*4.1. Global Asymptotic Stability of the Disease-Free Steady-State*

In this subsection, we assumed that R<sup>0</sup> < 1, and we show that the disease-free equilibrium (*S*0, *I*0, *u*0) is globally asymptotically stable. The proof is based on the use of the following auxiliary differential–difference system, for *t* > 0:

$$\begin{cases} \frac{dS^{+}}{dt}(t) &= \sigma - (\mu + K)S^{+}(t) + (1 - \theta)e^{-\mu \tau}u^{+}(t - \tau), \\\ u^{+}(t) &= -K S^{+}(t) + \theta e^{-\mu \tau}u^{+}(t - \tau), \end{cases} \tag{19}$$

with the same initial condition as the main system:

$$S^+(0) = S\_0 \quad \text{and} \quad u^+(t) = \wp(t), \ t \in [-\pi, 0].$$

By using a comparison principle, we obtain that *<sup>S</sup>*(*t*) <sup>≤</sup> *<sup>S</sup>*+(*t*) and *<sup>u</sup>*(*t*) <sup>≤</sup> *<sup>u</sup>*+(*t*) for all *<sup>t</sup>* <sup>&</sup>gt; 0. Furthermore, the system (19) has a unique equilibrium that corresponds to the disease-free equilibrium of the system (15). The basic reproduction number associated with (19) is also given by (18). The global asymptotic stability of the auxiliary system (19) was established in [11]. Indeed, by setting

$$\hat{S}(t) := S^+(t) - S^0, \quad \hat{u}(t) := u^+(t) - u^0$$

and considering the Lyapunov function *<sup>V</sup>* : <sup>R</sup> × C([−*τ*, 0], <sup>R</sup>) −→ <sup>R</sup>+,

$$V(S\_0, \varphi) = \frac{S\_0^2}{2} + \xi \int\_{-\tau}^0 \varphi^2(s) ds, \quad \text{with } \xi = \frac{1}{2K^2} \left[ \mu (1 - \theta^2 e^{-2\mu \tau}) + K \right] > 0,$$

we proved in [11] the following lemma.

**Lemma 2.** *Suppose that* R<sup>0</sup> < <sup>1</sup>*. Then, the unique steady-state* (*S*0, *<sup>u</sup>*0) *of the system* (19) *is globally asymptotically stable.*

Thanks to a comparison principle and the global attractivity of the set:

$$\begin{array}{rcl} \Omega\_{\mathfrak{t}} &:=& \{ (\mathcal{S}, I, \mathfrak{u}) \in \mathbb{R}^+ \times \mathbb{R}^+ \times \mathcal{C} ([-\mathfrak{r}, 0], \mathbb{R}^+): \\ & & 0 \le \mathcal{S} \le \mathcal{S}^0 + \mathfrak{e}, \ 0 \le \mathfrak{u}(s) \le \mathfrak{u}^0 + \mathfrak{e}, \ s \in [-\mathfrak{r}, 0] \}, \end{array}$$

for *ε* > 0 small enough, we are able to obtain the global asymptotic stability of the diseasefree equilibrium of (15). More precisely, the proof of the global asymptotic stability of the disease-free steady-state (16) is based on the following lemma.

**Lemma 3.** *Suppose that* R<sup>0</sup> < 1*. Then, for any sufficiently small ε* > 0*, the subset* Ω*<sup>ε</sup> of* <sup>R</sup><sup>+</sup> <sup>×</sup> <sup>R</sup><sup>+</sup> × C([−*τ*, 0], <sup>R</sup>+) *is a global attractor for the system* (15)*.*

**Proof.** The solutions of (15) satisfy, for all *t* > 0, the system:

$$\begin{cases} \begin{array}{rcl} S'(t) & \leq & \sigma - (\mu + K)S(t) + (1 - \theta)e^{-\mu \tau}u(t - \tau), \\ u(t) & = & KS(t) + \theta e^{-\mu \tau}u(t - \tau). \end{array} \end{cases}$$

By the comparison principle and the positivity of the solutions, we have 0 <sup>≤</sup> *<sup>S</sup>*(*t*) <sup>≤</sup> *<sup>S</sup>*+(*t*) and 0 <sup>≤</sup> *<sup>u</sup>*(*t*) <sup>≤</sup> *<sup>u</sup>*+(*t*), for all *<sup>t</sup>* <sup>&</sup>gt; 0, where (*S*+, *<sup>u</sup>*+) is the solution of the system (19). Lemma <sup>2</sup> implies that (*S*+(*t*), *<sup>u</sup>*+(*t*)) <sup>→</sup> (*S*0, *<sup>u</sup>*0) as *<sup>t</sup>* <sup>→</sup> <sup>+</sup>∞. This convergence means that the subset Ω*ε*, with *ε* > 0 small enough, is a global attractor for the system (15).

Lemma 3 allows us to restrict the analysis of the global stability of the disease-free steady-state (16) of (15) to the subset Ω*<sup>ε</sup>* with *ε* > 0 small enough.

**Theorem 1.** *Suppose that* R<sup>0</sup> < <sup>1</sup>*. Then, the disease-free steady-state* (*S*0, 0, *<sup>u</sup>*0) *given by* (16) *of the model* (15) *is globally asymptotically stable.*

**Proof.** We consider the solutions of System (15) in the subset Ω*ε*, with *ε* > 0 small enough. It is clear that the second equation of (15) implies that

$$I'(t) \le \beta(\mathcal{S}^0 + \varepsilon)I(t) - \mu I(t) = -\mu \left(1 - \frac{\beta(\mathcal{S}^0 + \varepsilon)}{\mu}\right)I(t).$$

Since <sup>R</sup><sup>0</sup> <sup>=</sup> *<sup>β</sup> μ <sup>S</sup>*<sup>0</sup> <sup>&</sup>lt; 1, we can choose *<sup>ε</sup>* <sup>&</sup>gt; 0 such that *<sup>β</sup>*(*S*<sup>0</sup> <sup>+</sup> *<sup>ε</sup>*) *μ* < 1. This implies that lim*t*→+<sup>∞</sup> *I*(*t*) = 0. Then, there exists *T<sup>ε</sup>* > 0 such that 0 ≤ *I*(*t*) ≤ *ε*, for all *t* > *Tε*. We then have, for *t* > *Tε*,

$$\begin{cases} \begin{array}{rcl} S'(t) & \geq & \sigma - (\mu + K)S(t) - \varepsilon \beta S(t) + (1 - \theta)e^{-\mu \tau}u(t - \tau), \\ u(t) & = & KS(t) + \theta e^{-\mu \tau}u(t - \tau). \end{array} \end{cases}$$

We again use the comparison principle to obtain *S*(*t*) ≥ *Sε*(*t*) and *u*(*t*) ≥ *uε*(*t*), for all *t* > *Tε*, where (*Sε*, *uε*) is the solution of the problem, for *t* > 0,

$$\begin{cases} S\_{\varepsilon}'(t) &= \sigma - (\mu + K)S\_{\varepsilon}(t) - \varepsilon \beta S\_{\varepsilon}(t) + (1 - \theta)e^{-\mu \tau} u\_{\varepsilon}(t - \tau), \\\ u\_{\varepsilon}(t) &= -K S\_{\varepsilon}(t) + \theta e^{-\mu \tau} u\_{\varepsilon}(t - \tau), \\\ S\_{\varepsilon}(0) &= \ S\_{0}, \ u\_{\varepsilon}(s) = \varrho(s), \quad \text{for } -\tau \le s \le 0. \end{cases} \tag{20}$$

We use the same techniques as for the proof of the lemmas 2 and 3 to show that(*Sε*(*t*), *uε*(*t*)) → (*S*<sup>0</sup> *<sup>ε</sup>* , *u*<sup>0</sup> *<sup>ε</sup>*) as *<sup>t</sup>* → +∞, with (*S*<sup>0</sup> *<sup>ε</sup>* , *u*<sup>0</sup> *<sup>ε</sup>*) the steady-state of System (20). In fact, (*S*<sup>0</sup> *<sup>ε</sup>* , *u*<sup>0</sup> *<sup>ε</sup>*) is given by the expression:

$$(S\_{\varepsilon}^{0}, \mu\_{\varepsilon}^{0}) := \left(\frac{\sigma(1 - \theta \varepsilon^{-\mu \tau})}{\mu\_{\varepsilon} + K - (\mu\_{\varepsilon}\theta + K)\varepsilon^{-\mu \tau}} \mid \frac{K\sigma}{\mu\_{\varepsilon} + K - (\mu\_{\varepsilon}\theta + K)\varepsilon^{-\mu \tau}}\right),$$

with *με* := *μ* + *εβ*. Then, there exists *T*˜ *<sup>ε</sup>* > *T<sup>ε</sup>* > 0 such that, for *t* > *T*˜ *ε*,

$$S^0\_\varepsilon - \varepsilon \le S(t) \le S^0 + \varepsilon \quad \text{and} \quad \mathfrak{u}^0\_\varepsilon - \varepsilon \le \mathfrak{u}(t) \le \mathfrak{u}^0 + \varepsilon.$$

As *<sup>ε</sup>* → 0, we have *<sup>S</sup>*<sup>0</sup> *<sup>ε</sup>* → *<sup>S</sup>*<sup>0</sup> and *<sup>u</sup>*<sup>0</sup> *<sup>ε</sup>* → *<sup>u</sup>*0. Then, we obtain

$$\lim\_{t \to +\infty} S(t) = S^0 \quad \text{and} \quad \lim\_{t \to +\infty} \mu(t) = \mu^0.$$

Recall that (*S*0, 0, *u*0) is locally asymptotically stable. Then, it is globally asymptotically stable.
