# A Jacobian involving probabilities

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?

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Your strategy is unsound from its inception since, except in some degenerate cases such as $\lambda=0$ or $(I_1,I_2)$ independent, the derivative of $E(I_k)$ is not a function of $(E(I_1),E(I_2))$. Hence the functions $f_k$ simply do not exist.
Without informations on the dependence structure of $(I_1,I_2)$, one cannot go much further. A remark, though, is that $U=E(I_1)-E(I_2)$ solves $U'=-(\lambda+\mu)U$ hence, for every $t\geqslant0$, $E(I_1)(t)-E(I_2)(t)=\mathrm e^{-(\lambda+\mu)t}(E(I_1)(0)-E(I_2)(0))$, and in particular, $E(I_1)(t)-E(I_2)(t)\to0$. This is only natural since every fixed point of the differential system is on the line $E(I_1)=E(I_2)$. Thus, the stability of the system depends on the behaviour of $E(I_1I_2)$ when $E(I_1)=E(I_2)$ or when $E(I_1)\approx E(I_2)$.
Thanks. I have a question. What is wrong in writing: $f_1(E(I_1),E(I_2))=-\mu E(I_1)+\lambda E(I_2)-\lambda (\rho \sigma_1\sigma_2+E(I_1)E(I_2))=-\mu E(I_1)+\lambda E(I_2)-\lambda(\rho \sqrt{E(I_1)(1-E(I_1))E(I_2)(1-E(I_2))}+E(I_1)E(I_2))$ – Bravo Nov 18 '12 at 17:12
Don't the functions have arguments only in $(E(I_1),E(I_2))$ then? Is there any fundamental violation in doing so? – Bravo Nov 18 '12 at 17:14
Only that $\rho$ is not a function of $(E(I_1),E(I_2))$. Unless you assume otherwise--and then we are back to the last sentence of my post. – Did Nov 18 '12 at 17:49