Theory and Modern Applications
Table 2 Global dynamics of system (1)–(2) summarized from Sects. 3, 4, and 5
From: Rich dynamics of a Filippov avian-only influenza model with a nonsmooth separation line
Case | Condition | Equilibrium | Main result |
|---|---|---|---|
\(S_{1}^{*}< S_{2}^{*}<\frac{I_{T}}{\xi }\) | \(I_{T}< I_{2}^{*}< I_{1}^{*}\) | \(E_{1}^{V}, E_{2}^{R}\) | (II) |
\(S_{1}^{*}< S_{2}^{*}<\frac{I_{T}}{\xi }\) | \(I_{2}^{*}< I_{T}< I_{1}^{*}\) | \(E_{1}^{V}, E_{2}^{V}, E_{p1}\) | (III) |
\(S_{1}^{*}< S_{2}^{*}<\frac{I_{T}}{\xi }\) | \(I_{2}^{*}< I_{1}^{*}< I_{T}\) | \(E_{1}^{R}, E_{2}^{V}\) | (I) |
\(S_{1}^{*}<\frac{I_{T}}{\xi }<S_{2}^{*}\) | \(I_{T}< I_{2}^{*}< I_{1}^{*}, \xi <\frac{I_{2}^{*}}{S_{2}^{*}}\) | \(E_{1}^{V}, E_{2}^{R}\) | (II) |
\(S_{1}^{*}<\frac{I_{T}}{\xi }<S_{2}^{*}\) | \(I_{T}< I_{2}^{*}< I_{1}^{*}, \xi >\frac{I_{2}^{*}}{S_{2}^{*}}\) | \(E_{1}^{V}, E_{2}^{V}, E_{p2}\) | (III) |
\(S_{1}^{*}<\frac{I_{T}}{\xi }<S_{2}^{*}\) | \(I_{2}^{*}< I_{T}< I_{1}^{*}, \xi < H_{1}\) | \(E_{1}^{V}, E_{2}^{V}, E_{p1}\) | (III) |
\(S_{1}^{*}<\frac{I_{T}}{\xi }<S_{2}^{*}\) | \(I_{2}^{*}< I_{T}< I_{1}^{*}, \xi >H_{1}\) | \(E_{1}^{V}, E_{2}^{V}, E_{p2}\) | (III) |
\(S_{1}^{*}<\frac{I_{T}}{\xi }<S_{2}^{*}\) | \(I_{2}^{*}< I_{1}^{*}< I_{T}\) | \(E_{1}^{R}, E_{2}^{V}\) | (I) |
\(\frac{I_{T}}{\xi }< S_{1}^{*}< S_{2}^{*}\) | \(I_{T}< I_{2}^{*}< I_{1}^{*}, \xi <\frac{I_{2}^{*}}{S_{2}^{*}}\) | \(E_{1}^{V}, E_{2}^{R}\) | (II) |
\(\frac{I_{T}}{\xi }< S_{1}^{*}< S_{2}^{*}\) | \(I_{T}< I_{2}^{*}< I_{1}^{*}, \frac{I_{2}^{*}}{S_{2}^{*}}<\xi <\frac{I_{1}^{*}}{S_{1}^{*}}\) | \(E_{1}^{V}, E_{2}^{V},E_{p2}\) | (III) |
\(\frac{I_{T}}{\xi }< S_{1}^{*}< S_{2}^{*}\) | \(I_{T}< I_{2}^{*}< I_{1}^{*}, \xi >\frac{I_{1}^{*}}{S_{1}^{*}}\) | \(E_{1}^{R}, E_{2}^{V}\) | (I) |
\(\frac{I_{T}}{\xi }< S_{1}^{*}< S_{2}^{*}\) | \(I_{2}^{*}< I_{T}< I_{1}^{*}, \xi <\frac{I_{1}^{*}}{S_{1}^{*}}\) | \(E_{1}^{V}, E_{2}^{V}, E_{p2}\) | (III) |
\(\frac{I_{T}}{\xi }< S_{1}^{*}< S_{2}^{*}\) | \(I_{2}^{*}< I_{T}< I_{1}^{*}, \xi >\frac{I_{1}^{*}}{S_{1}^{*}}\) | \(E_{1}^{R}, E_{2}^{V}\) | (I) |
\(\frac{I_{T}}{\xi }< S_{1}^{*}< S_{2}^{*}\) | \(I_{2}^{*}< I_{1}^{*}< I_{T}\) | \(E_{1}^{R}, E_{2}^{V}\) | (I) |