.
(b), (d), (f), and (h) are the corresponding plots of percentage of infected cells, which are infected with the Vpr+ genotype.

S. Holte, M. Emerman / Mathematical Biosciences 166 (2000) 69±84

77

Fig. 2 shows the cellular trajectories for a variety of combinations of perturbations of these
three mechanisms. All trajectories shown in Fig. 2 include a source of uninfected target cells.
Fig. 2(a) and (b) depict the results of adding 105 uninfected cells every hour without perturbing the
infected cell death-rate or the infectious viral half-life. In this case, the cells infected with the Vpr)
genotype are the competitive winners.
When we manipulated the infected cell death-rate so that the life expectancy (the inverse of the
linear contribution to the death-rate) of chronically infected cells decreased from 100 to 25 h
(approximate in vivo condition) while adding 105 cells per hour and keeping the infectious viral
half-life set at 6 h, we obtain the results shown in Fig. 2(c) and (d). In this scenario, the two
genotypes coexist. A more signi®cant decrease in the life expectancy of the cells infected with the
Vpr) genotype does result in complete domination by cells infected with Vpr+ genotype, but not
until the life expectancy of Vpr) infected cells is less than 20 h.
The results of perturbing the infected cell life expectancy as described in the previous paragraph keeping the infectious viral half-life set at 6 h, and adding uninfected cells every 24 h
rather than every hour are shown in Fig. 2(e) and (f). In this scenario, the model predicts that by
decreasing the source from once an hour to once a day, the population of cells infected with
Vpr+ has the selective advantage. This suggests that when the populations of target cells is in
some way limited but maintained, the Vpr+ genotype wins the competition for the resource of
uninfected cells.
Finally, we considered a perturbation of all three parameters of interest simultaneously. We
included a source of 105 uninfected target cells per hour, infected cell death-rates so that the life
expectancy of cells chronically infected with the Vpr) genotype is 25 h, and a perturbation to the
viral half-life by reducing it from 6 h to 20 min (approximate in vivo condition). The results are
shown in Fig. 2(g) and (h). These ®gures indicate that in this scenario, the most similar to the in
vivo situation, the population of cells infected with the Vpr+ genotype dominates completely after
25 days.
3.2. Bifurcation results
Our mathematical model can be used to predict parameter values where bifurcations in the
stability of the zero equilibria of cells infected with HIV-1 that is either Vpr+ or Vpr) occur, i.e.,
to determine values of the biological parameters where the competitive advantage switches from
one genotype to the other. Thus, to further assess the Vpr+/) competition, we produce bifurcation
diagrams for these equilibria values as functions of the life expectancy of cells chronically infected
with the Vpr) genotype and the source of uninfected cells. We assessed these bifurcations in the
situations when viral half-life is constant at 6 h (in vitro condition) and 20 min (in vivo condition).
To perform the bifurcation analysis, we iterated the function de®ning the model 50 000 times, and
used the last 50 iterates for each value of the parameter of interest to construct the diagrams. In all
cases, we obtained equilibrium conditions for the system after 50 000 iterations.
The bifurcation diagrams for the parameter which represents the infected cell death-rate are
shown in Fig. 3, where the stable equilibrium for cells infected with the Vpr+ genotype is depicted with a bold curve. Fig. 3(a) and (b) represent the situation when the infectious viral halflife is 6 h and Fig. 3(c) and (d) represent the situation when the infectious viral half-life is 20
min. In both cases, we set the source of uninfected cells to 105 every hour. Fig. 3(a) and (b)

78

S. Holte, M. Emerman / Mathematical Biosciences 166 (2000) 69±84

Fig. 3. Bifurcation diagram for chronically infected Vpr) infected cell life expectancy with target cell source of 100 000
every hour and viral half-life 6 h (a) and (b) and 20 min (c) and (d). Bold plots represent equilibria for the Vpr+ genotype. To perform the bifurcation analysis, we iterated the function de®ning the model 50 000 times, and used the last
50 iterates for each value of the infected cell death-rate to construct the diagrams. With a sustained source of target
cells, the Vpr+ genotype attains the competitive advantage when the life expectancy of infected cells is less that 20 h (if
viral half-life is 6 h) or 30 h (if viral half-life is 20 min). If the viral half-life is 6 h (in vitro condition) there may be a small
window of coexistence for the two genotypes. When the viral half-life is 20 min this window is signi®cantly reduced.

indicate that when the viral half-life is 6 h (in vitro condition) and the life expectancy for cells
chronically infected with the Vpr) genotype is less than 20 h, then the Vpr+ genotype dominates. If the life expectancy for cells chronically infected with the Vpr) genotype is between 20
and 30 h, then our model suggests that the two genotypes would coexist. When that life expectancy is greater than 30 h the Vpr) genotype dominates. Fig. 3(c) and (d) represent the
situation where the viral half-life is set to 20 min (in vivo condition). The competition outcome
is quite similar to the situation where viral half-life is 6 h, although the window of coexistence is
signi®cantly smaller.
A similar analysis is conduced by varying the source of uninfected cells and the results are
depicted in Fig. 4. For that analysis, we assume that the life expectancy for cells chronically infected with the Vpr) genotype is 25 h, which is close to the estimates of life expectancy for infected
cells in vivo [3]. We vary the infected cell source from 103 to 106 per hour. We assessed the situation when the viral half-life was set to 6 h (Fig. 4(a) and (b)) and when it was set to 20 min (Fig.
4(c) and (d)). Fig. 4(b) indicates that when the viral half-life is 6 h (in vitro condition) and a small