Mathematical analysis of pre-exposure prophylaxis on HIV infection.

A mathematical analysis of the model is performed to determine the boundary conditions that determine the stability of the steady states. HIV infection has been one of the most dramatic pandemics the world's population has ever faced. Awareness of PrEP, effectiveness of PrEP and acceptability of PrEP use are some of the various challenges concerning the approach of PrEP as a preventive measure [3].

We focus and validate the model analysis on two factors that influence the use of PrEP, i.e. PrEP awareness and PrEP effectiveness. We ensure the well-posedness of our model so that we can trust the validation of the results derived from the models. We perform a model analysis focusing on an important threshold parameter used as a measure to determine HIV progression in the PrEP-using population.

Literature review

HIV infection

HIV phases of infection progression

Basic notions and definitions

Equilibrium point of a system of equations
Local and global stability of an equilibrium point
Spectral radius of a matrix
Basic reproduction number

Asymptotic stability of x∗0 of the system (1.1) can also be derived when all the eigenvalues of the Jacobian matrix of the system (1.1) are negative or have negative real parts. This dimensionless positive number is one of the most useful parameters in mathematical analysis of epidemiological models. The role played by the reproduction number is to provide information about the spread and the possibilities of eradicating the infection.

Without loss of generality, let us consider the system (1.5) as a biological model defined in a domain Ω such that Ω is positively invariant. Let fk be the kth component of f, so that a=X. 1.9) The local stability of the equilibrium point X∗ =0 is confirmed by the signs a and b. We analyze the model (2.1) by calculating the basic reproduction number of the model using van den Driessche and Watmough's approach [16] and study its (local) stability by its equilibrium points.

Case 1: Birth rate balanced by death rate with no AIDS deaths

Nondimensionalization of the model

Positivity and boundedness of solutions of the model

Ω is positively invariant for the system (2.8) and there exists a constant M>0 such that all solutions starting with Ω satisfy s, i, a≤ M for all large t.

Stability analysis of the equilibrium points

Disease free equilibrium point (DFE)
Stability analysis of the disease free equilibrium
Endemic equilibrium point (EEP)
Stability analysis of the endemic equilibrium point

V−1 = (vjk) is a matrix where the entry (j, k) is the average duration of the infected individuals in compartmentj. To analyze the stability of the DFE, we evaluate the Jacobian matrix of the system of equations (2.8) at E0. The disease-free equilibrium point E0 is locally asymptotically stable if all eigenvalues of the Jacobian matrix JE0 are negative or have negative real parts, and unstable if at least one eigenvalue is positive or has a positive real part.

The disease-free equilibrium point E0 is locally asymptotically stable when R0 <1 and unstable when R0 >1. For 0 < R0 <1, the disease-free equilibrium point E is the only equilibrium that exists and is locally asymptotically stable in Ω. For R0 > 1, the DFE E0 becomes unstable and a new equilibrium point, the endemic equilibrium point (EEP) exists and is locally asymptotically stable in Ω.

Case 2: Birth rate different from death rate with no AIDS deaths

Comparison and observations

Case 3: Birth rate different from death rate with deaths due to AIDS

Disease free equilibrium point
Stability analysis of the disease free equilibrium point
The endemic equilibrium point
Stability analysis of the endemic equilibrium point

We can see that all the eigenvalues have negative real parts whena0 >0, that is, whenRδ<1. The disease-free equilibrium point Eδ in the system of equations (2.52) is locally asymptotically stable when Rδ <1, and unstable when Rδ>1. Therefore, by the Routh-Hurwitz criterion for stability [17], all the eigenvalues of JEδ∗ are negative or have negative real parts.

Remarks

Model description

We consider that some susceptible individuals who are at high risk of HIV infection take PrEP, but some do not. We assume that a proportion γ of susceptible individuals is on PrEP and the remainder (1−γ)S is not on prophylaxis (γ can be considered a measure of awareness of PrEP use, so 0 ≤ γ ≤ 1) . However, only a percentage σ of Sp is protected by PrEP (σ is the measure of effectiveness of drugs used as PrEP, with 0≤σ ≤1).

As a result (1−σ)Sp are exposed to the risk of HIV infection due to PrEP failure. Individuals who become infected due to PrEP failure move to a new infected class Ip while those who do not take PrEP and become infected move to infected class I. Individuals from class I and Ip progress to the AIDS class, denoted A, with constant rates ρ1 and ρ2 respectively.

We assume that the individuals recruited into the population N are susceptible individuals, so that the compartment S increases with a constant recruitment rate π.

Positivity and boundedness of solutions of the model

PrEP model analysis

Disease free equilibrium point and basic reproduction number

The matrix of the next generation F V−1 of the system (3.2) is given by. 3.9) The characteristic equation is given by.

Influence of PrEP on R 0

We find that when Rn is less than the threshold Rn∗, PrEP administration to individuals in the community may not affect the infection rate. When Rn is above the threshold value R∗n, the PrEP reduces the basic reproduction number. It is clear that when protection against PrEP improves, that is, when σ goes to one, only the second term of R0 goes to zero.

This means that increasing the effectiveness of PrEP in the community will not guarantee total eradication of the infection, but will certainly reduce the infection rate. When the response to PrEP awareness increases, that is, when γ goes to unity, the infection is also not eradicated. However, when both orσandγ simultaneously trend toward unity, the number of secondary infections decreases and tends toward zero as time progresses.

This means that a combined PrEP awareness and effectiveness strategy, if implemented effectively, can lead to effective HIV control. However, control status is unstable as drugs wear off over time and individuals may respond to consciousness at different rates. For maximum benefits, adherence to PrEP should be ensured and PrEP should be combined with other HIV control strategies.

Mathematical observations and biological interpretation

Following the same analysis as for equation (3.4), we define the feasible range of systems (S1) and (S2). From what has gone before in Section 2.2.2, we can claim that the regions Ω1 and Ω2 are positively invariant and attractive. The basic reproduction number of systems (S1) and (S2), calculated using the techniques of van den Driessche and Watmough [16], leads to.

This confirms that the basic reproduction number of the main model (3.2), R0, is a linear combination of R1 and R2, which are the reproduction numbers of models (S1) and (S2). We can see that the main model (3.2) is split into two models (S1) and (S2), and it follows that its basic reproduction number is a linear combination of each of both models (S1) and (S2). We call min{κR1,(1−κ)R2} the slow base reproduction number and max{κR1,(1−κ)R2} the fast base reproduction number.

Thus the parameter κ (0< κ≤1) is called the slow-fast parameter of the parameter-bound model. If the basic reproduction number cannot be divided as in (3.44), we call the model a compact model. Although some compact models present the required feature of linear combination, they cannot always be divided into two sub-models.

Analysis of the parameter-related model for epidemiological disease is important in that it reveals that control of community infection may require a balance of more than one intervention strategy. Thus, in our case, control will take place by simultaneously monitoring both the fast basic reproductive number and the slow basic reproductive number until the level where the epidemic can be effectively controlled. A parametrically coupled model allows us to determine a group of individuals in a community that is more susceptible to infection.

In order to eradicate the infection, measures are needed to provide more PrEP education to individuals who have a rapid baseline reproductive number and to administer more effective PrEP.

Local stability of the disease free equilibrium point

To eradicate the infection, measures are needed to provide more PrEP education to individuals with the rapid reproductive rate and who administer a more effective PrEP drug. 3.46). The Routh-Hurwitz criterion for stability [17] is satisfied if R0 < 1, therefore all eigenvalues of J(Epo) have a negative real part if R0 < 1. The disease-free equilibrium point Epo of the system of equations (3.2) is locally asymptotically stable if R0 < 1 and unstable if R0 >1.

This is a necessary condition for stability, but one must be careful when using this result for biological interpretation.

Global stability of the disease free equilibrium point

This implies that, independently of the values of the initial conditions, all solutions of the system (3.53) converge to the equilibrium point Xo. The DFE Epo for the system of equations (3.2) is globally asymptotically stable when R0 <1 and unstable R0 >1. The coordinates of the endemic equilibrium point Ep∗ = (S∗, Sp∗, I∗, Ip∗, A∗) for the system of equations (3.2) are obtained using the method of Lunguet al.

Substituting the expressions for I∗, Ip∗ and A∗ into. 3.70) Then λ∗1 = 0 corresponds to the disease-free equilibrium point and one of the nonzero (positive) solutions of equation (3.69), which are obtained through the solutions of the following quadratic equation obtained from equation (3.69). To analyze the local stability of the endemic point of the model (3.2), we use the multiple center theory [14]. The central manifold theory states that the stability of a steady state under the initial system is determined by its stability under the constraint of the system on the central manifold [20].

This means that the majority of individuals are either infected or have developed AIDS. Caution must be exercised when using this strategy, as much of the contribution to infection still comes from the non-PrEP-infected individuals. However, due care must be taken to ensure strict adherence to PrEP drug use to maintain the effectiveness of the strategy.

In this case, the rapid reproduction number switches from the non-PrEP group to the PrEP group. To investigate how PrEP use may affect the progression of HIV infection in the community, we developed a model of HIV/AIDS and performed an analysis of different steady states of the model. Boundary conditions for the stability of steady states were determined using the basic reproduction number.

The basic reproduction number for the PrEP model depends on the rate at which individuals use PrEP (PrEP awareness) and the rate at which PrEP protects individuals (PrEP efficacy). PrEP intervention with low PrEP use and low PrEP efficacy reduces the rate of progression but remains an ineffective strategy for eradicating the infection. Mbewu, South Africa's experience with the closure of the cellulose sulfate microbicide trial, Journal of PLoS Med.