Estimating the impact of vaccination on acute simian-human immunodeficiency virus/simian immunodeficiency virus infections.
ABSTRACT The dynamics of HIV infection have been studied in humans and in a variety of animal models. The standard model of infection has been used to estimate the basic reproductive ratio of the virus, calculated from the growth rate of virus in acute infection. This method has not been useful in studying the effects of vaccination, since, for the vaccines developed so far, early growth rates of virus do not differ between control and vaccinated animals. Here, we use the standard model of viral dynamics to derive the reproductive ratio from the peak viral load and nadir of target cell numbers in acute infection. We apply this method to data from studies of vaccination in SHIV and SIV infection and demonstrate that vaccination can reduce the reproductive ratio by 2.3- and 2-fold, respectively. This method allows the comparison of vaccination efficacies among different viral strains and animal models in vivo.
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ABSTRACT: Cytotoxic T-lymphocytes (CTLs) recognize viral protein fragments displayed by major histocompatibility complex molecules on the surface of virally infected cells and generate an anti-viral response that can kill the infected cells. Virus variants whose protein fragments are not efficiently presented on infected cells or whose fragments are presented but not recognized by CTLs therefore have a competitive advantage and spread rapidly through the population. We present a method that allows a more robust estimation of these escape rates from serially sampled sequence data. The proposed method accounts for competition between multiple escapes by explicitly modeling the accumulation of escape mutations and the stochastic effects of rare multiple mutants. Applying our method to serially sampled HIV sequence data, we estimate rates of HIV escape that are substantially larger than those previously reported. The method can be extended to complex escapes that require compensatory mutations. We expect our method to be applicable in other contexts such as cancer evolution where time series data is also available.Frontiers in Immunology 01/2013; 4:252.
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ABSTRACT: Mathematical modeling of virus dynamics has provided quantitative insights into viral infections such as influenza, the simian immunodeficiency virus/human immunodeficiency virus, hepatitis B, and hepatitis C. Through modeling, we can estimate the half-life of infected cells, the exponential growth rate, and the basic reproduction number (R0). To calculate R0 from virus load data, the death rate of productively infected cells is required. This can be readily estimated from treatment data collected during the chronic phase, but is difficult to determine from acute infection data. Here, we propose two new models that can reliably estimate the average life span of infected cells from acute-phase data, and apply both methods to experimental data from humanized mice infected with HIV-1.Theoretical Biology and Medical Modelling 05/2014; 11(1):22. · 1.27 Impact Factor
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ABSTRACT: HIV-1 often evades cytotoxic T cell (CTL) responses by generating variants that are not recognized by CTLs. We used single-genome amplification and sequencing of complete HIV genomes to identify longitudinal changes in the transmitted/founder virus from the establishment of infection to the viral set point at 1 year after the infection. We found that the rate of viral escape from CTL responses in a given patient decreases dramatically from acute infection to the viral set point. Using a novel mathematical model that tracks the dynamics of viral escape at multiple epitopes, we show that a number of factors could potentially contribute to a slower escape in the chronic phase of infection, such as a decreased magnitude of epitope-specific CTL responses, an increased fitness cost of escape mutations, or an increased diversity of the CTL response. In the model, an increase in the number of epitope-specific CTL responses can reduce the rate of viral escape from a given epitope-specific CTL response, particularly if CD8+ T cells compete for killing of infected cells or control virus replication nonlytically. Our mathematical framework of viral escape from multiple CTL responses can be used to predict the breadth and magnitude of HIV-specific CTL responses that need to be induced by vaccination to reduce (or even prevent) viral escape following HIV infection.Journal of Virology 08/2011; 85(20):10518-28. · 4.65 Impact Factor
JOURNAL OF VIROLOGY, Dec. 2008, p. 11589–11598
Copyright © 2008, American Society for Microbiology. All Rights Reserved.
Vol. 82, No. 23
Estimating the Impact of Vaccination on Acute Simian-Human
Immunodeficiency Virus/Simian Immunodeficiency Virus Infections?
Janka Petravic,1Ruy M. Ribeiro,2Danilo R. Casimiro,3Joseph J. Mattapallil,4
Mario Roederer,5John W. Shiver,3and Miles P. Davenport1*
Complex Systems in Biology Group, Centre for Vascular Research, University of New South Wales 2052, New South Wales, Australia1;
Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, New Mexico 875452; Merck Research Laboratories,
West Point, Pennsylvania3; Department of Microbiology and Immunology, Uniformed Services University of Health Sciences,
Bethesda, Maryland 208244; and ImmunoTechnology Section, National Institute of Allergy and Infectious Diseases,
National Institutes of Health, Bethesda, Maryland 208925
Received 28 July 2008/Accepted 8 September 2008
The dynamics of HIV infection have been studied in humans and in a variety of animal models. The standard
model of infection has been used to estimate the basic reproductive ratio of the virus, calculated from the
growth rate of virus in acute infection. This method has not been useful in studying the effects of vaccination,
since, for the vaccines developed so far, early growth rates of virus do not differ between control and vaccinated
animals. Here, we use the standard model of viral dynamics to derive the reproductive ratio from the peak viral
load and nadir of target cell numbers in acute infection. We apply this method to data from studies of
vaccination in SHIV and SIV infection and demonstrate that vaccination can reduce the reproductive ratio by
2.3- and 2-fold, respectively. This method allows the comparison of vaccination efficacies among different viral
strains and animal models in vivo.
Human immunodeficiency virus (HIV) infects approxi-
mately 0.5% of the world population and is a major cause of
morbidity and mortality worldwide. A vaccine for HIV is ur-
gently required, and a variety of vaccine modalities have been
tested in animal models of infection. A number of these studies
have shown protection in monkey models of infection, al-
though the ability of the vaccine to protect appears to vary with
the viral strain and animal model used (8). The recent failure
of a large vaccine study in humans (1) suggests that further
understanding of the basic dynamics of infection and the im-
pact of vaccination are required in order to understand the
variable efficacies of vaccination in different infections.
The initial ability of HIV to propagate within the host is
determined by the abundance of target cells (e.g., CD4?T
lymphocytes) the virus can infect in order to produce progeny,
by the replicative capacity of the virus, and by how cytopathic
it is to infected cells. At later stages of the disease, in addition
to changes in the target cell availability, there may also be
changes in virus-specific properties, such as the ability of the
virus to reproduce and the survival of productively infected
cells as a result of changes in immune pressure and viral evo-
lution. These parameters may be variable among individuals
and within one individual over time and affect the impact of
vaccination. In order to compare the efficacies of vaccination
strategies, we need a quantitative measure of the factors that
influence virus replication.
In this work, we focus on the basic reproductive ratio (R0) as
a measure of vaccine efficacy in the acute phase of infection. In
epidemiology, R0measures the potential for the spread of an
epidemic and is defined as the average number of people
infected by one infected individual in a susceptible population.
If this number is below 1, the disease will not spread in a
population. Therefore, the knowledge of R0allows one to
estimate the fraction of a population that needs to be vacci-
nated in order to eradicate the disease. Analogously, in host-
pathogen dynamics, the basic reproductive ratio is defined as
the number of infected cells generated by one infected cell
during its lifetime at the start of infection, i.e., before any
depletion of target cells. In order for the infection to spread
within the host, this number has to be larger than 1; otherwise,
the infection will be cleared before it has a chance to spread.
The basic reproductive ratio will depend on the ability of virus
to infect cells (“infectivity”), on the rate of virion production by
infected cells, on the lifetime of infected cells, and on the rate
at which free virus is cleared. The aim of vaccines is therefore
to create virus-specific immunity that changes any of these
parameters and thus decreases the basic reproductive ratio of
the virus, preferably below 1 (so that infection does not spread
beyond the initially infected cells). The importance of R0in
HIV vaccination, in order to assess how much protection a
particular vaccine can achieve, has been stressed previously
(11). Specifically, a variety of different animal models have
been used to study HIV vaccination. They differ in a number of
ways, particularly in the virulence and cell tropism of the virus.
Comparison between models may be difficult because of dif-
ferences in the underlying dynamics of infection. For example,
it may be more difficult to prevent CD4?T-cell depletion in a
very virulent animal model than in other infections, because of
the higher underlying growth rate. Thus, a vaccine may appear
efficacious in one model but ineffective in another, despite the
fact that it may have the same impact on viral growth. Mea-
surement of R0allows comparison of the effectiveness of dif-
* Corresponding author. Mailing address: Complex Systems in Bi-
ology Group, Centre for Vascular Research, University of New South
Wales 2052, New South Wales, Australia. Phone: 612 9385 2762. Fax:
61-2-9385 1389. E-mail: email@example.com.
?Published ahead of print on 17 September 2008.
at UNIVERSITY OF NEW SOUTH WALES on November 11, 2008
ferent vaccines within a given animal model, as well as across
different models. It could be a very useful and common stan-
dard to compare different vaccination protocols.
The basic reproductive ratio can be determined from the
rate of exponential growth of virus in the initial period, during
which the target cell levels are almost constant (14, 20, 29).
However, in monkeys infected with the same type of simian
immunodeficiency virus (SIV) there is in fact very little varia-
tion in the basic reproductive ratio determined from the initial
exponential growth between controls and vaccinees, despite
the variety of outcomes later in infection. Different factors
identified as possible predictors of disease progression (29),
such as the peak viral load, target cell nadir, decay rate of virus
following peak viremia, set point viral load, and chronic target
cell levels, all correlate poorly or not at all with the differences
in the reproductive ratio determined from the primary expo-
nential growth phase (29). In particular, it is not possible to
assess the efficacy of cytotoxic-T-lymphocyte (CTL)-based vac-
cines on the basis of the basic reproductive ratio, because there
is no effect of vaccination on viral growth before approximately
10 days postinfection, which is at the end of the exponential
growth period (4, 6).
We have recently shown the existence of a strong correlation
between the viral load at peak and the target cell depletion in
the acute phase (7, 32). Using this correlation, one can show
that vaccination results in the reduction of the peak viral load
and of acute CD4?T-cell depletion, thus improving the chronic-
The dependence of the nadir in the CD4?T-cell count on
the viral peak can be obtained from the standard model of
virus dynamics (7, 32). Here, we show that this relationship
between the viral peak and the number of target cells 1 week
after the peak (corresponding approximately to the minimum
number of target cells in primary infection) is parameterized
by the basic reproductive ratio of the virus. In other words, the
decrease in the peak viral load leads to less target cell deple-
tion at nadir, and both are a consequence of a lower basic
reproductive ratio. Thus, we can in principle determine the
basic reproductive ratio from experimental data on the viral
peak and target cell nadir. We show that this relationship is
indeed supported by experimental data from CXCR4-tropic
simian-human immunodeficiency virus (SHIV) infection (for
viral loads and CD4?T-cell counts in peripheral blood) and
for CCR5-tropic SIV infection data (for plasma viral loads and
memory CD4?T-cell depletion in the gut).
We show that the reproductive ratio estimated from the viral
peak and the target cell nadir, which we call the “reproductive
ratio at the peak,” is significantly lower than the basic repro-
ductive ratio estimated from the exponential growth. In addi-
tion, we found that in vaccinated animals the reproductive
ratio at the peak is on average twofold lower than in control
animals. We attribute this difference to the cellular immune
response appearing before the peak viral load, around day 10
of infection, and changing the properties of the virus and
infected-cell dynamics (e.g., decreasing the lifetime of infected
cells through the cytolytic function of CTLs or changing infec-
tivity or virus production through the release of cytokines).
Thus, we propose that the “reproductive ratio at the peak,” a
measurement that includes information on both the viral peak
and the target cell nadir, can be useful as a standard to com-
pare vaccine protocols.
MATERIALS AND METHODS
In the CXCR4-tropic virus study, 35 rhesus macaques (Macaca mulatta) were
vaccinated with a variety of regimens, consisting of SIV gag-containing plasmid
DNA (with different adjuvants), modified vaccinia virus Ankara, and adenovirus
type 5 vectors, as previously reported (27). Animals were challenged intrave-
nously at 6 weeks (study A) or at 12 weeks (study B) after the final boost with 50
monkey infectious doses of SHIV-89.6P. Viral loads and CD4?T-cell counts
were monitored in peripheral blood every 2 to 4 days until 4 weeks after infection
and then weekly. The results of the two studies were combined for the purpose
of this work.
In the CCR5-tropic virus study, 20 rhesus macaques were challenged intrave-
nously with 100 monkey infectious doses of uncloned SIVmac251. Six of these
animals received prior vaccination with plasmid DNA encoding SIV envelope,
Gag, and Pol, and were boosted with recombinant adenovirus encoding the same
antigens. Plasma and jejunum tissue samples were collected at various time
points by biopsy or at necropsy, and tissue CD4?T-cell percentages and plasma
viral loads were determined (17, 18).
Reproductive ratio in the initial exponential growth phase of
disease. The standard model of virus dynamics (15, 21) de-
scribes the relationships between the change in the number of
uninfected cells, T, that are targets for the virus; infected cells,
I; and free virus particles, V, in a given volume of blood or
dt? ? ? dTT ? ?VT(1)
dt? ?VT ? ?I(2)
dt? pI ? cV(3)
The parameters ?, the production rate of new target cells, and
dT, the loss rate of uninfected cells, describe the disease-free
target cell dynamics. The disease-free equilibrium number of
target cells is equal to T0, which is equal to ?/dT. The infectiv-
ity, ?, characterizes the rate at which a virus particle infects a
target cell, and ? is the death rate of infected cells; ? ? ? dT.
Free virus is produced by infected cells at rate p and is cleared
at rate c. In this model, the immune response is assumed to be
constant, and its effects are contained in the virus parameters
?, c, ?, and p. All the parameters are assumed to be constant
in time (28). The basic reproductive ratio, R0, for this model is
as follows (15, 21):
The virus infection can spread only if R0is ?1. In this case, the
viral load generally increases to the peak and then decays, in
order to finally reach the steady-state value, V*. Target cells
drop to the minimum and then partly recover and settle at the
steady-state value, T*, which is equal to T0/R0. The standard
model has been used to investigate different aspects of HIV/
SIV/SHIV infection, such as the turnover of infected cells (7,
12, 31), the clearance rate of virus (23), and the “burst size” of
infected cells (3, 13).
11590PETRAVIC ET AL.J. VIROL.
at UNIVERSITY OF NEW SOUTH WALES on November 11, 2008
The basic reproductive ratio of a virus is determined from
the initial rate of exponential expansion, r0, during which the
number of target cells is approximately constant (14, 20, 25).
Thus, if we know the initial exponential growth rate of a virus
(r0' 1/V dV/dt) and the death rate of infected cells (?), we can
r0? ??R0? 1?
In Fig. 1A, we show viral-load data during the first weeks of
infection from the Shiver et al. study (27), in which 21 out of 35
rhesus macaques were vaccinated with seven different SIV
gag-containing vaccines, and all 35 were challenged intrave-
nously with CXCR4-tropic SHIV-89.6P. The initial exponen-
tial growth rate and the decay rate of virus after the peak were
remarkably similar between monkeys, irrespective of whether
they were vaccinated. The results of the statistical analysis are
summarized in Fig. 1C and D and Table 1.
Equation 5 indicates how to measure R0from the initial
growth rate of virus and the death rate of infected cells. The
initial growth rate of virus (r0) is directly estimated from the
experimental data. The death rate of infected cells (?) can be
estimated from the rate of decline of the viral load after the
peak in each animal. Using this approach, we estimate that R0
is 2.4 for vaccinated animals (95% confidence interval [CI], 2.3
to 2.6) and almost the same, 2.5, for control animals (95% CI,
2.3 to 2.7). The two values are not statistically different (Mann-
Whitney, P ? 0.391).
On the basis of the similarity in virus growth and decay rates
between control and vaccinated animals, we (4, 6) and others
(24) concluded that the cellular (CD8?T-cell) immune re-
sponse does not emerge before at least 10 days postinfection
(reviewed in reference 5). This is why it has not been possible
to assess the effects of vaccination from the reproductive ratio
of the virus calculated from its growth rate within this initial
period of 10 days.
Equation 5 considerably underestimates R0because it ne-
glects the delay between the time when a cell becomes infected
and the time when it starts producing the virus. In a model with
FIG. 1. Dynamics of acute SHIV-89.6P infection. The time courses of the viral load (A) and CD4?T-cell numbers (B) are shown over the first
40 days of SHIV infection for control (black lines) and vaccinated (red lines) animals. The initial viral growth rates (r0) (C) and viral decay rates
(?) (D) after the peak viral load of SHIV-89.6P do not differ significantly between controls (black dots) and vaccinees (red dots). However, the
peak viral load (E) and the nadir CD4?T-cell number (F) are significantly different between controls and vaccinees. In panels C to F, the gray
rectangles represent the confidence intervals.
TABLE 1. Summary of statisticsa
? (per day)c
Tmin(no. of cells/?l)e
Control VaccinatedControl VaccinatedControl Vaccinated ControlVaccinated
0.866–1.12 0.913–1.107.75–8.286.92–7.49 1.13–57.6137–339
aFor initial exponential growth rate, r0; decay rate of virus after the peak, ?; peak viral load, VP; and the target cell nadir, Tmin, in vaccinated and unvaccinated
bMann-Whitney P value, 0.83.
cMann-Whitney P value, 0.90.
dMann-Whitney P value, 0.00018.
eMann-Whitney P value, 0.00015.
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a fixed intracellular delay, ? (and assuming that all infected
cells survive the delay period) (10), R0is found from the initial
growth rate as follows (15, 25):
R0? ?1 ? r0/??er0?
With a delay of 1 day (22), we obtain an R0of 9.9 for vacci-
nated animals (95% CI, 8.6 to 11) and an R0of 11 for control
animals (95% CI, 8.6 to 12), again not different for the two
groups and consistent with previous studies of the basic repro-
ductive ratio in SIV infection (20).
As demonstrated above, the traditional approach to measur-
ing R0in vaccinated animals shows no significant difference in
R0between controls and vaccinees, despite clear differences in
infection outcomes. Thus, we aimed to develop a new measure
of viral growth that can quantify vaccine effects.
Virus peak and target cell nadir in the standard model of
virus dynamics. In contrast to the exponential growth and
decay rates of the viral load, the peak of the viral load and the
nadir of CD4?T-cells were significantly different between con-
trol and vaccinated animals (Fig. 1B, E, and F and Table 1).
Since the peak viral load and level of depletion of CD4?T cells
were important factors that were clearly affected by vaccination
and were correlated with outcome, we aimed to develop a
measurement of viral growth and vaccine efficacy based on
them. Obtaining an expression for the peak viral load or nadir
target cell numbers is quite difficult using the standard model
(equation 1 to equation 3), since they are dependent on a large
number of other parameters. Thus, we needed to simplify the
standard model to identify which parameters can be ignored
because in practice they have negligible influence on the peak
viral load and nadir targets. Before infection, target cells are at
a constant level, which implies that the daily production (?)
and death (dTT0) are balanced, and it is thought that ?1% of
cells turn over in this way each day. Thus, early in infection,
while there is very little depletion in target cell numbers (T ?
T0) (Fig. 2), we can neglect those two terms in equation 1
because they balance each other. Around the peak of viremia,
there is massive infection of target cells (?VT becomes very
large), and up to 99% of cells disappear over a period of 1 to
2 weeks. During this period, the normal production and loss of
target cells (? and dT) are much smaller than the loss due to
infection (i.e., ?VT ? ? ? ? dTT), and we can safely ignore the
effects of normal turnover on the target cell number. That is,
we can simplify the standard model to the following (7):
dt? ? ?VT(7)
dt? ?VT ? ?I (8)
dt? pI ? cV. (9)
In this reduced model, the viral load always vanishes at long
times (i.e., infection is always cleared if there is no natural loss
and replacement of target cells), with target cells steadily de-
clining toward a low steady-state value, Tmin. However, we can
further justify our approximation by noting that the profile of
viral-load and target cell dynamics during primary infection
(up to the target cell nadir, days 18 to 20 in Fig. 2) is only mildly
affected. As expected, the exponential growth rate of the virus
is the same as in the full standard model, and the viral load has
a maximum, VP, with a value very close to the peak of the
standard model (Fig. 2A).
Figure 2 also shows that the steady-state target cell number,
Tmin, in the reduced model is slightly lower, but still a very
good approximation for the uninfected target cell nadir of the
standard model in equation 1 to equation 3 if around 1% of
CD4?T cells turn over each day (i.e., for the parameters ? ?
10 cells/?l/day and dT? 0.01/day  usually attributed to
CD4?T cells in HIV/SIV/SHIV infection). The condition for
equation 7 to be a good approximation to the full model, that
is, ?VT ? ? ? ? dTT, is violated only very close to the nadir of
target cells, as the viral load decreases, and that is why we
obtain a good estimate of the nadir from this approximation.
This is true whenever the normal replacement or turnover of
CD4?T cells is relatively small.
Instead of uninfected target cells, the quantity that is usually
measured experimentally is the total number of uninfected and
infected cells (T ? I). Figure 2B shows that the minimum of
FIG. 2. A reduced model of viral dynamics. The time courses of virus growth and decay (A) and CD4?T-cell (target cell) decline in primary
infection (B) are shown, obtained by numerical integration of equation 1 to equation 3 and equation 7 to equation 9. (A) The peak viral loads for
the reduced model (?, dT? 0) and the standard model (? ? 10 cells/?l/day; dT? 0.01 day?1). (B) The nadir numbers of uninfected CD4?T cells
are indicated for the reduced model and the standard model. The total CD4?T-cell number (infected plus uninfected) is also indicated for the
standard model. The other parameters were as follows: ? ? 8 ? 10?8ml/copy/day, ? ? 0.8 day?1, P ? 103?l/ml ? 500 copies/cell/day, and c ?
20/day. The initial inoculum was 50 copies/ml.
11592PETRAVIC ET AL.J. VIROL.
at UNIVERSITY OF NEW SOUTH WALES on November 11, 2008
the total T-cell number is again very close to the minimum of
target cells. Therefore, we can compare the experimental nadir
target cell counts to the steady-state number of uninfected
cells, Tmin, in the reduced model.
An advantage of the reduced system in equation 7 to equa-
tion 9 is that one can estimate VPand Tminanalytically. We
shall use the analytical results as estimates for the viral peak
and the target cell nadir in the full standard model.
The basic reproductive ratio of the reduced model, R0, is still
given by equation 4 (with T0being an arbitrary initial target cell
number). The peak of viremia, as a function of R0, is as follows
(see the Appendix for the derivation):
which is valid for the reduced model (equation 7 to equation 9)
and a good approximation for the standard model, with low
target cell replacement and loss (equation 1 to equation 3).
Tminis found as the solution of the equation (see the Ap-
The fraction of uninfected target cells at the nadir in the full
standard model, approximately equal to Tmin/T0in equation
11, depends only on the basic reproductive ratio, R0. The
solution is illustrated in Fig. 3. For an R0of ?5, the value of
Tmin/T0is practically zero and the depletion at nadir is almost
100%. In Fig. 3, we plotted both Tmin/T0(the ratio at the target
cell nadir) and T*/T0(the ratio in chronic infection, i.e., in the
steady state, given by 1/R0) as functions of R0, showing that
Tminis always (for all R0) less than the steady-state target cell
Relationship between the virus peak and the target cell
nadir in the model. In the context of acute infection, i.e., the
time between the virus peak and the target cell nadir, we shall
call the expression in equation 4 the reproductive ratio at the
peak (and use the symbol RP) instead of the basic reproductive
ratio, in order to stress that the parameter values in this period
may have changed from those of initial infection. Thus, one
should replace the symbol R0by RPin equation 10 and equa-
tion 11. For a very low target cell replacement rate ? and loss
rate dT, the depletion (D) at the target cell nadir (D ? 1 ?
Tmin/T0[from equation 11]) depends only on the reproductive
ratio at the peak, RP, since this is the only free parameter in the
equation. The peak viral load (equation 10) depends on RP
and, in addition, on the ratio ?/?. This means that, for viruses
with different characteristic infectivities, ?, or different cyto-
toxicities and cellular immune responses resulting in different
death rates of infected cells, ?, the relationship between the
virus peak and the target cell minimum will be described by a
family of parallel sigmoid curves (Fig. 4A). RPvaries along
each curve with the characteristic ratio ?/?. Increases in infec-
tivity shift the curve to the left, while decreases in infectivity
shift it to the right.
Reducing the reproductive ratio at the peak for a given virus
decreases the peak viral load. The reduction of the peak by,
say, 0.5 log unit will have little effect on target cell depletion for
very high viral loads (when RPis high, there is a plateau at
nearly 100% depletion [Fig. 4]), but the same reduction can
significantly reduce CD4?T-cell depletion and change the
prognosis for lower viral loads (Fig. 4B).
The relationship between the peak viral load and the CD4?
T-cell nadir in experimental infection. The analysis above pro-
FIG. 3. The target cell nadir, Tmin, is lower than the steady-state
value, T*, for all values of R0.
FIG. 4. (A) The curve describing the relationship between the logarithm of the peak viral load and target cell depletion at the nadir has a
universal S shape and moves left or right only with the change in log(?/?). The constant ?/? is assumed to be characteristic of the virus type. Points
higher on the same curve have higher reproductive ratios at the peak RP. (B) The effect of lowering the virus peak depends on the basic
reproductive ratio and the peak viral load (i.e., on the value of ?/? and the peak viral load, which are likely characteristic of a particular virus).
For example, reducing the peak viral load by 0.5 log10copies/ml has a negligible impact on CD4?T-cell depletion in the shaded area to the right,
because depletion remains near 100%. However, for an infection with a lower R0and a lower peak viral load, the same reduction in the peak viral
load reduces CD4?T-cell depletion significantly (shaded area to left).
VOL. 82, 2008ESTIMATING VACCINATION IMPACT ON ACUTE SHIV/SIV INFECTION11593
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vided a theoretical relationship between the peak viral load
and the CD4?T-cell nadir. We then investigated whether this
relationship was observed in experimental models of HIV in-
fection. As an illustration, we present the SHIV-89.6P data
(27) for the peak of virus and the depletion at the CD4?T-cell
minimum. The advantage of considering a CXCR4-tropic virus
is that it infects “all” CD4?T cells, so that the target pool in
peripheral blood is well defined and can be measured. In ad-
dition, for the total number of CD4?T cells, it is safe to
assume that the net replacement rate, ?, from the external
source (thymus) and, consequently, the loss rate, dT, are low
(i.e., that less than 1% of the normal uninfected CD4?T-cell
number is replaced from the thymus per day).
In Fig. 5A we show the measured values of depletion of
CD4?T cells at the nadir, defined as (1 ? Tmin/T0) ? 100%,
against the peak viral load for the control and vaccinated
animals. The theoretical relationship between the virus peak
(equation 10) and the CD4?nadir (Tminin equation 11) for
each value of the reproductive ratio RPdepends on ?/?. We
have shown in our earlier work that ? and ? do not vary much
between individual monkeys, whether vaccinated or not.
Briefly, the death rate of infected cells is the main determinant
of the slope of the viral load after the peak, and it does not
differ between controls and vaccinees (Fig. 1A and D) (4, 6).
Also, we have estimated the infectivity of SHIV-89.6P in rhe-
sus macaques on the same data set and found that there is no
significant difference between vaccinated and unvaccinated an-
imals (7). As explained in Fig. 4A, the shape of the theoretical
sigmoid curve, D(VP), cannot be changed by fitting, since it is
parameterized only by RP. It is universal for all diseases that
are well described by the standard model, with low replace-
ment and loss rates of target cells. However, by shifting the
curve along the x axis, we can determine the best-fit value of
the factor ?/? that minimizes the sum-of-squares deviation of
experimental data from the theoretical curve (Fig. 4A). Since
the quantities described by both axes have errors, the squared
deviation of each point is defined as the sum of squares of the
deviations in x and y directions from the closest point on the
theoretical curve (a form of type 2 regression). We found that
the best-fit value of log(?/?) is 7.53 (?/? ? 3.39 ? 107copies/
ml) with a CI of 7.45 to 7.65 found by bootstrapping. The
correlation coefficient (r2) was 0.985.
In our previous work (7), we estimated infectivity from the
relationship between the peak viral load and the number of
CD4?T cells 1 week after the peak, using the experimental
viral-load timeline and the first two equations of the reduced
model, equation 7 and equation 8. The procedure required
advance knowledge of the death rate of infected cells, ?, which
we estimated from the maximum decay rate of the viral load
after the peak as follows: ? ? 0.84/day. This led to the best-fit
average infectivity, ?, of 4.4 ? 10?8ml copy?1day?1(95% CI,
between 3.4 ? 10?8and 5.6 ? 10?8ml copy?1day?1). Our
present estimate of ?/? as 3.39 ? 107copies/ml with a ? of
0.84/day would give a lower best-fit infectivity of 2.48 ? 10?8
ml copy?1day?1for the same death rate of infected cells.
However, since the death rate of infected cells used in the
previous work was the minimum estimate of the actual death
rate, it is possible that both ? and our best-fit infectivity are
Analyzing infection by CCR5-tropic viruses is complicated
because the number of target cells (CCR5?CD4?T cells) is
more difficult to measure. However, in the gastrointestinal
tract, most CD4?T cells express CCR5 coreceptor, so most
gut CD4?T cells are targets for CCR5-tropic viruses. In recent
studies, Mattapallil et al. (17, 18) measured CD4?depletion in
the jejunum in 20 rhesus macaques infected with CCR5-tropic
SIVmac251, 6 of which were previously vaccinated.
Nine animals were euthanized before the viral load reached
the first peak, so they could not be analyzed. Figure 5B con-
tains data points from the remaining 11 animals (7 control
animals and 4 vaccinated animals). The longitudinal data for
each animal contain only three results of gut biopsies with
percentages of CD4?T-cells out of all CD3?lymphocytes, and
the lowest percentage was always the last point. We considered
this point to be close to the target cell nadir.
We assumed slow turnover of memory CD4?T cells prior to
infection, consistent with the lack of reconstitution of these
cells during antiretroviral therapy (19). The decay rate of
SIVmac251-infected cells has been independently estimated as a
? of 1.49/day (2). By means of the same procedure using equa-
tion 7, equation 8, and the experimental viral load, we esti-
mated the infectivity as a ? of 1.45 ? 10?7ml copy?1day?1, so
that ?/? was 1.02 ? 107(32).
The line corresponds to the value of log(?/?), which is equal
to 7.20 (?/? ? 1.58 ? 107copies/ml), which minimizes the
sum-of-squares deviation of experimental data from the theo-
FIG. 5. Fitting theoretical curves to experimental data. Best-fit curves for the relationship between CD4?T-cell depletion at the nadir and the
peak of virus (from equation 10 and equation 11) for SHIV-89.6P data (7, 27) (A) and for SIVmac251data (B) (17, 18, 32). The solid lines are the
best-fit curves obtained by optimization of the value of ?/?.
11594PETRAVIC ET AL.J. VIROL.
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retical curve [the CI for log(?/?) is 7.09 to 7.59; the correlation
coefficient, r2, is 0.874). The scatter of the experimental points
around the theoretical curve is much larger than for SHIV
infection in Fig. 5A. One reason for the larger scatter is the
infrequent sampling in the longitudinal data (each animal had
only one biopsy before challenge and one postchallenge and a
necropsy), which probably caused imprecision in the estimates
of the minimum. One would need in principle several samples
in the 2- to 3-week period after the peak in order to correctly
estimate the nadir. Another reason for the scatter could be that
the depletion was calculated from the change in the fraction of
CD4?T cells (out of all CD3?cells) instead of the CD4?
T-cell count used in the model. The two ways of calculating
depletion would be equivalent only if the number of CD3?
lymphocytes stayed constant during the infection. The in-
creased scatter of experimental data compared to SHIV infec-
tion can also be observed in the curve fit in reference 32.
Vaccination reduces the reproductive ratio at the peak of the
virus. The fitting of the model to experimental data suggests
that the theoretical curves are a good predictor of the relation-
ship between the viral load and CD4?T-cell depletion in
SHIV/SIV infection. Thus, we went on to use this approach to
estimate the impact of vaccination on the reproductive ratio at
peak. We first applied this approach to the experimental data
on SHIV89.6P infection. Figure 5A shows that vaccinated an-
imals have on average significantly lower peak viral loads (vac-
cinated logVP? 7.19 [95% CI, between 6.92 and 7.47]; unvac-
cinated logVP? 8.03 [95% CI, 7.78 and 8.28]; Mann-Whitney,
P ? 0.0004) and lower CD4?T-cell depletion at minimum
(vaccinated Dmin? 81.7% [95% CI, 71.4 and 92.1%]; unvac-
cinated Dmin? 97.4% [95% CI, 95.2 and 99.7%]; Mann-Whit-
ney, P ? 0.0003), so we expect the average reproductive ratio
at the peak to be lower in vaccinated animals. The results are
summarized in Fig. 6.
The change in the reproductive ratio from R0to RPis as-
sumed to be the result of the change in the virus-dependent
parameters due to the onset of cellular immunity a few days
before the peak of the viral load, i.e., before significant deple-
tion of target cells (5). The relationship between the virus peak
and the CD4?T-cell nadir in SHIV-89.6P is consistent with
slow replacement of uninfected cells, so that in this case, the
depletion at nadir can be considered to be completely deter-
mined by the peak reproductive ratio, RP, of the virus (equa-
tion 11). From the range of the nadir CD4?T-cell depletions
and peak viral loads, we can determine the range of the repro-
ductive ratios at the peak of SHIV-89.6P in the infected ani-
mals in the Shiver et al. study (18). Therefore, we estimate the
range of RPfrom the range of peak viral loads and the line of
best fit in Fig. 5A, assuming ?/? ? 107.53. The obtained range
of RPis between 1.5 and 19.2. The peak reproductive ratio, RP,
for vaccinated animals is 3.81, with a 95% CI between 2.3 and
4.6. For unvaccinated animals, RPis 8.87 with a 95% CI be-
tween 4.8 and 11.0. Vaccinated animals have on average a
significantly lower peak reproductive ratio (Mann-Whitney,
P ? 0.0007).
Since the 21 animals in the Shiver study were vaccinated with
seven different types of vaccines, all we can conclude is that, in
general, the effect of vaccination is to reduce the peak repro-
ductive ratio of the virus. The amount by which RPis reduced
with respect to the average in control animals would reflect the
effectiveness of a particular vaccine.
We obtained similar reduction in the reproductive ratio at
the peak for SIVmac251in the jejunum tissues of the macaques
in the Mattapallil et al. study (17, 18). The vaccinated animals
had an RPvalue of 1.8, and control animals had an RPvalue of
3.6; however, the difference did not reach significance (Mann-
Whitney, P ? 0.073) due to the small sample size (11 animals).
Vaccination trials have been used to compare the effective-
ness of different vaccines in a number of monkey species, and
using a wide variety of vaccine modalities. The efficacies of
different vaccines can then be ranked by comparison of their
abilities to preserve CD4?T-cell numbers or reduce peak viral
loads. Such comparisons can be misleading at times, since in
viruses with a higher R0, it is much more difficult to prevent
CD4 T-cell depletion than in viruses with low R0. Thus, the
ideal metric for ranking vaccines would allow comparison of
efficacies both within one infection model and across multiple
infection models. Moreover, such a metric should also allow
consideration of whether a given vaccine may be more effective
in more virulent infections (with high R0) or less virulent in-
fections, in order to predict which will perform best in HIV.
Previous approaches to comparing viral virulence have mea-
sured the basic reproductive ratios (R0) of different infection
models. The basic reproductive ratio contains all model pa-
rameters in the initial exponential growth phase of the virus.
While R0may be useful in comparing different animal models
of infection, it is not useful in quantifying the impact of vacci-
nation. In particular, it does not reflect the effects of the cel-
lular immune responses (or, consequently, the effects of vac-
cination) in primary infection, because in current vaccines
these effects become evident only at the end of the early ex-
pansion period (4, 5).
In order to overcome this limitation, we used a modeling
approach to develop a new method of estimating both the
FIG. 6. Measuring the impact of vaccination. The values of the
basic reproductive ratio (R0) and the reproductive ratio at peak (RP)
were estimated for SHIV-89.6P infection. Distributions of values of
the basic reproductive ratio, R0, obtained from the initial exponential
growth rate with a fixed intracellular delay of 1 day show no difference
between control (solid circles) and vaccinated (solid diamonds) ani-
mals. The values of the peak reproductive ratio, RP, (estimated from
the virus peak and the target cell nadir) are reduced approximately
twofold in vaccinated animals (open diamonds) compared to controls
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at UNIVERSITY OF NEW SOUTH WALES on November 11, 2008
virulence of infection and the impact of vaccination, which we
term the reproductive ratio at peak. We first showed that the
relationship between the virus peak and the target cell nadir in
the acute phase of the CXCR4-tropic SHIV infection and
CCR5-tropic SIV infection is consistent with the standard
model of virus dynamics. This relationship allowed us to esti-
mate the decrease in reproductive capacity of the virus caused
by vaccination. We have defined the reproductive ratio at the
peak as the same function of the standard model parameters as
in the basic reproductive ratio but evaluated in the period
between the viral peak and the CD4?T-cell nadir. In this
period, CTL response is the main factor responsible for the
change of parameters. We found that in both SHIV and SIV
infections, vaccination on average reduced the reproductive
ratio at the peak between 2 and 2.5 times. This was not suffi-
cient to clear the virus. In order to suppress the infection
during the first few weeks, vaccines should decrease the repro-
ductive ratio below unity, i.e., approximately eightfold for
SHIV and fourfold for SIV. In the absence of CTL responses,
the basic and the peak reproductive ratios would probably be
the same (this is in principle testable in macaques with anti-
CD8 treatment). However, our method cannot determine spe-
cifically which parameters have changed as a result of the CTL
response, only that the compound quantity RPhas changed
considerably as a result of vaccination.
The disadvantage of using the reproductive ratio at the peak
as a measure of immune pressure is that the method requires
knowledge of the target cell numbers, which is straightforward
to measure in blood for CXCR4-tropic viruses but may require
sampling of mucosal sites for CCR5-tropic viruses, like HIV
(9). In the present study, we have also focused on CD4?T cells
as the major targets for HIV infection; however, macrophages
and other cells may also play a minor role (26, 30). With this
limitation, the method would be applicable to a variety of viral
The basic reproductive ratio at peak permits comparisons of
vaccine efficacies between studies performed in different ani-
mal models of infection. In addition, it also has the inherent
threshold of unity in its definition. The idea of reducing the
basic reproductive ratio of the virus below unity by inducing
specific immune response has been present in the context of
HIV vaccination for a long time (11). However, we believe that
we present the first method that allows evaluation of the im-
pact of vaccination using such a concept, in this case, the
“reproductive ratio at the peak.”
Derivation of equation 10 for virus peak. Let us assume that
at the start of infection, t ? 0, we have an initial target cell
number T0and an initial viral load V0, with an initial number
of infected cells, I0, equal to 0. If the peak viral load, VP, occurs
at time tPand the target and infected cell numbers at tPare TP
and IP, respectively, then
dt? pIp? cVP? 0(A1)
If we assume that V and I peak at the same time, which is true
when c is ? ??I, then from
dt? ?VPTP? ?Ip? 0 (A3)
it follows that
?p? T* ?T0
where T* is the steady-state target cell number in the full
standard model, equation 1 to equation 3. The lag between the
peak of infected cells and the viral load becomes negligible in
the limit ? ? ? c. This is justified in the case of HIV/SIV/SHIV
infection, where the parameter estimates are as follows: ? ?
1.0/day and c ? 20/day (16, 23).
dt? ? ?VT(A5)
and equations A1 and A3 from t ? 0 to t ? tPand using
equations A2 and A4, we get the following expression for the
c ? ?T0?1 ?
c ? ?V0?
Since ? is ? ?c and the initial virus concentration in blood, V0,
is usually much lower than T0, the first term on the right-hand
side of equation A6 can be neglected compared to the second
term. The peak viral load then depends only on the constant
parameters of the system (Fig. 7A). Since ? is ? ?c, from equa-
tion A6 we obtain the approximate expression for the virus
Derivation of equation 11 for the target cell minimum. In the
infinite time limit, all three derivatives, equation 7 to equation
9, must vanish. This means that the steady-state values of both
V and I are zero, while the steady-state value of target cells,
Tmin, is undetermined.
We can find Tminby integration. We substitute the expres-
sion for the viral load from equation A5,
V ? ?1
into the second term on the right-hand side of equation A1 and
integrate over time from zero to infinity. Taking into account
that the infection is always cleared, i.e., that the viral load
vanishes in the long time limit, we obtain the following result:
Next, we integrate both sides of equation A3 and substitute the
expression A9 for the integral of infected cells. We note that
11596PETRAVIC ET AL.J. VIROL.
at UNIVERSITY OF NEW SOUTH WALES on November 11, 2008
the integral on the left-hand side of equation A3 vanishes at
both limits (at t ? 0 and at t ? ?), leading to
After substituting equation A10 into equation A5 and integrat-
ing from zero to infinity, we finally obtain the equation for
? 1 (A11)
In deriving equation A11, we have again neglected the initial
viral load in the inoculum (V0? 0).
This work was supported by the James S. McDonnell Foundation
21st Century Research Award/Studying Complex Systems and the Aus-
tralian National Health and Medical Research Council. M.P.D. is a
Sylvia and Charles Viertel Senior Medical Research Fellow. R.M.R.
was supported by grant P20-RR18754 from the National Institutes of
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