= Solution
Put $Z=\sum_iY_i(\infty)$ and $\mathbf1=(1,\ldots,1)^T$. Sum the bounds in part (a) over the <vertices>, using <linearity of expectation> and the <Tonelli theorem> for the nonnegative series:
$$
\boxed{\mathbb E Z\leq
\mathbf1^T\sum_{k=0}^\infty(\beta A)^kX(0)}.
$$
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>.
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