*3.1. Well-Posedness of the Model*

The problem (4) and (5) is a coupled system of nonlinear differential and difference equations with discrete delay. We directly derive the following proposition about the well-posedness of the model.

**Proposition 1.** *For any nonnegative initial condition* (*S*0, *I*0, *ϕ*)*, where ϕ is a nonnegative continuous function on* [−*τ*, 0]*, there exists a unique nonnegative solution to Problem* (4) *and* (5) *defined on* [0, +∞)*. Moreover, this solution is uniformly bounded on* [0, +∞)*.*

**Proof.** We first solve the system (4) and (5) on the interval [0, *τ*]. In this case, since *u* is completely defined, the Cauchy–Lipschitz theorem gives the existence and uniqueness of the solution on [0, *τ*], and this solution is nonnegative. Then, we repeat this method on each interval of type [*kτ*,(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>)*τ*] with *<sup>k</sup>* <sup>∈</sup> <sup>N</sup>. Thus, we obtain a unique nonnegative solution on the interval [0, +∞). Furthermore, using (6) and the fact that *S*(*t*) + *I*(*t*) + *u*(*t*) ≤ *S*(*t*) + *I*(*t*) + *P*(*t*) = *N*(*t*), we deduce that (*S*, *I*, *u*) is uniformly bounded on [0, +∞).
