**4. Global Stability Analysis**

We consider in this section the case:

$$f(\mathbb{S}) = \mathbb{S}.$$

The model becomes

$$\begin{cases} S'(t) &= \sigma - \beta I(t)S(t) - \mu S(t) - KS(t) + (1 - \theta)e^{-\mu \tau}u(t - \tau), \\\ I'(t) &= \beta I(t)S(t) - \mu I(t), \\\ u(t) &= \quad KS(t) + \theta e^{-\mu \tau}u(t - \tau), \end{cases} \tag{15}$$

with the initial condition:

$$S(0) = S\_0, \quad I(0) = I\_0 \quad \text{and} \quad \mu(t) = \varphi(t), \ t \in [-\tau, 0].$$

The function Φ given by (8) becomes

$$\Phi(S^\*) = \left[1 + \frac{K(1 - e^{-\mu\tau})}{\mu(1 - \theta e^{-\mu\tau})}\right] S^\*.$$

The corresponding disease-free equilibrium is

$$\mathcal{S}\left(S^{0}, I^{0}, \mu^{0}\right) := \left(\frac{\sigma\left(1 - \theta e^{-\mu\tau}\right)}{\mu + K - (\mu\theta + K)e^{-\mu\tau}}, \ 0, \ \frac{K\sigma}{\mu + K - (\mu\theta + K)e^{-\mu\tau}}\right),\tag{16}$$

and the endemic steady-state is

$$(S^\*, I^\*, \mu^\*) = \left(\frac{\mu}{\beta}, \frac{\sigma}{\mu} - \Phi\left(\frac{\mu}{\beta}\right), \frac{K\mu}{\beta(1 - \theta e^{-\mu\tau})}\right). \tag{17}$$

The basic reproduction number becomes

$$\mathcal{R}\_0 := \frac{\beta}{\mu} \Phi^{-1} \left( \frac{\sigma}{\mu} \right) = \frac{\beta}{\mu} S^0 = \frac{\beta \sigma (1 - \theta e^{-\mu \tau})}{\mu^2 (1 - \theta e^{-\mu \tau}) + \mu K (1 - e^{-\mu \tau})}. \tag{18}$$

A model similar to (15) was studied in [11] as a generalization of a Kermack–McKendrick SIR model with an age-structured protection phase. We established the global asymptotic stability of the two steady-states. We showed that if R<sup>0</sup> < 1, then the disease-free steadystate is globally asymptotically stable, and if R<sup>0</sup> > 1, then the endemic steady-state is globally asymptotically stable. We constructed quadratic and logarithmic Lyapunov functions to establish this global stability. In the following subsections, we also present results on the global asymptotic stability of the two steady-states. We used similar techniques as those used for the proofs of the global stability results in [11].
