Time dependent Classification problem
From the description of the problem, you can just choose whatever classification algorithm you like to classify the need of intervention at  $t_{0}$.

The general rule for choosing a classifier for this is to start simple--e.g., by using nearest neighbor and iterate to more powerful classifiers until you get enough accuracy.

If additional measurements at $t_{0}$ can be made a classifier can be used to determine the necessity of intervention without this additional data, then based on the probability output of this classifier and a threshold the additional measurements can be triggered. Based on this measurements another probability can be output which hopefully uses more informative features.

If you care about $\Delta t$, you should think about wether there is a real time dependency. Does the need of intervention depend upon the series of classifications? Is the order of determining interventions as necessary important for you? 

In my opinion, you do not have to care about ordering for this application.
Rather, I would suggest pooling classifications for one instance according to the most frequent label or something like that.