Dynamical Behavior of a New Epidemiological Model

Figure 4

Taking 𝑏=6, 𝛼=1, 𝑑=0.4, 𝑘=3, 𝑐=1, 𝑢₁=0.8, and 𝑢₂=0.5, then 𝑅₀=20.4545>1, 𝑝₁=−6.9925<0, 𝑞₁=−8.2363<0, and 𝑟₁=−0.4091<0. And set the initial condition (𝑆(𝜃),𝐼(𝜃),𝐷(𝜃))=(1,2,2), 𝜃∈[−𝜏,0].

(a) When 𝜏=4, then the solution (𝑆(𝑡),𝐼(𝑡),𝐷(𝑡)) converges to the endemic equilibrium 𝐸_∗=(4.4138,5.0136,5.726) as 𝑡→∞

(b) When 𝜏=4.8924, then a periodic orbit appears in the small neighborhood of the endemic equilibrium 𝐸_∗=(4.4138,5.0136,5.726)