For the simple case, where there is no intent of mutating the pointee, &'a T becomes Arc<T> within your structs.
So, where you used to create a value x of type T,
let x: T = ...;
You now need to wrap the value in an Arc:
let x: Arc<T> = Arc::new(...);
And now instead of giving a borrow of x, like &x, you give an Arc::clone(&x).
The code becomes slightly more cumbersome, but you get rid of references this way 
If you want to ever be able to mutate a value with this pattern, you need to wrap it in a RwLock (or a Mutex) before wrapping it in the Arc: Arc<RwLock<T>>.
Sharing it is still done using the .clone() from the outer Arc, but now reading the pointee requires calling .read().unwrap(), and read/writing to it requires .write().unwrap().
For a brief summary of these wrapper types, see this post