You have three options.
Option 1 - Your solution, where the user must install a new payed app every month (code on how to uninstall an app follows).
Option 2 - Make a server/authentication solution, where the app pings a server to ensure the user has payed for that month.
Option 3 - Make your own version on Apple's in-app purchases, where the user can say, pay via Paypal every month to keep the app running.
Code to uninstall apps:
Intent intent = new Intent(Intent.ACTION_DELETE);
String packageName = "com.example.app.package";
Uri uri = Uri.fromParts("package", packageName, null);
intent.setData(uri);
startActivity(intent);
Your app would need the android.permission.DELETE_PACKAGES permission to run the above code.