In order to apply our model to real data, we chose the French MSM population, and later, we focused more specifically on the high-risk individuals within this group. We define here the high-risk MSM population as multi-partner gay men, as well as sexual intercourse while using drugs, usually called chemsex, or new generations not inclined to use any protection anymore (mainly condoms). Of course, this reduces our study to a small part of the population, but, because of their having the highest rate of infectivity, they belong to the target of PrEP treatments at the infectious disease center in France.
It is important at this stage to remind that HIV is not curable and that PrEP does not have a direct influence on the decreasing number of the infected compartment over time, which is mainly due to natural death (with a life expectancy for treated persons as high as non-infected ones) or, more rarely now, thus neglected here for the French group, untreated AIDS individuals. However, PrEP policy in the population may have an important impact on the infectivity rate. Since PrEP’s efficiency can be measured in a long-term range and using incidence, we simulated over 15 years (that is, 180 months) for a forecast of this prevention strategy over time.
5.1. Choice of the Parameters
In Table 1
, we give the parameters used in Model (1
), as well as their meaning and values for the general and high-risk MSM community in France estimated from [2
], assuming that
of the the population considered in the dataset of the study are MSM.
PrEP treatment in France started in 2016, and since the SARS-CoV-2 epidemic drastically modified sexual habits (through successive lockdowns), we considered the population data until 2019 only. Then, after, we simulated our model based on the estimated parameter values for the following 15 years. Note here that we did not take the SARS-CoV-2 nor the monkey pox epidemic into account for three reasons: first, because these periods are a bias for our model and need to be carefully adjusted with updated parameter values; second, to the best of our knowledge, the official French data related to these past two years have not been released yet; finally, we preferred keeping our model as simple as possible for this present work. A more complex approach will be the object of a future work.
], information between 2016 and 2019 was collected every 6 months. We remind here that, at the end of a 3-month period, the patient decides whether to give up the treatment or to continue. This is the reason why our simulations below were computed on a monthly basis.
Initial conditions for functions
were chosen to be January, the 1st of 2016, according to [2
Because of the delay equation for u
, the function
was adjusted with a cubic spline interpolation of 60% of each total number of the last column in Table 2
. Note that
is the same for both populations’ (general MSM and its high-risk subset) simulations because we assumed here that most of the French PrEP users belong to the high-risk MSM group.
Furthermore, on the official website of UNAIDS (https://www.unaids.org/en/regionscountries/countries/france
) (accessed on 13 September 2022), we obtained that 170,000 individuals were infected with HIV in France in 2016. We assumed that a majority belong to the MSM group, but not the entire number (indeed, a large portion of new cases was reported to be foreign heterosexuals, among other subcases). This is why we took
90,000. It is important to remind here that this value may not be the exact one, since the number of HIV infections detected is always below the real number of HIV infections (HIV testing is not compulsory, and thus, several individuals may carry the virus for years without knowing it).
Finally, when simulating the MSM high-risk population, the estimated susceptible number would be between 3000 and 50,000 depending on the literature. Thus, we took
40,000 according to [2
]. According to French data (https://vih.org/20190328/stabilite-des-chiffres-du-vih-en-france/
) (accessed on 13 September 2022), French infections have been stable since 2010. On average, there are 6500 contaminations per year, and infections within the MSM population represent 43%. Then, we chose
According to the official French data (https://www.insee.fr/fr/statistiques/2383440
) (accessed on 13 September 2022),
at the country scale. We chose this value for the French MSM removal rate (death, as well as removed from sexual life). For the high-risk MSM,
is considered much larger because a high-risk sexual life style does not last longer than the average MSM one. Thus,
in this subgroup.
The 2016–2019 semester official French datasets are given in Table 3
Thanks to Table 2
and Table 3
, we are able to assess the values of functions
for the years 2016–2019. Indeed, for each semester, we obtained a different
by using the following formula coming from its definition:
We assumed that this value is the same for every month of the semester. Then, we chose
, per month, as follows:
is the initial condition for the susceptible compartment S
as the average value of all the previous
from Table 4
As mentioned in Section 2
, we assumed that
is the solution of a logistic equation given by:
is the carrying capacity and r
exponential growth. We remind that the explicit form of
is then written by the following expression:
is the initial condition depending on the population type. Using the data for
given in Table 4
, we summarize our results in Table 5
We assumed a Hill function for the dynamics of the function f
the saturation of the Hill function,
the abscissa of the inflection point, and n
the intensity of the slope. We summarize the values of these parameters in Table 6
One of our goals was to estimate the HIV epidemic
for French MSM. We used the package in the R®
language untitled R0 (https://www.rdocumentation.org/packages/R0/versions/1.2-6
) (accessed on 13 September 2022) and, precisely, the function est.R0.SB
, which estimates
using a Bayesian approach following the idea developed in [13
]. Thanks to our data of new HIV infections ([14
]), we obtained
for the normal MSM French population.
The two remaining parameters, and , were estimated to keep equal to . We decided to choose ; in other words, each month, 3000 individuals might join the susceptible compartment (by reaching the sexual consent age or deciding to join the MSM group). Thus, was estimated for the French general MSM group.
5.2. Numerical Simulations for the General French MSM
In Figure 2
, protected (P
) in green (graph on the right) keeps track of the exact data (crosses) for the first semesters, then reaches a threshold due to the French policy of regulating PrEP users. Indeed, the treatment is fully taken care of by the health insurance in France. Thus, a threshold needs to be set up (estimated at 60,000 MSM individuals here). With this threshold, it seems already that, within 15 years, the number of infected drops and keeps on decreasing. The number of susceptibles (blue curve on the left of the graph) keeps on growing with a lower slope between 50 and 100 months due to the increase of the protected and then the threshold. However, even with the increase of the susceptibles, the infected remain low. The protected compartment plays the role of a reservoir to prevent the HIV epidemic from growing back.
Just as a comparison point, we simulated our model with a function f
defined as identity (and not a Hill function anymore) (see Figure 3
). We easily observed that the graph of the protected population does not fit the data in the first months with the same realistic parameter sets. Then, it seems that f
needs to be more complex than identity for the case of French MSM, and the Hill function seems to be validated here.
In Figure 4
, we plot the incidence with or without PrEP treatment. It seems obvious that without the PrEP reservoir (in blue), the incidence keeps on growing with no chance for the HIV epidemic to be controlled. On the other hand, with the PrEP compartment (in red), it is possible to keep the incidence at a low level (and even to decrease it significantly for the first years). The slight increase at the end (after Month 125) is due to the threshold of the number of PrEP users imposed in our model and likely used by the French health insurance policy.
In Figure 5
, we depict the effect of the variation of
as a function of
. It appears clearly here that, as
increases (that is, the number of PrEP users rises),
declines drastically. This is, however, not linear, and a plateau may be reached after a certain threshold (above
), which indicates that the flow of new PrEP users does not need to expand drastically. Indeed, even at
, we obtain
, which is quite satisfactory. Augmenting
, but not as fast as the first values of
5.3. Numerical Simulations for High-Risk Population
In this subsection, we focus our attention to the French MSM high-risk population. We remind that this MSM subgroup consists of individuals with multi-partner intercourse, sex relationships while using drugs (chemsex), etc. The HIV risk of infection is thus much higher than the rest of the MSM population. This is one of the reasons why this subgroup is the PrEP treatment target for the French health insurance. The protected compartment in this particular case plays a major role as a reservoir. This was clearly confirmed in our model simulations, as shown in Figure 6
, where the number of susceptibles clearly drops as the number of protected individuals increases. Note that the infected population rises, but for the first time, reaches a threshold. This means that barely any new infected cases appear. We remind here that the model is an
and not an
, that is no recovered individuals can appear since
is a disease with a lifetime treatment.
In Figure 7
, the incidence evolution is plotted with or without PrEP treatment. The reservoir effect is even more explicit there in the sense that, with PrEP, the incidence declines to reach a low plateau, while without, it keeps on growing indefinitely.
Finally, as mentioned before, for the MSM high-risk population, the
in France is higher than 1. Figure 8
reveals that, as the number
of new PrEP users increases,
not only decreases below one, but also keeps on falling off. Here, the critical value of
, which keeps
below one, is
. According to our data,
, as we assume the maximum threshold has been reached. The decay appears, however with a smaller slope, which means, as in the previous subsection, that, after a certain threshold (
here), very large values of
do not influence the reduction of
This reservoir effect is so important that, if for any reason, there is the decision to stop PrEP treatment for this MSM group, the rise in new infected individuals would be inevitable, as shown in Figure 9