 Solution

Put $Z={\sum }_i{Y}_i(\infty )$ and $1=(1,\dots ,1{)}^T$. Sum the bounds in part (a) over the vertices, using linearity of expectation and the Tonelli theorem for the nonnegative series:The bound remains valid if the series diverges, although it then gives no finite estimate. When the spectral radius of $\beta A$ is less than one, the series is a Neumann series, giving the resolvent bound for a discrete SIR epidemic.