Solving system of delay differential equations

Question

Are there any numerical methods for solving systems of delay differential equations with time-dependent delays? For example, I have a system:

$$\frac{dP_1}{dt} = f_1(t) P_2(t-\tau(t)) P_3(t)$$

$$\frac{dP_2}{dt} = f_2(t) P_3(t-\tau(t)) P_1(t)$$

$$\frac{dP_3}{dt} = f_3(t) P_1(t-\tau(t)) P_2(t)$$

I thought about applying method of steps together with Runge-Kutta method, but it leads to loss of information, because Runge-Kutta method requires values of RHS at $(x+1/2 h)$, where $h$ is a step.

So, I'm interested if any other methods except this one exist for solving such systems. And, if they exist, could you please tell me where I can read about them, because I didn't find useful information.

Answer 1

Have a look at the ddeint package.

Some further explanation can be found here.

It is not very fast, but very flexible, and coded in just a few lines on top of Scipy’s differential equations solver, odeint.