Android does not have a good architecture and it does not follow the MVC design pattern.
The best way to conceptualize Activity in MVC is as a View (with some controller logic) because it gets destroyed every time that a configuration change happens, such as a screen rotation or locale change, losing all its state.
The controller in this case would be the Application object, because it is your bridge to access the view (Activity and its GUI components) from code outside the Activity context. You have to use singletons so that there is only one Application object at a given time, otherwise there will be one Application object per process.
The Application object is not a good place to store the Model because its life cycle is detached from that of the Activities. It can get destroyed and recreated at any point of an Activity's life cycle (while the application is in the background) without causing it to restart, and then all its variables and references not assigned inside its onCreate() method will become null.
The Model therefore has to be stored on a GUI-less, headless fragment with setRetainInstance(true), that is linked to an activity, and has to be re-attached to it every time the activity is recreated. You have to use a fragment manager to ensure you are recovering the retained fragment and not creating it anew. This is also the best place to run background tasks, though you can also run them in the Application object. Never run tasks on an Activity as they will get destroyed on configuration changes.