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Mathematical Biosciences 169 (2001) 53±87
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Three stage AIDS incubation period: a worst case scenario
using addict±needle interaction assumptions
Fraser Lewis, David Greenhalgh *
Department of Statistics and Modelling Science, University of Strathclyde, Livingstone Tower, 26 Richmond Street,
Glasgow G1 1XH, UK
Received 12 October 1999; received in revised form 28 July 2000; accepted 22 September 2000

Abstract
In this paper we develop and analyse a model for the spread of HIV/AIDS amongst a population of
injecting drug users. We start o€ with a brief literature survey and review; this is followed by the derivation
of a model which allows addicts to progress through three distinct stages of variable infectivity prior to the
onset of full blown AIDS and where the class of infectious needles is split into three according to the
di€erent levels of infectivity in addicts. Given the structure of this model we are required to make assumptions regarding the interaction of addicts and needles of di€erent infectivity levels. We deliberately
choose these assumptions so that our model serves as an upper bound for the prevalence of HIV under the
assumption of a three stage AIDS incubation period. We then perform an equilibrium and stability analysis
on this model. We ®nd that there is a critical threshold parameter R0 which determines the behaviour of the
model. If R0 6 1, then irrespective of the initial conditions of the system HIV will die out in all addicts and
all needles. If R0 > 1, then there is a unique endemic equilibrium which is locally stable if, as is realistic, the

time scale on which addicts inject is much shorter than that of the other epidemiological and demographic
processes. Simulations indicate that if R0 > 1, then provided that disease is initially present in at least one
addict or needle it will tend to the endemic equilibrium. In addition we derive conditions which guarantee
this. We also ®nd that under calibration the long term prevalence of disease in our variable infectivity
model is always greater than in an equivalent constant infectivity model. These results are con®rmed and
explored further by simulation. We conclude with a short discussion. Ó 2001 Elsevier Science Inc. All
rights reserved.
Keywords: HIV; AIDS; Three stage infectivity; Equilibrium and stability analysis; Pessimistic model

*

Corresponding author. Tel.: +44-141 552 4400, ext. 3653; fax: +44-141 552 2079.
E-mail address: david@stams.strath.ac.uk (D. Greenhalgh).

0025-5564/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved.
PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 5 3 - 5

54

F. Lewis, D. Greenhalgh / Mathematical Biosciences 169 (2001) 53±87

1. Introduction and literature review
It is commonly thought that an individual infected with human immunode®ciency virus (HIV)
is not uniformly infectious throughout the whole acquired immune de®ciency syndrome (AIDS)
incubation period, instead the infectious period can be divided into three stages during which an
individual has respectively, very high, very low and intermediate infectivity [1±3]. While there have
been several studies of the e€ect of this on the sexual transmission of HIV, to the best of our
knowledge most previous studies of the spread of HIV/AIDS amongst drug users assume constant
infectivity throughout the incubation period. In this paper we extend a single stage infectivity
model due to Kaplan and O'Keefe [4], which has assisted with the development of needle exchange programs and legislation across the USA, to cater for a three stage infectious period. Due
to its practical impact we feel justi®ed in using this model as a basis for our investigation into the
e€ects of three stage infectivity on the spread of HIV via needle sharing.
We extend the Kaplan and O'Keefe model to investigate the e€ect of allowing addicts to progress through three stages of infectivity prior to the onset of full blown AIDS. We ®rst review some
of the background to the mathematical modelling of HIV/AIDS amongst populations of intravenous drug users and the case for including a three stage infectious period. In Section 2, we discuss
Kaplan and O'Keefe's model and its underlying assumptions. In Section 3, we extend this model to
allow addicts and needles to exist in three infectious states and derive the di€erential equations
which de®ne this extended model. In Section 4, we perform an equilibrium and stability analysis on
our extended model and examine the di€erences between the long term behaviour of this model and
the original Kaplan and O'Keefe model. There is a critical threshold parameter R0 which determines the behaviour of the three stage model and we discuss the interpretation of this parameter. In
Section 5, we examine numerical simulations of the three stage model and the Kaplan and O'Keefe

model in order to validate our previous mathematical results and to examine any di€erences in
dynamic behaviour between these models. A brief discussion concludes the paper.
The ®rst cases of AIDS were diagnosed in the early 1980s and soon after the infectious agent of
AIDS, HIV, was isolated. From the mid 1980s onwards the worldwide number of cases of HIV and
AIDS has risen dramatically. UNAIDS [5] claims that in 1998 alone AIDS accounted for 2.5 million
deaths worldwide with a further 5.8 million people newly infected with HIV. AIDS is present
throughout the world but the population groups worst a€ected vary substantially. For example the
majority of cases in North America and Western Europe have been in homosexual men and intravenous drug users, whereas in sub-Saharan Africa the spread is mostly through heterosexual
contact [5]. In recent years there has been a rapid increase in the number of cases of HIV in the
Ukraine, Belarus, Moldova and the Russian Federation, mostly among intravenous drug users [6].
Mathematical modelling of the spread of HIV and AIDS represents a large body of work,
however, the majority of this has concentrated on the sexual spread of the disease [7]. In recent
years the number of articles concerned speci®cally with modelling the spread of HIV among
populations of intravenous drug users has increased. Populations of intravenous drug users are
particularly vulnerable to HIV infection due to the common practice of sharing injection
equipment. To our knowledge the ®rst attempt at modelling the spread of HIV via needle sharing
among a population of intravenous drug users is due to Kaplan [8]. This is a pioneering paper and
has been a starting point for much of the literature concerned with modelling the spread of HIV
among intravenous drug users. Kaplan and O'Keefe [4], extend Kaplan's original model to allow

F. Lewis, D. Greenhalgh / Mathematical Biosciences 169 (2001) 53±87

55

addicts to clean (or bleach) injection equipment prior to use and to allow needles to be removed
from the population and be replaced by unused (and obviously uncontaminated) needles. We
discuss this model in detail in Section 2.
One of the main de®ciencies in the models discussed by Kaplan and Kaplan and O'Keefe is that
the population of intravenous drug users is treated as a single homogeneous group. In a large
metropolitan area such as New York City which has an estimated 200 000 drug users, [9], it is almost
inevitable that many `shooting galleries' will exist and each of these may have a di€erent composition of drug users (in terms of needle cleaning practices and the rate at which needles are shared).
Greenhalgh [10] extends Kaplan's original model to incorporate variability in the rate at which
addicts visit `shooting galleries' and the choice of `shooting gallery'. In addition it is no longer assumed that all addicts successfully clean needles prior to injection with the same probability.
Greenhalgh and Hay [7] examine a further potential de®ciency in Kaplan's original model. They
examine the relationship between the probability that an infectious needle is ¯ushed by a susceptible
addict and the probability that the susceptible addict is infected during this process. Kaplan assumes
that these two probabilities are independent, but intuitively the probability of infection should
increase if the needle is ¯ushed. In addition to incorporating a joint probability distribution between
the transmission probability of HIV and the probability that a needle is ¯ushed Greenhalgh and Hay
also allow infectious addicts to leave a needle virus free after use and examine the possibility that

addicts who discover that they are HIV positive stop or at least reduce their level of needle sharing.
So far we have discussed Kaplan's basic model and a number of more realistic extensions. We
now discuss other work not directly based on Kaplan's model. Heterogeneous mixing in addicts is
both more realistic and gives long term prevalence results which di€er from homogeneous models
[11]. Capasso et al. [12] discuss a deterministic model which assumes that addicts share needles in
`friendship groups'. They show that for the prevalence of disease to reach an endemic equilibrium
among the population the basic reproductive number must exceed unity. If the basic reproductive
number is less than or equal to unity then the disease will die out in all addicts and all needles.
Gani and Yakowitz [13] model the spread of HIV through the sharing of contaminated needles
amongst small groups of intravenous drug users who are friends or relatives (buddy-users). They
use a Markov chain model to examine the increase in the number of infectious users among stable
groups of addicts. Yakowitz [14] uses a stochastic simulation approach to model the transmission
of HIV among a population of drug users who meet on a periodic basis to share needles and inject
drugs. Allard [15] describes a mathematical model of the risk of infection from sharing injection
equipment. He uses a probabilistic (as opposed to dynamic) model which examines risk of infection from HIV each time an addict injects with a shared needle.
An analogy can be drawn between the spread of malaria through mosquito bites and the spread
of HIV through the sharing of contaminated drug injection equipment. Massad et al. [16] explore
this analogy and develop a new approach for the estimation of the basic reproductive number for
HIV among intravenous drug users. Blower et al. [17] use a data-based deterministic model to
examine the epidemiological consequences of heterosexual, intravenous drug use and perinatal

transmission in New York City, USA. This model consists of 34 ordinary di€erential equations
and a large number of behavioural parameters. Kretzschmar and Wiessing [18] examine the
spread of HIV among populations of drug users in the Netherlands using a stochastic simulation
model. They examine the frequency at which needles are shared and the social networks in which
sharing occurs. In addition they incorporate variability in the infectivity of addicts by assuming

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F. Lewis, D. Greenhalgh / Mathematical Biosciences 169 (2001) 53±87

that after initial infection addicts enter a brief period of very high infectivity after which infectivity
is very low until the development of full blown AIDS.
Peterson et al. [19] use a complex Monte Carlo simulation model to examine behavioural and
epidemiological e€ects of HIV infection among populations of intravenous drug users. Their
simulation model consists of three interacting sub-models: a model of HIV disease progression
within an infected individual; a model describing the heterogeneity of intravenous drug use within
needle sharing injecting communities; and a model of the social networks describing the pattern of
needle sharing in drug addicts. Seitz and M
uller [20] model the spread of HIV in the population at
large (including drug addicts, heterosexual and homosexual population groups) and speci®cally

examine the e€ect of moving from the assumption of constant infectivity to a three stage infectious period. They assume that the infectivity of an HIV positive individual has a so-called `bathtub' shape and ®nd that in comparison to constant infectivity this assumption greatly increases the
long term incidence of HIV and AIDS. Tan and Tang [21] formulate a stochastic model for the
HIV epidemic involving both sexual contact and intravenous drug use. They divide the population
of addicts into susceptible, infectious with HIV or full blown AIDS. In addition the incubation
period is divided into k sub-stages to allow for varying levels of infectivity. This stochastic model
also separates addicts into groups according to their sexual behaviour and frequency of drug use.
We have brie¯y outlined some of the literature concerned with modelling the spread of HIV
among intravenous drug using populations. We now discuss the case for using a three stage rather
than a single stage AIDS incubation period. Jacquez et al. [1] use an infectious period with three
sequential infectious stages: primary infection, asymptomatic and pre-AIDS, based on an original
seven stage model for CD4 count progression [22]. Estimates for the mean total duration of the
AIDS incubation period are around 10 years. Other estimates of the length of the incubation
period are a median of 9.8 years [23], and 10.5 years [24] and a mean of between 9 and 13.5 years
[3]. Koopman et al. [2] estimate that the mean duration spent in each infectious period is 1.5, 104
and 14.5 months for primary infection, asymptomatic and pre-AIDS stages, respectively. Other
articles contain broadly similar estimates [1,3]. The life expectancy of an individual on developing
full-blown AIDS is approximately 1 year [24].

2. Kaplan and O'Keefe model
The model which we investigate in this paper is developed from a model due to Kaplan and

O'Keefe [4] which is itself an extension of a model due to Kaplan [8]. We use the model featured in
the later paper as it incorporates a needle exchange program which has been demonstrated to be
an important measure in reducing the spread of HIV among intravenous drug users. Greenhalgh
and Hay [7] discuss Kaplan's model in detail. Kaplan describes a deterministic model and assumes
that the population amongst whom the disease is spreading is of size n, where n is large. The
following assumptions are also made [8]:
1. All sharing of drug injecting equipment occurs in shooting galleries. In the model a shooting
gallery is de®ned as a location where addicts sequentially rent the same drug-injection equipment. There are m shooting galleries (or equivalently m `kits' of drug-injection equipment
are in circulation) and addicts select shooting galleries (or `kits') at random. All addicts inject
once per visit to a shooting gallery.

F. Lewis, D. Greenhalgh / Mathematical Biosciences 169 (2001) 53±87

57

2. Each addict visits shooting galleries in accordance with a Poisson process with rate k, independently of the actions of other addicts.
3. Injection equipment always becomes infectious if it is used by an infected addict. When infectious injection equipment is used by an uninfected addict the act of injecting will replace the
infectious blood in the needle with uninfectious blood from the addict with probability h. When
this occurs the needle is said to have been `¯ushed'. Any uninfected addict who uses infectious
injection equipment is considered to be exposed to HIV.

4. Given exposure to HIV an addict becomes infected with probability a; a is the infectivity of
HIV via shared injection equipment. Sharing injection equipment is the only means by which
addicts may become infected.
5. Infectious addicts develop full blown AIDS according to a Poisson process with rate d, at this
stage addicts leave the sharing, injecting population. These addicts are immediately replaced by
susceptible addicts.
6. Infectious addicts depart the population for reasons other than developing full blown AIDS
(for example, due to death, treatment with methadone, or relocation) at rate l and are immediately replaced by susceptible addicts.
7. The random variability in the fraction of infected addicts and needles at time t is suciently
small to be ignored.
The Kaplan and O'Keefe extension to Kaplan's model additionally assumes that
8. An addict e€ectively cleans (or bleaches) the injection equipment immediately prior to use with
probability /.
9. Each needle is exchanged (or renewed) for an uninfected needle according to a Poisson process
with rate s.
We now state the equations which de®ne the model based on Assumptions 1±9. Let p…t† denote
the fraction of the population of addicts that are infected with HIV at time t (the prevalence of HIV
infection), and b…t† denote the fraction of the population of needles that are infected with HIV at
time t. De®ne the `gallery ratio' by c ˆ n=m, this represents the (constant) number of addicts per
needle in the population. The following di€erential equations describe the spread of the disease:

dp
ˆ …1 ÿ p†kba…1 ÿ /† ÿ p…l ‡ d†;
…1†
dt
and
db
ˆ …1 ÿ b†kcp ÿ bkc…1 ÿ p†…1 ÿ …1 ÿ h†…1 ÿ /†† ÿ bs:
…2†
dt
For Kaplan's original model [8] it was shown that an endemic solution is possible if and only if the
parameter R0 exceeds one, where
ka
R0 ˆ
:
…3†
…l ‡ d†h
This result easily extends to the Kaplan and O'Keefe model described by Eqs. (1) and (2) except
now we have that
ka…1 ÿ /†
R0 ˆ

;
…4†
…l ‡ d†…h^ ‡ s^†

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F. Lewis, D. Greenhalgh / Mathematical Biosciences 169 (2001) 53±87

where h^ ˆ 1 ÿ …1 ÿ h†…1 ÿ /† and s^ ˆ s=…kc†. Kaplan [8] showed that the parameter R0 has a
natural biological interpretation as the total expected number of secondary infections caused by a
single infectious addict during his or her entire infectious lifetime, on entering into a population of
uninfectious needles and susceptible addicts. Again this interpretation also extends to the expression in Eq. (4). Note that the threshold parameter R0 is, as usual, a key parameter determining
whether the disease will establish itself. We expect the epidemic to take o€ if R0 > 1, and to die out
if R0 6 1.

3. Three stage infectivity model
We have outlined the single stage infectivity model due to Kaplan and O'Keefe, we now discuss
extending this model to include a three stage AIDS incubation period. First we extend the single
stage model to allow addicts to ¯ow through each of the three infectious stages. This is
straightforward and involves replacing Assumption 5 in Kaplan and O'Keefe's model with:
(5a) After initial infection an addict is de®ned to be acutely infectious and enters the asymptomatic stage according to a Poisson process with rate d1 ;
(5b) Asymptomatic addicts enter the pre-AIDS stage according to a Poisson process with rate
d2 ;
(5c) Pre-AIDS addicts enter the full blown AIDS stage according to a Poisson process with rate
d3 , at this stage addicts leave the sharing, injecting population. These addicts are immediately
replaced by susceptible addicts.
The e€ect of these additional assumptions is to break up Eq. (1) into three sequential classes, p1 ; p2
and p3 , representing the prevalence of stage one, stage two and stage three infected addicts, respectively. Note that we have assumed that addicts must progress through each infectious stage in
turn and therefore superinfection cannot occur. This assumption is usually made in HIV models
with variable infectivity, [1±3,18±20,23]. Medical evidence supporting this assumption is discussed
in [1]. We now derive the di€erential equations which de®ne the spread of HIV among an intravenous drug addict population where addicts progress through three stages of infectivity prior
to the onset of full blown AIDS. The number of stage one infected addicts at time t ‡ Dt
ˆ fnumber of stage one addicts at time tg
‡ f…number of uninfected addicts at time t†
 …fraction of addicts who inject in ‰t; t ‡ Dt† with an infectious needle
which is not cleaned prior to use and where transmission of HIV occurs
in a single injection†g
ÿ fnumber of stage one infected addicts who progress into stage two
infectivity or leave the sharing; injecting population in ‰t; t ‡ Dt†g:
Thus
np1 …t ‡ Dt† ˆ np1 …t† ‡ n…1 ÿ p1 …t† ÿ p2 …t† ÿ p3 …t††kDtb…t†a…1 ÿ /†
ÿ np1 …t†Dt…l ‡ d1 † ‡ o…Dt†:

F. Lewis, D. Greenhalgh / Mathematical Biosciences 169 (2001) 53±87

59

Subtracting np1 …t† from both sides, dividing by nDt and letting Dt ! 0 we deduce that
!
3
X
dp1
ˆ 1ÿ
pi kba…1 ÿ /† ÿ p1 …l ‡ d1 †:
dt
iˆ1

The number of stage two infected addicts at time t ‡ Dt
ˆ fnumber of stage two addicts at time tg

‡ fnumber of stage one addicts who enter the stage two infectious
class in ‰t; t ‡ Dt†g
ÿ fnumber of stage two addicts who enter the stage three infectious
class or leave the sharing; injecting population in ‰t; t ‡ Dt†g:
Thus
np2 …t ‡ Dt† ˆ np2 …t† ‡ np1 …t†d1 Dt ÿ np2 …t†…l ‡ d2 †Dt ‡ o…Dt†:
Subtracting np2 …t† from both sides, dividing by nDt and letting Dt ! 0 we deduce that
dp2
ˆ d1 p1 ÿ …l ‡ d2 †p2 :
dt
Similarly
dp3
ˆ d2 p2 ÿ …l ‡ d3 †p3 :
dt
We have now extended the single stage model to allow addicts to move through three infectious
stages prior to the onset of full blown AIDS. We are assuming that the infectivity of addicts in
each of the three infectious stages is di€erent. Hence, we need to adjust the single population of
needles in Kaplan and O'Keefe's model to re¯ect this, (since it is the infectivity of addicts which
determines the infectivity of a shared needle). The most natural way to divide the single population of infectious needles is to split this into three sub-populations, each corresponding to the
three infectious stages of the addicts. Hence the ®rst sub-population contains (previously uninfectious) needles which have been used by addicts in stage one infectivity and have therefore an
HIV viral load proportional to that of the blood in the addict. Similarly the second and third subpopulations correspond to (previously uninfectious) needles used by addicts in stage two and
stage three infectivity, respectively. We now have three types of infectious needles in our model
and therefore need to replace ba in dp1 =dt with b1 a1 ‡ b2 a2 ‡ b3 a3 , where bi is the prevalence of
stage i infectivity among needles and ai is the probability of HIV transmission from a stage i
needle in a single injection.
We have now adjusted the addict equations in the Kaplan and O'Keefe model, it now remains
for us to incorporate three types of infectious needles. This is a much more dicult task as to
construct a model with both three types of infectious addicts and three types of infectious needles
we now are required to determine the outcomes of addict±needle interactions. For example we
require the outcome of the event where an addict in stage one infectivity uses a needle in stage two
infectivity. If the addict ¯ushes the needle then the HIV viral load left in the needle after use will be
comparable to that of the stage one addict and hence the needle should move from the stage two

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F. Lewis, D. Greenhalgh / Mathematical Biosciences 169 (2001) 53±87

to the stage one infectious population. Alternatively if the needle is not ¯ushed then the HIV viral
load in the needle may be more likely to remain close to stage two infectivity, in which case the
needle remains in the stage two population. More precisely we must specify for i; j; k ˆ 0; 1; 2; 3
what fraction pijk of needles initially in infectious stage i are left in stage k after use by an addict in
stage j. This gives 64 potential needle±addict interactions. However for 16 of these cases the
answer is obvious. If the initial infectious stage of the needle is equal to the infectious stage of the
addict then the ®nal infectious stage of the needle must be the same as the initial stage.
It is dicult to determine the remaining 48 pijk probabilities. It is clear that important factors in
the outcome of each interaction are di€erences in HIV viral load between the di€erent infectious
stages, the volume of addict's blood which is drawn into a needle and the volume of blood already
in the needle from the previous user. Unfortunately there are no empirical data to aid with the
problem of estimating these probabilities pijk . Research has been carried out to ascertain the relative HIV viral load in human blood during each stage of infectivity [25,26]; however, to the best of
our knowledge this is the extent of the data. Jacquez et al. [1] and Hyman et al. [3] claim that viral
loads in infectious stages one, two and three are approximately in the ratio, 100:1:10. While these
data are useful they do not assist directly in determining the outcome of any of the addict±needle
interactions as we can only guess at the di€erence between the volume of blood drawn into a needle
and the volume of residual blood left behind in the needle after an addict has used it.
While it is dicult to determine individual pijk terms it is straightforward to choose addict±
needle interaction assumptions which give rise to a set of pijk terms which will be more pessimistic
than would realistically be the case. By pessimistic we mean that the incidence rate and prevalence
of the disease should be higher than reasonably expected. In this way we can establish an upper
bound for the prevalence of HIV among intravenous drug users under the assumption of a three
stage infectious period. Kaplan and O'Keefe make two assumptions in their model relating to the
way addicts and needles interact. Firstly they assume that h ˆ 0 and hence a susceptible addict
cannot render an infectious needle virus free. This in itself was chosen as a deliberately pessimistic
assumption. Experimental evidence shows that HIV can still be isolated from syringes in which
the infected blood has been greatly diluted [27]. This provides some supporting evidence for the
hypothesis that `¯ushing' of infected needles never occurs and indirectly supports the model assumptions made in this paper. Secondly Kaplan and O'Keefe assume that an uninfectious needle
always becomes infectious after use by an infectious addict. We now generalise these assumptions
to mean ®rstly that an addict of a lower infectivity class than the needle being used cannot alter
the viral load in this needle, and secondly that a needle of a lower infectivity class than the addict
always adopts the infectivity characteristics of the addict. Excluding needle cleansing and exchange this implies that through use a needle becomes more infectious until it ends up in the
highest infectivity level. Therefore, these addict±needle interaction assumptions are more pessimistic than we would expect to occur in practice and hence a model constructed using them will
provide an upper bound for the prevalence of HIV. As in Kaplan and O'Keefe's model [4] it is
useful to have a model which is pessimistic about endemic levels of HIV and AIDS as we can be
reasonably con®dent that any control measures based on this model will eliminate the disease in
practice if they do so in our model and any cost estimates based on the pessimistic model will
exceed the likely real costs. We have that stage one (primary infection) is more infectious than
stage three (pre-AIDS) which in turn is more infectious than stage two (asymptomatic). Hence we
can now derive the needle equations in our upper bound three stage model.

F. Lewis, D. Greenhalgh / Mathematical Biosciences 169 (2001) 53±87

The number of infected stage one needles at time t ‡ Dt
ˆ fnumber of stage one infectious needles at time tg
‡ f…number of non-stage one needles at time t†
 …fraction of syringes used by stage one infected addicts in ‰t; t ‡ Dt††g
ÿ f…number of stage one infected needles at time t†
 …fraction of needles used and successfully cleaned prior to use
by non-stage one addicts in ‰t; t ‡ Dt††g
ÿ fnumber of stage one infectious needles exchanged in ‰t; t ‡ Dt†g:
Thus
b1 …t ‡ Dt† ˆ mb1 …t† ‡ mkDtcp1 …t†…1 ÿ b1 …t†† ÿ mkDtc/…1 ÿ p1 …t††b1 …t† ÿ mb1 …t†sDt ‡ o…Dt†:
Subtracting mb1 …t† from both sides, dividing by mDt and letting Dt ! 0 we deduce that
db1
ˆ kc…1 ÿ b1 †p1 ÿ b1 …1 ÿ p1 †/kc ÿ b1 s:
dt
The number of infected stage two needles at time t ‡ Dt
ˆ fnumber of stage two infectious needles at time tg
‡ f…number of uncontaminated needles at time t†
 …fraction of needles used by stage two infected addicts in ‰t; t ‡ Dt††g
‡ f…number of stage three and stage one needles at time t†
 …fraction of needles used and cleaned prior to use by stage two addicts
in ‰t; t ‡ Dt††g
ÿ f…number of stage two infected needles at time t†
 …fraction of needles used by stage one or stage three addicts in ‰t; t ‡ Dt††g
ÿ f…number of stage two infected needles at time t†
 …fraction of needles used and cleaned prior to use by uncontaminated
addicts in ‰t; t ‡ Dt††g
ÿ fnumber of stage two infectious needles exchanged in ‰t; t ‡ Dt†g:
Thus
mb2 …t ‡ Dt† ˆ mb2 …t† ‡ mkDtcp2 …t† 1 ÿ

3
X

!

‡ mkDtc/b1 …t†p2 …t† ‡ mkDtc/b3 …t†p2 …t†
!
3
X
ÿ mkDtc…p1 …t† ‡ p3 …t††b2 …t† ÿ mkDtc/ 1 ÿ
pi …t† b2 …t†
iˆ1

bi …t†

iˆ1

ÿ mb2 …t†sDt ‡ o…Dt†:

Subtracting mb2 …t† from both sides, dividing by mDt and letting Dt ! 0 we deduce that

61

62

F. Lewis, D. Greenhalgh / Mathematical Biosciences 169 (2001) 53±87

!
!
3
3
X
X
db2
pi ÿ b2 s:
bi p2 ‡ b1 p2 /kc ‡ b3 p2 /kc ÿ b2 p3 kc ÿ b2 p1 kc ÿ b2 kc/ 1 ÿ
ˆ kc 1 ÿ
dt
iˆ1
iˆ1

The number of infected stage three needles at time t ‡ Dt

ˆ fnumber of stage three infectious needles at time tg
‡ f…number of uncontaminated and stage two needles at time t†
 …fraction of needles used by stage three infected addicts in ‰t; t ‡ Dt††g
‡ f…number of stage one needles at time t†
 …fraction of needles used and cleaned prior to use by stage three addicts
in ‰t; t ‡ Dt††g
ÿ f…number of stage three infected needles at time t†
 …fraction of needles used by stage one addicts in ‰t; t ‡ Dt††g
ÿ f…number of stage three infected needles at time t†
 …fraction of needles used and cleaned prior to use by uncontaminated
or stage two addicts in ‰t; t ‡ Dt††g
ÿ fnumber of stage three infectious needles exchanged in ‰t; t ‡ Dt†g:
Thus
mb3 …t ‡ Dt† ˆ mb3 …t† ‡ mkDtcp3 …t†…1 ÿ b1 …t† ÿ b3 …t†† ‡ mkDtc/b1 …t†p3 …t† ÿ mkDtcb3 …t†p1 …t†
ÿ mb3 …t†kDtc/…1 ÿ p1 …t† ÿ p3 …t†† ÿ mb3 …t†sDt ‡ o…Dt†:
Subtracting mb3 …t† from both sides, dividing by mDt and letting Dt ! 0 we deduce that
db3
ˆ kcp3 …1 ÿ b1 ÿ b3 † ‡ kc/b1 p3 ÿ kcb3 p1 ÿ b3 kc/…1 ÿ p1 ÿ p3 † ÿ b3 s:
dt
Hence the system of di€erential equations which describe the spread of the disease are
!
3
X
dp1
ˆ 1ÿ
pi k…b1 a1 ‡ b2 a2 ‡ b3 a3 †…1 ÿ /† ÿ …l ‡ d1 †p1 ;
dt
iˆ1

…5†

dp2
ˆ d1 p1 ÿ …l ‡ d2 †p2 ;
dt

…6†

dp3
ˆ d2 p2 ÿ …l ‡ d3 †p3 ;
dt

…7†

db1
ˆ kc…1 ÿ b1 †p1 ÿ b1 …1 ÿ p1 †/kc ÿ b1 s;
dt
!
3
X
db2
ˆ kc 1 ÿ
bi p2 ‡ b1 p2 /kc ‡ b3 p2 /kc ÿ b2 p3 kc
dt
iˆ1
!
3
X
pi ÿ b2 s;
ÿ b2 p1 kc ÿ b2 kc/ 1 ÿ
iˆ1

…8†

…9†

F. Lewis, D. Greenhalgh / Mathematical Biosciences 169 (2001) 53±87

63

and
db3
ˆ kcp3 …1 ÿ b1 ÿ b3 † ‡ kc/b1 p3 ÿ kcb3 p1 ÿ b3 kc/…1 ÿ p1 ÿ p3 † ÿ b3 s
…10†
dt
with suitable initial conditions: 0 6 p1 …0†, p2 …0†, p3 …0†, b1 …0†, b2 …0†, b3 …0†, p1 …0† ‡ p2 …0† ‡ p3 …0†
6 1 and b1 …0† ‡ b2 …0† ‡ b3 …0† 6 1.
We have formally derived the equations which de®ne our upper bound three stage infectivity
model, we now investigate the behaviour of the solutions to this system of di€erential equations.
In particular we are interested in the conditions necessary for the disease to die out or persist in
the population.

4. Equilibrium and stability results
In this section we examine the behaviour of our upper bound three stage infectivity model and
use analytical results to illustrate key properties. We are primarily interested in whether the long
term behaviour of the three stage model is similar to that of the Kaplan and O'Keefe model.
Greenhalgh and Hay [7] showed that for the model used by Kaplan [8], a critical threshold parameter exists which de®nes the long term behaviour of this model. This threshold result also
extends directly to Kaplan and O'Keefe's model. We now wish to determine the long term behaviour of the three stage infectivity model.
De®ne the region D in R6 by D ˆ ‰0; 1Š6 . The system de®ned by di€erential equations (5)±(10)
starts in the region D. The right-hand sides of these equations are di€erentiable with respect to
p1 ; p2 ; p3 ; b1 ; b2 and b3 with continuous derivatives, and the corresponding vector points into D on
its boundary except at the origin, which is clearly an equilibrium point. It is straightforward to
show using standard techniques [28] that Eqs. (5)±(10) with initial conditions in D, have a unique
solution that remains in D for all time. De®ne


k…1 ÿ /†
a2 d1
a3 d1 d2
:
…11†
‡
a1 ‡
R0 ˆ
…l ‡ d1 †…^
l ‡ d2 …l ‡ d2 †…l ‡ d3 †
s ‡ /†
Note also that the time scale on which addicts inject is of the order of days whereas that of the
other epidemiological and demographic processes is measured in years and is a lot slower. We
de®ne My to be the matrix:
2

l ‡ k…1 ÿ /†
6
0
6
6
6
ÿd
3
6
6 ÿ…1 ÿ pH †k…a1 ÿ a3 †…1 ÿ /†
6
4
ÿ…1 ÿ pH †ka2 …1 ÿ /†
ÿ…1 ÿ pH †k…a3 ÿ a2 †…1 ÿ /†

ÿd1
l ‡ d1 ‡ d2
ÿd1
0
0
0

0
ÿd2
l ‡ d3
0
0
0

3
ÿkc…1 ÿ bH
ÿkc…1 ÿ bH …1 ÿ /†† ÿkc…1 ÿ bH
1 …1 ÿ /††
1‡3 …1 ÿ /††
7
ÿkc…1 ÿ bH
0
ÿkc…1 ÿ bH
1 …1 ÿ /††
1‡3 …1 ÿ /†† 7
7
H
7
ÿkc…1 ÿ b1 …1 ÿ /††
0
0
7:
7
kc…/ ‡ s^ ‡ …1 ÿ /††
0
0
7
5
0
kc…/ ‡ s^ ‡ …1 ÿ /††
0
0
0
kc…/ ‡ s^ ‡ …1 ÿ /††

The following theorem is the main result of the paper; it indicates that if R0 6 1 we expect the
disease to die out, whereas if R0 > 1 we expect the disease to take o€.

Theorem 4.1. If R0 6 1 the system of equations (5)±(10) has a unique equilibrium solution where the
disease has died out in both addicts and needles. Whatever the initial state the disease will die out in

64

F. Lewis, D. Greenhalgh / Mathematical Biosciences 169 (2001) 53±87

all addicts and all needles. If R0 > 1, and provided that disease is initially present, then there exists
 > 0 and g > 0 such that for i ˆ 1; 2; 3, pi …t† >  and b1 …t† ‡ b2 …t† ‡ b3 …t† >  for t > g; moreover
there now also exists a unique endemic equilibrium solution. If, as appears realistic, the time scale on
which addicts inject is much shorter than that of other epidemiological and demographic processes
then the endemic equilibrium is locally stable. If …l ‡ d1 ‡ d2 †…l ‡ d3 † > d1 d2 , det My > 0, R0 > 1
and disease is present initially, then both the fraction of infected addicts and infected needles tend to
their unique endemic equilibrium values.
Proof. See Appendix A. 
4.1. Interpretation of R0
Having shown that the parameter R0 is a critical threshold parameter we now examine the
biological interpretation of this parameter. Consider a single newly infected addict entering a
population at the disease-free equilibrium containing only susceptible addicts and uninfectious
needles. It is straightforward to derive an expression for the expected number of secondary infections caused by this single infected addict. The initial infection process can be broken down
into two distinct phases: ®rstly the disease passes from the single infectious addict to an uninfectious needle, secondly this needle (which is now infectious) passes on the disease to a susceptible addict. We ®rst derive the expected number of each type of infectious needle a single
infectious addict will create during his or her entire infectious lifetime. We then derive the expected number of addicts each of these three types of infectious needle will infect.
Addicts progress through three infectious stages, during each stage an addict will leave needles
infectious. Addicts inject at rate k per unit time and spend on average 1=…l ‡ d1 † time units in
stage one. An addict progresses from stage one to stage two with probability d1 =…l ‡ d1 † and
spends on average 1=…l ‡ d2 † time units in this stage. Similarly an addict progresses from stage
two to stage three with probability d2 =…l ‡ d2 † and spends on average 1=…l ‡ d3 † time units in this
stage. Hence on average an addict creates
k
l ‡ d1
stage one infectious needles,
kd1
…l ‡ d1 †…l ‡ d2 †
stage two infectious needles, and
kd1 d2
…l ‡ d1 †…l ‡ d2 †…l ‡ d3 †
stage three infectious needles during his or her entire infectious lifetime. We determine how many
infections are caused by each type of infectious needle until it is rendered virus free (in other words
either exchanged, ¯ushed or cleaned). Consider a single stage one infectious needle, we want to
®nd the expected number of addicts infected by a single type one needle which we shall denote
E1 ˆ E (addicts infected by a single type one needle). To ®nd this value we ®rst condition on the
outcome of the next event, that of a needle being rendered virus free (cleaned or exchanged) before

F. Lewis, D. Greenhalgh / Mathematical Biosciences 169 (2001) 53±87

65

the next user injects with it. We partition this event into two, either the needle is rendered virus
free before the next injection or it is not. Let Y denote the number of addicts infected by a single
needle, X1 denote the event that the needle is rendered safe before the next injection, and X2 the
event that the needle is still infectious at next injection. Therefore we have that
E1 …Y † ˆ E1 …Y j X1 †P …X1 † ‡ E1 …Y j X2 †P …X2 †:
If the needle is rendered safe prior to the next injection then the infected needle has infected zero
addicts, thus E1 …Y j X1 † ˆ 0. The event X2 corresponds to the needle being neither cleaned nor
exchanged prior to use. The probability of this event is kc…1 ÿ /†=…kc ‡ s†, hence
E1 …Y † ˆ E1 …Y jX2 †

kc…1 ÿ /†
:
kc ‡ s

We now explore E1 …Y jX2 † by conditioning on the next event, that of a susceptible addict injecting
with an infectious needle. This event has only two outcomes (since we are assuming that an infectious needle is never ¯ushed by a susceptible addict). An addict is infected by the needle with
probability a1 or remains susceptible with probability 1 ÿ a1 . Therefore,
E1 …Y † ˆ

kc…1 ÿ /†
…a1 ‡ E1 …Y ††:
kc ‡ s

…12†

s ‡ /†, where s^ ˆ s=kc. Following a similar
Hence solving for E1 ˆ E1 …Y † gives E1 ˆ ……1 ÿ /†a1 †=…^
argument for stage two and three infectious needles we ®nd that E2 ˆ ……1 ÿ /†a2 †=…^
s ‡ /† and
s ‡ /†. We now have the expected number of addicts infected by a single stage
E3 ˆ ……1 ÿ /†a3 †=…^
one, two and three infectious needle. Putting these expectations together with the expected
number of each type of needle an addict creates during his or her entire infectious lifetime gives


k…1 ÿ /†
a2 d1
a3 d1 d2
:
…13†
‡
a1 ‡
…l ‡ d1 †…^
l ‡ d2 …l ‡ d2 †…l ‡ d3 †
s ‡ /†
This expression corresponds to the total number of secondary infectious addicts infected by the
original single infectious addict and hence is the basic reproductive number and the threshold
parameter for the three stage model. It can also be interpreted as the long-term average of the
number of secondary needles infected by a single infectious needle entering a large population
where all of the addicts and needles are uninfected.
4.2. Discussion
Theorem 4.1 demonstrates that moving from a single stage AIDS incubation period to a three
stage AIDS incubation period does not a€ect the qualitative behaviour of Kaplan's basic model. In
both this model and its extension to three stage infectivity we have that if R0 6 1, then HIV will die
out in the population and if R0 > 1 (and disease is initially present) then HIV will spread among the
population until a unique endemic equilibrium prevalence level is attained. We now turn our attention to the e€ect on the long term prevalence of HIV (the endemic equilibrium) of moving from
single stage to three stage infectivity. Seitz and M
uller [20] assert that a three stage infectious period
greatly increases the level of HIV among a population compared to assuming a single stage
infectious period. We now investigate whether this is the case by comparing the endemic solution of

66

F. Lewis, D. Greenhalgh / Mathematical Biosciences 169 (2001) 53±87

the Kaplan and O'Keefe model with that of our upper bound three stage infectivity model. We
choose to compare our model with the predictions of Kaplan and O'Keefe's model as this is a well
established recognised model for predicting the spread of HIV and AIDS amongst drug users.
The infection process occurs in two stages, ®rstly infectious addicts pass on the virus to previously uninfectious needles. Secondly the amount of virus passed on to the needle population by
addicts then infects new susceptible addicts. In order to identify the e€ect of splitting the AIDS
incubation period into three distinct stages we need to compare the Kaplan and O'Keefe model
with our three stage model where the only di€erence between these models is the move from single
to three stage infectivity. It seems reasonable to suppose that the relative infectivity of an infectious needle should be proportional to viral load of an infectious addict (since it is an addict's
blood which makes the needle infectious). Therefore we expect that in the Kaplan and O'Keefe
model the average cumulative viral load during the entire infectious lifetime of an addict will be
ja
;
l‡d
where j is a constant and …1=d† represents the AIDS incubation period. In our three stage model
the corresponding average cumulative viral load is


a1
a2 d1
a3 d1 d2
‡
‡
;
j
l ‡ d1 …l ‡ d1 †…l ‡ d2 † …l ‡ d1 †…l ‡ d2 †…l ‡ d3 †
where …1=d1 † ‡ …1=d2 † ‡ …1=d3 † ˆ …1=d†. Again we are treating ai , the infectivity of a state i infectious needle, as proportional to the viral load in a stage i infectious addict. As the cumulative
viral load experienced by an addict over his or her infectious lifetime is an actual real biological
quantity (which could in theory be measured) we calibrate the models so that the average value of
this quantity is the same in both of them. Hence in order to identify the e€ect of three stage
infectivity we require the calibration
a
a1
a2 d1
a3 d1 d2
‡
ˆ
‡
:
l ‡ d l ‡ d1 …l ‡ d1 †…l ‡ d2 † …l ‡ d1 †…l ‡ d2 †…l ‡ d3 †

…14†

To satisfy Eq. (14) we must adjust at least one of our model parameters, natural choices here
would seem to be a1 , a2 and a3 since these represent the di€erent levels of infectivity in our three
stage model. We assume that a1 ˆ f1 a2 and a3 ˆ f3 a2 and hence to satisfy Eq. (14) we estimate all
model parameters (including f1 and f3 ) with the exception of a2 and solve to ®nd the value of a2
which calibrates our models. Our proportionality assumption then ensures that the average
amount of virus passed on to needles is the same in both models (at least if we assume that all
needles are initially uninfected). Hence Eq. (14) ensures that the average amount of virus transferred by an infectious addict to needles at the disease-free equilibrium is the same and that the
basic reproduction number R0 given by Eq. (4) with h ˆ 0 as in Kaplan and O'Keefe's model and
Eq. (11) in our model is also the same. As the quantity R0 has a natural interpretation (as the
expected number of secondary addicts infected from a single infected addict entering the diseasefree equilibrium population) a sensible calibration method should ensure that this is the same in
both models. Any other calibration method would imply that two (measurable) biological
quantities, namely the cumulative viral load of an addict throughout his or her average infectious
lifetime and R0 had di€erent values between the two models. We feel that this fact makes our

F. Lewis, D. Greenhalgh / Mathematical Biosciences 169 (2001) 53±87

67

calibration method the best possible. As the average infectivity in needles depends on the prevalences it is not possible to calibrate the models so that the average infectivity in needles is always
the same whatever the prevalences.
The following theorem shows the relationship between the long term prevalence in our single
stage and three stage models.
Theorem 4.2. Under calibration the long term prevalence of HIV in both addicts and needles is
lower in the Kaplan and O'Keefe Model (with no flushing) than in the upper bound three stage infectivity model.
Proof. See Appendix B. 
To summarise, we have shown that (under calibration) the long term prevalence of disease is
increased by allowing addicts and needles to exist in three di€erent infectious stages when addicts
and needles interact in accordance with the addict±needle interaction assumptions in Section 3.

5. Simulations
We now use simulations to validate our results in Theorem 4.1. We wish to verify that our three
stage model does indeed have globally stable equilibria with R0 ˆ 1 as the critical threshold point.
In addition we wish to compare the Kaplan and O'Keefe model (with no ¯ushing) with our three
stage model in order to get an idea of the size of the di€erence in long term prevalence between these
models and any di€erences in dynamic behaviour. Before we can simulate either the Kaplan and
O'Keefe model or our three stage model we ®rst need to estimate the parameters in these models.
We do not possess our own source of data from which to estimate the parameters required.
Instead we rely on parameter estimates from existing published work. In the simulations we use
the following estimates: k ˆ 246:22 per year [4]; c ˆ 0:90798 addicts per needle [4,31]; a2 ˆ 0:0011
and a ˆ 0:0060 (derived from the method used to estimate a in [4]); a1 ˆ 100a2 and a3 ˆ 10a2 [3];
l ˆ 0:1333 per year [29]; / ˆ 0:64 [4,30]; s ˆ 15:53 per year [31]; d1 ˆ 8:0 per year, d2 ˆ 0:1154
per year, and d3 ˆ 0:8276 per year [2] and h ˆ 0 [4].
There is a variety of estimates for the infectivity ratios f1 and f3 in the literature. The values
used of f1 ˆ 100 and f3 ˆ 10 agree roughly with those used by Jacquez et al. [1] and Hyman et al.
[3]. Koopman et al. [2] use f1 ˆ 200 and f3 ˆ 76. A two-stage incubation period was used by
Kretzschmar and Wiessing [18] who take f1 ˆ 500 and Seitz and M
uller [20] (f1 ˆ 169). Thus, our
values are in broad agreement with those in the recent literature (except Peterson et al. [19] who
use f1 ˆ 5 and f3 ˆ 3).
We now simulate our three stage model using the above set of parameter estimates, using the
expression for R0 in our three stage model we have that R0 ˆ 2:92. Fig. 1 shows the upper bound
three stage model simulated over 10 years. At time zero we have assumed that one percent of the
total population of addicts are in stage one infectivity, at this time no other addicts or needles are
infectious. The ®gure shows the progress of each type of infectious addict and needle over time. It
is clear that the fraction of addicts infected in each stage reaches a steady state as does the fraction
of infected needles in each stage. The steady state values in these simulations are

68

F. Lewis, D. Greenhalgh / Mathematical Biosciences 169 (2001) 53±87

H
H
H
H
H
H
…pH
1 ; p2 ; p3 ; b1 ; b2 ; b3 † ˆ …0:017; 0:549; 0:066; 0:024; 0:562; 0:088†, corresponding to p ˆ 0:633
H
and b ˆ 0:675. Due to a lack of space we do not report any other simulations relating to
the stability of the endemic equilibrium in our three stage infectivity model. However, simulations
for a variety of di€erent parameter estimates and initial conditions suggest that when R0 > 1 and
disease is present initially the prevalence of disease tends to the unique endemic equilibrium solution, as we expect from Theorem 4.1.
We now simulate the upper bound three stage model using the same set of parameter estimates
as in Fig. 1 except now / ˆ 0:85 which gives R0 ˆ 0:94. Fig. 2 shows simulations of the total
fractions of infected addicts and infected needles in the upper bound three stage model simulated
over 120 years. At time zero the population is at the endemic steady state shown in Fig. 1. It is
clear from the ®gure that the disease dies out in all addicts and needles and after about 110 years

Fig. 1. Upper bound three stage model when R0 > 1.

Fig. 2. Upper bound three stage model when R0 < 1.

F. Lewis, D. Greenhalgh / Mathematical Biosciences 169 (2001) 53±87

69

Fig. 3. Single stage infectivity versus upper bound three stage infectivity.

the model reaches the disease free equilibrium. Again due to a lack of space we do not report any
other simulations relating to the global stability of the disease free equilibrium in our three stage
infectivity model. However, simulations for a variety of di€erent parameter estimates and initial
conditions suggest that when R0 6 1 the disease dies out in all addicts and all needles.
We now simulate the Kaplan and O'Keefe model with no ¯ushing and our upper bound three
stage infectivity model. In order that we have a fair comparison we require that Eq. (14) is satis®ed. We achieve this by adjusting the value of a2 to a2 ˆ 0:001227, and we still have that
a1 ˆ 100a2 ˆ 0:1227, a3 ˆ 10a2 ˆ 0:0123 and a ˆ 0:0060. This gives a common value for R0 of
3.22 in both these models. Fig. 3 shows simulations of the Kaplan and O'Keefe model with no
¯ushing and our upper bound three stage infectivity model using the parameter estimates previously outlined. We assumed that initially a fraction 0.01 of all addicts are infectious (and these
add