Let $Z$ be a random variable taking values in $\{ 0 , \ldots , M \}$. Show that:
$$\mathbf { E } [ Z ] = \sum _ { ( s , i , r ) \in E } \left( \sum _ { k = 0 } ^ { M } k \mathbf { P } \left( Z = k \mid \left( \tilde { S } _ { n } , \tilde { I } _ { n } , \tilde { R } _ { n } \right) = ( s , i , r ) \right) \right) \mathbf { P } \left( \left( \tilde { S } _ { n } , \tilde { I } _ { n } , \tilde { R } _ { n } \right) = ( s , i , r ) \right).$$