Does “risk reduction” reduce the risk? - modelling HIV spread among men who have sex with men

26-th ECMI Modelling Week
Final Report

19.08.2012—25.08.2012
Dresden, Germany

Chapter 11

Does “risk reduction” reduce the risk? - modelling HIV spread among men who have sex with men
Helene Alpfjord
Engineering Mathematics, Lund University,
Lund, Sweden.
Marco Cioffi
Department of Mathematics, University of Milan,
Milan, Italy.
Ingrid Johansson
Engineering Mathematics, Lund University,
Lund, Sweden.
Verena Schmid
Department of Mathematics, Technische Universitaet Dresden,
Dresden, Germany
Rui Sequeira
Department of Mathematics, University of Coimbra,
Coimbra, Portugal

Instructor: Daniel Simpson
Norwegian University of Science and Technology,
Trondheim, Norway

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Abstract
HIV is spreading fast among men who have sex with men (MSM), thus it is an important issue to study and model in order to prevent further spreading. We look at different strategies used by MSM in order to reduce the risk of getting infected or infecting others. Our model simulating future infections, depending on the risk reduction strategies we included, was based on an epidemic model. We considered the population to be divided into three groups; healthy, infected with diagnosis and infected without diagnosis.
As expected the spreading can be controlled or decreased by using risk reduction strategies such as condom use and medical treatment. The efficiency of strategic positioning, the choice of the sexual position in order to decrease the risk of infection, and serosorting, the choice of the partner in order to decrease the risk of new infections, depends on knowledge of HIV status, which in turn depends on the testing rate and disclosure.
The knowledge of efficiency of different risk reduction strategies can be used when developing information campaigns and health care campaigns with MSM as target audience.

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Modelling HIV spread among men who have sex with men

11.1

Introduction

The most used mathematical models for disease spreading are based on flu transmitting models and other air-transmitting diseases. Also well known models exist for food or water transmitted diseases. The sexually transmitted infections such as HIV are rarely modelled by mathematical models. The model presented in this report simulates the spreading among men who have sex with men (MSMs). This group is interesting to model since it is a high risk group, due to the relatively high risk of getting infected by anal intercourse compared to other sexual activities. To focus on MSMs is also a way to narrow down the project and keep it at a manageable level and to have more available data.

11.1.1

Purpose

The purpose with our model is to answer the question: “Does ‘risk reduction’ really reduce the risk?” By the term “risk reduction” we mean different strategies one can adopt in order to decrease the risk of getting infected.
The strategies we have considered are:
• Condom use - how the rate of condom usage affects the spreading.
• HIV testing rate - how the rate of getting tested influences.
• Use of antiretroviral treatment (ART) - ART is a medicine you can take if you are HIV positive. It both keeps you healthier and decreases your risk of spreading the infection further.
• Strategic positioning - This means that you take different positions during anal intercourse depending on your own and your partner’s HIV status. If you have sex with an HIV positive partner, then the risk of getting infected is higher if you are receptive, especially if your partner ejaculates inside you, compared to the risk if you are the insertive one.
• Disclosure and serosorting - Disclosure is simply that you and your partner tell your presumed HIV statuses to each other. When you know the status of your partner you can choose to serosort, which means that you only have sex with people who have the same HIV status as you. If you are an HIV negative man you choose to only have sex with other HIV negative men and vice versa. Due to this, the rate of condom use is is less when serosorting has taken place compared to when it has not.

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11.2
11.2.1

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Model
Our models

The first step while working on this project was to choose a way to solve it.
Our first idea was to model the problem with an epidemic system.
Our population is the MSM population (men who have sex with men) and in the first model we have chosen to divide the population into two classes: susceptibles and infected. A susceptible is a member of our population who is at risk of becoming infected by HIV, while the meaning of infected is obvious.
We knew that this choice was too simplistic but we did it just to have a base on which to build improvements.
Here are the equations: dS = A − δS S − αS S − βS dt dI
= βS − δI I − αI I dt (11.1)

We have assumed unit time of 6 months, this is because six months is the conservative window period to be sure of HIV test (this will be very important in the second model where we introduce a testing rate).
The parameter A is the susceptible inflow rate, δ is the death rate (of course, it may be different for the 2 classes), α is the aging out rate (because we are considering only the sexual active part of the population, as before it may be different for the 2 classes).
At last β is the infection rate, in our work this has been the constant that we studied more.

In our second and last model we decided to add another class, the undiagnosed class (infected without knowing it). This is obviously more realistic, in fact to be HIV-positive and know it are two very different things.
This figure show the division into classes of our population.

We can see the 3 classes, the inflows (aging in) and outflows (aging out and deaths), and the flows from the class of susceptibles to the unknows, and that from unknows to the infected.
Here are the equations:

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Modelling HIV spread among men who have sex with men

Figure 11.1: Compartment model

dS
= A − δS S − αS S − βS + δ(U + D) dt dU
= βS − γU − δU − αU U dt dD
= γU − αD D − δD dt (11.2)

All parameters have been chosen in order to have constant population over time: d (S + U + D) = 0 dt A,δ,α and β have the same meaning as before, the new parameter γ is an
HIV testing rate (table 11.1), which controls the passage between the second and the third class. The parameters are obtained from various statistical reports on the MSM population of Sydney.
We have chosen 7 different infection categories, as we can see from this picture, showing exactly how we assumed were formed the classes.
Each one has a different probability depending on different factors, then we use them to calculate the rate of infection (so β ≈ 7 βi ). To give an
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example, β5 is the probability of infection from casual partners who were not asked about their HIV status and who are infected but undiagnosed.
The formula for β5 is the following one: β5 = 1 − ((1 − Pcu,U )nu )(1 − Pcu,P Q)np for this case the values are:
• Pcu,U = Probability to have sex with an undiagnosed casual partner without asking about him HIV-status and without condom = 9.4×10−3

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Figure 11.2: Tree of the infection categories
• Pcu,P = Probability to have sex with an undiagnosed casual partner without asking about him HIV-status with condom= 6.1610−4
• nu = Mean of sex acts (per time unit) in P u,U case = 1.348
• np = Mean of sex acts (per time unit) in P u,P case =2.56
• Q = 1 - Condom efficiency = 1 − 0.95 = 0.05
Then summarize how we found the betas (we will try to summarize all the information in the notation):
(u)ndiagnosed or (d)iagnosed, (U )nprotected or or (c)asual

P(r)egular

(P )rotected

the meaning is “probability to have sex with . . . ”
1

1

u,U u,P • β1 = 1 − (1 − Pr )nu (1 − Pr Q)np

2

2

4

4

6

6

d,U d,P • β2 = 1 − PART (1 − RiskRedART )(1 − (1 − Pr )nu )(1 − Pr Q)np
3

3

• β3 = 1 − (1 − Pcu,U )nu (1 − Pcu,P Q)np
• β4 = 1 − PART (1 − RiskRedART )(1 − (1 − Pcd,U )nu )(1 − Pcd,P Q)np
5

5

• β5 = 1 − (1 − Pcu,U )nu (1 − Pcu,P Q)np
• β6 = 1 − PART (1 − RiskRedART )(1 − (1 − Pcu,U )nu )(1 − Pcu,P Q)np ser ser

• β7 = 1 − (1 − Pcu,U,Seros )nu (1 − Pcu,P,Seros Q)np

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Modelling HIV spread among men who have sex with men

11.3

Results

We tested our model described in (11.2) with the data for Sydney which we either got from the literature directly or estimated from various tables. The used parameters are given in table 11.1, the results are shown in figure 11.3.
PARAMETER
VALUE testing rate
0.26
ART rate
0.75
# casual partners randomly, Unif[0;2]
# regular partners randomly, Unif[0.8;1.2]
# sex acts with regular partners
20
condom efficiency
0.95
probabilities to get infected depending on position insertive 0.0062 receiving with ejaculation
0.0143
receiving without ejaculation
0.0065
rates of positions when condom is used insertive 0.47 receiving with ejaculation
0.53
receiving without ejaculation
0
rates of positions without condom for undiagnosed regular partners insertive 0.51 receiving with ejaculation
0.24
receiving without ejaculation
0.25
rates of positions without condom for diagnosed regular partners insertive 0.79 receiving with ejaculation
0.12
receiving without ejaculation
0.09
rates of positions without condom for undiagnosed casual partners insertive 0.72 receiving with ejaculation
0.17
receiving without ejaculation
0.11
rates of positions without condom for diagnosed casual partners insertive 0.79 receiving with ejaculation
0.12
receiving without ejaculation
0.09
rates of positions without condom for casual partners, not asked for their status insertive 0.46 receiving with ejaculation
0.21
receiving without ejaculation
0.33

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rates of condom use undiagnosed regular partners diagnosed regular partners undiagnosed casual partners diagnosed casual partners casual partners, not asked for their status factor of reduced condom use after serosorting rate of disclosure rate of serosorting
# of newly active MSMs every time step death rate of HIV-positive men death rate of HIV-negative men
# of uninfected at t = 0
# of undiagnosed infected at t = 0
# of diagnosed infected at t = 0
Table 11.1: Data from Sydney

Figure 11.3: Model with Data from Sydney

0.635
0.8186
0.737
0.85
0.663
0.5
0.4
0.33
1000
0.01
0.00
86000
2000
12000

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Modelling HIV spread among men who have sex with men

There, as in all following pictures, the blue line represents the percentage of healthy people among the population, the green line the percentage of infected that are not tested yet and the red line the percentage of the infected, that are also tested.
The model was run for 50 years (t=100). It gives us about 500 newly diagnosed people each year. This overestimates the true values from Sydney
(about 250-300 people per year). Possible reasons are, that in reality the spread of HIV has not stabilized yet, that some of the parameters had to be estimated and the given homogeneity assumption: In the model, the whole population is assumed to get involved in risky behaviour, where as in reality the population is divided in at least two groups, the ones who are involved in risky behaviour and those who are not. Therefore, in reality, this second group has a much smaller likelihood to get infected. The model just averages over the whole population.
In general you can see a slow increase in the percentage of infected for the used parameter values.
We used this model to test effectivity of the risk reduction methods described in (11.1.1).

11.3.1

Testing Rate

In Sydney we have a testing rate of about 26%, which means the people get tested about every second year. In figure (11.4) you can see, that if the testing rate would fall down to 10%, which means testing every fifth year, the percentage of infected, especially the percentage of the undiagnosed infected, increases extremly.
In figure 11.5 on the other side you can see, that if testing would be increased to 40% (testing every 15 month), the percentage of infected people could get at a constant level.
At this point it should be mentioned, that we do not see a reasonable aim to look for a total extinguish of HIV since it would take a long time period
(a few hundred years). In the meantime a lot of changes in the behaviour and thinking of the people might occur while our model was build based on the actual situation in Sydney.

11.3.2

ART

Figure 11.6 shows the percentages of the three groups described in 11.3 if no ART would be given. It clearly shows the importance of this treatment, not only for the wellbeing of the person who takes it, but also for possible partners and the whole community.
The rate of ART in Sydney is 0.75. As discussed for the general Model, this leads to slowly increasing percentage of infected. To get to a stable,

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Figure 11.4: Model with testing rate set to 10%

Figure 11.5: Model with testing rate set to 40%

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Modelling HIV spread among men who have sex with men

Figure 11.6: Model with ART rate set to 0%

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even slightly decreasing percentage, you would only need an increase up to
0.85. (See figure 11.7.)

Figure 11.7: Model with ART rate set to 85%

11.3.3

Condom Use

The actual average condom use in Sydney is about 73%. As you would expect, an decrease in the condom use increases the number of infected persons. In figure 11.8 we modeled a condom use of 67%. Even this small decrease has a strong effect on the outcome.
But you also need only small increases to get a stabilized situation.
Therefore, see figure 11.9 with an average condom use of 80%.

11.3.4

Strategic Positioning

In the general model for Sydney, strategic positioning is included in form of the rates given in table 11.1. To see the effect of strategic positioning on the spread of HIV we excluded it from the model. Therefore we set the rates for all possible positions to the rates that we used for acts with casual partners with a condom, which reflect what the men said to be their

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Modelling HIV spread among men who have sex with men

Figure 11.8: Model with condom use set to 67%

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Figure 11.9: Model with condom use set to 80%

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Modelling HIV spread among men who have sex with men

favourite positions. As you can see in figure 11.10, taking out strategic positioning increases the percentage of infected men the population, so strategic positioning is also a way to decrease the spread of HIV.

Figure 11.10: Model without strategic positioning

11.3.5

Serosorting

The use of serosorting, that is choosing partner based on HIV status, as a risk reduction method is not very efficient according to our model. If the testing rate is high, then serosorting reduces risk slightly (with a few percent), but the testing rate itself is a much more significant factor. When testing rates are low serosorting is very risky, since no condom is used with undiagnosed casual partners. The figure 11.11 shows the relations between testing rates on the x-axis, the probability of serosorting on the y-axis and the percentage of healthy people in the population on the z-axis. Here full disclosure is assumed, as well as no condom use at all when being told that the partner is HIV negative.

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Figure 11.11: Percentage of healthy people in the population as a function of the testing rate and the probability of serosorting.

11.3.6

Disclosure

A high rate of disclosure is reducing the spread of HIV, together with 33% serosorting and 73.7% condom use when being told that your casual partner is negative. About 5% less infections occur when having full disclosure compared to having no disclosure. When there is no disclosure no serosorting can be performed. The figure 11.12 below shows the percentage of healthy people with different disclosures. The red line indicates no disclosure, black circles 40% disclosure and the green line indicates full disclosure.

11.4

Future expansions of the model

There are several ways of expanding the model, which easily come to mind.
The following expansions could be considered in a more advanced model.
• Group sex. A high risk group is people engaging in group sex. According to reports 19% of newly diagnosed MSMs had engaged in group sex in the last six months before diagnosis. Common meeting points are internet communities and gay bars, so this group can rather easily be targeted for education.
• Dividing the susceptible group into subgroups depending on gay community engagement. The sexual behavior differs a lot due to this factor.
• Dividing the diagnosed group into subgroups depending on disease stage. The viral load and thus the infection risk vary a lot. A newly infected person or a person diagnosed with AIDS, the last stage of
HIV, is far more contagious than someone in another disease stage.

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Modelling HIV spread among men who have sex with men

Figure 11.12: Model with different rates of disclosure.

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• PEP/PrEP. Post-exposure prophylaxis and pre-exposure prophylaxis are treatments used before or immediately after risky encounters. A correct use of these medicines reduces the infection risk significantly.
• Drug use combined with sexual contacts. Both an intravenous infection risk and riskier sexual behavior due to drug use are important parts of understanding the HIV spread.
• Widen the population. Including other groups in the model is an expansion, however a rather complicated one.
• Co infections. Other sexually transmitted diseases can increase the risk of HIV infections.

11.5

Discussion and conclusion

Different risk reduction methods used today are efficient together, and small improvements in each of them would all together decrease the spread. Condom use is a very important factor, since it is a cheap and easy way to prevent HIV spread. Strategic positioning and serosorting are methods that work quite well when there is good knowledge about the true HIV status.
This assumes a rather high testing rate. Serosorting is unfortunately often combined with a decreased use of condoms, which becomes counterproductive.
Increasing the testing rate is not an efficient method in itself. The costs are rather high and it is time consuming, both for the individuals taking the test and for the health care system.
The disclosure rate is probably a very difficult issue to influence, due to the taboo of HIV in our society.
The rather simple model we implemented works quite well considering its simplicity. We believe the model captures the most important aspects of
HIV spreading among MSMs in western societies. The number of infected individuals that our model predicts is a bit high compared to example data from Sydney. One way of handling that is to divide the susceptible group into two groups, depending on their level of risky sexual behaviour.