subtree $T_{\xi}^{-\gamma}.$ Then, $$\begin{aligned}
&\Pr\left({\bf X}([0,t],T_w^{-u})|t^I_u,t^R_u\right)\\
&=\Pr\left({\bf X}([0,t],T_w^{-u}\backslash T_{\xi
}^{-\gamma})|t^I_u,t^R_u\right)\\
&\times \Pr\left({\bf X}([0,t], T_{\xi
}^{-\gamma})|t^I_{\gamma},t^R_{\gamma}\right)\\
&=\Pr\left({\bf X}([0,t],T_w^{-u}\backslash T_{\xi
}^{-\gamma})|t^I_u,t^R_u\right)(1-q)^{t^R_{\gamma}-t^I_{\gamma}}\label{eqn:opt}\\
&\leq (1-q)\label{eqn:optrequirement},\end{aligned}$$ where equation (\[eqn:opt\]) holds because $\xi$ remained to be susceptible during the time slots at which $\gamma$ was in the infected state and (\[eqn:optrequirement\]) holds because $t^R_{\gamma}-t^I_{\gamma}\geq1.$ The maximum value of $\Pr\left({\bf X}([0,t],T_w^{-u})|t^I_u,t^R_u\right)$ can be achieved in the sample path in which $u$ was infected and then recovered in the next time slot so that $w$ was vulnerable to infection only in one time slot. Furthermore, $$(1-q)^{t^I_u-1}q(1-p)^{t^R_u-t^I_u-1}p$$ is maximized when $t^{I}_u=1,t^{R}_u=2$ i.e., $u$ was infected at the first time slot and recovered in the second time slot. Therefore, if $X_u(t)=R,$ $$\begin{aligned}
&\Pr({\bf X}([0,t],T_u^{-v_r})| X_{v_r}(t)=I) \leq qp(1-q)^{|{\cal C}(u)|}. \label{eq: k=0-R}\end{aligned}$$

[**Step 1.c**]{} Define ${\bf X}^*[0,t]$ to be the optimal solution to $$\max_{{\bf X}[0,t]\in {\cal X}(t)} \Pr({\bf X}[0,t]).$$ For $t=1,$ since all $u\in {\cal C}(v_r)$ are in the susceptible state, $$\begin{aligned}
\Pr\left({\bf X}^*([0,t]\right)=(1-p)(1-q)^{|{\cal C}(v_r)|}.\end{aligned}$$ For $t\geq 2,$ according to (\[eq: k=0-S\]) and (\[eq: k=0-R\]), $$\begin{aligned}
&\Pr\left({\bf X}^*([0,t]\right)\\
&=(1-p)^t \prod_{u \in {\cal
C}(v_r)}\max\left\{(1-q)^{t},qp(1-q)^{|{\cal
C}(u)|}\right\}\label{eqn:t2}.\end{aligned}$$ Note that $t$ is fixed in this optimization problem and (\[eqn:t2\]) is a none-increasing function of $t$. Since $|{\cal
C}(u)|\geq 1$, $$\begin{aligned}
&\Pr\left({\bf X}^*([0,2]\right)\\
&\leq (1-p)^2 \max\left\{(1-q)^{2|{\cal
C}(v_r)|},\left(qp(1-q)\right)^{|{\cal 