Theorem 4.3
  1. If \(R_{0}\) ≤ 1, then the infection free-equilibrium\({P_{u}}^{*}=(\frac{\lambda}{d},0,0)\) is globally asymptotically stable for (1) in the positive octant \(\mathbb{R}_{+}^{3}\).
  2. If \(R_{0}>1\), then, the endemic equilibrium point\({P_{e}}^{*}=(\frac{\lambda}{d}\frac{1}{R_{0}},\ \frac{dc}{\text{kσ}}\left(R_{0}-1\right),\ \frac{d}{k}\left(R_{0}-1\right)\)) is globally asymptotically stable for the system (1) in the positive octant \(\mathbb{R}_{+}^{3}\)
Where \(R_{0}\) is the basic reproduction ratio, given by (5)