$ sudo apt-get update
$ sudo apt-get install python-pyspatialite 

python-pyspatialite is installed in your system.

Make ensure the python-pyspatialite package were installed using the commands given below,

$ sudo dpkg-query -l | grep python-pyspatialite *

You will get with python-pyspatialite package name, version, architecture and description in a table

python-pyspatialite package basic information: