Using a min function within scipy-integrate.ODESolver

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I had a question regarding using a function such as min() when defining a function to be solved using odeint from scipy.integrate. This term solves for the minmial between x (the variable I am solving for) and some other fixed parameter s. Because the output of min (x , s) may change as time goes on, I was wondering if odeint accounts for this - as the solution that I receieve differs from the trajectory that I expect would solve this system. I'm not sure if this is the problem, but it is the only thing I can think of. Any insight or help is appreciated, thank you!

The ODE solution is meant to be an average of stochastic sample paths - I have verified that the sample paths converge to what I should expect analytically, but the solution to the ODE using this solver does not seem to agree with this answer

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