Table 3 Equilibrium point stability analysis.

From: Allocation strategy of medical supplies during a public health emergency: a tripartite evolutionary game perspective

Equilibrium point

Eigenvalues of Jacobian matrix

The symbol of real part

Stability

Condition

λ1

λ2

λ3

E1 (0, 0, 0)

\({\alpha V}_{2}-{\alpha V}_{1}-A\)

\({\alpha H}_{2}-{\alpha W}_{2}\)

\({H}_{5}-{H}_{4}+{K}_{1}+{K}_{2} +{R}_{2}+T+\alpha {P}_{2}\)

(× , −, +)

Unstable point

–

E2 (1, 0, 0)

\({\alpha H}_{2}-{\alpha W}_{1}\)

\(A+{\alpha V}_{1}-{\alpha V}_{2}\)

\({H}_{5}-{H}_{4}+{K}_{1}+{R}_{2}+T+\alpha {P}_{1}\)

(−, × , +)

Unstable point

–

E3 (0, 1, 0)

\({\alpha W}_{2}-{\alpha H}_{2}\)

\(-A\)

\({H}_{5}-{H}_{4}-B+{K}_{2}+{R}_{2}\)

(−, −, −)

ESS

(1)

E4 (0, 0, 1)

\({K}_{2}-A-{\alpha V}_{1}+{\alpha V}_{2}\)

\(B+\alpha {H}_{2}+\alpha {P}_{2}-{\alpha W}_{2}\)

\({H}_{4}{-H}_{5}-{K}_{1}-{K}_{2}\) \(-{R}_{2}-T-\alpha {P}_{2}\)

(+ , + , −)

Unstable point

–

E5 (1, 1, 0)

\(A\)

\({\alpha W}_{1}-{\alpha H}_{2}\)

\({H}_{5}-{H}_{4}+{R}_{2}\)

(+ , + , ×)

Unstable point

–

E6 (1, 0, 1)

\({{\alpha H}_{2}+\alpha {P}_{1}-\alpha W}_{1}\)

\(A-{K}_{2}+{\alpha V}_{1}-{\alpha V}_{2}\)

\({H}_{4}{-H}_{5}-{K}_{1}-{R}_{2}-T-\alpha {P}_{1}\)

(+ , −, −)

Unstable point

–

E7 (0, 1, 1)

\({K}_{2}-A\)

\({\alpha W}_{2}-{\alpha H}_{2}-\alpha {P}_{2}-B\)

\({B+{H}_{4}-H}_{5}-{K}_{2}-{R}_{2}\)

(+ , −, ×)

Unstable point

–

E8 (1, 1, 1)

\(A-{K}_{2}\)

\({{H}_{4}-H}_{5}-{R}_{2}\)

\({\alpha W}_{1}-\alpha {P}_{1}-{\alpha H}_{2}\)

(−, −, −)

ESS

(2)

  1. “ + ” means the symbol of the real part is positive, “−” means the symbol of the real part is negative, and “×” means the symbol of the real part is uncertain.
  2. (1) \({\alpha W}_{2}-{\alpha H}_{2}<0\),\({H}_{5}-{H}_{4}-B+{K}_{2}+{R}_{2}<0\); (2) \(A-{K}_{2}<0\), \({{H}_{4}-H}_{5}-{R}_{2}<0\), \({\alpha W}_{1}-\alpha {P}_{1}-{\alpha H}_{2}<0\)
Back to article page