Route Connectivity for Network Dataset Creation

[JusticeOseiAkoto]

Hello all,

I am looking to create a network dataset to determine service area of various facilities and services. I have a path and road layer (as line features) and a point feature as the connecting node where the path and road meet. The problem is that when I use all three layer in the creation of the network dataset, the result doesn't show the paths and the roads connected hence, makes the network dataset near inaccurate. 

Is there a way I could make the paths and the roads be connected before building the network dataset or some settings to do in the network dataset properties before building the network? I have attached images for context (Green line: Road; Red line: Path; Grey line: network dataset; points: connecting nodes)