*4.2. Global Asymptotic Stability of the Endemic Equilibrium*

In this subsection, we assumed that

R<sup>0</sup> > 1,

where R<sup>0</sup> is given by (18), and we studied the global asymptotic stability of the endemic steady-state (17). Let us define

$$
\bar{S}(t) := S(t) - S^\* \quad \text{and} \quad \bar{u}(t) := u(t) - u^\*.
$$

Then, the system (15) becomes, with *βS*∗ = *μ*,

$$\begin{cases} \begin{array}{rcl} \mathcal{S}'(t) &=& -(\mu + K)\mathcal{S}(t) - \beta \mathcal{S}(t)I(t) - \beta \mathcal{S}^\* I(t) + \beta \mathcal{S}^\* I^\* + (1 - \theta)e^{-\mu \tau} \tilde{u}(t - \tau), \\\ I'(t) &=& \beta \mathcal{S}(t)I(t), \\\ \tilde{u}(t) &=& K \mathcal{S}(t) + \theta e^{-\mu \tau} \tilde{u}(t - \tau). \end{array} \end{cases} \tag{21}$$

The endemic steady-state (17) of the system (15) becomes (0, *I*∗, 0) (as a steady-state of the system (21)). We then obtain the following result.

**Theorem 2.** *Let us suppose that* R<sup>0</sup> > 1*. Then, the endemic steady-state* (*S*∗, *I*∗, *u*∗) *given by* (17) *of the model* (15) *is globally asymptotically stable.*

**Proof.** The proof of this theorem is based on the use of the following Lyapunov function *<sup>V</sup>* : <sup>R</sup> <sup>×</sup> <sup>R</sup><sup>+</sup> × C([−*τ*, 0], <sup>R</sup>) <sup>→</sup> <sup>R</sup><sup>+</sup> defined by

$$V(S\_0, I\_0, \varphi) = \frac{S\_0^2}{2} + \xi \int\_{-\pi}^0 \rho^2(s)ds + S^\* \left[I\_0 - I^\* - I^\* \ln\left(\frac{I\_0}{I^\*}\right)\right],$$

with *<sup>ξ</sup>* <sup>=</sup> <sup>1</sup> 2*K*<sup>2</sup> *<sup>K</sup>* <sup>+</sup> *<sup>μ</sup>*(<sup>1</sup> <sup>−</sup> (*θ*2*e*−2*μτ*) . First, we note that the function *G* : (0 + ∞) → [0 + ∞) defined by

$$G(I\_0) = I\_0 - I^\* - I^\* \ln\left(\frac{I\_0}{I^\*}\right), \quad I\_0 > 0,$$

satisfies *G*(*I*0) ≥ 0 for all *I*<sup>0</sup> > 0 and *G*(*I*0) = 0 if and only if *I*<sup>0</sup> = *I*∗. Then, we have *V*(*S*0, *I*0, *u*0) = 0 if and only if (*S*0, *I*0, *u*0)=(0, *I*∗, 0). We set

$$\mathcal{V}(t) := V(\bar{\mathcal{S}}(t), I(t), \mathfrak{u}\_t), \quad t \ge 0,$$

where (*S*˜(*t*), *I*(*t*), *u*˜*t*) is the solution of (21). Then, we obtain

$$\begin{array}{rcl} \bar{V}'(t) &=& -a\bar{S}^2(t) + b\bar{S}(t)\bar{u}(t-\tau) - c\bar{u}^2(t-\tau) - \beta I(t)\bar{S}^2(t),\\ &\leq& \frac{b^2 - 4ac}{4c}\bar{S}^2(t),\\ &=& -\kappa \bar{S}^2(t), \end{array} \tag{22}$$

with

$$\begin{cases} a &= \mu + K - \xi K^2, \\ b &= (1 - \theta)e^{-\mu \tau} + 2\xi K \theta e^{-\mu \tau} > 0, \\ c &= \xi (1 - \theta e^{-\mu \tau})(1 + \theta e^{-\mu \tau}) > 0, \\ \kappa &= (4ac - b^2)/4c > 0. \end{cases}$$

We conclude that the function *<sup>t</sup>* → *<sup>V</sup>*(*S*˜(*t*), *<sup>I</sup>*(*t*), *<sup>u</sup>*˜*t*) is nonincreasing. Then, we obtain

$$\lim\_{t \to +\infty} V(\mathcal{S}(t), I(t), \vec{u}\_t) = \inf\_{s \ge 0} V(\mathcal{S}(s), I(s), \vec{u}\_s) =: V^\* \ge 0.$$

By integrating (22) from 0 to *t* > 0, we obtain

$$\propto \int\_0^t \bar{S}^2(s)ds \le V(\bar{S}(0), I(0), \mathfrak{a}\_0) - V(\bar{S}(t), I(t), \mathfrak{a}\_t).$$

It is clear that the limits on both sides exist when *t* → +∞, and we have

$$\lim\_{t \to +\infty} \int\_0^t \bar{S}^2(s) ds \le \frac{1}{\kappa} \left[ V(\bar{S}(0), I(0), \bar{u}\_0) - V^\* \right]. \tag{23}$$

Furthermore, the first equation of System (21) implies that *<sup>t</sup>* → *<sup>S</sup>*˜ (*t*) is uniformly bounded. Then, the function *<sup>t</sup>* → *<sup>S</sup>*˜(*t*) is uniformly continuous. We conclude from (23) that lim*t*→+<sup>∞</sup> *<sup>S</sup>*˜(*t*) = 0, and the fluctuations lemma implies that there exists a sequence (*tk*)*k*∈N, *tk* → +<sup>∞</sup> such that lim*k*→+<sup>∞</sup> *<sup>S</sup>*˜ (*tk*) = 0. On the other hand, using the difference equation satisfied

by *u*˜, we also have lim*t*→+<sup>∞</sup> *u*˜(*t*) = 0. Then, the first equation of System (21) implies that lim*k*→+<sup>∞</sup> *<sup>I</sup>*(*tk*) = *<sup>I</sup>*∗. We also have from the expression of *<sup>V</sup>* that

$$\lim\_{t \to +\infty} G(I(t)) = \frac{V^\*}{S^\*}.$$

Furthermore, the continuity of the function *<sup>G</sup>* implies that lim*k*→+<sup>∞</sup> *<sup>G</sup>*(*I*(*tk*)) = *<sup>G</sup>*(*I*∗) = 0. Then, *V*<sup>∗</sup> = 0. This means that lim*t*→+<sup>∞</sup> *G*(*I*(*t*)) = 0. The properties of the function *G* imply that lim*t*→+<sup>∞</sup> *I*(*t*) = *I*∗. We conclude that the endemic steady-state (*S*∗, *I*∗, *u*∗) is globally asymptotically stable.

We assumed, for the first time, that the rate of the start of PrEP treatment was constant and that *f*(*S*) = *S*. Thus, the model admits two equilibrium points, the disease-free equilibrium and the endemic equilibrium. By considering R<sup>0</sup> as a function of the delay, we showed that, if R<sup>0</sup> < 1, the disease-free equilibrium is globally asymptotically stable and that, if R<sup>0</sup> > 1, then it is the endemic equilibrium that becomes globally asymptotically stable.
