Table 3 Stability analysis of equilibria based on Jacobian matrix.

From: Evolutionary game analysis of the longitudinal integration of electronic health record based on prospect theory

Equilibrium point

Eigenvalues of Jacobian matrix

\(\:{\lambda\:}_{1}\)

\(\:{\lambda\:}_{2}\)

\(\:{\lambda\:}_{3}\)

real part sign

\(\:{E}_{1}\left(\text{0,0},0\right)\)

\(\:V\left(-{C}_{p}\right)\)

\(\:V\left(-{C}_{h}\right)+V\left({M}_{h}\right)\)

\(\:{P}_{g}+V\left({M}_{g}\right)+{P}_{h}+V\left(-{C}_{g}\right)\)

\(\:(-,U,U)\)

\(\:{E}_{2}\left(\text{1,0},0\right)\)

\(\:V\left({M}_{p}\right)+V\left(-{C}_{p}\right)\)

\(\:-V\left({M}_{h}\right)-V\left(-{C}_{h}\right)\)

\(\:V\left({M}_{g}\right)-{R}_{h}+V\left(-{C}_{g}\right)\)

\(\:(U,U,U)\)

\(\:{E}_{3}\left(\text{0,1},0\right)\)

\(\:-V\left(-{C}_{p}\right)\)

\(\:{P}_{h}+V\left({M}_{h}\right)+V\left(-{C}_{h}\right)\)

\(\:{P}_{g}-{R}_{p}+V\left({M}_{g}\right)+V\left(-{C}_{g}\right)\)

\(\:\left(+,U,U\right)\)

\(\:{E}_{4}\left(\text{0,0},1\right)\)

\(\:{R}_{p}+V\left(-{C}_{p}\right)\)

\(\:-{P}_{g}-{P}_{h}-V\left({M}_{g}\right)-V\left(-{C}_{g}\right)\)

\(\:{P}_{h}+{R}_{h}+V\left(-{C}_{h}\right)+V\left({M}_{h}\right)\)

\(\:(U,U,U)\)

\(\:{E}_{5}\left(\text{1,1},0\right)\)

\(\:-V\left({M}_{p}\right)-V\left(-{C}_{p}\right)\)

\(\:-{P}_{h}-V\left(-{C}_{h}\right)-V\left({M}_{h}\right)\)

\(\:V\left({M}_{g}\right)-{R}_{h}+V\left(-{C}_{g}\right)\)

\(\:(U,U,U)\)

\(\:{E}_{6}\left(\text{1,0},1\right)\)

\(\:V\left(-{C}_{p}\right)+V\left({M}_{p}\right)\)

\(\:{R}_{h}-V\left({M}_{g}\right)-V\left(-{C}_{g}\right)\)

\(\:-{P}_{h}-{R}_{h}-V\left(-{C}_{h}\right)-V\left({M}_{h}\right)\)

\(\:(U,U,U)\)

\(\:{E}_{7}\left(\text{0,1},1\right)\)

\(\:-{R}_{p}-V\left(-{C}_{p}\right)\)

\(\:{P}_{h}+{R}_{h}+V\left({M}_{h}\right)+V\left(-{C}_{h}\right)\)

\(\:{R}_{p}-{P}_{g}-V\left({M}_{g}\right)-V\left(-{C}_{g}\right)\)

\(\:(U,U,U)\)

\(\:{E}_{8}\left(\text{1,1},1\right)\)

\(\:-V\left(-{C}_{p}\right)-V\left({M}_{p}\right)\)

\(\:{R}_{h}-V\left({M}_{g}\right)-V\left(-{C}_{g}\right)\)

\(\:-{P}_{h}-{R}_{h}-V\left(-{C}_{h}\right)-V\left({M}_{h}\right)\)

\(\:(U,U,U)\)

\(\:{E}_{9}\left({x}_{1},{y}_{1},{z}_{1}\right)\)

\(\:{\lambda\:}_{1}=-{\lambda\:}_{2}=\)

\(\frac{-{R}_{h}*\sqrt{{\text{z}}_{1}*{\text{x}}_{1}*\left(1-{\text{z}}_{1}\right)\left(V\left({M}_{g}\right)-{R}_{h}+V\left(-{C}_{g}\right)\right)*{\left({P}_{g}+{R}_{h}-{R}_{p}\right)}^{2}}}{{P}_{g}*{R}_{h}-{R}_{h}*{R}_{p}+{R}_{h}^{2}}\)

\(\:-N/({P}_{g}*{R}_{h}-{R}_{h}*{R}_{p}+{R}_{h}^{2})\)

\(\:(+,-,+)\)

  1. \(\:{x}_{1},{y}_{1},{z}_{1}\:\)are the coordinates of the corresponding equilibrium points; \(\:\:{E}_{9}\) is meaningful only when \(\:V\left({M}_{g}\right)+V\left(-{C}_{g}\right)>{R}_{h},{P}_{g}+{R}_{h}\ne\:{R}_{p}\:\); N is a complex expression consisting of parameter terms representing the net effect term of government intervention on the dynamics of patient strategies, \(\:N>0\); U denotes sign uncertainty.
Back to article page