Figure 3.

(a)–(d) show that system (7) has a disease-free equilibrium E ₀(50,0, 0), which is unstable, and an endemic equilibrium E _∗(13.6954573045,1.8542772633,1.9865335214), which is locally asymptotically stable. In this case, A = 10, β = 0.3, ε = 1.2, d = 0.05, μ = 0.2, υ = 0.4, r = 0.1, α = 0.8, and k = 2.