Fig. 1

Indirect response model structure employed to capture population dynamics of responses to different types of mental healthcare services. The structural model is set up to take session events as unit inputs into a latent service compartment. The choice of using a unit input as 'mass' into the latent space is arbitrary and can be challenged with the data over the course of time to gauge improvements in the fit of the data. Each service compartment has a first-order elimination rate constant (\({k}_{out, *}\)), which is a convenient starting point for modeling as it ensures values greater than zero. Each service compartment is combined into a combined latent treatment 'mass' (TRT), which is used to drive inhibition of depressive symptoms. The function driving the inhibition of symptom generation is a type of Hill function with a capacity/sensitivity term (\({S}_{50}\)) driving the duration of successful inhibition of symptoms. The proposed model also makes the simplifying assumption that the maximum effect term in the usual Hill function be fixed to 1 and is therefore omitted in the expression. The interested technical reader may refer to Additional file 1: S1 for additional details of the model. For this work, only members who enter treatment with severe depression were included. Depression symptom dynamics (“Dep” compartment) are modeled as having zero-order production and first-order elimination. The values of this compartment are patient-reported symptom severity scores. A further simplifying assumption is made so that \({r}_{out,Dep}={r}_{in,Dep}\), which allows the change in symptoms to be capped at the severe (\(Dep=1\)) level