1 Dept of Epidemiology, University of Michigan
2 Dept. of Biostatistics, University of Michigan
3 Dept. of Physiology, University of Michigan
4 Dept. of Mathematics, University of Michigan
5 Dept. of Economics, University of Michigan
6 School of Public Policy, University of Michigan
7 Dept. of Industrial and Operations Engineering, University of Michigan
8 Dept. of Electrical Engineering and Computer Science, University
of Michigan
This work was supported by a grant from the Office of the Vice President for Research at the University of Michigan.
Early HIV infection makes an even greater contribution to the
sexual spread of virus than would be predicted on the basis of
the high virus levels during that period. The disproportionate
effect of early infection occurs because: 1) infected individuals
encounter new sexual partners at a higher rate during early infection
than they do later on, and 2) the individuals infected by someone
with early infection are more likely to infect others. This last
effect is often ignored in weighing the importance of controlling
transmissions from early HIV infection. We demonstrate its importance
by using a transmission model that contains realistic aspects
of sexual contact patterns. In particular, the model we examine
has sexual partnership rates that rise and then fall after coitarche.
Moreover, partnerships are formed preferentially between individuals
having similar times since coitarche. If only 20% of total virus
available for transmission is allocated to early infection and
73% to late infection, the model shows that preventing all transmissions
during early infection prevents an epidemic while preventing all
transmissions from late stage infection does not slow the initial
rise of an epidemic and reduces endemic levels of infection by
only about half. Late in an epidemic, when infection prevalence
has leveled off, preventing transmissions during early infection
results in the eradication of infection while preventing transmissions
during late infection reduces endemic levels only by half. Our
analysis thus gives credence to the possibility that the HIV epidemic
can be stopped by vaccines which reduce contagiousness during
early infection even if they fail to prevent infection.
Keywords: Human Immunodeficiency Virus, Primary Infection, Transmission, Epidemiological Models
The period of increased viral levels during early HIV infection
begins with very high levels of virus during primary infection
(1,2) and persists for
some months after an immune response is present as viral levels
fall to a set point (3). During epidemics
in which infection prevalence rises quickly over a couple years,
the rate of that rise is largely dependent upon the rate of transmission
during early HIV infection (4,5). That is
not surprising as early in an epidemic there are few individuals
in advanced stages of infection. Later in the epidemic, however,
fewer infected individuals will be in the early stage of infection.
Therefore it might seem that early infection would lose its dominant
role in transmission dynamics. Under the assumption of random
or proportionate mixing that is the case (4). However, proportionate
mixing, by assuming partners are chosen independently of one's
status, misses a major reason why early HIV infection is so crucial
to transmission dynamics: partners infected by individuals with
early infection differ from the partners infected by individuals
with later stages of infection. They are more likely to spread
infection.
We present here a model which demonstrates dominance of the early infection stage even during the later years of an HIV epidemic and even when much more virus is available for transmission later in infection than early in infection. The model includes sexual life histories for individuals exhibiting: a) a rising and then falling rate of new partnership formation over time, and b) a preference by individuals to form partnerships with others of their own age.
METHODS
The model:
The model consists of a set of ordinary differential equations
(presented in the appendix) describing deterministic flows between
continuous population compartments of homosexually active individuals.
Insertive and receptive roles are not distinguished. We chose
such a population for the sake of simplicity. The general phenomenon
we illustrate also occurs in heterosexual populations. Not distinguishing
gender or insertive versus receptive roles means that the population
is divided along only two dimensions: infection stage and the
duration of sexual activity or "sexual age". The arrangement
of flows between model compartments is shown in Figure 1.
The model has the following characteristics:
Choosing Transmission Probability Values:
To argue the importance of early infection, we chose lower transmission
probabilities on during early infection than is indicated by recent
reports on total contagiousness. This choice is supported by analyses
of observed infection patterns which indicate that if early infection
is as short as in our model, transmission probabilities must be
200 to 1000 times greater in early infection than they are in
later stages (5,8). We use be
= 0.2 for the transmission probability during early infection
since that value generates an epidemic compatible with the one
observed in the San Francisco Hepatitis B cohort study (9). (A
longer period of early infection, with a correspondingly lower
transmission probability, produces results that are essentially
identical to those presented here.) be
= 0.2, however, represents a plausible lower limit in light of
data on the frequency with which transfusion infected males subsequently
transmitted infection to their female partners (10). When these
males resumed sex within 60 days of transfusion, 28% of their
female partners were infected.
On the other hand, low transmissibility during middle infection
is supported by the same study (10) which observed almost no transmission
upon subsequent prospective follow up of transfusion-infected
individuals at longer times after initial infection. Low transmissibility
during middle infection is further supported by studies which
combined transmission probabilities across all stages of infection
(11-16), as well as by studies of discordant pairs (15-17). Most
of the first group of studies estimate average transmission probabilities
to be between 0.001 and 0.01 (11,12,14-16). Since transmission
probabilities in early and late infection are higher than those
of middle infection, the transmission probabilities during middle
infection are not higher than these estimates. The one study that
estimated a higher overall transmission probability was the one
most strongly influenced by transmissions during early infection
(13). Studies of discordant pairs assess transmission probabilities
more specifically for middle infection (14-17). They show these
probabilities to be extremely low. Thus, we assume a transmission
probability of bm
= 0.001 during middle HIV infection.
The late stage begins as much as a year before the onset of AIDS,
when transmission probabilities have been noted to increase by
factors of from 5 to 15 (18,19). To insure that any early infection
effects we might observe are not due to underestimation of transmission
risks during the later stages of infection, we assume that late
transmission probabilities rise 76 fold to 0.076 per partnership.
Our choice of transmission probabilities is consistent with biological
evidence as well as the population evidence cited above. Although
virus turnover rates are high during middle infection (20,21),
virus levels are two to three orders of magnitude higher during
primary infection (1,2). Measurements of virus production reviewed
by Ho (22) are also consistent with our parameter values. The
rate of virus production can be high during middle infection and
yet the amount of virus available for transmission can be low
because virus survival is low in the presence of an immune response.
Computer Implementation of Model:
We constructed and numerically solved the model using STELLA II™
(High Performance Systems,
Lyme, NH). We used the Runge-Kutta-4 integrator with a 0.2 month
time step. A simple version of the STELLA II™ model
can be downloaded.
RESULTS
Epidemic patterns and the effects of reduced transmission potential in different stages.
Figure 2 shows the incidence
of new HIV infection generated by numerical solution of the model.
The sharp peak of infection seen around year 3 occurs only in
the presence of both age-peaking of contact rates and age-preferred
mixing. If either of these elements are lacking, infection levels
rise very slowly. Age-peaking of partnership formation rates and
age-preferred mixing allow the third age group, the one with the
high contact rate, to act as a core group in which the basic reproduction
number during primary infection exceeds the threshold value of
one. As a result, an epidemic takes off quickly and is then sustained
in the broader population.
The top curve of Figure 3a shows the
prevalence of infection, with an equilibrium prevalence
of 47.4%. The other curves in Figure 3a illustrate the sensitivity
of the prevalence to elimination of contagiousness during early
or late stages of infection before an epidemic begins. Whereas
completely eliminating transmissions from early infection stops
all transmission, eliminating transmissions from the late stage
of infection slows the rise of the epidemic only slightly and
only cuts the endemic level of infection approximately in half.
Figure 3b shows that eliminating transmission from the early or
late stages of infection after the epidemic has become endemic
has the same effect as eliminating these transmissions before
an epidemic begins.
This dominant effect of early infection is not sensitive to the
value of any particular parameter. For example, when bm,
the transmission probability in the middle stage is raised from
0.001 to 0.0045, or when bl,
the transmission probability in the late stage of infection is
raised from 0.076 to 0.14, eliminating the contagiousness of primary
infection still stops the epidemic. This dominance of primary
infection also persists when the the total time spent forming
new partnerships is increased from 16 to 22 years.
Reproduction number:
The behavior of the model can be understood by considering the
basic reproduction number as viewed from either an individual
or a population perspective. The former is widely used by STD
and HIV researchers. A population perspective, however, helps
to explain the results in Figure 3.
We denote the "individual" basic reproduction number
as R0I and the population basic reproduction
number as R0P. R0I is
the average number of secondary infections arising from an infected
individual over the entire course of infection, when all individuals
contacted are susceptible. On the other hand, (R0P
- 1) is the "force" behind the initial growth of an
epidemic. If R0P < 1, an epidemic will
not take off and endemic transmission will not be sustained. R0P
is generally a complex but (in theory) well defined function of
contact rates and transmission probabilities between groups of
individuals having different partnership formation rates (23-25).
R0I and R0P have identical
values under the following assumptions: 1) contagiousness is constant
over the course of infection, 2) all individuals, regardless of
age, leave the population at the same rate, 3) everyone in the
population has the same contact rate, 4) partnerships are formed
randomly in proportion to their availability. Under these assumptions
R0I = R0P = CßD,
[1]
where
C = the rate of new sexual partnership formation of each individual,
ß = the probability of transmission per partnership, and
D= the average time an infected individual circulates in the population.
This average duration is determined both by the rate of recovery
from disease and by the rate of exit from the population for other
reasons, such as death.
In the four sections below we consider the effect on R0I and R0P of the realistic assumptions in the model:
A) three stages of infection, each having distinct transmission probabilities,
B) an aging process with finite duration of partnership formation
C) age-peaked contact rates, and
D) age-preferred mixing.
Effects of staged infection on R0:
Anderson and May (4) have examined how staging of infection affects
transmission dynamics. Given stages of infection in a fixed population
with homogenous contact rates,
R0I = R0P = C{ßeDe
+ ßmDm + ßlDl}.
[2]
This formulation separates the biological part of R0
from the sexual contact part. The biological part is proportional
to the amount of virus available in transmissible body fluids
multiplied by the time that this virus is available. This, in
turn, is proportional to ßiDi
which we define as the "cumulative contagiousness" for
stage of infection "i". With the parameters shown in
Table 1, the early stage contributes ßeDe
= 0.3 or 20% (0.3 / (0.3 + 0.1 + 1.1)) of the total contagiousness;
the middle stage contributes ßmDm
= 0.1 or 6.67% of the total; and the late stage contributes ßlDl
= 1.1 or 73.3% of total cumulative contagiousness. Using the average
contact rate of C = 2, the overall R0I =R0P
= 2*1.5 = 3. Note that each stage contributes to R0I
in the same proportion that it contributes to cumulative contagiousness.
Effect of Sexual Age on R0:
The model population consists of only those individuals who are
forming new sexual partnerships. Individuals go through an aging
process after which they cease new sexual partnership formation.
When they do so, they effectively leave the population and shorten
the average time spent in the different stages of infection. For
example, if half of the infected population ceases new partnership
formation before they reach the late stage of infection, then
the average duration of that stage will be less than half of the
biological duration of that stage. The middle and late stages
are more affected than the early stage for three reasons. First,
since a person goes through the stages serially in their order,
everyone who misses part of early infection misses all of the
later stages. Second, the later the stage of infection, the older
the individual, and the more likely they are to cease new sexual
partnership formation. Finally, relatively few people leave the
population during the first stage, not only because it is first,
but also because it is the shortest stage.
Let de, dm and dl
denote the effective lengths of the three stages. It is these
durations that apply now to the formula for R0.
Still assuming that all age groups have the same contact rate
C, the formula for R0 is:
R0I = R0P = C{ßede
+ ßmdm + ßldl}.
[3]
Note in comparing the top two thirds of Table 1 that there is
almost no change in the length of the first stage: 1.5 months
to 1.49 months, a much bigger percent change in the second stage:
104 months to 69.37 months, and a large percent change in the
late stage: 14.5 months to 6.14 months. Table 1 shows that, with
constant contact rates across age groups and proportionate mixing,
R0 = 0.60 + 0.13 + 0.93 = 1.66. The resulting
epidemic rises slowly to an endemic level of 49.5%. Eliminating
transmissions from the late stage of infection leaves an R0
from early and middle infection of only 0.73 and therefore eliminates
all transmission. Eliminating transmission from early stage infection
leaves an R0 from middle and late of 1.06,
which reduces the endemic level of infection to 10.3%.
The Effects of Age-Peaked Contact Rates on R0:
Contact rates that depend upon age can be used to define an "effective"
contact rate if there is no aging process, if there is a single
stage of contagiousness, and if mixing is proportionate (1,26).
In the absence of an aging process, this can also be done for
preferred mixing (24). These effective contact rates can be substituted
for C in equation [1]. This, however, causes R0P
to be a peculiar type of average of the R0I
calculated for each individual (1,24-26). To avoid this complicated
averaging process, we confine the following discussion to effects
on R0I.
In the presence of an aging process, the rate at which one forms
new partnerships during different stages of infection depends
upon the age at which one is infected. To obtain R0I
in this case, we first calculate the expected number of new partnerships
that an individual introducing infection to the population would
make during each stage of infection. In stage i, this expected
number is cidi where ci
is the average (across all age groups) of the new partnership
formation rates of individuals in stage i of infection. Again
we only consider individuals who introduce infection and assume
that introductions are proportionate to new partnership formation
rates. The model produces the results shown in Table 1. The total
R0I can then be calculated by summing the
contributions from the different stages:
R0I = cedeße
+ cmdmßm
+ cldlßl
[4]
From part 3 of Table 1, we see that the total R0I
= 0.92 + 0.16 + 0.86 = 1.94. The epidemic takes off slowly
but eventually reaches a prevalence of 54%, higher than the 49.5%
reached without age-peaked mixing since R0I
= 1.94 is greater than the 1.66 of the previous case. Eliminating
transmissions during early infection produces R0
= 1.02 and reduces the endemic infection prevalence to 3.5%. Eliminating
transmissions during late infection produces R0
= 1.08 and reduces the endemic infection prevalence to 8.5%.
The Effects of Age-Preferred Mixing on R0:
We have seen that adding both an aging process and age-peaked
contact rates to our model increases the effect of early infection.
Figure 3 and the sensitivity analyses presented earlier demonstrate
that adding age-preferred mixing to the model generates even more
dominance for early infection. Let us consider why this is so.
With proportionate mixing, new partners are recruited from a common
pool that ignores infection stages or age, and in particular assumes
that whom one contacts is independent of one's own status. In
the age-preferred model, however, individuals in the very active
age group 3 will preferentially transmit to others in that age
group. That preferential mixing, together with high new partnership
formation rates, make individuals in this age group more likely
to be recently infected than individuals in other age groups.
Thus recently infected individuals are more likely to transmit
to individuals with high new partnership formation rates than
individuals in later stages of infection. When individuals reach
the later stages of infection, they are older and more likely
to transmit to other older individuals who have lower rates of
new partnership formation.
In this situation, the relationship between R0I and R0P is complex (23-26) and the data used to calculate R0P must specify not only the age and contact rates of the individuals making contact, but of the individuals they contact as well.
DISCUSSION
We have presented a model in which each individual has several
times more virus available for transmission during late HIV infection
than during early infection. Yet eliminating contagiousness during
early infection can stop transmission entirely whereas eliminating
contagiousness during late infection has a much smaller effect.
The effects of early HIV infection are seen in both the early
and late years of an HIV epidemic. This amplified influence of
early HIV occurs because the importance of each stage of infection
is only partially determined by the potential of the individual
in that stage to transmit. It is also determined by the potential
of the individual to whom infection is transmitted to carry on
or amplify chains of transmission.
The model we present is intended to enhance understanding of the
forces spreading infection through populations and thereby improve
public health decisions. It is not intended to quantify
the effects of eliminating or reducing transmission during early
HIV infection. For that purpose, models with more detail describing
contact patterns and more precise parameter estimates are needed.
We believe, however, that more complete modeling with better parameter
estimates will demonstrate that the real world importance of early
infection is even greater than that demonstrated in our model.
Two arguments support this view.
First, we have chosen conservative parameter values for the proportion
of contagiousness that occurs during early infection. When contagiousness
is defined as transmission probability times duration of infection,
analyses of observed epidemic patterns suggest that more than
50% of contagiousness during HIV infection occurs during early
infection (5,8). Moreover, virus levels during early infection
are more likely to translate into transmission risk because early
virus is less under attack by the immune system and has had less
time for "short sighted evolution" which adapts it to
a particular host rather than to transmission between hosts (27).
Second, some details of contact patterns not included in our model
are likely to further amplify the importance of primary infection.
One of these is transient partnership formation in core groups.
This might occur when either a long term partnership breaks up
or the social environment otherwise facilitates new partnership
formation. Once individuals are past the early stage of HIV infection,
they are likely to be out of those core groups. On the other hand,
serial monogamy might decrease transmissions during early infection
if it generates an interval between partnerships that spans the
period of early infection.
Implications for control of the HIV epidemic.
The dominant role of early infection in transmission dynamics
emphasizes the urgent need to change behavior and reduce transmission
risks before infection is detected. After infection is detected
it may be too late to prevent those key transmissions which sustain
viral circulation. Vaccines offer the promise of reducing those
key transmissions during early infection even if the immunity
stimulated by the vaccine does not prevent infection. All that
is needed is an immune response to the vaccine that controls early
infection in the same way that the immune response to natural
infection does.
The analysis presented in this paper extends our previous work
on the potential of vaccines to control HIV transmission (28).
It addresses doubts raised about the potential impact of reducing
transmission during early infection (29). Vaccine trials where
the unit of analysis is couples rather than individuals can estimate
the effect of vaccines on reducing transmission during early HIV
infection (30). While the inability of current vaccines to prevent
infection may not justify standard vaccine trials, the analysis
presented here supports the undertaking of vaccine trials where
the prevention of transmission between sexual partners is analyzed.
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