This article presents a mathematical model designed to simulate the impact of targeted interventions aimed at preventing HIV transmission among female sex workers (FSWs) and their clients, while also analyzing their effects on the health of the general population. The compartmental model distinguishes between high-risk populations (FSWs and their clients) and low-risk populations (sexually active men and women in the general population), and links prevention efforts in high-risk groups to the evolution of the epidemic in the low-risk population. The fundamental properties of the model, such as the positivity of solutions and the boundedness of the system, have been verified, and the basic reproduction number R₀ has been calculated. Finally, the stability of the model was studied using Varga’s theorem and the Lyapunov method. Simulation results show that targeted prevention among FSWs and their clients reduces HIV incidence in the general population. This framework provides a valuable tool for guiding policymakers in the design of effective strategies to combat the epidemic, especially relevant in the context of suspension of USAID funding.

Health systems are constantly evolving in order to improve the services offered to the population. When implementing these policies, it is important to be able to assess their impact on population’s health status. In the context of HIV, several prevention and treatment strategies have been defined to effectively control the spread of the disease [1]. According to UNAIDS, new HIV infections have been reduced by 59% since the peak in 1995 [2] and the current prevalence if 0.7% globally. However, the epidemic is most prevalent among key populations (KP), including female sex workers (FSW), men who have sex with men (MSM), injectable drug users (IDU), and prison mates [2]. As matter of fact, the prevalence of HIV infection was 3.6 times higher among sex workers, 10.7 times higher among MSMs, 7 times higher among IDUs, 14.7 times higher among transgender people and twice among the prison mates [3]. In Sub Saharan Africa, key populations accounted for 51% of new infections in 2021 [2]. Considering this new situation which represents a significant threat to disease control, a completely new response plan has been put in place by UNAIDS with KPs at the center of this new strategic plan [4].

Several mathematical models have been developed by researchers to provide a better understanding of the disease transmission in key population groups and to measure the effect of prevention and treatment strategies targeting them. In Kenya, for example, Omondi et al. [5] developed a mathematical model of HIV transmission between sex workers and injecting drug users to assess the effect of combination of Pre-Exposure Prophylaxis (PrEP) and antiretroviral therapy (ART) on the spread of HIV. In Cote d’Ivoire, Maheu-Giroux et al. [6] developed an age-stratified dynamic model of sexual and vertical transmission of HIV among the general population, FSWs and MSM. Their model was calibrated on detailed prevalence and intervention data (ART and condoms). Geidelberg et al. [7] looked at the role of PrEP on the spread of HIV among FSWs in Cotonou (Benin Republic). The authors also used a compartmental model involving PrEP and ART in high-risk (FSWs and clients) and low-risk populations. In Burkina Faso, Low et al. [8] formulated an initial mathematical model to assess the impact of combining ARTs and condoms on FSWs’ health status.

So far, there is no standardised approach to directly visualising how interventions targeting key populations can affect the general population that does not interact directly with them. The aim of the model proposed in this study is therefore to describe and simulate the potential benefits of HIV control programs for key populations, taking into account the indirect effects on the general population. This research extends the initial study by Low et al. [8]. To illustrate our approach at the national level, we examine the specific situation of Burkina Faso, a landlocked country in West Africa where HIV prevalence varies widely among demographic groups and identified key populations. In 2019, the national census reported a population of 20.5 million, currently estimated to be close to 23.5 million [9]. HIV prevalence in the general population is estimated at 0.6% [10], with specific rates of 0.8% for women and 0.5% for men aged 15-49 years. Among key populations, prevalence is estimated at 5.4% among sex workers, 1.9% among men who have sex with men, 1.3% among prisoners and 1.0% among injecting drug users [11].

The work in this paper is organized as followed: in Section 2, the model describing the transmission of HIV between high and low risk individuals is formulated. In Section 3 the mathematical analysis of the model is given. Application with numerical simulation and the model simulation results are presented in Sections 4. Finally, the discussion and conclusion are presented in Section 5.

In this section, a mathematical model is formulated. It describes the HIV transmission between two different risk groups, namely sex workers and their clients considered to be at high risk of infection and the general sexually active population considered to be at low risk of infection. This model is an extension of a model studied in [12] and is based on the modelling approach given in [5, 13]. The transition between these risk groups is assumed to exist in this model. The model has twelve compartments. S, I, and A represent the level of infection according to a classical SI model. These are respectively the susceptible, the infectious and the AIDS class in the model. The type of group $\{\mathrm{s}\mathrm{w},\mathrm{ }\mathrm{c},\mathrm{ }\mathrm{m},\mathrm{ }\mathrm{f}\}$represents the category of the participants which are respectively the FSWs, their clients, the general male population and the general female population. The infection rate $\lambda$is determined from the average annual number of sexual partners ($\psi$), the probability of HIV transmission during vaginal sexual intercourse ($p_M$ for male and $p_F$ for female) with a partner in infectious (I) or AIDS (A) state. The parameter$\mathrm{\Gamma }$ will permit to modify the risks of HIV sexual transmissions. Infection rates in at-risk populations are given by

The parameter $\pi$ is added to consider the effect of preventive intervention targeting FSWs and their clients. The model assumes that all risk groups grow at the same rate over each period. The population sizes at time t for the risk groups are:

It is important to note that the intervention is implemented in the group of high-risk individuals. The full list of model parameters is summarized in table 1.

The flow and interactions between the compartments are described in figure 1. In this figure, black arrows indicate the evolution of individuals in the course of the infection while red arrows indicate sexual infection across groups.

Figure 1

A compartmental representation of the model for HIV transmission.

The above description gives the following non-linear ordinary differential equations: For sex workers and their clients

For men and women in general population

The system of equations in (1) — (2) is subject to the following initial conditions$,\mathrm{ }{\mathrm{S}}_{\mathrm{s}\mathrm{w}}\left(0\right)\mathrm{ }\ge \mathrm{ }0,\mathrm{ }{\mathrm{I}}_{\mathrm{s}\mathrm{w}}\left(\right(0)\mathrm{ }\ge \mathrm{ }0,{\mathrm{A}}_{\mathrm{s}\mathrm{w}}(\left(0\right)\mathrm{ }\ge \mathrm{ }0,{\mathrm{S}}_{\mathrm{c}}\left(0\right)\mathrm{ }\ge \mathrm{ }0,{\mathrm{I}}_{\mathrm{c}}\left(0\right)\mathrm{ }\ge \mathrm{ }0,$
 Ac(0) ≥ 0, Sm(0) ≥ 0,   Im(0)) ≥ 0,  Am(0) ≥ 0, Sf(0) ≥ 0,  If(0) ≥ 0 and ${\mathrm{A}}_{\mathrm{f}}\left(0\right)$≥ 0.

3.1. Positivity and boundedness properties

In this subsection, we determine the equilibrium points, and the basic reproduction number associated with System (1) — (2).

Lemma 3.1. If $S_{sw}\left(0\right)$, $I_{sw}\left(0\right)$, $A_{sw}\left(0\right)$, $S_c\left(0\right)$, $I_c\left(0\right)$, $A_c\left(0\right)$, $S_m\left(0\right)$, $I_m\left(0\right)$, $A_m\left(0\right)$, $S_f\left(0\right)$, $I_f\left(0\right)$, and $A_f\left(0\right)$ are non negative, then so are $S_{sw}\left(t\right),$ $I_{sw}\left(t\right)$, $A_{sw}\left(t\right)$, $S_c\left(t\right)$, $I_c\left(t\right)$, $A_c\left(t\right)$, $S_m\left(t\right)$, $I_m\left(t\right)$, $A_m\left(t\right)$, $S_f\left(t\right)$, $I_f\left(t\right)$, and $A_f\left(t\right)$ for all time $t>0$. Moreover, ${lim}_{t\to +\infty }SupN\left(t\right)\le \frac{\Lambda }{\mu }$.

Proof 3.1. For that, we first use the contradiction that the state variable $S$ is nonnegative for all $t\le 0$. Let $e\left(t\right)=min\{S_{sw}\left(t\right),I_{sw}\left(t\right),A_{sw}\left(t\right),S_{cO}\left(t\right),I_c\left(t\right),A_c\left(t\right),S_m\left(t\right),I_m\left(t\right),A_m\left(t\right),S_f\left(t\right),I_f\left(t\right),A_f\left(t\right)\}$ and let us suppose that there is at least one $t_1>0$ such that $e\left(t_1\right)=0$ and $e\left(t_1\right)>0$ for all $t\in \left(0,t_1\right)$. Therefore, if $e\left(t\right) = S_{sw}\left(t\right)$ then each state of model (1) $\&$ (2), is positive and from the first equation of (1) $\&$ (2), we have

It then follows that

which leads to a contradiction.

Similar proof can be given for the other state variables. Thus, any solution of system (1) — (2) is nonnegative for $t\ge 0$. Moreover, the total number of the population N(t) at any time is governed by

Where $\Lambda ={\Lambda }_{sw}+{\Lambda }_c+{\Lambda }_f+{\Lambda }_m$.

Thus, for the initial conditions $0\le N\left(0\right)\le \frac{\Lambda }{\mu }$, by using Gronwall inequality [14, 15], we get

Hence, system (1) — (2) defines a dynamical on

3.2. Equilibrium points and basic reproduction number

In this subsection, we determine the equilibrium points, and the basic reproduction number associated with System (1) — (2).

3.2.1. Equilibrium points

Let ε = (S_sw, I_sw, A_sw, S_c, I_c, A_c, S_f , I_f , A_f , S_m, I_m, A_m) be the equilibrium point of System. This corresponds to the values where the equations (1) — (2) are all null.

Let ε₀ and ε^∗ be respectively the disease-free and endemic equilibrium points of System (1) — (2). The disease-free equilibrium corresponds to the case were there is not infected individual. In this case, we have ${\mathrm{I}}_{\mathrm{s}\mathrm{w}}=0,{\mathrm{A}}_{\mathrm{s}\mathrm{w}}=0,{\mathrm{I}}_{\mathrm{c}}=0,{\mathrm{A}}_{\mathrm{c}}=0,{\mathrm{I}}_{\mathrm{f}}=0,{\mathrm{A}}_{\mathrm{f}}=0,{\mathrm{I}}_{\mathrm{m}}=0,{\mathrm{A}}_{\mathrm{m}}=0$ and the disease-free equilibrium given by:

We design by ${\mathrm{I}}_{\mathrm{s}\mathrm{w}}^{\mathrm{*}},{\mathrm{A}}_{\mathrm{s}\mathrm{w}}^{\mathrm{*}},{\mathrm{I}}_{\mathrm{c}}^{\mathrm{*}},{\mathrm{A}}_{\mathrm{c}}^{\mathrm{*}},{\mathrm{I}}_{\mathrm{f}}^{\mathrm{*}},{\mathrm{A}}_{\mathrm{f}}^{\mathrm{*}},{\mathrm{I}}_{\mathrm{m}}^{\mathrm{*}}$ and ${\mathrm{A}}_{\mathrm{s}\mathrm{w}}^{\mathrm{*}}$ the infectious at endemic equilibrium points, so

With

By putting the right hand side of system (1) — (2) to zero, and keeping each state variable different from zero ($S_{sw}\ne 0$, $I_{sw}\ne 0$, $S_c\ne 0$, $I_c\ne 0$, $I_f\ne 0$, $I_f\ne 0$, $S_m\ne 0$ and $I_m\ne 0$).

We also make this assumption:

3.2.2. The Basic reproduction numbers

The threshold parameter R₀ is defined as the average cases of secondary infections generated by a single infectious individual in a completely susceptible population during his/her period of infectiousness [16, 17]. In our model the infected classes correspond to states I_sw, A_sw, I_c, A_c, I_f, A_f, I_m and A_m. Thus, we can rewrite system (1) — (2) as

where $\chi =\left({\mathrm{I}}_{\mathrm{s}\mathrm{w}},{\mathrm{A}}_{\mathrm{s}\mathrm{w}},{\mathrm{I}}_{\mathrm{c}},{\mathrm{A}}_{\mathrm{c}},{\mathrm{I}}_{\mathrm{f}},{\mathrm{A}}_{\mathrm{f}},I_m,A_m\right)$; $\mathcal{F}$ is the rate of appearance of new infections in each class, and $\mathcal{V}$ is the rate of transfer of individuals out of (for positive values) or into (for negative values) compartment by all other means. Hence, in our case, we have that

and

The jacobian matrix of $\mathcal{F}$ and $\mathcal{V}$ at the disease-free equilibrium ${\epsilon }_0$ are given by

where:

and

respectively.

The matrix $\mathrm{V}$ is invertible (non-zero determinant) and its inverse ${\mathrm{V}}^{-1}$ is defined by:

So, the next-generation matrix $\mathrm{F}{\mathrm{V}}^{-1}$ is given by:

Let’s put

The basic reproduction ratio is given by ${\mathcal{R}}_0=\mathrm{\rho }\left(-\mathrm{F}{\mathrm{V}}^{-1}\right)$, the spectral radius of the next-generation matrix $\mathrm{F}{\mathrm{V}}^{-1}$. In our case, we have that

Where:

3.3. Stability of equilibrium points

In this subsection, we prove the global stability of disease-free equilibrium ${\mathrm{\epsilon }}_0$, when ${\mathcal{R}}_0<1$ and the global stability of endemic equilibrium ${\mathrm{\epsilon }}^{\mathrm{*}}$, when ${\mathcal{R}}_0>1$.

Lemma 3.2. The disease-free equilibrium ${\epsilon }_0$ of system (1) — (2) is globally asymptotically stable, when ${\mathcal{R}}_0<1$ and unstable if ${\mathcal{R}}_0>1$.

Proof 3.2. Let us consider the infected classes $\chi =\left(I_{sw},A_{sw},I_c,A_c,I_f,A_f,I_m,A_m\right)$. By the equations corresponding to these states, we have the linearization system at ${\epsilon }_0$ given by:

the matrix associated to the linearized system (11) is given by:

With

It is important to note that $\mathrm{A}$ has all off diagonal entries non-negative. This implies that $\mathrm{A}$ is a Metzler matrix. The matrix A can be decomposed as follows $\mathrm{A}={\mathrm{F}}_1+{\mathrm{V}}_1$, with: