Analysis of the Mathematical Model for the Spread of Pine Wilt Disease

Figure 2

(a) Time series diagram of the system (4), where the blue and green continuous curves express the number of pines which is normal and susceptible 𝑆(𝑡) and pine which is infected Bursaphelenchus xylophilus 𝐼(𝑡), respectively. And the red continuous curves expresses the number of Monochamus alternatus which carries Bursaphelenchus xylophilus Nickle. The pine wilt disease will not break out, where 𝑏=0.25,𝜇=0.05,𝛽=0.006,𝑎=0.003,𝛼=0.002,𝛿=0.01 and 𝑘=0.05. In this case, ℛ₀=0.7746 and ℛ_𝑐=1.7321. (b) Time series diagram of the system (4), where the blue and green continuous curve express the number of pines which is normal and susceptible 𝑆(𝑡) and pine which is infected Bursaphelenchus xylophilus 𝐼(𝑡), respectively. And the red continuous curve express the number of Monochamus alternatus carry Bursaphelenchus xylophilus Nickle. the endemic equilibrium 𝐸^∗(𝑆^∗,𝐼^∗,𝑌^∗) is globally asymptotically stable, where 𝑏=0.25,𝜇=0.0005,𝛽=0.006,𝑎=0.003,𝛼=0.002,𝛿=0.008 and 𝑘=0.05. In this case, ℛ₀=8.6603 and ℛ_𝑐=17.3205. (c) Time series diagram of the system (4), where the blue and green continuous curves express the number of pines which is normal and susceptible 𝑆(𝑡) and pine which is infected Bursaphelenchus xylophilus 𝐼(𝑡), respectively. And the red continuous curve expresses the number of Monochamus alternatus which carries Bursaphelenchus xylophilus Nickle. The disease-free equilibrium 𝐸₀(𝑆₀,0,0) of the system (4) is globally asymptotically stable, where 𝑏=0.25,𝜇=0.5,𝛽=0.006,𝑎=0.003,𝛼=0.002,𝛿=0.008 and 𝑘=0.05. In this case, ℛ₀=0.2739 and ℛ_𝑐=0.5477.

(a)

(b)

(c)