Variation in HIV-1 set-point viral load: Epidemiological
analysis and an evolutionary hypothesis
Christophe Fraser†‡, T. De ´irdre Hollingsworth†, Ruth Chapman†§, Frank de Wolf†¶, and William P. Hanage†
†Department of Infectious Disease Epidemiology, Faculty of Medicine, Imperial College London, London W2 1PG, United Kingdom;§Infectious Diseases
Epidemiology Unit, London School of Hygiene and Tropical Medicine, London WC1E 7HT, United Kingdom; and¶HIV Monitoring Foundation, Academic
Medical Centre of the University of Amsterdam, 1105 AZ, Amsterdam, The Netherlands
Communicated by Stanley Falkow, Stanford University, Stanford, CA, September 10, 2007 (received for review February 9, 2007)
The natural course of HIV-1 infection is characterized by a high
degree of heterogeneity in viral load, not just within patients over
time, but also between patients, especially during the asymptom-
atic stage of infection. Asymptomatic, or set-point, viral load has
been shown to correlate with both decreased time to AIDS and
epidemiological impact of heterogeneity in set-point viral load. By
analyzing two cohorts of untreated patients, we quantify the
relationships between both viral load and infectiousness and the
duration of the asymptomatic infectious period. We find that,
because both the duration of infection and infectiousness deter-
mine the opportunities for the virus to be transmitted, this sug-
gests a trade-off between these contributions to the overall
transmission potential. Some public health implications of varia-
tion in set-point viral load are discussed. We observe that set-point
viral loads are clustered around those that maximize the transmis-
sion potential, and this leads us to hypothesize that HIV-1 could
have evolved to optimize its transmissibility, a form of adaptation
to the human host population. We discuss how this evolutionary
hypothesis can be tested, review the evidence available to date,
and highlight directions for future research.
cohort studies ? life-history ? mathematical model ? trade-off ? virulence
density in other body compartments and to viral replicative
capacity is unclear (1–3), it has a proven track record in the
prognosis of patients (4, 5) and has more recently been shown to
predict the probability of transmission between discordant cou-
ples (6, 7). Viral load is heterogeneous both within patients over
of infection, viral loads fluctuate around a steady set-point value,
which varies up to 1,000-fold between patients (4, 5).
Although much work has considered the significance of
primary infection in HIV-1 transmission (e.g., refs. 8 and 9), it
is not known which set-point viral loads have the greatest
epidemiological impact, in terms of leading to the greatest
number of infections over the lifetime of the host. Such infor-
of targeting prevention efforts at patients with certain subsets of
viral loads: high, low, or intermediate. Although it is clear that
patients with higher viral loads will be more infectious (6, 7), it
is also well known that, untreated, these individuals have a
poorer prognosis and, hence, will have fewer lifetime opportu-
nities for transmission (4, 5). The epidemiological impact of
different viral loads will hence be determined by the interplay
between these two antagonistic processes.
The aim of this study is to use available data on infectiousness
(6, 7, 9) and duration of infection (5, 10) to determine, in general
terms, the epidemiological impact of cross-sectional variation in
viral load. We estimate the product of infectiousness and dura-
tion of infection, which we term the transmission potential. This
is the mean number of persons one index case can potentially
iral load, a measure of the density of virus particles in
peripheral blood, is an imperfect but important measure of
infect over their whole asymptomatic period, estimated as a
function of set-point viral load.
We quantify the transmission potential as a function of
set-point viral load and find that it is maximized for intermediate
viral loads, which we observe are also the most common among
untreated patients. Although individuals with high viral loads
are the most infectious in the short term, the total contribution
to infection of those with intermediate viral loads is found to be
larger because of the longer duration of asymptomatic infection.
The consequences for public health and an evolutionary hypoth-
esis arising from this observation are discussed.
Set-Point Viral Load and the Duration of Asymptomatic Infection.
HIV-1 viral load follows a characteristic U-shape during the
course of untreated infection, highest at the start (primary stage)
and end (late stage) of infection, whereas lower and relatively
steady levels are maintained for a variable number of years
during asymptomatic infection. Levels measured in peripheral
blood during the asymptomatic period are very variable, ranging
from 1,000 to 1 million viral copies per milliliter, and this
quantity is positively correlated with viral levels in other body
compartments (1–5). We aim to study its relation to the duration
of infection using a flexible parametric model in a robust
To provide good quantitative detail, we focus on the Amster-
dam seroconverters cohort, where homosexual men were re-
cruited prospectively to study the incidence and natural history
of HIV-1 infection from 1982 onwards and followed for many
years; the cohort has been described elsewhere (5). We censor
all observations after 22 November 1993, the date when the first
protease inhibitor was used in this cohort, to avoid biases caused
by the availability of effective treatment and ignore all treatment
effects before this date; our sample size is 123 men, followed for
504 person-years. We determine set-point viral loads as the
geometric mean viral load in the interval between the end of
primary infection (defined as 6 months after first seropositive
sample) and the first AIDS-defining event (CDC type C) or
censoring, whichever occurs first. The distribution of these
set-points is plotted in Fig. 1, as well as the distribution of viral
loads collected from a Zambian cohort (7), referred to in more
We describe the duration of the asymptomatic stage of
Author contributions: C.F., F.d.W., and W.P.H. designed research; C.F., T.D.H., and R.C.
performed research; C.F. contributed new reagents/analytic tools; C.F., T.D.H., and R.C.
analyzed data; and C.F., T.D.H., R.C., and W.P.H. wrote the paper.
The authors declare no conflict of interest.
Freely available online through the PNAS open access option.
‡To whom correspondence should be addressed at: Department of Infectious Disease
Epidemiology, Faculty of Medicine, Imperial College London, St Mary’s Campus, Norfolk
Place, London W2 1PG, United Kingdom. E-mail: firstname.lastname@example.org.
This article contains supporting information online at www.pnas.org/cgi/content/full/
© 2007 by The National Academy of Sciences of the USA
October 30, 2007 ?
vol. 104 ?
no. 44 ?
infection by a flexible parametric model described in Methods.
The model describes the change in the mean duration as a
function of viral load and also allows for variability in the
duration, given a value of the set-point viral load. The best-fit
model is shown in Fig. 2. This demonstrates a pattern of decline
in the duration of asymptomatic infection with increasing viral
load with, as expected, more uncertainty in the estimates for
atypically high or low viral loads. Estimates of the mean duration
of asymptomatic infection range from 15.6 years [95% confi-
dence interval (c.i.), 9.4–31.3] for a set-point viral load of 1,000
viral copies per milliliter, through 9.7 years (95% c.i., 7.7–12.9)
for 10,000 copies, 4.9 years (95% c.i., 4.1–6) for 100,000 copies,
to 2.1 years (95% c.i., 0.9–3.7) for 1 million copies. Because these
estimates rely on the use of parametric forms to extrapolate to
extreme viral loads, we tested the use of a very general survival
function, the generalized gamma distribution, but this did not fit
the data significantly better (P ? 0.95). We also considered
allowing for the possibility that the mean duration could plateau
to a low nonzero value at high viral loads, but this also failed to
significantly improve the fit (P ? 0.18) [see Methods and Methods
in supporting information (SI) Text].
Infectiousness and Viral Load. Several studies have empirically
estimated the rates of HIV transmission in stable heterosexual
partnerships as a function of HIV-1 load (6, 7). Because these
studies have focused on demonstrating the significance of the
association rather than the functional relationship, we reana-
lyzed the data from the Zambian study (7), using a flexible
parametric function to describe the dependence of the annual
transmission rate within a partnership on viral load. The best-fit
Notably, we find that the transmission rate tends to reach a
plateau at high viral loads. To confirm that this was not an
artifact of our parametric assumptions, we plot the directly
inferred transmission rate for eight groups of subjects classed in
increasing octiles of viral load in Fig. 3B, where it is clearly seen
five octiles. Estimates of the annualized transmission rate within
(95% c.i., 0.001–0.084) for a set-point viral load of 1,000 viral
copies per milliliter, through 0.132 year?1(95% c.i., 0.08–0.223)
for 10,000 copies, 0.279 year?1(95% c.i., 0.223–0.343) for
100,000 copies, to 0.313 year?1(95% c.i., 0.233–0.471) for 1
million copies. Fig. 3A also shows the transmission rate inferred
(copies per milliliter of peripheral blood) is plotted for untreated individuals
in the Amsterdam Seroconverters Cohort (black bars) and the Zambian Trans-
mission Study (7) (gray bars). The bars represent bins 0.5 log10wide and are
labeled by their midpoint viral load.
mean duration, in years, of the asymptomatic stage of infection is estimated
as a function of viral load. (A) Best-fit and 95% confidence interval estimates
the asymptomatic period are shown for patients grouped into quartiles of
viral load (jagged lines, with crosses showing censored patients), along with
model predictions (smooth lines).
Set-point viral load and duration of asymptomatic infection. The
within a stable discordant partnership is estimated as a function of viral load.
(A) Best-fit and 95% confidence interval estimates of the transmission rate
based on a parametric model fitted to the data of ref. 7. The data from the
Rakai study (6) are shown for comparison (dashed line). (B) We also plot the
transmission rate as a function of the geometric mean viral load for subjects
grouped into ascending octiles of viral load for these data. This shows strong
evidence for saturation of the transmission rate at high viral loads.
www.pnas.org?cgi?doi?10.1073?pnas.0708559104 Fraser et al.
from a cohort of HIV serodiscordant subjects in Rakai, Uganda
(6), which are consistent with these estimates. Both these studies
(6, 7) involved a degree of counseling to reduce unprotected sex,
so transmission rates within uncounseled partnerships could be
somewhat higher, although the dependence on viral load would
be similar. These two studies of transmission also recorded very
different rates of unprotected sex despite observing similar
transmission rates, an observation that motivated our choice of
focusing on the transmission rate per unit time rather than the
more conventional but apparently less reliable choice of report-
ing the transmission probability per unprotected sex act.
Transmission Potential. One way of summarizing the epidemio-
logical contribution of individuals with different set-point viral
loads is to estimate the expected number of people infected over
their entire infectious lifespan. Many factors can affect this,
including host behavior, coinfections, and the state of epidemic
itself, because opportunities for transmission are reduced when
prevalence is already high. We define a quantity, which we call
the transmission potential, as the average number of people
potentially infected over the duration of the whole infectious
period, in circumstances where most people are uninfected, for
an infected individual with a particular viral load; where the rate
of partner change is sufficiently high that it does not limit
transmission, and where the transmission rate within partner-
ships is similar to that reported by the two cohorts studied here.
We relax the second, simplifying but not crucial assumption
regarding the partner change rate in Methods in SI Text. We
average over all cofactors affecting transmission apart from
set-point viral load. The transmission potential relates to the
better known basic reproduction number R0 discussed below,
which is obtained by specifying a sexual mixing model and
averaging over the distribution of set-point viral loads.
A particular concern at this stage is that we have analyzed data
on the duration of asymptomatic infection as a function of viral
load from a population of Dutch homosexual men infected with
HIV-1 subtype B, whereas we have considered data on infec-
tiousness from two populations of Zambian and Ugandan het-
erosexual individuals infected with mixed subtypes of virus.
However, there is some evidence that the relation between viral
load and duration of asymptomatic infection is relatively inde-
pendent of subtype, population, or setting. One study of sero-
converting women in Uganda found that the time to AIDS
country cohorts (11), whereas the prevalence of general symp-
toms that may or may not be attributable to HIV infection
(defined as WHO stage 2 and 3 events) is much higher (12). In
SI Fig. 6, we show a direct side-by-side comparison of survival
rates between the Amsterdam seroconverters analyzed here and
a cohort of untreated female commercial sex workers in Nairobi,
Kenya, followed since seroconversion (10); these show similar
survival rates for individuals in similar viral load classes. In
situations where this equivalence between populations and set-
tings holds, our method provides a good estimate of the trans-
mission potential for untreated infection in a heterosexual
The contribution of the asymptomatic infection stage of
infection to the transmission potential depends on set-point viral
load and is the product of the transmission rate and the duration
of infection, plotted in Fig. 4. At low viral loads, the transmission
potential is limited by low infectiousness, whereas at high viral
loads where infectiousness is maximized, the transmission po-
tential is limited by the short duration of the infectious period.
Key to this shape is the observation that infectiousness does not
appear to increase rapidly for viral loads over ?100,000 copies
per milliliter. The inference is robust to variation in the choice
of parametric model used for estimation (SI Fig. 7).
The periods of highest viral load are found at the start of
infection, during a brief period of uncontrolled replication
before the host immune system gains temporary control and,
later, during the development of AIDS. These are also the
periods at which the host will be most infectious (9). However,
these periods are very short, and, within them, viral loads differ
little between hosts, whereas there is great heterogeneity in
set-point viral load. Therefore whatever the contribution of
primary and end-stage infection to transmission potential (ex-
plored in detail in T.D.H., R. M. Anderson, and C.F., unpub-
in transmission potential among individual hosts are determined
by set-point viral load. The transmission potential for these
stages is 0.67 (0.32–1.23 95% c. i.) for primary infection and 0.50
(0.31–0.96 95% c. i.) for pre-AIDS/AIDS. However, the assump-
tion that partner change is frequent enough for it not to be a
limiting factor is not likely to remain valid during these short
periods of high infectiousness, and thus the transmission poten-
tial of these stages is less likely to be realized than the trans-
mission potential of asymptomatic infection. Estimates for a
variety of parameters in a simple ‘‘serial monogamy’’ scenario
are explored in SI Figs. 9A and 10A.
Strengths and Frailties of the Transmission Potential Analysis. We
analyzed large, well studied cohorts using robust statistical
methods. The models used for inference are sufficiently flexible
that analyses are unlikely to be too dependent on the precise
parametric forms chosen (see Methods in SI Text for some
sensitivity analysis). The main assumption, that both infectious-
ness and duration of infection are, respectively, increasing and
decreasing monotonic functions of viral load, is well supported
and biologically plausible.
A significant limitation of our analysis that could be addressed
in future work is that we had only data available from different
sources for estimating the different parameters. We noted that
the relation between the duration of asymptomatic infection and
set-point viral load is similar in some different settings (10–12),
but this may not be universal. For example, one study reported
rapid disease progression in subtype D infections not explained
by higher than expected viral loads.?A further concern is the
?Laeyendecker, O., Li, X., Arroyo, M., McCutchan, F., Gray, R., Wawer, M., Serwadda, D.,
Progression in Rakai, Uganda, Abstract 44LB, 13th Conference on Retroviruses and
Opportunistic Infections, February 5–8, 2006, Denver, CO, www.retroconference.org/
2006, accessed April 27, 2007.
expected number of people one case could infect over the whole course of
asymptomatic infection, based on random contacts with susceptible individ-
uals. It is the product of the transmission rate and the mean duration of the
asymptomatic period (Figs. 2A and 3B) and is shown plotted with 95%
confidence intervals as a function of viral load.
Transmission potential. The transmission potential is defined as the
Fraser et al.
October 30, 2007 ?
vol. 104 ?
no. 44 ?
possible covariance between duration of infection and infec-
tiousness, which would imply that the transmission potential,
defined as the average of the product of duration and infec-
tiousness, might not be well approximated by the product of the
averages estimated here. Such a situation could arise because of
on factors that independently affect survival or infectiousness,
such as the dependence of infectiousness on sexual risk behavior,
would not be problematic. Resolution of these concerns could be
addressed by direct estimation of the transmission potential
within a single patient cohort.
Consequences of Variation in Set-Point Viral Load for Public Health.
Comparison of Figs. 1 and 4 shows that individuals with com-
mon, intermediate viral loads have the largest transmission
potential. For current public health initiatives based on the mass
deployment of antiretroviral therapy, we suggest that to attempt
to maximize indirect population benefits by singling out those
with the highest viral loads for treatment would be misguided (a
strategy explored but not advocated in ref. 13), because it is
actually the majority of patients with intermediate viral loads
who ultimately cause the most infections. Although indirect
population benefits of mass therapy are possible and desirable,
treatment protocols in areas of limited resource should use other
inclusion criteria, such as clinical need, likelihood of treatment
adherence, or sexual behavior.
As for future mass interventions based on, for example,
imperfect vaccines, immunotherapy, or microbicides, this frame-
work offers a simple tool for predicting so-called ‘‘perverse
outcomes’’ (14, 15). If the intervention reduces patients’ viral
load in such a way as to increase their transmission potential on
average, then incidence will increase, not decrease. An inter-
vention that reduces viral loads from high to intermediate levels
and is therefore beneficial to the individual may nevertheless
benefit. The ultimate outcome of any intervention that changes
the distribution of viral loads can be predicted by calculating the
change in the mean transmission potential. These conclusions
are based on the epidemiological analysis of variation in set-
point viral load and are independent of the evolutionary discus-
sion that follows.
A Hypothesis: The Evolution of HIV-1 Virulence. The viral load that
maximizes the transmission potential is 4.52 log10 copies per
milliliter (Fig. 4), close to the observed means of 4.36 and 4.74
for the Dutch and Zambian cohorts, respectively (Fig. 1). Viral
loads during the asymptomatic period are clustered around
values that maximize the transmission potential of the virus. Is
it possible that this is not coincidence but, rather, an outcome of
natural selection acting on HIV-1 to maximize opportunities for
onwards transmission? This would suggest that HIV-1 conforms
to the classical adaptive virulence model: Seen from the per-
spective of the virus, a negative correlation between infectious-
ness and duration of infection could be interpreted as a trade-off
between two viral life-history traits, with natural selection
leading to an optimal balance in this trade-off (16, 17).
This adaptive virulence model for HIV-1 results in a number of
clear predictions that could be regarded as tests of the hypothesis.
loads needs to be consistent with an evolutionary interpretation of
the life-history tradeoff in the transmission potential. Our analysis
supports this. Second, the hypothesis predicts that set-point viral
loads in transmitter and recipient will be correlated. If a trait is
heritable, the conclusion that natural selection can act on it follows
automatically. Conversely, if this is not the case, it is impossible for
natural selection to act on a trait, no matter what its relationship to
fitness might be.
No studies have satisfactorily addressed the question of her-
itability in viral load to date. What evidence there is, direct and
indirect in terms of other indicators of heritability in viral traits,
is reviewed in Discussion in SI Text. We leave this as an open,
testable prediction generated by our study.
Although there is no consensus on the dominant mechanisms
evolution (e.g., ref. 18) to pathological host immune activation
(e.g., refs. 19 and 20). Viral load has been implicated as a
measure of viral replication, which itself may regulate the rate of
cell destruction, and mathematical models can capture the
relation between set-point load and duration of the incubation
period of AIDS (e.g., ref. 21). The adaptive-virulence model
does not negate the role of host immune or environmental
that viral genetic factors modulate this progression, leading to
marginally more or less severe distributions of outcomes in
More specifically, the model predicts that the viral population
will eventually become dominated by the ‘‘strain’’ with the
largest basic reproduction number R0, defined as the number of
individuals infected by a typical index case in a totally susceptible
population (16, 17). However, in an emerging epidemic, some
strains could initially spread faster before being replaced by
others with higher R0[so-called r-selection (22)]. We attempt to
disentangle these processes as follows.
Consider a transmission model with multiple hypothetical
viral strains. Each strain is characterized by a distribution of
set-point viral loads (representing the effects of host, environ-
mental and chance variability), but some strains have an inher-
ited tendency to produce slightly higher or lower viral loads, on
average. We use an ‘‘age-of-infection’’ framework (23) to cal-
culate the basic reproduction number, R0, and the initial expo-
nential rate of spread, r0, of strains characterized by their mean
set-point viral load. The details of the calculation, of assumed
sexual mixing, and sensitivity to the parameters are presented in
Methods in SI Text. R0is maximized for a mean set-point viral
load of 4.34 log10copies per milliliter, whereas the exponential
log10copies per milliliter. These predicted ‘‘optimal’’ values are
close to the mean viral loads observed in the cohorts (Fig. 5).
Because both the negative relation between viral load and
duration of infection and the positive relation between viral load
and infectiousness can be understood in simple biological terms,
the tradeoff between these in producing the peaked transmission
potential curve does not, per se, suggest viral adaptation. How-
duction number R0(solid line) and the initial epidemic exponential growth
rate (dashed line) are estimated as functions of the mean set-point viral load
of a hypothetical viral ‘‘strain.’’ The viral load values that maximize these
cohorts are shown as circles (open for Zambia and filled for Amsterdam
Optimal viral loads in a simple transmission model. The basic repro-
www.pnas.org?cgi?doi?10.1073?pnas.0708559104Fraser et al.
ever, viral adaptation does provide a natural explanation for the
relatively good agreement between the calculated optimal and
observed distribution of viral loads.
An area requiring further development is the study of inte-
grated models for exploring multilevel selection for understand-
ing the differential roles of selection for viral replication at the
cell-cell level (within the host) and host-host level (involving
transmission). Although it may be thought that high viral
turnover and mutation rate would favor within-host adaptation
[so-called short-sighted evolution (24, 25)], these factors do not
seem, in practice, to lead to any erosion of infectiousness during
HIV-1 infection (9). There is also a need to explain the diversity
of virus–host patterns for lentiviruses, to identify the determi-
nants of virulence in related lentiviruses (such as HIV-2 and the
simian (SIV) ancestors of HIV-1 and HIV-2 in chimpanzees and
sooty mangabeys, respectively), and their relation to infectious-
ness and survival. A more detailed discussion of these challenges
is included in Discussion in SI Text.
To summarize, we have quantified the transmission potential of
HIV-1 as a function of set-point viral load and have found that
the most common set-point viral loads result in nearly optimal
transmissibility over the lifetime of the host. Crucial to these
analyses were the availability of good long-term longitudinal
data and the use of robust statistical methods to parameterize
the dependence of infectiousness and duration of infection on
and in different settings. We have hypothesized that this situa-
tion could have arisen because of adaptive evolution of HIV-1
to maximize transmission between humans, although the agree-
ment between observed viral loads and the maximum of the
transmission potential could also, of course, be an interesting
coincidence. The phenomenon of adaptive virulence, if verified,
would have practical consequences in terms of the potential for
public health interventions to impact on virulence (26). There
may be as yet unidentified viral genetic factors that modulate the
severity of infection.
We have explored this evolutionary hypothesis, developed
testable predictions, and highlighted conceptual challenges.
Testing for the existence of differences in viral load or virulence
between populations and testing whether viral load is a trait
heritable from one infection to the next are questions that could
be answered with simple study designs. More detailed predic-
tions and tests could be devised with dynamical epidemic models
of HIV evolution. A specific challenge is predicting the time
scale and outcome of natural selection acting in conflicting
directions for within- and between-host viral replication. The
identification of human genetic factors that determine the
severity of HIV-1 infection has caused much excitement, and, to
date. human genes have been shown to account for ?10% of
variability in disease progression rates (27). This leaves consid-
erable scope for identifying other sources of variation, of which
viral genetic factors have been underexplored.
The Amsterdam Seroconverters Cohort. Patients were recruited
from 11 January 1982 onwards and followed at quarterly inter-
vals thereafter (5). To minimize biases associated with the use of
treatment, we included only data collected before 22 November
1993, when protease inhibitors were first introduced, resulting in
a sample size n ? 123.
Viral load was measured by quantitative PCR from frozen sera.
taken for the first 6 months after first seropositive sample were
excluded, as were viral load measurements taken after the first
AIDS-defining event (CDC type C). Set-point viral load was
determined as the geometric mean of these measurements.
Asymptomatic Duration as a Function of Viral Load. We start by
proposing the following decreasing Hill function for the duration
of the asymptomatic period D(V) as a function of the set-point
viral load V, such that D(V) ? Dmax (D50)Dk/[VDk? (D50)Dk],
where Dmaxis the maximum duration in years, D50is the viral
load at which the duration is half its maximum, and Dkis the
steepness of the decrease in duration as a function of viral load
To estimate the full profile of durations, we proposed that the
probability a person is still asymptomatic at time T after primary
infection is given by a gamma distribution with mean D(V) and
shape parameter ?. The cumulative ‘‘survival’’ probability is
S(V, T) ? ?(?, ?T/D(V))/?(?), where ? denotes the standard
gamma function and ? the lower, incomplete gamma function.
Our data consists of set-point viral loads vifor patients who
progressed to AIDS or were censored after a time ti, taken to
start 6 months after the first positive test, to exclude primary
infection. An indicator variable Iiis defined such that Ii? 0 if the
patient was censored and Ii? 1 if the patient developed AIDS.
The log-likelihood for this survival analysis is ¥i[Iiln[S(vi, ti)] ?
(1 ? Ii)ln[?S?(vi, ti)]], where S?(V, T) is the probability density
function corresponding to S.
The parameter values which maximize this likelihood are
Dmax? 25.4 years, D50? 3,058 copies per milliliter peripheral
blood, Dk? 0.41 and ? ? 3.46. Confidence intervals for the
duration were estimated at a specific viral load, V* say, by
treating D(V*) as a parameter, recasting Dmaxas a function of
this and by using the likelihood ratio method to determine 95%
c.i.s for D(V*). The procedure was iterated for values of V*
over a range, as shown in Fig. 2A. We consider the effect of
using other parametric models in SI Fig. 7.
Infectiousness as a Function of Viral Load. Because of substantial
inconsistencies in the reported frequency of unprotected sex acts
(6, 7, 28) despite consistent seroconversion rates (Fig. 3A), we
decided to formulate our model of infectiousness in terms of an
infection hazard (probability per unit of time or rate) rather than
as a probability per unprotected sex act. We thus introduced an
increasing Hill function for infectiousness ?(V) a function of
viral load V, ?(V) ? ?maxV?k/[V?k? (?50)?k], where ?maxis the
maximum infection rate per annum, ?50is the viral load at which
infectiousness is half its maximum, and ?kis the steepness of the
increase in infectiousness as a function of viral load. The
probability p(T, V) that a person is infected after a time T of
exposure to an infected person is p(T, V) ? 1 ? exp(??(V)T).
Given our data (from the Zambian transmission study),
consisting of set-point viral loads vifor index cases in couples
observed for a mean duration ?, and an indicator variable Ii
defined such that Ii? 0 if the partner was not infected and Ii?
1 if the partner was infected, then the log-likelihood is ¥i[Ii
ln[p(?, vi)] ? (1 ? Ii)ln[1 ? p(?, vi)]]. Ideally we would have the
duration of observation of each couple, but these data were not
made available to us. We verified by simulation that this was
unlikely to introduce systematic biases in our estimate.
The parameter values which maximize this likelihood are
?max ? 0.317 per year, ?50 ? 13,938 copies per milliliter of
peripheral blood and ?k ? 1.02. Confidence intervals were
estimated as for D(V) above.
We also considered a more general formula allowing for a
minimum infection rate ?min, i.e., ?(V) ? ?min? (?max? ?min)V?k/
[V?k? (?50)?k], but this did not improve the quality of fit (p ? 0.5
by one-sided likelihood ratio, the best-fit value was ?min? 0).
Fideli et al. (7) separate the Zambian data between male and
female index cases, and we repeated our analysis allowing for
separate parameters for male-to-female and female-to-male
Fraser et al.
October 30, 2007 ?
vol. 104 ?
no. 44 ?
transmission. Although the best-fit curves looked different Download full-text
(with a higher Hill coefficient for female-to-male transmis-
sion), the model did not fit significantly better (P ? 0.67 based
on likelihood ratio).
Transmission Potential. The transmission potential TP(V) is de-
fined in the main text as the product of the infection rate during
asymptomatic infection and the duration of asymptomatic in-
fection, i.e., TP(V) ? ?(V)D(V). Confidence intervals for the
transmission potential were estimated as above.
We thank R. M. Anderson, N. M. Ferguson, N. C. Grassly, B. G. Spratt,
and V. Mu ¨ller for useful discussions. We gratefully acknowledge funding
from the Wellcome Trust (R.C., T.D.H., F.d.W., and W.P.H.) and the
Royal Society (C.F.) and the assistance provided by the Amsterdam
Cohort Studies on HIV/AIDS.
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