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IMA Journal
of
Mathematics
Applied
in
Medicine
&
Biology (1986)
3,
229-263
A Preliminary Study of the Transmission Dynamics of the Human
Immunodeficiency Virus (HIV), the Causative Agent of AIDS
R. M. ANDERSON, G. F. MEDLEY
Parasite
Epidemiology Research Group, Department
of
Pure
and Applied
Biology, Imperial
College,
University
of
London, London SW7 2BB
R.
M. MAY
Biology Department, Princeton University, Princeton,
N.J.
08544 U.S.A.
AND
A. M.
JOHNSON
Academic Department
of
Genito-Urinary Medicine,
The
Middlesex Hospital
Medical School, James Pringle House,
The
Middlesex Hospital,
London
WIN 8AA
[Received
23
October
1986 and in
revised form
12
November
1986]
The paper describes some preliminary attempts
to
formulate simple mathematical
models
of the
transmission dynamics
of HTV
infection
in
homosexual
com-
munities.
In
conjunction with
a
survey
of the
available epidemiological data
on
HTV infection
and the
incidence
of
AIDS,
the
models
are
used
to
assess
how
various processes influence
the
course
of the
initial epidemic following
the
introduction
of
the virus. Models
of
the early stages
of
viral spread provide crude
methods
for
estimating
the
basic reproductive rate
of the
virus, given
a
knowledge
of
the incubation period
of
the disease (AIDS)
and the
initial doubling
time
of the
epidemic. More complex models
are
formulated
to
assess
the
influence
of
variation
in the
incubation period
and
heterogeneity
in
sexual
activity.
The
latter factor
is
shown
to
have
a
major effect
on the
predicted pattern
of
the
epidemic; high levels
of
heterogeneity decrease
its
magnitude. Areas
of
biological uncertainty, future research needs,
and
public health implications
are
discussed.
Keywords
HTV;
AIDS; transmission dynamics; variable incubation periods;
heterogeneity
in
transmission; basic reproductive rate
of
infection; doubling time
of
an
epidemic; serologjcal survey.
1.
Introduction
BASIC
RESEARCH
on the
biology
of the
virus responsible
for
acquired immuno-
deficiency syndrome
in
humans (AIDS)
and the
pathology associated with
infection
has
proceeded
at a
rapid rate since
the
discovery
of the
aetiological
agent
in 1983 by
Barre-Sinoussi
et al.
(1983)
and
Gallo
et al.
(1983). Competing
This paper
was
presented
at the IMA
Conference
on "The
Mathematical Theory
of
Biological
Systems" held
in
Oxford, 7-9 July, 1986.
229© Oxford University Press
1986
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230 R. M. ANDERSON ET AL.
claims by the French and American research groups over who first isolated the
causative agent of AIDS created some confusion over the correct terminology for
the virus (named Human T-Lymphotrophic Virus type in (HTLV-HI) by Gallo et
al. (1983) and Lymphadenopathy Associated Virus (LAV) by Barre-Sinoussi
et al. (1983). A recent report by the International Committee on Taxonomy of
Viruses, however, endorsed the name 'human immunodeficiency virus' (HIV) to
resolve the controversy regarding priority of discovery (Coffin, 1986). This new
name describes the host and a major biological property of the virus.
A striking illustration of the pace of research in this area is provided by a
recent paper by Hahn et al. (1986) that describes genetic variation in the AIDS
virus (HTV) via genomic analysis, molecular cloning, and nucleotide sequencing.
These authors compared the nucleotide, and deduced amino acid sequences of
the gene encoding the extracellular envelope glycoprotein of the virus (the gene is
called env) in four to six isolates obtained from each of three patients over a
1-year
to 2-year period. Hahn et al.'s (1986) study highlights the hypervariability
of env relative to the remainder of the viral genome, and pinpoints localized
regions of hypervariability within the gene. Genetic changes among different
viruses result largely from duplications, insertions, or deletions of short stretches
of nucleotides, as well as from an accumulation of nucleotide point mutations.
Such variability appears to be correlated with antigenicity and might also give rise
to viruses with altered virulence or tissue tropism. The detail of such research
must be viewed in the context of the relatively short period that has elapsed since
the virus was first isolated (since 1983).
In marked contrast to our current understanding of the molecular biology and
genetic structure of the aetiological agent is the present state of the epidemiologi-
cal research that is concerned with virus transmission and persistence within
human communities (Peterman, Dotman, & Curran, 1985; Anderson & May,
1986).
Little is known at present concerning the basic epidemiological parameters
that characterize virus transmission dynamics. Among the unknowns are such
factors as the duration of the latent period of infection (the time from infection to
the point when the host is infectious to other members of the population), the
proportion of infected people who will go on to exhibit end-stage disease (i.e.
'full blown' AIDS), the duration of viral persistence within the host following
infection, and the infectious period. The lack of knowledge in this area creates
many difficulties in the design of effective control policies, in assessing future
trends in the incidence of AIDS, and in planning provision of health-care
facilities.
This paper describes a preliminary study of the transmission dynamics of HTV
based on analyses of the quantitative epidemiological data that are available, and
on the formulation of simple deterministic mathematical models of viral spread
and persistence. It is organized as follows. Section 2 provides a brief background
to past studies of the dynamics of sexually transmitted diseases (STDs) and the
factors that distinguish the transmission dynamics of such infections from the
more frequently studied directly transmitted viral infections such as measles.
Section 3 examines the available epidemiological data and attempts to derive
parameter estimates for the central processes that control viral transmission.
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TRANSMISSION OF THE HUMAN IMMUNODEFICIENCY VTRUS 231
Sections 4 and 5 detail the formulation of various deterministic models of viral
spread starting with the early stages of an epidemic (Section 4) and moving on
to more complex formulation describing heterogeneity in exposure to infection
and variable incubation periods (Section 5). Section 6 provides a general
discussion of future research needs and current problems in model formulation
and analysis.
2.
Sexually transmitted diseases (STDs)
Most STDs have certain characteristics that cause their epidemiology to be
somewhat different from infections such as measles and rubella (German
measles), which have received the bulk of attention in the literature concerned
with mathematical epidemiology (e.g. Anderson & May, 1985). First, for
infections such as gonorrhoea, syphilis, and genital herpes, only sexually active
individuals need to be considered as candidates in the transmission process. In
contrast with simple 'mass action' transmission models for measles, a doubling of
the hosts' population density does not tend to increase the rate at which new
infections are produced by infectious people. Second, the carrier phenomenon, in
which certain individuals harbour asymptomatic infection, is important for many
STDs.
In the case of gonorrhoea, for example, many women are virtually
asymptomatic and so do not seek treatment and remain active spreaders of
infection for relatively long periods of time. The carrier state may well be of
particular importance in the transmission of the AIDS virus, since many
individuals appear to be seropositive (they possess antibodies specific to the HTV
antigens) but they do not show overt symptoms of infection.
Third, many STDs induce little or no acquired immunity following recovery
from infection, so that most individuals who have'been treated or who have
spontaneously recovered rejoin the susceptible class (e.g. gonorrhoea). In the
case of HTV, the situation is probably more complex since, even in those
individuals who are seropositive but without symptoms of AIDS, the virus
probably persists within the host. Viral persistence following HTV infections may
well be lifelong, since the retroviruses (the group to which HTV belongs) are
characterized by their ability for long-term persistence within the body of the host
(Weiss, 1982). HTV has a complex life cycle that includes a chromosomally
integrated proviral DNA stage that has the potential for indefinite persistence.
Fourth, the transmission of most STDs is characterized by a high degree of
heterogeneity generated by great variability in sexual habits among individuals
within a given community.
This set of characteristics—virtual absence of a threshold density of hosts for
disease-agent persistence, long-lived carriers of infection, absence of lasting
immunity, and great heterogeneity in transmission—give rise to infectious
diseases that are well adapted to persist in small low-density aggregations of
people.
Past work on mathematical models of STDs is largely centred on two
infections, namely Hepatitis B and gonorrhoea. The work of Cooke & Yorke
(1973) on models for gonorrhoea provided the impetus for a series of papers in
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232 R. M. ANDERSON ET AL.
the late 1970s and early 1980s describing models which take account of the
characteristics described above. The recent monograph of Hethcote & Yorke
(1984) on the dynamics and control of gonorrhoea provides an excellent summary
of this body of research and is exemplary for the way the models are grounded on
data and how the conclusions are aimed at public health workers in a way that
emphasizes the ideas and not the mathematical details.
This body of research provides a useful starting point for the development of
mathematical models for AIDS. There is, however, a series of features of HIV
transmission that are unique amongst STDs and hence necessitate the develop-
ment of new models. These will be described in Section 4. First we turn to the
epidemiological data that are currently available to guide model formulation and
to derive estimates of the central parameters of transmission.
3.
Epidemiological data
The last few years have seen an enormous explosion in the volume of published
papers concerned with the biology and the epidemiology of the AIDS virus. In
this section we focus on the small fraction of this literature that contains
quantitative data on the course of infection within individual patients and the
spread of the infection within communities.
3.1 Infection and Disease
Infection with HTV results in a number of immunological abnormalities of
widely varying severities in different patients. In patients with severely impaired
lymphocyte function, serious disease may result from infection by opportunistic
parasites, the development of cancers, and other serious conditions. Not all those
infected, however, show severe immunodeficiency, and hence it is important, in
the context of disease surveillance, to define tightly (as far as is possible) the
symptoms associated with a case of AIDS. The case definition is as follows. 'A
disease, at least moderately indicative of a defect in cell-mediated immunity,
occurring in a person with no known cause for diminished resistance to that
disease' (CDC, 1982). The definition requires specific evidence for the oppor-
tunistic infection or cancer, but it does not require direct evidence of im-
munodeficiency. The diseases considered indicative of underlying im-
munodeficiency are many and varied (Peterman, Drotman, & Curran, 1985) but
the most frequent in cases of AIDS are Pneumocystis carini, Systemic Candidi-
asis,
and Kaposi's sarcoma.
3.2 Risk Groups in Human Communities
Persons at increased risks of acquiring HIV infection include homosexual and
bisexual men, intravenous (IV) drug abusers, persons transfused with contamin-
ated blood or blood products, heterosexual contacts of persons with HTV
infection, and children born to infected mothers. HTV is transmitted through
sexual contact, perinatal exposure to infected blood or blood components, and
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TRANSMISSION OF THE HUMAN IMMUNODEFICIENCY VIRUS 233
TABLE 1
AIDS: 'high
risk'
groups (in rank order of
importance as
judged by total
case
reports
in the USA up to January 1986)
Patient group
(1) Homosexual/bisexual men,
not IV drug users
(2) IV drug users
(3) Homosexual/bisexual men,
IV drug users
(4) Transfusion recipients
(5) Heterosexual contact with HTV positives
(6) Children with HTV positive parent(s)
(7) Haemophilia patients
Number of
cases
10,600
2,766
1,310
261
182
175
124
perinatal transmission from mother to neonate (CDC, 1985c). Virus has been
isolated from blood, semen, saliva, tears, breast milk, and urine, and is likely to
be isolated from other body fluids, secretions, and excretions. Epidemiological
evidence has as yet only implicated blood and semen in transmission. Studies of
nonsexual household contacts of AIDS patients indicate that casual contact with
saliva and tears does not appear to result in transmission. Transmission to normal
household contacts of infected persons has not been detected when the household
contacts have not been sexual partners or have not been infants of infected
mothers. The kind of non-sexual person-to-person contact that generally occurs
among workers, clients, or others in the workplace does not seem to pose a risk
for transmission of HTV. The epidemiology of HTV infection is thus somewhat
similar to that of Hepatitis B (HBV) infection. The high-risk groups for HTV
infection are very similar to those for HBV infection (CDC, 1985c). An indication
of relative risk between different groups of people is reflected in Table 1 which
records the numbers of reported cases of AIDS in the USA up to 13 January 1986
in different groups such as homosexuals and IV drug abusers (CDC, 1986).
Among the group at greatest risk (homosexual men), recent research suggests
that frequent and receptive anal intercourse with many partners predisposes to
HIV infection (Goedert et al., 1984).
3.3 Incidence of AIDS
Since October 1980, when the first cases of AIDS were reported in the United
States (some cases in the USA have since been traced back to 1978) the infection
has rapidly spread to Europe and to many other regions of the developed world.
The situation in Africa is more complex, and it is still uncertain how long the
disease has been present on this continent. However, evidence is accumulating
that heterosexual transmission is the predominant mode of spread. The incidence
in many of these regions (defined as reported cases per annum) has risen
exponentially with, as yet, no signs of turnover in the epidemic. The case reports,
particularly in the USA, provide a clear picture of the magnitude and seriousness
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234R. M. ANDERSON ET AL.
1979 1980 1981 1982 1983 1984 1985*
Year
FIG.
1. Reported cases of AIDS per annum (incidence) and reported deaths from AIDS in the United
States over the period 1979-1985 (the 1985 figure only includes cases up to November while the
deaths for 1985 are unavailable at present). The reported deaths for the USA are thought to be an
underestimate of the true figure (CDC, 1985a,b,c, 1986).
of the epidemic (Figs 1,2, and 3). The doubling time in incidence, defined as td,
differs among different countries and among different risk groups of people.
However, as recorded in Tables 2 and 3, the variance around estimated doubling
times for different countries is fairly small. On average, td is between 8 and 10
months in the early stages of the epidemic.
3.4 Serology:
Changes
in the
Proportion
Infected
with
HIV Through Time
Accurate serodiagnostic tests for HTV infection are now available. A series of
cohort studies of different at risk groups (in particular, homosexual males, IV
drug abusers, and transfusion patients) have provided good data on changes in
the proportion seropositive through time. Data from two such studies, one in
England (London) (Weber et a/., 1986; Came et al., 1985) and one in the USA
(San Francisco) (CDC, 1985a) are recorded in Fig. 4. Doubling times for the
proportion seropositive in the early stages of epidemics in different countries
(Table 4) agree well with those derived from case reports (compare Table 4 with
Table 2).
The relationship between the presence of antibodies specific to HIV antigens
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TRANSMISSION OF THE HUMAN IMMUNODEFICIENCY VIRUS235
1979 1980 19811982 1983
Year
1984 1985
FlG. 2. Reported cases of AIDS per annum and deaths from AIDS in the United Kingdom over the
period 1979-1985. Data from the Public Health Laboratory Service, Communicable Disease
Surveillance Centre.
and the course of infection within an individual patient is unclear at present.
Research on seroconversion in patients who have received infected blood
products or transplant organs from infected patients suggest that antibodies are
normally detectable between 40 and 60 days after infection (L'Age-Stehr et al.,
1985).
Current thinking is that thereafter they will persist and be detectable for
life.
It will, of course, be many decades before this assumption can be rigorously
tested by reference to long-term studies of infected patients. In the study of
L'Age-Stehr et al. (1985) of four transplant patients who received kidneys from
infected donors, antibody titres continued to rise for 12 months following the
transplant of the infected organ.
Seropositivity seems to be almost invariably associated with the presence of the
virus.
Wong-Staal & Gallo (1985) reported that greater than 80% of seropositive
patients possess HIV in their peripheral blood.
3.5 Incubation Period
The incubation period is defined as the interval between the point of
acquisition of the infection and the point of appearance of symptoms of
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236R.
M.
ANDERSON
ET AL.
ouu-
400-
300-
200-
100-
0-
[77] Males
O Females
7]
7
/
7
7
7
7
7
7
7
7
7
7
7
7
7
/
a.
/
7
/
/
/Yl
rJ
'/T1
r7i
0-
1- 5- 10- 15- 20- 30- 40- 50- 60+
Age group (years)
FIG.
3.
The age and sex
distribution
of
reported cases
of
AIDS
in 18
European Countries
up to
June
1985 (Data source
CDC,
1985b) (Austria, Belgium, Czechoslovakia, Denmark, Finland, France,
Federal Republic
of
Germany, Greece, Iceland, Italy, Luxemburg, Netherlands, Norway, Poland,
Spain, Sweden, Switzerland, United Kingdom).
TABLE
2
Doubling time
td in
AIDS incidence
(in the
early stages
of
the
epidemic)
Country
Fed. Rep.
Germany
Australia
Canada
Austria
Spain
Sweden
Switzerland
Italy
England
USA
Average
Period
1980-85
1983-85
1981-85
1983-85
1982-85
1983-85
1983-85
1983-85
1982-85
1981-85
Doubling
time
(m)
8-8
4-8
9-3
15-6
7-9
8-0
9-9
50
6-6
9-2
8-5
Rate
(r/yr)
0-94
1-73
0-89
0-53
105
1-04
0-84
1-66
1-27
0-90
108
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TRANSMISSION
OF
THE HUMAN IMMUNODEFICIENCY VIRUS
237
TABLE
3
Doubling time
td in
AIDS incidence
by
risk groups,
USA (in the
early stages
of
the epidemic)
Group
Homosexual/bisexual men,
IV drug abusers
Homosexual/bisexual men,
non-IV drug abusers
IV drug abusers
Heterosexual contacts
Total
Doubling
time
(m)
9-5
91
8-25
7-85
9-2
Cases (1986)
599
5009
1429
100
8661
%
of
total
6-9
65-4
16-5
1-1
1001
Data from Peterman, Drotman
&
Curran (1985), CDC (1985a)
80
70-
60-
50-
40-
30-
20-
10-
77!
San Francisco
E3 London
YX
YX
YX
YX
1978
1979 1980 1981 1982
Year
1983
1984 1985
FIG.
4.
The
prevalence
of
antibodies
to
HTV antigens
in
a
cohort
of
homosexual and bisexual men
in
San Francisco, U.S.A. over
the
period 1978-85 (CDC, 1985a).
The
figures
for
London
in
1982 and
1984
are
from
a
study
of
homosexuals attending
a
STD clinic (Came
a
al.,
1985).
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238
R- M-
ANDERSON
ET AL.
TABLE
4
Doubling time
ta
from
cohort
HIV
serology
in
the
early stages
of
the
epidemic
Country/City
San Francisco
New York
London
London
Switzerland
Italy
Cohort
Homosexual
Homosexual
Homosexual
rVdrug
IV drug
IVdrug
Period
1978-80
1982-83
1982-84
1983-85
1983-84
1980-83
Rate
(r/yr)
0-73
0-77
0-84
0-72
0-96
0-55
Doubling
time
(m)
11-3
10-7
9-9
11-5
8-6
15-2
Data from CDC (1985a), Came
et
al.
(1985), Mortimer
et
al.
(1985)
full-blown AIDS. Studies
of
blood-transfusion-associated cases
of
AIDS
in the
United States have provided good data
on the
distribution
of the
incubation
period
in
fairly large samples
of
patients.
The
studies
of
Peterman, Drotman,
&
Curran (1985)
and Lui et
al. (1986),
for
example, focused
on
194 cases
of
possible
transfusion-associated AIDS.
The
distribution
of the
incubation periods
in
this
sample
is
displayed
in Fig. 5. The
mean
of
this distribution
is 30
months,
but a
high variance
is
shown
in
this sample
of
patients.
A simple deterministic model describing
the
incubation
of
AIDS
is as
follows.
Let
y(t)
denote
the
proportion
of a
cohort
of
patients
(all of
whom were infected
with
HTV at
time
t = 0) who
have AIDS
at
time
t. If we
assume that
the
rate
of
conversion from seropositivity
to
'full blown' AIDS
is v(t) at
time
t
from
the
point
of
infection, then
the
rates
of
change
of y(t) and x(i) (the
proportion
who
do
not
have AIDS;
x + v = 1) are
given
by
^=-u(0*(0, ^ = f('M0- (3-la,b)
The initial condition is, of course, x(0) = 1 and y(0) = 0. We assume for simplicity
that all infected members of the cohort eventually develop AIDS. A simple
assumption concerning the form of the function v(t) is that it is linear with an
intercept at zero (y(t) = at). A rationale to support this assumption would be that
the progressive impairment of the patient's immune system through time from the
point of infection with HTV results in a linear rise with time in the probability that
an opportunistic infection or cancer develops in that patient. With these
assumptions the solutions of equations (3.1) are
l-x(t).
(3.2a,b)
As illustrated
in
Fig.
5, the fit of
this very simple model
to the
data
of
Peterman,
Jaffe
et al.
(1985) (presented
as
incidence
of
AIDS
in the
sample
of
transfusion-
infected patients
per
month)
is
remarkably good. Interestingly, equation (3.2a)
is
identical
to the
'hazard function'
of the
Weibull distribution with probability
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TRANSMISSION OF THE HUMAN IMMUNODEFICIENCY VIRUS239
I
2 4 6
Years after infection (seroconversion)
Fio.
5. The proportion of diagnosed cases of AIDS (proportion of all cases diagnosed over a 10-year
period) in each 6-month interval from the point of seroconversion for antibodies to HTV (data from
Petennan, Drotman, & Curran, 1985). The crosses denote observed values and the squares denote
values predicted by the model defined in (3.2) in the main text with a = 0-237 yr"1. Note that these
data give a biased (over-) estimate of a (see Lmetal., 1986).
density function
with the parameters ar and a2 equal to 2 and (iar)i respectively (Cox & Oakes,
1984).
The Weibull distribution provides an excellent empirical description of the
data recorded in Fig. 5, as has recently been noted by Lui et
al.
(1986). However,
the simple model defined by (3.1) provides a more biologically orientated
description of the observed pattern and may thus be of more general application
in the formulation of models of the transmission dynamics of HTV.
3.6 The Proportion of Infecteds who Develop AIDS
A number of different studies of cohorts of infected patients from various
high-risk groups have provided information about the proportion of infecteds that
develop AIDS. At present, these studies only cover periods of 3 to 7 years and
hence the proportions derived from them are probably underestimates of the true
picture. As documented in Table 5, the estimates range from around 34% to as
low as 8%. In the most extensive study, namely that of 6875 homosexual and
bisexual men who attended a San Francisco City Clinic (USA), roughly one third
of men infected over 5 years have developed AIDS (CDC, 1985a). This is an
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240
R- M. ANDERSON ET AL.
TABLE 5
% Seropositive who develop AIDS over a three-year period
(data
from Goedert et
al.
(1986))
Study Group Percentage
USA, Manhattan: Homosexuals 34-2
USA, Washington: Homosexuals 17-2
Denmark: Homosexuals 80
USA, Queens: IV Drug Abusers 12-5
USA, Hersley: Haemophilia Patients 12-8
indication that a fraction of those infected may not necessarily develop severe
immunodeficiency as a result of viral invasion. The reasons for this pattern are
unclear at present, although the recent study of Hahn et al. (1985) of genetic
variation in HTV over time in patients with AIDS may suggest that different
strains of the virus induce different severities of symptoms of disease. One
particular aspect of this study is especially remarkable: genetic variation in viruses
isolated at different times from the same patient is less than that observed
between viral strains isolated from different patients (Hahn et al., 1986). This
seems puzzling in light of the observation that the 35 patients who formed the
basis of this study were homosexual men from HTV endemic regions who had
hundreds, and in some cases thousands, of different sexual partners over 1 to 2
year periods. On the one hand, extensive genomic heterogeneity of independent
HIV viruses is common, yet, on the other hand, each patient appeared to be
infected with only a very limited number of predominant viral forms. It is
conceivable that immunological or non-immunological events that occur after the
initial infection with HIV lead to protection from subsequent HIV infections
(Hahn et al., 1986).
3.7 Survival of AIDS Patients
Once AIDS is diagnosed, the survival period of patients is relatively short, on
average being a matter of a few months to a few years. The mean survival period
is commonly quoted as being between 9 and 12 months (Peterman, Drotman, &
Curran, 1985) although this depends to some extent on the risk group and the
opportunistic infection or cancer that is acquired by the patient. Patients with
Kaposi's sarcoma have a better prognosis than patients with opportunistic
infections (Moss et al., 1984; Marasca & McEvoy, 1986). The median survival
from diagnosis in the study of Marasca & McEvoy was 21 months for 44 cases
presenting with Kaposi's sarcoma, and twelve months for 124 cases with other
diseases.
3.8 Latent and Infectious
Periods
Relatively little is known at present concerning the latent and infectious periods
of HIV infection. The course of virus expression and antibody response from the
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TRANSMISSION OF THE HUMAN IMMUNODEFICIENCY VIRUS 241
time of infection through the incubation period and during different stages of
clinical disease have not yet been clearly established (Esteban et al., 1985).
Patients with AIDS appear to have less virus in their blood than 'healthy' carriers
or people with AIDS-related complex (Wong-Staal & Gallo, 1985). The serum
level of antibody to HTV is high in most patients by the time clinical symptoms
are recognized, and these antibody titres are sometimes reduced to low and
barely detectable levels in advanced stages of disease (Salahuddin et al., 1984).
The period that elapses from initial infection to the point where an infected
person is infectious to other susceptibles (the latent period) is thought to be of the
order of a few days to a few weeks. Virus is detectable in blood, semen, and
other secretions and excretions, a matter of days to weeks after initial infection
(data from transfusion-associated cases).
Whether the presence of detectable levels of virus infection indicates infec-
tiousness, however, is not clear at present. Similarly, the duration of infectious-
ness is not yet known. For those patients who proceed to develop full-blown
AIDS (with an average incubation period of around 4-5 years) the infectious
period is probably a little less than the mean incubation time (to account for a
short latent period). Once symptoms of AIDS are diagnosed, such patients are
effectively removed from circulation and hence cease to contribute to viral
transmission within the community at large. In the case of infected (serologically
positive) but apparently 'healthy' patients (certain of whom have been seroposi-
tive for periods approaching 8 years) the situation is more uncertain. As stated
earlier, seropositivity implies persistent viral infection (probably lifelong) and
hence such individuals do in principle constitute a pool of potentially infectious
people. However, it may be that their infectiousness is somewhat less than
individuals who go on to develop 'fully fledged' AIDS.
3.9 Sexual Activity
There are many distinct subgroups within the total population at high risk of
HTV infection. Even within such subgroups, such as male homosexuals, the risk
of acquiring infection will depend in part on the behaviour of an individual. In the
following section, our attention is focused on the largest high-risk group, namely,
homosexual males. In this sector of the population, the primary factors that
determine the likelihood of acquiring infection appear to be the degree of sexual
activity (denned as the number of different sexual partners per unit of time) and
the nature of sexual practice (Goedert et al., 1984). Data on these factors are
understandably sparse at present and rarely quantitative in nature.
Two unpublished studies of sexual activity amongst homosexuals in London
and one published study of a similar at risk group in San Francisco provide some
data on the frequency distribution of the different sexual partners per unit of time
(C.
Carne & I. Weller, unpub; T. MacManus, unpub; McKusick et al., 1985).
This information is presented in Figs 6 and 7 and Table 6. A striking feature of
the most detailed quantitative study (C. Carne & I. Weller unpub.) is the high
average of the frequency distribution (defined as the mean number of different
partners per month) and the very large variance (variance » mean) (Table 6).
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242R. M. ANDERSON ET AL.
5 6 7 8 10
Number erf partners12 15 16 20 40
FIG.
6. Frequency distribution of the number of sexual partners per month among a sample of
homosexual males attending a London STD clinic in January, 1986 (I. Weller & C. Came,
unpublished data obtained from interviews). The members of the sample were highly selected and
came from a high-risk group (see main text). (Mean = 4-28 m"1, variance =
57-9).
The expected
values are from the best-fit negative-binomial probability model which provides a poor fit to observed
trends.
The data were collected, however, from a highly selected sample of patients with
either persistent generalized lymphadenopathy (PGL) or AIDS, partners of
patients with PGL or AIDS, or homosexual men with more than 10 partners in
the last 3 months. As such, the mean is probably much higher than the mean of a
randomly drawn sample from the total homosexual population of London.
However, it is interesting to note that a similarly high mean was recorded by
McKusick et
al.
(1985) in the San Francisco study (Table 6). This latter study also
revealed evidence for a decrease in sexual activity over the period November
1982 to November 1986 as a result of wide publicity of the factors that determine
the risk of acquiring HIV infection (Table 6).
The frequency distribution recorded by Carne & Weller is poorly described by
the negative binomial probability model (Fig. 6). A slightly better, although not
satisfactory fit is obtained with the gamma distribution (Fig. 8).
4.
Models with homogeneous mixing
This section and the next two turn to the formulation of a series of
mathematical models to mirror the transmission dynamics of HTV infection within
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TRANSMISSION OF THE HUMAN IMMUNODEFICIENCY VIRUS243
60-
50-
40-
20-
10-
A)ln=1292
EZ) London n=523
K3 OOL n=744
1-5 6-50
Male partners/year51-100101-1000+
FIG.
7. Frequency distribution of the number of sexual partners of homosexuals (per year) drawn
from London and outside London (OOL) (T. MacManus, unpublished data).
communities of homosexual males. By restricting attention to this single group we
keep the models relatively simple, while still accounting for 70-80% of the known
cases (Table 1).
We start by considering a very simple framework in which the population mixes
homogeneously before moving on to more complex heterogeneous
—
mixing
models in the following section. The reason for adopting this oversimplified
framework initially is to help clarify how various observed factors influence the
dynamics of disease spread and persistence. Throughout this and the next section
we assume that: (1) susceptibles acquire infection via sexual contact with
infectious people; (2) patients with AIDS are effectively withdrawn from
circulation in the population such that they do not generate new cases of HIV
TABLE 6
Sex partners I month for homosexual males
attending
STD Clinics
San Francisco, USALondon, England
Month/YearMeanVarianceMeanVariance
Nov 1982
May 1983
Nov 1983
May 1984
Nov 1984
1985
6-8
—
4-8
3-9
2-64-756-7
Data from McKusick et
a!.
(1985), Weller & Carne (unpub.)
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244R. M. ANDERSON ET AL.
68 10 12 15 16 20 40 61
Number of partners
FIG.
8. The fit of the gamma distribution to the frequency distribution data of Weller & Came
presented in Fig. 6.
infection; (3) infected people are infectious for a period 1/u time units, after
which a proportion p proceeds to develop AIDS while the remaining fraction
1
- p
passes into a 'seropositive' but non-infectious class, and (4) the latent period of
infection is so short in relation to the incubation that it can be ignored. These
assumptions are portrayed diagrammatically in a flow chart in Fig. 9a.
4.1 The Early
Stages
of
the
Epidemic
We first consider a closed population of fixed size N in which the density of
suscepdbles and infectious people (homosexual men) at time / are denoted by
X(t) and Y(t) respectively. We ignore most of what happens following infection
(i.e.
whether individuals progress to AIDS or not) and simply denote the rate of
movement out of the infectious class as v, where 1/v denotes that average
incubation period, such that a proportion p of those leaving the infectious class
(e.g. pvX) enters the AIDS patient class. AIDS-related deaths are ignored in the
early stages of the epidemic.
The model is defined as
at (4.1a,b)
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TRANSMISSION OF THE HUMAN IMMUNODEFICIENCY VIRUS245
(a)
AIDS A
disease-induced death
Seroposltive Z
non-infectious
(b)Susceptible X
kcC\-p)
Infectious V
AIDS-A
I-
disease-induced death
Infectious /
Seropositive Z
non-infectious
Fio.
9. (a) Flow diagram of the different model structure described in (4.1) in the text. Assumes
homogenous mixing and a constant rate, v, of leaving the incubating (infectious) class, (b) Equations
(4.11) divide the incubating/infectious class into two subgroups (those who develop AIDS and those
who do not) in order to introduce a variable incubation period, v{x). Note that all individuals in each
subgroup suffer a background mortality rate ft.
Here, c is the average number of sexual partners (the nature of this average is
investigated in more detail in the following section) and
A
is the probability of
acquiring infection from a randomly chosen partner. The term is proportional to
the transmission probability
/3
and to the probability Y/N of a given partner being
infected:
A
= pY/N. (4-2)
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246
R. M.
ANDERSON
ET AL.
In the early stages
of
the epidemic,
X^N
and (4.1b) and (4.2) give
^~Wc-v)Y.
(4.3)
That is, the number
of
infectious people
is
given
by
Y{t)~Y(O)exp[(Pc-v)t],
(4.4)
where the incidence
of
diagnosed AIDS
is
simply pvY(t).
Equation
(4.3)
gives
the
doubling time
td of the
epidemic during
its
early
stages:
td
=
(ln2)/(fk-v). (4.5)
For this very simple model,
the
basic reproductive rate
Ro of
HTV infection
is
defined
as the
number
of
secondary infections produced,
on
average,
by one
primary infection
in
a
wholly susceptible population. From (4.3),
Ro
is given
as
Ro
=
PcD,
(4.6)
where D is the average incubation period;
D
= 1/v. Equation (4.5) may therefore
be expressed
in
terms
of Ro,
where
-l)
(4.7)
Gearly, Ro
>
1
for
the
infection
to
trigger
an
epidemic. Equation (4.6) gives
an
explicit formula
for
the change in average sexual habits among homosexuals that
is needed
to
reduce transmission
of
HIV
infection below this threshold:
the
distribution
of
sexual habits needs
to
change such that
the
new mean activity
£
obeys
e<c/R0.
(4.8)
The main function
of
this very simple model
is to
provide
a
crude method
of
estimating
the
magnitude
of Ro
given information
on the
doubling time
td (in
seropositives
or
cases
of
AIDS—see Tables
2
and 4)
of
the epidemic during
its
initial phase
of
growth. Equation (4.7) suggests that
Ro
lies in the range 5-6 for
a
doubling time
of
around
9
months
and
an
incubation period
in
the
range 4—5
years.
More generally, (4.5) gives
fie
=*
1
yr"1, independent
of
the estimate
of t;
(provided £)
*»
several years); then
we
have
the
rough relation RomD (with
D
measured
in
years).
4.2 The Epidemic in a
Population
with Recruitment
The model defined
in
Section 4.1 above
can be
extended
to
consider
the
full
course
of the
epidemic
in a
population
of
size
N(t) at
time
t
subject
to
immigration
at
a
rate A and
a
natural mortality rate
n
(in the absence
of
infection
the steady state population size N* =
Alfi).
We assume that those with AIDS die
at
a
rate
d
(where
l/d
= 9 months— 1 year). We denote susceptibles, infectious,
AIDS patients, and non-infectious seropositives as
X(t), Y(i), A(t),
and
Z(i),
at
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TRANSMISSION
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247
time t, respectively.
dX dY
—
=
A-(iX-XcX, —
=
kcX-(y
+
n)Y, (4.9a,b)
at at
^
=
pvY-(d
+ fi)A,
^=(l-p)vY-pZ
(4.9c,d)
where A
is as
denned
in (4.2) and
N(t)
=
X(t)
+ Y{t)
+
A(t) + Z(t). Analytical
and numerical studies reveal that, provided
Ro>
1, the system settles
to a
steady
state with HTV infection maintained within the population. The approach
to
this
equilibrium, however,
is
oscillatory
in
character, where the interepidemic period
T is given very approximately (see Anderson & May, 1982)
by
T~2n[(L/R0)D]m.
(4.10)
Here,
D is the
average incubation
(=
infectious) period
and L is the
sexually
active life expectancy
of a
homosexual
in the
absence
of
HTV infection once
he
has joined the homosexually active population (say 18-50 years). With
Ro
values
in
the
range
of 4-5 and
with
D set at 4-5
years, equation (4.10)
(or,
more
generally, with
the
rough relation
R0~D
noted
at the end of the
previous
section) gives the very crude prediction
of
damped epidemics every 30-40 years.
But the parameters characterizing social behaviour (such
as A
and
c,
and thence
Ro)
are
unlikely
to
remain unchanging on such
a
time scale.
Numerical studies
of the
model defined
by
(4.9) give
a
clearer picture
of the
pattern
of the
initial epidemic following
the
introduction
of
HTV into
a
totally
susceptible homosexual community.
A
summary
of
the parameter values used
in
these studies is presented in Table 7; the population size was set
at
100 000 with
a
life expectancy (1/^)
of
32 years
in the
absence
of
infection. Two simulations
of
the trajectories
of
AIDS incidence (/annum/10 000) and the proportion seroposi-
tive (y(t)
=
[A(t) +
Y(t)
+ Z(t)]/N(t))
in the
population through time
are re-
corded
in
Figs. 10 and 11. Note that the model predicts that incidence will reach
its maximum value 12-15 years after the arrival
of
the infection (with
Ro
values
in
the range 4-5.0). Also note that
the
predicted change
in the
proportion
TABLE
7
HIV
infection,
crude parameter
values
Parameter Estimate
Latent period Days
to a few
weeks
Incubation period 4-5-5-0 years
(+?)
Infectious period 4-5-5-0 years
(?)
Seroconversion
A few
weeks
Life expectancy
of
AIDS patient
9
months
-
1
year
Doubling time (initial)
8-10
months
Proportion seropositive
who develop AIDS 10%
-
30%
(+?)
Average
sex
partners/month
2-6
Variance
in
sexual habits Variance/mean ratio,
5-20?
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248R. M. ANDERSON ET AL.
28
FIG.
10. Temporal predictions of the homogeneous-mixing model with recruitment of susceptible*
((4.9) in the text). The graph denotes temporal changes in the proportion seropositive ( x) and the
incidence of AIDS ( +) (incidcnce/year/10000). Parameter values: Ro = 515, D = 5 yr, d = 1 yr"1,
p = 0-3, N(0) = 100 000, ft = (1/32) yr"1, A = 1333-3 yr-1.
seropositive over the first 8 years is similar to that observed (compare Fig. 4 with
Figs.
10 and 11). In short, despite the simplicity of the model and its many
shortcomings, predicted patterns crudely mirror observed trends in homosexual
communities.
4.3
Variable
Incubation Periods
As discussed in an early section that examined the available epidemiological
data, the incubation period of AIDS varies widely between infected individuals.
It was shown (see Fig. 5) that a reasonable approximation of the observed pattern
is given by a model in which the incubation period is depicted as a linear function
of the time T that a person has been incubating the infection (v(x) = ax, see
(3.1a)).
The model defined in (4.9) can be easily modified to take account of
this biological property. We define Y(t, x) as the number of infectious people who
will develop AIDS that have been incubating the virus for x time units at time t,
and I{t) as the number of infectious people who will not develop AIDS (they are
assumed to have an average duration of infectiousness of 1/y time units) (see Fig.
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TRANSMISSION
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28
Fio.
11. Identical to Fig. 10 except
RQ
= 70.
9b).
The new model
is
(4.11c)
Here,
A
= pY/N,
where
and
the
initial
and
boundary conditions
of
(4.11b)
are
defined
as
with
Y
the initial number
of
infectious people (=1 throughout the paper).
A
clear
picture
of
the influence
of a
variable incubation period
is
provided
via
numerical
studies
of the
model defined
in
(4.11),
and the
comparison
of
these predictions
with those
of
the simpler model
in
which the incubation period
v is
assumed
to be
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250R. M. ANDERSON ET AL.
28
Years
FIG.
12. Temporal solution of the homogeneous-mixing model with recruitment and variable
incubation ((4.11) in the text). Proportion seropositive to HTV (x) and the incidence of AIDS
(incidence/year/10 000) ( + ) are recorded on the same graph. Parameter values are as defined in Fig.
10 with y = 0-2 and a
=•
0-0628 (chosen to give a mean incubation period of 5 years).
a constant and independent of the duration of infection. One such comparison is
presented in Fig. 12, where the parameter of the function
U(T)
was chosen to give
a mean incubation period of five years. Note that, for a fixed value of RQ,
variable incubation does not greatly influence the time to maximum incidence
(compare with Fig. 10 for the simpler model with constant v). Its main effect is to
alter the shape of the epidemic curve in such a way that the rise in cases of
'full-blown' AIDS follows the patterns set by the rise in seropositivity (i.e.
incidence of infection), but with a fairly pronounced delay. This contrasts with
the simpler model with constant u(l/u = 4-5 years), in which there was a less
marked lag between the rise in seropositivity and the rise in AIDS cases. In
summary, therefore, the inclusion of an incubation period that is itself sharply
defined produces an epidemic curve for AIDS cases that tends initially to track
the seropositivity curve with a marked delay, and that tends to have a relatively
high proportion of the total number of AIDS cases centred around the peak of
the curve.
5. Models with heterogeneity in sexual activity
We now turn to consider an important complication, namely, the transmission
of HIV in a population in which sexual activity (denned as the number of
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TRANSMISSION OF THE HUMAN IMMUNODEFICIENCY VIRUS 251
different partners per unit of time) varies greatly among individuals (such that the
variance in activity greatly exceeds the mean level).
5.1 The Early
Stages
of
the
Epidemic
As for the homogeneous mixing model, we first examine the early stages of the
epidemic to assess how heterogeneity in sexual activity influences the estimation
of the basic reproductive rate Ro. The simplest approach to this problem is to
divide the total at-risk population of N individuals into subgroups of Nt
individuals who have an average of i sexual partners per unit of time:
A/,
= NP(i)
where P(i) is the proportion of the population in the ith class. The number of
susceptible and infectious individuals in the ith class at time t are defined as X,(t)
and Y,(t) respectively. The rate of acquiring infection is taken to be i (number of
partners) times A (probability that a randomly chosen partner infects the
susceptible person). Equations (4.1) generalize to
dX, dY,
—- =
-ikXh
-1
=
ikX, - vY,.
(5.
la,b)
at at
We assume, in the first instance, that v is constant and independent of the
duration of infection. The infection probability A depends on /3 and on the
probability that a partner is infectious. Weighting partners by their degree of
sexual activity (i.e. proportionate mixing), we have
A(0 =
/32'Y,(')/S'N,(0- (5-2)
Equation (5.1a) can be integrated directly to give (R. M. May et al., unpub.)
*,(<) = tyexpHV(')], (5-3)
where
rp(t)={'k(g)dg.
Jo
Substituting (5.3) into (5.1b), and summing to produce a differential equation for
k(t),
we arrive at the result
f
=
A(/82
iV'^o/S iPQ)
- v). (5.4)
In the early phases of the epidemic, Xt =
N,
and
xp =
0, whereupon (5.4) gives
Here, m is the mean number of partners and o2 is the variance. We therefore
have exponential growth in the number of infected (as mirrored by temporal
changes in A(f)) with growth rate /3c
—
v, where the c in (4.3) of the equivalent
homogeneous-mixing model is now, in the heterogeneous-mixing model, defined
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252
R. M.
ANDERSON
ET AL.
as
c
=
m
+
o^/m. (5.5)
Note that
the
crude estimation procedure
for the
basic reproductive rate
of
HTV
infection, based
on the
data
for the
doubling time
of the
epidemic
(see
(4.5)),
remains
the
same, except that
c is now as
defined
in
(5.5).
In
this equation,
the
mean
and
variance
of
the distribution
of
sexual activity
can be
defined
in
terms
of
the parameters
of a
suitable probability distribution that mirrors
the
empirical
data (see Figs.
6, 7, and
8).
For
instance,
the use of a
gamma distribution
for P(i)
permits
a
good deal
of
analytical insight
(R. M. May et ai,
unpub.).
The model
can be
further generalized
to
take account
of the
likelihood that,
upon acquiring
HIV
infection, different individuals fall into different classes
of
subsequent epidemiological histories
(or the
presentation
of
different symptoms
of disease resulting from infection). Some
may
remain infectious
for
relatively
short periods before exhibiting 'fully fledged' AIDS
or
AIDs-related complica-
tions (ARC), while others
may
remain infectious
(or
non-infectious)
for
long
periods
of
time without suffering symptoms
of
disease
(the
non-AIDS seroposi-
tive individuals).
Such
a
range
of
possibilities, perhaps based
on the
mode
of
acquisition
of
infection,
the
viral strain that first infects
an
individual,
or
biological characteris-
tics
of the
individual,
can be
formally incorporated
in the
models
by
letting there
be
K
different kinds
of
infectious individuals. Equation (5.1a) remains
the
same
but (5.1b) becomes
&Yi,k(t)ldt=fkik{t)Xi-vkYi,k.
Here,
Ylk is the
number
of
infectious individuals
of the
infection class
k
(k = 1,. .. , K) in the ith
class
of
sexual activity;
fk is the
proportion
of
infections that pass
to the )tth
infectious class
and vk is the
rate
of
movement
out
of this infectious class.
The
different infectious classes may transmit infection with
different efficiencies
(per
unit
of
time)
so
that
(5.2)
should
be
generalized
to
The numerical analysis
of
such more general models
is
relatively straight-forward
(given data
on the
distributions
of
sexual activity
and
infectious classes),
but the
main problem
is the
proliferation
of
parameters. Exploration
of the
properties
of
this kind
of
model must await
the
acquisition
of
more detailed epidemiological
data.
5.2 The Epidemic
in
a Population with Recruitment
To describe
the
full course
of the
initial epidemic following invasion
we
require
a more complicated model than that outlined
in
Section
5.1. For
mathematical
convenience
we
employ
a
continuous framework
for the
definition
of
variation
in
sexual activity
(in
Section
5.1, a
discrete framework
was
employed
(Cox &
Anderson,
in
preparation)).
We
define
X(t, s), Y(t, s), and I(t, s) as the
number
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TRANSMISSION
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253
of susceptibles,
the
number
of
infectious people
who
develop AIDS,
and the
number
of
infectious people
who
do not
develop AIDS
at
time
t
with sexual
activity
s
(the
number
of
different partners
per
unit
of
time), respectively.
It is
assumed that each individual maintains
the
same activity status
as a
susceptible
and
as an
infectious person.
We
again assume that
a
proportion
p of
infectious
persons proceeds
to
'fully fledged' AIDS
and a
proportion
1-p
moves
to a
non-infectious seropositive state
(see
Fig.
9b). The
model
is of the
form:
dX
— (t, 5)
= A(s)
-
sX{t, s)A(r)
-
pX(t, s),
(5.6a)
at
^(t, s) =psX(t, s)k(t)
-
(i; + fi)Y(t, s),
(5.6b)
^(t, s)
= (1
-p)sX(t, s)k(t)
-(u
+ n)I,
(5.6c)
where
dA
f°° AZ r°°
—
=
u| Y(t,s)ds-(d
+
n)A,
—
=
vj I(t,s)ds-pZ, (5.6d,e)
A(0 =
fi
fs[Y(t, s)
+
I(t,
s)] ds
/fsN(t,
s)
ds.
(5.7)
Jo
/
Jo
Here
A(t),
Z(t), \i, v, y,
and
d
are as
defined
in
Section
4.2 (see Fig. 9b). The
term
A(s)
denotes recruitment
to
the
homosexual population
of
individuals with
sexual activity
s, and N(t, s)
denotes
the
total number
of
persons with activity
s.
From (5.6a),
the
number
of
susceptibles
x{t, s)
is
X(t, s)
=
XQ(S)
exp
(-(it
-1
sk(u) du) +
A(s)j
exp (~n(t
-
x)
-
J sk(u)
duj dx,
(5.8)
where XJis)
is the
number
of
susceptibles
of
activity
s at
time
t
=
0.
If
we
define
Y(t)
as the
total number
of
infectious individuals
at
time
t,
where
=
£
[Y(t,S)
+
I(t,s)]ds,
(5.9)
then (5.6b)
can be
reduced
to
a
differential equation
for Y(t):
~
X(t)fsX(t,
s)
ds
-
(v +
n)Y(t).
(5.10)
dt
Jo
If
we
assume that
the
distribution
of
sexual activity
is
gamma
in
form with mean
m
and
variance
m2/6,
then (5.10)
can be
expressed
in
terms
of
the parameters
of
the distribution
(m and
6).
The properties
of
this
can be
explored using numerical procedures.
The
scale
of
the computational problem, however,
can be
reduced somewhat
by
employing
a
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254
R. M.
ANDERSON ET
AL.
discrete approximation to the continuous gamma distribution of the number of
sexual partners per unit of time. A series of sexual-partner classes (of total n) can
be defined (e.g. those who have 0-1, 1-5, 5-10, 10-50, 50-100, and 100 plus
partners
yr"1),
and the mean number s, of partners for the ith class and the mean
recruitment rate A, of susceptibles into this class may then be calculated from
gamma distributions with defined means (over all classes) and variances. The
equations describing the dynamics within each sexual-partner class (with subscript
/) are as defined in (4.11) with A and c replaced by A, and sh respectively. The
rate of infection for the discrete sexual-partner class appproximation of (5.6)-
(5.7) is defined as
= fit,
s,[Y(t,
0
+
I(t,
0]/i
s,N(t,
i).
One advantage of this approach to the numerical solution of equations (5.6) to
(5.10) is that the discrete approximation to the gamma distribution more
accurately reflects the data that are currently available on the sexual habits of
homosexuals (see Figs. 7 and 8).
Numerical studies of the discrete sexual-partner class model provide some
insight into the manner in which the degree of heterogeneity in sexual activity
amongst homosexuals influences the form of the epidemic. Predictions of the
influence of changes in the variance-to-mean ratio {mid) of the distribution of
sexual activity on the incidence of AIDS and the proportion seropositive in the
population (y(t)
=
[Y(t)+A(t)
+
Z(t)]/N(t) where N(t)
=
ft N(t, s)
ds)
are pre-
sented in Fig. 13.
The graph shows clearly that increasing degrees of heterogeneity in sexual
activity decrease the magnitude of the epidemic of AIDS and reduce the
maximum level of seropositivity for HIV infection attained during the course of
the epidemic. A very high variance-to-mean ratio implies that the population
contains a small fraction of highly sexually active individuals who are removed
rapidly from the infectious pool (either by death or by passage into the
non-infectious seropositive class). As such, the rapid removal of these individuals
reduces the total magnitude of the epidemic. The value of the variance-to-mean
ratio has little influence on the time period that elapses from invasion to the peak
in AIDS cases. The general point illustrated by these numerical simulations is the
important influence of heterogeneity in sexual activity on the pattern and
magnitude of the epidemic.
FIG.
13.
Temporal solutions
of an
approximation
to the
heterogeneous-mixing model with recruit-
ment
of
susceptibles ((5.6)
to
(5.10) plus explanation
in the
main text).
The
graphs show
(a)
proportion seropositive
to HTV and (b)
incidence
of
AIDS
per
year
for
four different variance
to
mean ratios
(m/8) of the
gamma distribution
of
sexual activity:
1 (top
line),
5, 10, 20
(bottom line).
Parameter values:
Ro"5, D
-5yrs, u
=
0-2yr-\
d=
lyr"1,
p
-0-3,
N(0)
•= 100
000,
n~l/32yr~\
Six sexual partner classes were defined
in the
discrete approximation
of the
gamma distribution:
0-1,
1-5,
5-10,
10-50, 50-100,
and 100+
partners
yr~\ The A, and s,
were chosen from gamma
distributions with
the
mean number
of
partners fixed
at 5
yr"1
and
variances
of 5, 25, 50, and 100 (to
give
the
variance
to
mean ratios defined above).
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256
R. M.
ANDERSON
ET AL.
5.3
Variable
Incubation
Periods
in
Heterogeneous
Mixing Models
The model defined
by
(5.6)-(5.7)
can
easily
be
extended
to
encompass
a
variable incubation period
of the
form denned
in
(3.1b)
and
(3.2a)
(e.g.
v(x)
= ax). We
define
Y{t, s, r) as the
number
of
infectious individuals
(who go
on
to
develop AIDS)
at
time
t of
sexual characteristics
5 who
have been
incubating
the
infection
for x
time units with
the
further definition:
Y(t, s) =
jf
Y(t, s,
x)
dx,
Y'(t,
x)
=
|"
Y(t, s,
x)
ds, I(t)
=
f
I(t,
s) ds.
The new model is
9X
— (t,
s) = A(s) -
sX{t,
s)k(t) - nX(t, s),
(5.lla)
C7J
^(t, s, x)
+
j-^(t,
s, x)
=
~[v(x)
+
n]Y(t, s, x),
(5.11b)
j({t,
s)
= (1
-p)sX(t, s)k{t) - (y + fi)l(t, s),
(5.11c)
where
A(0
=
fi£
s[Y(t,
s) + I(t,
s)]dLs/jT
sN(t, s)
ds.
(5.12)
The boundary condition
for
(5.11b)
is
given
by
Y(t,s,O)=psX(t,s)X(t).
(5.13)
The model
is
complex
and
numerical work (again based
on
dividing
the
distribution
of
sexual partners into
a
series
of
discrete classes)
is
required
to
explore
its
properties. Given
the
insights generated
by the
inclusion
of
variable
incubation
in the
homogeneous-mixing model (4.11),
it is
hardly surprising that
the main effect
is to
delay
the
interval between invasion
and
peak incidence,
beyond that predicted
by the
heterogeneous mixing model with
v
constant. Also,
as
we saw
earlier, variable incubation makes
the
epidemic more 'spiky'
in
character (Fig.
14).
6. Discussion
The structures
and
assumptions incorporated
in the
models discussed
in
this
paper remain tentative
at
present. Much
is
unknown about
the
epidemiology
of
HIV,
and
refinements
and
alteration
to the
basic structure
of the
models will
be
required
in the
future,
as
further biological
and
epidemiological data become
available.
The
main reason
for
formulating
and
exploring such models
is to
investigate
how the
various parameters affect viral persistence
and
spread,
and to
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TRANSMISSION OF THE HUMAN IMMUNODEFICIENCY VIRUS257
28
Years
28
Fio.
14. Temporal solutions of the heterogeneous mixing model with recniitment of susceptibles and
variable incubation ((5.11)—(5.13) in the main text). Graphs (a) to (b) as defined in Rg. 13, but with
the mean incubation period set at 5 years (a = 0-0628 yr"1; see (3.2) in the text). The numerical
methods used to generate these graphs are the same as those used in Fig. 13 and defined in the main
text.
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258 R. M. ANDERSON ET AL.
help improve general understanding of the transmission dynamics of HTV
infection. We wish to stress that the models are not yet able to generate reliable
predictions concerning future trends in AIDS incidence.
The major biological unknowns concern the proportion of those infected with
HTV who will progress to develop AIDS, the duration of infectiousness of
infected individuals (and whether or not those who proceed to develop AIDS are
more or less infectious during the incubation period than those who do not), and
the duration of the latent period of infection. At present, reliable data on
incubation periods are only available from those infected via blood transfusions. It
is possible that such data severely underestimate the average incubation period
in those who acquire the virus via sexual contact. Recent work on the rate of
appearance of AIDS in HTV seropositive homosexuals (with unknown dates of
acquisition of infection) has shown that 34% of individuals had developed AIDS
over a three year period (Goedert et al., 1986). This study hints that a much
larger proportion of those infected than previously thought may proceed to
develop AIDS, and that the average incubation period may be considerably in
excess of 4-5 years (Lui et al., 1986). A 10 year average incubation period
has been suggested on the assumption that most of those infected with HIV will
eventually develop AIDS. If this is the case, and if asymptomatic individuals
remain infectious throughout the incubation period, then our crude estimates of
Ro
detailed in the first section must be revised upward. For example, if D = 10
with a doubling time for cases of infection in the early stages of the epidemic of 9
months, then (4.7) gives an Ro value of around 10. An illustration of the
consequences of this increase is presented in Fig. 15 where the predictions of the
homogeneous-mixing model with recruitment, for various values of D and p, are
displayed. Broadly speaking, a longer incubation period results in a more
drawn-out epidemic (a slower decline in cases following the peak in incidence).
Increasing the value of p naturally increases the magnitude of the epidemic. This
simple example well illustrates the problems involved in predicting the future
course of the current epidemic of HTV given the uncertainties surrounding the
biology and epidemiology of the virus.
The heterogeneous-mixing models demonstrate the importance of variability in
sexual activity as a determinant of the pattern of the epidemic. High variability
results in fewer cases and a lower proportion seropositive to HIV antigens. A
substantial fraction of the at high-risk population can remain uninfected, even for
moderately large Ro, if variability in sexual habits is sufficiently high. Active
people are essentially all quickly infected and hence relatively rapidly removed as
infectors before many of the less active individuals are infected. The very simple
heterogeneous model for the early stages of the epidemic provides a useful
illustration of how the mean and variance of sexual activity combine to determine
the magnitude of the force of infection within a community (see (5.5)). Most
importantly, this particular model indicates how heterogeneity can influence the
measurement and interpretation of key epidemiological parameters.For example,
the magnitude of Ro may be estimated from a knowledge of td and D. However, a
rough guide to the magnitude of the average transmission coefficient (the chance
that an infected partner passes on the infection) can only be obtained via a
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TRANSMISSION OF THE HUMAN IMMUNODEFICIENCY VIRUS259
11
10-
9-
8-
3
CO
atienl
ci
w
Q
~o
fe
umbi
z
7-
6-
5-
4-
3-
p „
1-
(a)
Years
FIG.
15. Temporal solutions of the homogeneous-mixing model with recruitment ((4.9) in the text)
showing the effect of varying the incubation (= infectious) period D and of varying the proportion p
of those infected who develop AIDS on the number of AIDS patients throughout the epidemic.
Parameter values as defined for Fig. 10 but with D = 5 yrs and Ro = 5-62 in graph (a), and D = 10 yrs
and Ro= 10-24 in graph (b). In both graphs, p = 0-3 (bottom line), 0-5, and 10 (top line).
at Bodleian Library on July 7, 2014http://imammb.oxfordjournals.org/Downloaded from
260 R. M. ANDERSON ET AL.
knowledge of the mean and variance of the distribution of sexual activity ((4.6),
(4.7),
(5.5)). Any assessment of how much sexual habits must change in order to
reduce Ro for HTV to less than unity depends not simply on changes in mean
activity but also on changes in the variance. A substantial reduction in both could
be achieved by limiting or changing the activities of the small proportion of highly
promiscuous homosexuals alone, irrespective of changes in the habits of the
majority of the population. For example, the unpublished data of Carne & Weller
and McManus (Figs. 7 and 8) suggest that the magnitude of the mean level of
activity is very much dominated by the activities of a small fraction of very highly
active individuals (with 400 or more partners per year).
Our preliminary analyses have centred on the homosexual community and we
have ignored other high-risk groups such as IV drug abusers and transfusion
recipients. The passage of the infection from homosexuals through bisexual men
to heterosexuals, and then between heterosexuals, implies that studies of the
overall transmission dynamics of HTV should ideally be based on a matrix of
transmission rates between the various risk groups in the total population. For
any two groups, the passage of infection between them is likely to be
asymmetrical, due, in part, to unequal population sizes. Data of sufficient
quantity to define such a matrix of transmission coefficients are unavailable at
present, but their collection should clearly be a priority in order to assess the likely
spread of the virus through the heterosexual community. Transmission between
heterosexuals is known to occur (via studies of the partners of heterosexuals
infected by blood transfusion), and it appears to be an efficient route for the
spread of the virus (Vogt et al., 1985; Des Jarlais et al., 1984). In certain parts of
Africa, heterosexuals appear to be the major risk group. This may be associated
with differences in sexual and other practices that facilitate virus transmission
(Biggar et al., 1986; Piot et al, 1984). To what extent the virus will be able to
persist within a wide cross-section of adult society in Europe and North America
remains to be seen.
However, it is possible that, following its recent establishment in heterosexual
population, HIV could spread rapidly and persist endemically. Whether the virus
can maintain itself within heterosexual communities depends ultimately on
whether the values of /3 (transmission probability) and c (effective average
number of sexual partners)—although lower than the corresponding values for
transmission among homosexual males—are nevertheless large enough to keep Ro
above unity. As far as the homosexual community is concerned, the prevailing
variability in sexual activity within a population will have a major influence on the
possible rate of spread of HIV, on the total magnitude of the initial epidemic, and
on the equilibrium endemic prevalence of infection.
Finally, it is worth considering briefly the future role of mathematical models in
the study of AIDS. An immediate problem that springs to mind is: 'why bother
with simple models, when those of great complexity, perhaps encapturing
transmission between many at-risk groups, can be explored by numerical means.'
At this stage of data collection and given current biological and epidemiological
knowledge, it seems sensible to concentrate on models that focus on what
information can and cannot be extracted from the sparse data available.
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TRANSMISSION OF THE HUMAN IMMUNODEFICIENCY VIRUS 261
Estimates of the doubling time can be made, and they yield crude measures of the
overall basic reproductive rate within given at-risk groups (in particular, the
dominant one—homosexual males). The information is not sufficient as yet to
determine how this overall rate is made up of contributions arising from
cross-transmission between risk groups. One hopes it will become so, as cohort
studies progress. Data on the peak incidence of AIDS in the initial epidemic
within a particular at-risk group will tell us about the proportion of infecteds who
go on to develop AIDS, the duration of the incubation period, and variability in
sexual activity. Similarly, the maximum prevalence of seropositivity to HTV
antigens will provide information on Ro and variability in sexual activity (May and
Anderson, 1987).
As the intensity of epidemiological research increases, simple models will be of
less value once the various unknowns and complications that have been hinted at
in this paper are defined and quantified. When this stage is reached, such models
should provide tools for predictive work to help assess future health-care and
service needs. The last year has seen increasing pressure on government and
health services in the U.K. to take preventative action to control the HIV
epidemic and to encourage behavioural changes. Models may help in determining
the magnitude and type of behavioural change which has to be achieved to
control the spread of infection.
Acknowledgements
We thank the M.R.C. for financial support of part of this research (to R. M.
A.) and Drs. I. Weller, C. Carne, and T. MacManus for access to unpublished
data on the sexual activity of homosexuals in London.
This article is based on a paper read at the EMA Conference on "The
Mathematical Theory of Biological Systems' held in Oxford, 7-9 July, 1986.
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