I am working with the following system of OD equations:
$\frac{dE(I_1)}{dt}=-\mu E(I_1)+\lambda E(I_2)-\lambda E(I_1I_2)=f_1(E(I_1),E(I_2))$ $\frac{dE(I_2)}{dt}=-\lambda E(I_1I_2)+\lambda E(I_1)-\mu E(I_2)=f_2(E(I_1),E(I_2))$
Here $I_1$ and $I_2$ are indicator random variables. Assume $\lambda,\mu>0$. I need to stability results wherein I face the Jacobian matrix:: $J=\left[ \begin{array} {}\frac{\partial f_1}{\partial E(I_1)} \frac{\partial f_1}{\partial E(I_2)}\\ \frac{\partial f_2}{\partial E(I_1)} \frac{\partial f_2}{\partial E(I_2)} \end{array}\right]$
I tried evaluating this Jacobian by writing $E(I_1I_2)=\rho \sigma_1\sigma_2+E(I_1)E(I_2)$ and then considering $\rho \sigma_1\sigma_2$ as a constant. Thus I evaluated $J$ as:
$J=\left[ \begin{array} {} -\mu-\lambda E(I_2) & \lambda-\lambda E(I_1)\\ \lambda-\lambda E(I_2) & -\mu-\lambda E(I_1) \end{array}\right]$
Given $I_1$ and $I_2$ are any two random variables that are not necessarily independent, is my evaluation of Jacobian sound? Are there any mathematical niceties that I have ignored in my calculation?