Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol169.Issue1.2001:

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 as-sumptions 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 parameterR0which determines the behaviour of the model. IfR061, then irrespective of the initial conditions of the system HIV will die out in all addicts and all needles. IfR0>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 ifR0>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).

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 ex-change 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 pro-gress 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 intra-venous 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

deter-mines 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.

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 compo-sition 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 as-sumed 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 in-fection from HIV each time an addict injects with a shared needle.

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 infec-tious period. They assume that the infectivity of an HIV positive individual has a so-called `bath-tub' 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 intoksub-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]:

2. Each addict visits shooting galleries in accordance with a Poisson process with ratek, indepen-dently of the actions of other addicts.

3. Injection equipment always becomes infectious if it is used by an infected addict. When infec-tious 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 probabilityh. 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 rated, 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 land are imme-diately replaced by susceptible addicts.

7. The random variability in the fraction of infected addicts and needles at timet 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. Letp…t†denote the fraction of the population of addicts that are infected with HIV at timet(the prevalence of HIV infection), andb…t†denote the fraction of the population of needles that are infected with HIV at timet. De®ne the `gallery ratio' bycˆn=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

dt ˆ …1ÿp†kba…1ÿ/† ÿp…l‡d†; …1†

and db

dt ˆ …1ÿb†kcpÿbkc…1ÿp†…1ÿ …1ÿh†…1ÿ/†† ÿbs: …2†

For Kaplan's original model [8] it was shown that an endemic solution is possible if and only if the parameterR0 exceeds one, where

R0ˆ

ka

…l‡d†h: …3†

This result easily extends to the Kaplan and O'Keefe model described by Eqs. (1) and (2) except now we have that

R0ˆ

ka…1ÿ/†

where h^ˆ1ÿ …1ÿh†…1ÿ/† and _^sˆs=…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 ex-pression in Eq. (4). Note that the threshold parameterR0is, as usual, a key parameter determining

whether the disease will establish itself. We expect the epidemic to take o€ ifR0 >1, and to die out

if R061.

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 asymptom-atic stage according to a Poisson process with rated1;

(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,

re-spectively. 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 in-travenous 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 timet‡_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† ˆnp1…t† ‡n…1ÿp1…t† ÿp2…t† ÿp3…t††kDtb…t†a…1ÿ/†

Subtractingnp1…t† from both sides, dividing by nDtand letting Dt!0 we deduce that

dp1

dt ˆ 1ÿ

X3 iˆ1

pi !

kba…1ÿ/† ÿp1…l‡d1†:

The number of stage two infected addicts at timet‡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† ˆnp2…t† ‡np1…t†d1Dtÿnp2…t†…l‡d2†Dt‡o…Dt†:

Subtractingnp2…t† from both sides, dividing by nDtand letting Dt!0 we deduce that

dp2

dt ˆd1p1ÿ …l‡d2†p2:

Similarly

dp3

dt ˆd2p2ÿ …l‡d3†p3:

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 popu-lation of infectious needles is to split this into three sub-popupopu-lations, each corresponding to the three infectious stages of the addicts. Hence the ®rst sub-population contains (previously unin-fectious) 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 sub-populations 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 replaceba in dp1=dtwithb1a1‡b2a2‡b3a3, wherebi 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.

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 ˆ0;1;2;3 what fractionpijkof needles initially in infectious stageiare left in stagekafter 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 48pijkprobabilities. 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 rel-ative 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.

The number of infected stage one needles at time t‡_Dt

ˆ fnumber of stage one infectious needles at timetg ‡ f…number of non-stage one needles at timet†

…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† ˆmb₁…t† ‡mk_Dtcp1…t†…1ÿb1…t†† ÿmkDtc/…1ÿp1…t††b1…t† ÿmb1…t†sDt‡o…Dt†:

Subtractingmb₁…t† from both sides, dividing bymDt and letting_Dt!0 we deduce that db₁

dt ˆkc…1ÿb1†p1ÿb1…1ÿp1†/kcÿb1s:

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

ˆ fnumber of stage two infectious needles at timetg ‡ f…number of uncontaminated needles at timet†

…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† ˆmb2…t† ‡mkDtcp2…t† 1ÿ

X3 iˆ1

b_i…t† !

‡mk_Dtc/b1…t†p2…t† ‡mkDtc/b3…t†p2…t†

ÿmk_Dtc…p1…t† ‡p3…t††b2…t† ÿmkDtc/ 1ÿ

X

3

iˆ1

pi…t† !

b₂…t†

ÿmb₂…t†s_Dt‡o…Dt†:

db2

dt ˆkc 1ÿ

X3 iˆ1

b_i !

p2‡b1p2/kc‡b3p2/kcÿb2p3kcÿb2p1kcÿb2kc/ 1ÿ

X3 iˆ1

pi !

ÿb₂s:

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† ˆmb3…t† ‡mkDtcp3…t†…1ÿb1…t† ÿb3…t†† ‡mkDtc/b1…t†p3…t† ÿmkDtcb3…t†p1…t†

ÿmb₃…t†k_Dtc/…1ÿp1…t† ÿp3…t†† ÿmb3…t†sDt‡o…Dt†:

Subtractingmb₃…t† from both sides, dividing by mDt and letting_Dt!0 we deduce that db3

dt ˆkcp3…1ÿb1ÿb3† ‡kc/b1p3ÿkcb3p1ÿb3kc/…1ÿp1ÿp3† ÿb3s:

Hence the system of di€erential equations which describe the spread of the disease are

dp1

dt ˆ 1ÿ

X

3

iˆ1

pi !

k…b₁a1‡b2a2‡b3a3†…1ÿ/† ÿ …l‡d1†p1; …5†

dp2

dt ˆd1p1ÿ …l‡d2†p2; …6†

dp3

dt ˆd2p2ÿ …l‡d3†p3; …7†

db₁

dt ˆkc…1ÿb1†p1ÿb1…1ÿp1†/kcÿb1s; …8†

db₂

dt ˆkc 1ÿ

X3 iˆ1

b_i !

p2‡b1p2/kc‡b3p2/kcÿb2p3kc

ÿb₂p1kcÿb2kc/ 1ÿ

X

3

iˆ1

pi !

and

db₃

dt ˆkcp3…1ÿb1ÿb3† ‡kc/b1p3ÿkcb3p1ÿb3kc/…1ÿp1ÿp3† ÿb3s …10†

with suitable initial conditions: 06p1…0†, p2…0†, p3…0†, b1…0†, b2…0†, b3…0†, p1…0† ‡p2…0† ‡p3…0† 61 and b1…0† ‡b2…0† ‡b3…0†61.

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 pa-rameter 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 be-haviour of the three stage infectivity model.

De®ne the region DinR6 _{by Dˆ ‰0;1Š}6_{. 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;b2andb3with continuous derivatives, and the corresponding vector points intoDon

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 inD, have a unique solution that remains inD for all time. De®ne

R0ˆ

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®neMy to be the matrix:

The following theorem is the main result of the paper; it indicates that if R061 we expect the

disease to die out, whereas if R0>1 we expect the disease to take o€.

Theorem 4.1. IfR061the system of equations(5)±(10)has a unique equilibrium solution where the

all addicts and all needles. IfR0 >1,and provided that disease is initially present, then there exists

>0 andg>0such that foriˆ1;2;3,pi…t†> andb1…t† ‡b2…t† ‡b3…t†> fort>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†>d1d2, detMy>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 in-fections 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 unin-fectious needle, secondly this needle (which is now inunin-fectious) passes on the disease to a suscep-tible 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 ex-pected 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 probabilityd2=…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 kd1d2

…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 ˆE (addicts infected by a single type one needle). To ®nd this value we ®rst condition on the

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. LetYdenote 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† ˆE1…Y jX1†P…X1† ‡E1…Y jX2†P…X2†:

If the needle is rendered safe prior to the next injection then the infected needle has infected zero addicts, thus E1…Y jX1† ˆ0. The event X2 corresponds to the needle being neither cleaned nor

exchanged prior to use. The probability of this event iskc…1ÿ/†=…kc‡s†, hence

E1…Y† ˆE1…YjX2†

kc…1ÿ/† kc‡s :

We now exploreE1…YjX2†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 in-fectious needle is never ¯ushed by a susceptible addict). An addict is infected by the needle with probabilitya1 or remains susceptible with probability 1ÿa1. Therefore,

E1…Y† ˆ

kc…1ÿ/†

kc‡s …a1‡E1…Y††: …12†

Hence solving forE1 ˆE1…Y†givesE1 ˆ ……1ÿ/†a1†=…^s‡/†, wheres^ˆs=kc. Following a similar

argument for stage two and three infectious needles we ®nd that E2 ˆ ……1ÿ/†a2†=…^s‡/† and

E3 ˆ ……1ÿ/†a3†=…^s‡/†. We now have the expected number of addicts infected by a single stage

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…1ÿ/† …l‡d1†…^s‡/†

a1

‡ a2d1 l‡d2

‡ a3d1d2 …l‡d2†…l‡d3†

: …13†

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 ifR061, then HIV will die

out in the population and ifR0>1 (and disease is initially present) then HIV will spread among the

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 pre-viously 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 infec-tious needle should be proportional to viral load of an infecinfec-tious 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;

wherejis a constant and …1=d† represents the AIDS incubation period. In our three stage model the corresponding average cumulative viral load is

j a1

l‡d1

‡ a2d1

…l‡d1†…l‡d2†

‡ a3d1d2

…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 in-fectious needle, as proportional to the viral load in a stage iinfectious 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 l‡dˆ

a1

l‡d1

‡ a2d1

…l‡d1†…l‡d2†

‡ a3d1d2

…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 bea1, a2 anda3 since these represent the di€erent levels of infectivity in our three

stage model. We assume thata1 ˆf1a2anda3 ˆf3a2 and hence to satisfy Eq. (14) we estimate all

model parameters (includingf1 andf3) with the exception ofa2 and solve to ®nd the value ofa2

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 trans-ferred by an infectious addict to needles at the disease-free equilibrium is the same and that the basic reproduction numberR0given by Eq. (4) withhˆ0 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

calibration method the best possible. As the average infectivity in needles depends on the pre-valences 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

in-fectivity 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 withR0 ˆ1 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:kˆ246:22 per year [4];cˆ0:90798 addicts per needle [4,31];a2 ˆ0:0011

andaˆ0:0060 (derived from the method used to estimateain [4]);a1ˆ100a2 anda3 ˆ10a2 [3];

lˆ0:1333 per year [29]; /ˆ0:64 [4,30];sˆ15:53 per year [31]; d1ˆ8:0 per year, d2 ˆ0:1154

per year, and d3ˆ0:8276 per year [2] andhˆ0 [4].

There is a variety of estimates for the infectivity ratios f₁ and f₃ in the literature. The values used off1 ˆ100 andf3ˆ10 agree roughly with those used by Jacquez et al. [1] and Hyman et al.

[3]. Koopman et al. [2] use f1 ˆ200 and f3ˆ76. A two-stage incubation period was used by

Kretzschmar and Wiessing [18] who takef1 ˆ500 and Seitz and Muller [20] ( f1 ˆ169). Thus, our

values are in broad agreement with those in the recent literature (except Peterson et al. [19] who usef1 ˆ5 andf3ˆ3).

We now simulate our three stage model using the above set of parameter estimates, using the expression forR0 in our three stage model we have thatR0 ˆ2:92. Fig. 1 shows the upper bound

…pH 1;p

H 2;p

H 3;b

H 1;b

H 2;b

H

3† ˆ …0:017;0:549;0:066;0:024;0:562;0:088†, corresponding to p H

ˆ0:633 and bH

ˆ0:675. 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 whenR0 >1 and

disease is present initially the prevalence of disease tends to the unique endemic equilibrium so-lution, 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 /ˆ0:85 which gives R0 ˆ0:94. 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 whenR0>1.

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 whenR061 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 sat-is®ed. We achieve this by adjusting the value of a2 to a2 ˆ0:001227, and we still have that

a1 ˆ100a2ˆ0:1227, a3 ˆ10a2ˆ0:0123 and aˆ0:0060. 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 pre-viously outlined. We assumed that initially a fraction 0.01 of all addicts are infectious (and these addicts are all in stage one infectivity in the three stage model), initially no other addicts or needles are infectious. It is clear that even though both models share the same R0 value the three stage

model reaches equilibrium very quickly after only a few years. It is also apparent that the endemic level of disease in the three stage model is greater than in the constant infectivity model. Other simulations (not illustrated) suggest that for a wide variety of di€erent parameter estimates the upper bound three stage infectivity model always has a higher long term prevalence level than the single stage Kaplan and O'Keefe model (with no ¯ushing). This supports Seitz and Muller's as-_ sertion that three stage infectivity increases the long term endemic prevalence, although the dif-ference does not appear to be as great as they suggest.

6. Conclusions and discussion

In this paper we extended an original single stage infectivity model by Kaplan and O'Keefe [4] to incorporate a three stage infectious period. This extended model allows addicts to progress through three di€erent levels of infectivity prior to the onset of full blown AIDS and splits the class of infectious needles into three separate classes, one for each type of infectious addict. In a

model with three types of infectious addicts and three types of infectious needles it is necessary to make assumptions about how addicts and needles of di€erent infectivity levels interact. We noted that no data are available to assist with these assumptions and as such we adopted assumptions which were deliberately biased so that our three stage infectivity model gives rise to the worst possible prevalence of disease which would occur. In this way we constructed an upper bound three stage infectivity model for the spread of HIV/AIDS among intravenous drug users. We then used analytical results to illustrate key properties of our model. We showed that there is a key threshold parameter which determines whether HIV will eventually die out or become endemic among the population. As with many epidemiological models this critical threshold parameter can be interpreted as the basic reproductive number. We showed thatR0 ˆ1 is the critical threshold

point between the endemic and disease free states. Finally we examined the e€ect of moving from single stage to three stage infectivity. We showed analytically that the upper bound three stage model always gives rise to a higher endemic level of HIV than in the single stage infectivity case. It is tempting to argue that since our three stage model is simply a generalisation of the addict± needle interactions in the single stage model then these models are directly comparable and hence since the three stage model has a higher long term prevalence then it is reasonable to conclude that we have shown that moving from single stage to three stage infectivity increases the prevalence of disease. While it is true that the three stage model always has a greater long term prevalence than in the single stage model it is wrong to conclude that this is evidence that a three stage infectious period increases the prevalence of disease. This is because while our three stage model is indeed a generalisation of the addict±needle interactions in the single stage model it is not the only one possible and as previously discussed our particular form of generalisation is more extreme than would be expected to occur. If our three stage model had a lower long term prevalence than in the single stage model then this would have been good evidence that three stage infectivity decreases the prevalence of HIV. However, this is not the case and the most we can conclude from com-paring our three stage model with the Kaplan and O'Keefe model is that moving to a three stage infectious period may result in an increase in the prevalence of HIV.

In other publications [1,2], it was shown that for an infectious disease with variable infectivity, the contact patterns in the population greatly in¯uence the epidemic outcome. For example, if there are longer lasting partnerships within which risk behaviour takes place, an infected person may spend all his or her contacts during the highly infectious stage with one partner. Also heterogeneity in risk behaviour and non-random mixing might amplify or dampen the e€ects of variable infectivity [11]. One of the major drawbacks of the model is the assumption that the population size remains constant so that infected drug users who leave the population because they develop AIDS are immediately replaced by susceptible drug users. Although it is unrealistic a similar assumption is often made in epidemiological modelling [4,7,8,32]. However relaxing this assumption has little e€ect on the model discussed in our paper. We followed Greenhalgh [10] and assumed that new susceptible drug users enter the population at a rate proportional to nm _{[29]. We found that}

similarly to Greenhalgh [10], the equilibrium equations in the variable population size model are identical to those for the constant population size model with an additional equilibrium popu-lation size equation. The threshold value and endemic equilibrium values of p

i and

Having examined a three stage infectivity model which is based on only one particularly ex-treme set of addict±needle interactions assumptions the next natural step would seem to be to develop a model which can incorporate a general addict±needle interaction structure. For example a model could be constructed using thepijkprobabilities discussed earlier, where rather than assign a ®xed value to each of these probabilities they would simply remain unspeci®ed model param-eters. This would result in a much more complex model than that discussed in this paper; however this model could incorporate a wide range of di€erent addict±needle interaction assumptions. Although it is very dicult to obtain accurate data to estimate the probabilitiespijkif this could be done it might provide a more satisfactory way to answer the question of whether incorporating a three stage infectious period increases the prevalence of HIV.

Acknowledgements

We are grateful to the referees for their helpful comments. F.L. also wishes to thank the EPSRC for a studentship to support this work.

Appendix A

Proof of Theorem 4.1. The proof of Theorem 4.1 requires a number of stages and supporting

results. Note that we always assume that/, the probability that an addict cleans a needle prior to use, is strictly less than unity. First of all we show the equilibrium results. Suppose thatpH

1,p

3, respectively we ®nd that

bH

Now consider the equilibrium equation obtained by setting dp1=dtˆ0,

bH

is a multiplicative factor in each of bH 1, b solution to Eq. (A.4) and consequently there always exists a disease free equilibrium solution. The other (non-zero) solutions must satisfy

when R061 and hence we have a unique endemic solution when R0 >1 and no endemic

equi-librium when R061. We do not attempt to derive the roots explicitly but instead focus on the

conditions necessary for the existence of only a single root in …0;1†. Let

F…pH

† ˆ g1…l‡d1† …1ÿpH

†k…1ÿ/†: …A:6†

It is clear thatF…pH

† is strictly monotone increasing for pH

2 …0;1†. We have that

† is strictly monotone decreasing in pH

for pH

† is strictly monotone increasing in…0;1† and G…pH

† is strictly monotone decreasing in …0;1†. Moreover for small and positive we have that F…1ÿ†>G…1ÿ†.

Now consider the initial conditions of F and G, we have three distinct cases, ®rstly, if

G…0†>F…0†, then the two functions must cross in…0;1†and hence Eq. (A.5) has a unique strictly positive root, pH

2 …0;1†. Secondly, if G…0†<F…0†, then the functions never cross in …0;1† and Eq. (A.5) has no root in…0;1†. Finally, ifG…0† ˆF…0†Eq. (A.5) has a single root pH

unique strictly positive solution. We now show that for this case not only ispH

2 …0;1†but also bH

2 …0;1† and hence we have a unique feasible endemic solution. Using Eqs. (A.1)±(A.3) we have that

bH

Therefore, it follows immediately that bH

2 …0;1† ifpH

2 …0;1†, thus whenG…0†>F…0† we have that pH

2 …0;1† and bH

2 …0;1† and hence we have a unique endemic solution when …G…0†=F…0††>1, where

which simpli®es toR0. Hence we have shown that ifR0>1 there is a unique endemic equilibrium

…pH ;bH

† withpH

>0 andbH

>0 in addition to the disease-free one, whilst ifR061 there is only

the disease-free equilibrium.

We now prove global stability of the disease-free equilibrium for R061. We shall show that

p…t† !0 and b…t† !0 as t! 1. We prove this result in several stages. The key stage in the

Proof.From Eq. (6) we have

Hence

. This provides a contradiction and completes the proof.

Corollary A.1. Ifp1

Proof.Using Eq. (7) and following the method of Lemma A.1 we ®nd thatp1

3 6…d2p12 †=…l‡d3†.

Hence we can express each needle equation in a form similar to that of the addicts equations, therefore it follows similarly to the proof of Lemma A.1 that

where1 ˆ

Following the same method as in Lemma A.1 we ®nd that

p1₁ 63‡

where3 is an arbitrarily small positive constant and

4 ˆ3‡

However, since 4 is an arbitrary positive constant we can choose

4 ˆ

This provides a contradiction and hencep1

1 ˆ0 provided 1PR0P0.

This completes the proof of global stability of the disease-free equilibrium when 1P_R0P0. We now show that this equilibrium is unstable whenR0 >1.

We wish to show that at least one eigenvalue of J has a strictly positive real part. Using the Routh±Hurwitz conditions it is sucient to show that the constant term,a6, in the characteristic

equation of J, x6_‡a

1x5‡a2x4‡a3x3‡a4x2‡a5x‡a6ˆ0; is strictly negative. It is

straightforward to show that a6ˆ …kc/‡s† 3

…l‡d1†…l‡d2†…l‡d3†…1ÿR0†, hence if R0 >1,

thena6<0 and the result follows.

We now show that if disease is initially present andR0 >1 there exists a ®xed >0 andT‡>0

such that

p1Pp H

1; p2Pp H

2; p3Pp H

3 and b1Pb H

1 for alltPT

‡_{: …A:12†}

Letp1…t† ˆinfnPtp1…n†, this is monotone increasing astincreases. Hence, given >0 there exists

t1…† such that p1…t†Pp1;1ÿ for all tPt1…†, where p1;1ˆlim inft!1p1…t†. We need the

fol-lowing corollaries to Lemma A.1:

Corollary A.2. If p2;1ˆlim inft!1p2…t†, then p2;1P…d1p1;1=…l‡d2††:

Proof.The result follows using a similar method to Lemma A.1.

Corollary A.3. If p3;1ˆlim inft!1p3…t†, then

p3;1P

d1d2p1;1

…l‡d2†…l‡d3†

:

Proof. Using Eq. (7) and following the method of Lemma A.1 we ®nd that

p3;1P…d2p2;1=…l‡d3††; and the result follows using Lemma A.1.

Corollary A.4. If b_1;1ˆlim inft!1b1…t†, then b1;1P…kcp1;1=…kc‡s††.

Proof.Using Eq. (8) we have that …db₁=dt†Pkcp1ÿ …kc‡s†b1: Therefore,

d

dt‰b1exp‰…kc‡s†tŠŠPp1kcexp‰…kc‡s†tŠ

and from the form of this, arguing as in the proof of Lemma A.1, the result follows.

If we can show thatp1;1>0, then from Corollaries A.2±A.4 we have thatp2;1>0,p3;1>0

andb1;1>0. We must havep1Pp1;1ÿmfortPt…m†, wheremis arbitrarily small, hence choosing

mˆ1

2p1;1and suciently small we havep1Pp H

1 for alltPT‡ for someT‡ >0, and similarly

the other results in (A.12) hold. Hence (A.12) holds ifp1;1>0.

Lemma A.2. Provided that at least one ofp1…t†,p2…t†,p3…t†,b1…t†,b2…t†andb3…t†is strictly positive

at tˆ0, then p1…Dt†>0, p2…Dt†>0, p3…Dt†>0, b1…Dt†>0, b2…Dt†>0, and b3…Dt†>0 for Dt

small and positive.

Proof.We need to consider four separate initial conditions:b…0† ˆ0;p…0†>0;b…0†>0;p…0† ˆ0;

expan-sion of the appropriate model equations abouttˆ0 it is easy to show thatp…Dt†>0,b…Dt†>0 and 1ÿp…_Dt†>0 for _Dt small and positive for each of these initial conditions, a similar result then follows for each model component.

From Lemma A.2 we have that there exists ®xed , where 1> >0 such that if _Dt is small

which case (A.12) follows directly by the above remark. Next suppose thatp1;1<12p H

starts to drop beneath the level1 2p

2 just after t1. We now show that if p1 becomes small, then all components eventually

become small.

Hence if Dis small and positive andt is suciently large then the result follows.

Corollary A.5. There exists a time T2 >0 such that if t0‡T1‡T2 <t1, then for all

t2 ‰t0‡T1‡T2;t1Š, 0<p3<…1₂‡2D†p H

3, where D and are small and T2 depends only on the

Proof.Similar method to Lemma A.3 starting with

and arguing as in the proof of Lemma A.3,

b1…t†6

……1=2† ‡D†pH 1 ^

s‡/ …for tsufficiently large say tPt0‡T3†: However, we cannot replacepH

1=…s^‡/† by b

1 ift is suciently large.

From Eq. (9) we have that fortP_t0‡_T1

From Eq. (10) we have that

3 ˆ

We have shown that if p1 approaches zero then all components must also approach zero. We

now show that p1 cannot become arbitrarily small. We do this by showing that t1ÿt0 can be

bounded above by a ®xed ®nite value and hence p1 is not below 1₂pH1 long enough to become

arbitrarily close to zero. Now eitherp1 is below1₂p H

1 long enough for all components to become

small orp1 increases past1₂pH1 before all components become small. Hence we have either that

…i† t1Pt0‡max‰T1;T1‡T2;T3;T1‡T4;T1‡T2‡T5Š;

or

…ii† t1 <t0‡max‰T1;T1‡T2;T3;T1‡T4;T1‡T2‡T5Š:

We wish to show that t1ÿt0 <T, where T is a ®xed ®nite value dependent only on the model

parameters,andD. In case (ii) we already have this result, we now show this case must always be true by obtaining an upper bound for t1ÿt0 in case (i). We use the fact that the disease free

equilibrium is unstable when R0 >1 to show that p1 cannot become arbitrarily small. We ®rst

need the following result:

Corollary A.6. LetF1…x; †be annth degree polynomial inxand. Denote the(possibly complex)

roots of F1…x; † ˆ0 by xj for jˆ1;. . .;n. Then each xj is defined and continuous in in a

neighbourhood of ˆ0.

Proof. If F1…x; † is a polynomial then this function of two complex variables is analytic in a

neighbourhood of (0,0), and the result follows directly from Corollary 6.6 in Ref. [33].

Lemma A.4. If p1…t† drops to below 1₂pH1 at time t0, then p1…t† returns to 1₂pH1 by at least time

t1yˆt0‡max‰T1;T1‡T2;T3;T1‡T4;T1‡T2‡T5;t2‡T6Š. ty1ÿt0 is finite and only depends on D;

and the model parameters.

Proof.Suppose that2 is real and positive and 1P2P0 and consider the matrix

When 2 ˆ0,J…0† ˆJ, the linearised stability matrix about the disease-free equilibrium for the

coordinate system x0ˆ …p1;p2;p3;b1;b;b1‡3†. Suppose that the eigenvalues of J…2† are x1…2†,

x2…3†, x3…2†, x4…2†, x5…2† and x6…2†. For M large and positive we have that J…2† ‡MI is a

non-negative irreducible matrix. Using Lemma 2.1, from [34], the characteristic equation of this matrix has a simple root equal to its spectral radius. The eigenvalues of J…2† ‡MI are

M‡x1…2†, M‡x2…2†, M‡x3…2†, M‡x4…2†, M‡x5…2† and M‡x6…2†. Hence if

M‡x1…2†is the spectral radius ofJ…2† ‡MI, thenM‡x1…2†is real and all other eigenvalues of

J…2† ‡MI have smaller real parts. Hence x1…2† is real and the other eigenvalues of J…2† have

smaller real parts. In particular this is true for2ˆ0. Moreover from Corollary A.6 we have that

the roots of the characteristic equation of J…2† are continuous functions of 2, hence

x1…2† !x1…0† as 2 !0. We have already shown that x1…0†>0 if R0 >1. Therefore by

choosing2small enough we can ensure thatx1…2†>0. Without loss of generality we can assume

that 1> 2 >0. We can choose small enough so that 1₂p

where T6 depends only on ;D and the model parameters. We already know that provided

t06t6t1 then after a timet0‡max‰T1;T1‡T2;T3;T1‡T4;T1‡T2‡T5Š,

This completes the proof of Lemma A.4.

We have shown that the ®rst timep1 drops below 1₂pH1 it must return back to this level by a

(®xed and ®nite) duration of most T later. This result is easily extended to cover any time p1

dp1

dt P ÿ …l‡d1†p1:

Integrating we deduce that

p1P

1 2p

H

1 exp‰ÿ…l‡d1†…tÿ~t0†Š;

P 1

2p H

1 exp‰ÿ…l‡d1†TŠ;

where T is a ®xed duration dependent only on , D and the model parameters. Since

1 2p

H

1 exp‰ÿ…l‡d1†TŠis strictly positive we have thatp1;1>0. Hence we have that (A.12) is true

by the remark on p. 76.

We now examine local stability of the endemic equilibrium. We do not show this directly as this would require examining the roots of a sixth-order polynomial. Instead we show that a model which is a close approximation to that in Eqs. (5)±(10) (and has the same endemic equilibrium solution) has a locally stable endemic equilibrium whenR0>1. The basic idea is that the timescale

on which the addicts inject is much faster than that of the other epidemiological and demographic processes. Hence in the timescale on which the bs change it is reasonable to treat the pis as constant, piˆpyi; iˆ1;2;3.

If we treat the pis as constants then we can show thatb1;b2 and b3 approach their respective

equilibrium values as time becomes large. We have that

db₁

dt ˆkc p

y

1

h

ÿb₁…py₁‡ …1ÿp₁y†/‡^s†i: …A:13†

If by₁ ˆpy₁=…py₁‡ …1ÿpy₁†/‡^s† is the equilibrium solution to Eq. (A.13), then for b₁…t†<by₁; b₁…t†is increasing and forb₁…t†>by₁; b₁…t†is decreasing. It is straightforward to show thatb₁…t† !by₁ ast! 1. Similarly from the equations

db_1‡3

dt ˆkc p

y

1‡3

h

ÿb_1‡3…py_1‡3‡ …1ÿpy_1‡3†/‡_^s†i …A:14†

and

db

dt ˆkc p

y

h

ÿb…py‡ …1ÿpy†/‡^s†i; …A:15†

we deduce that

b_1‡3…t† !b₁y_‡3 ˆpy_1‡3=…py_1‡3‡ …1ÿpy_1‡3†/‡_^s†

and

b…t† !byˆpy=…py‡ …1ÿpy†/‡^s† ast! 1:

Hence foriˆ1;2;3,b_i…t† !by_i ast! 1, whereby₁ is given by Eqs. (A.1)±(A.3) withp

i replaced by py_i.

We can express b₁a1‡b2a2‡b3a3 as b1…a1ÿa3† ‡ba2‡b1‡3…a3ÿa2†. Substituting in the

dp1

For realistic parameter values we expect that the `addict-only' equations are a good approxi-mation to the full model and simulations con®rm this. A similar argument is used by Kaplan and O'Keefe [4] to calculate the reduction in HIV incidence due to the introduction of a needle ex-change and by Kaplan [31] to justify assuming that HIV prevalence amongst addicts is constant in a model examining only the dynamics of HIV amongst needles. It is possible to make this ap-proximation argument rigorous by showing that ifkcis large compared with the other parameters of the model apart froms, (includingka1;ka2andka3), then three of the roots of the characteristic

equation of the full model at the endemic equilibrium tend to negative in®nity as kc goes to in-®nity and the other three are close to the roots of the characteristic equation of the `addict-only' model at the endemic equilibrium. We now show that the endemic equilibrium of the `addict-only' model (which is the same as that of the full model) is locally stable.

a1ˆl‡d2‡l‡d3ÿJ11; …A:22†

a2ˆ …l‡d2†…l‡d3† ÿJ11…l‡d2‡l‡d3† ÿd1J12 …A:23†

and

a3ˆ ÿJ11…l‡d2†…l‡d3† ÿd1d2J13ÿd1J12…l‡d3†: …A:24†

Using Eqs. (A.19)±(A.24), (A.1), (A.7) and (A.8) we ®nd that

a1Pl‡d1‡l‡d2‡l‡d3ÿk…1ÿ/†…1ÿp†…^s‡/†

>0; arguing similarly to before:

We now wish to show that a1a2ÿa3>0. We have that

a2 >0. Hencea1a2>a3and all the Routh±Hurwitz conditions are satis®ed. Therefore the

`addict-only' approximation to the full model has a locally stable endemic equilibrium when R0 >1.

We now examine a sucient condition for global stability in our full model. Suppose that disease is initially present. Consider the coordinate system, …p_~;p_~2;p~3;b~1;b~;b~1‡3†, where b~iˆ

sucient conditions for the positivity of all principal minors ofMy. The ®rst condition is satis®ed for most parameter values in the literature. If also detMy>0 then by Theorem 1 of Berman and Herschowitz [35],Myis a non-singular M-matrix. Hence there exists a positive diagonal matrixP such thatW…†TP‡PW…† ˆ ÿQ, whereQis positive de®nite, [36]. Thenvˆ~yT_{P~yis a Lyapunov}

dv

dtˆ~y

T_{W…y†P‡PW…y†}T

†~y6~yT…W…†P‡PW…†T†~yˆ ÿ~yTQ~y:

Therefore, if R0>1 and disease is initially present the system will tend to the unique endemic

equilibrium. This completes the proof of Theorem 4.1.

Appendix B

Proof of Theorem 4.2. Write pH

S to denote the value of p

H

for the single stage model with no ¯ushing and pH

P to denote the value of p

H

in the three stage model. From the proof of the equilibrium results we have thatpH

for the three stage model is the unique positive solution to

g₁…l‡d1†

We have that the left-hand side of Eq. (B.1) is strictly increasing and the right-hand side is strictly decreasing in pH

S the left-hand side of Eq. (B.1) is

and from the calibration equation (14) we have that

a1g1‡a2g2‡a3g3 ˆ

ag1…l‡d1†

…l‡d† : …B:7†

Hence the left-hand side of Eq. (B.1) atpH

S is strictly less than the right-hand side of Eq. (B.1) at pH

This completes the proof of Theorem 4.2.

References

[1] J.A. Jacquez, J.S. Koopman, C.P. Simon, I.M. Longini Jr., Role of the primary infection in epidemics of HIV infection in gay cohorts, J. AIDS 7 (1994) 1169.

[2] J.S. Koopman, J.A. Jacquez, G.W. Welch, C.P. Simon, B. Foxman, S.M. Pollock, D. Brath-Jones, A.L. Adams, K. Lange, The role of early HIV infection in the spread of HIV through populations, J. AIDS 14 (1997) 249. [3] J.M. Hyman, J. Li, E.A. Stanley, The di€erential infectivity and staged progression models for the transmission of

HIV, Math. Biosci. 155 (1999) 77.

[4] E.H. Kaplan, E. O'Keefe, Let the needles do the talking! Evaluating the New Haven needle exchange, Interfaces 23 (1993) 7.

[5] UNAIDS, AIDS epidemic update: December 1998, UNAIDS Joint United Nations Programme on HIV/AIDS, 1998.

[6] UNAIDS, Drug Use and HIV/AIDS, UNAIDS statement presented at the United Nations General Assembly Special Session on Drugs 1999, UNAIDS Joint United Nations Programme on HIV/AIDS, 1999.

[7] D. Greenhalgh, G. Hay, Mathematical modelling of the spread of HIV/AIDS amongst injecting drug users, IMA J. Math. Appl. Med. Biol. 14 (1997) 11.

[8] E.H. Kaplan, Needles that kill: modeling human immunode®ciency virus transmission via shared drug injection equipment in shooting galleries, Rev. Inf. Diseases 11 (1989) 289.

[9] E. Drucker, AIDS and addiction in New York city, Am. J. Drug Alcohol Abuse 12 (1986) 165.

[10] D. Greenhalgh, E€ects of heterogeneity on the spread of HIV/AIDS among intravenous drug users in shooting galleries, Math. Biosci. 136 (1996) 141.

[11] D. Greenhalgh, Bounds on endemic disease levels, risks and basic reproductive numbers in heterogeneous models for HIV/AIDS amongst injecting drug users, J. Biol. Systems 5 (1997) 151.

[12] V. Capasso, F. Sicurallo, M. Villa, Mathematical models for HIV transmission in groups of injecting drug users via shared drug equipment, J. Biol. Systems 3 (1995) 747.

[14] S.J. Yakowitz, From a microcosmic IDU model to a macrocosmic HIV epidemic, in: E.H. Kaplan, M.L. Brandeau (Eds.), Modeling the AIDS Epidemic: Planning Policy and Prediction, Raven, New York, 1994, p. 365.

[15] R. Allard, A mathematical model to describe the risk of infection from sharing injection equipment, J. AIDS 3 (1990) 1010.

[16] E. Massad, F.A.B. Coutinho, H.M. Yang, H.B. Decarvalho, F. Mesquita, M. Burattini, The basic reproductive ratio of HIV among intravenous drug users, Math. Biosci. 123 (1994) 227.

[17] S.M. Blower, D. Hartel, H. Dowlatabadi, R.M. Anderson, R.M. May, Drugs sex and HIV: a mathematical model for New York city, Philos. Trans. R. Soc. London B 321 (1991) 171.

[18] M. Kretzschmar, L.G. Wiessing, Modelling the spread of HIV in social networks of injecting drug users, AIDS 12 (1998) 801±811.

[19] D. Peterson, K. Willard, M. Altmann, L. Gatewood, G. Davidson, Monte-Carlo simulation of HIV infection in an intravenous drug user community, J. AIDS 3 (1990) 1086.

[20] S.T. Seitz, G.E. Muller, Viral load and sexual risk. Epidemiological and policy implications for HIV/AIDS, in: E.H. Kaplan, M.L. Brandeau (Eds.), Modeling the AIDS Epidemic: Planning, Policy and Prediction, Raven Press, New York, 1994, p. 461.

[21] W.Y. Tan, S.C. Tang, A stochastic model of the HIV epidemic involving both sexual contact and IV drug use, Math. Comput. Modelling 17 (1993) 31.

[22] I.M. Longini Jr., W.S. Clark, L.I. Gardner, J.F. Brundage, The dynamics of CD-4‡_{T-lymphocyte decline in}

HIV-infected individuals: a Markov modelling approach, J. AIDS 4 (1991) 1141.

[23] I.M. Longini, W.S. Clark, M. Haber, R. Horsburg, The stages of HIV infection: waiting times and infection transmission probabilities, in: C. Castillo-Chavez (Ed.), Mathematical and Statistical Approaches to AIDS Epidemiology, Lecture Notes in Biomathematics, vol. 83, Springer, New York, 1989, p. 111.

[24] J.C.M. Hendriks, G.A. Satten, E.J.C. van Ameijden, H.A.M. van Druten, R.A. Coutinho, G.J.P. van Greinsven, The incubation period to AIDS in injecting drug users estimated from prevalent cohort data accounting for death prior to an AIDS diagnosis, AIDS 12 (1998) 1537.

[25] J.-P. Allain, Y. Laurian, D.A. Paul, F. Verroust, M. Leuther, C. Gazengel, D. Senn, M.-J. Larrieu, C. Bosser, Long term evaluation of HIV antigen and antibodies to p24 and gp41 patients with haemophilia: Potential clinical importance, New Engl. J. Med. 317 (1987) 1114.

[26] J.W. Ward, D.A. Deppe, S. Samson, H. Perkins, P. Holland, L. Fernando, P.M. Feorino, P. Thompson, S. Kleinman, J.R. Allen, Risk of HIV infection from blood donors who later developed AIDS, Ann. Intern. Med. 106 (1987) 61.

[27] E.H. Kaplan, Private communication, 1996.

[28] J.K. Hale, Ordinary Di€erential Equations, Wiley, New York, 1969.

[29] J.P. Caulkins, E.H. Kaplan, AIDS impact on the number of intravenous drug users, Interfaces 21 (1991) 50. [30] D. Goldberg, M. Frischer, S.T. Green, A. Taylor, N. McKeganey, Probability of HIV transmission among

injecting drug users in Glasgow, Unpublished manuscript, 1995.

[31] E.H. Kaplan, Probability models of needle exchange, Oper. Res. 43 (1995) 558.

[32] R.M. Anderson, R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University, Oxford, 1991.

[33] S.N. Chow, J.K. Hale, Methods of Bifurcation Theory, Wiley, New York, 1982. [34] A. Nold, Heterogeneity in disease-transmission modelling, Math. Biosci. 52 (1980) 227.

[35] A. Berman, D. Herschkowitz, Matrix diagonal stability and its implications, SIAM J. Alg. Disc. Meth. 4 (1983) 377.