Quadratic form of nonlinear differential equations

Question

I've got nonlinear differential equations like this: \begin{equation} \vec{y}={\begin{pmatrix} S\\E\\I\\R \end{pmatrix}},\vec{f}(x,\vec{y})=\frac{\partial \vec{y}}{\partial t}={\begin{pmatrix} -10^{-6}SI\\10^{-6}SI-0.4E\\0.4E-0.5I\\0.5I \end{pmatrix}} \end{equation} In class my TA tell us we can rewrite it in quadratic from: $Y_{n \times 1}^{\prime}=f\left(x, Y_{n \times 1}\right)=B_{n \times 1} Y_{n \times 1}^{T} A_{n \times n} Y_{n \times 1}+C_{n \times n} Y_{n \times 1}$,so we can get it derivative for further calculating,But I don't have an idea how to get Matrix B and A,is there any limit or I can just write a special form which I guess out?

Answer 1

\begin{align} \frac{d}{dt}\begin{bmatrix} S\\ E\\ I\\ R \end{bmatrix} \, = \, \begin{bmatrix} -10^{-6}\\ \,\,\,\,\,10^{-6}\\ 0\\ 0 \end{bmatrix} \Big[ S \,\, E \,\, I \,\, R \Big]\begin{bmatrix} 0 & 0 & \frac{1}{2} & 0\\ 0 & 0 & 0 & 0\\ \frac{1}{2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} S\\ E\\ I\\ R \end{bmatrix} \,+\, \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & -0.4 & 0 & 0\\ 0 & \,\,\,\,0.4 & -0.5 & 0\\ 0 & 0 & \,\,\,\,0.5 & 0 \end{bmatrix} \begin{bmatrix} S\\ E\\ I\\ R \end{bmatrix} \end{align}