Discrete-time treatment number

We introduce the discrete-time treatment number of a graph, in which each vertex is in exactly one of three states at any given time-step: compromised, vulnerable, or treated. Our treatment number is distinct from other graph searching parameters that use only two states, such as the firefighter problem or Bernshteyn and Lee's inspection number. Vertices represent individuals and edges exist between individuals with close connections. Each vertex starts out as compromised; it can become compromised again even after treatment. Our objective is to treat the entire population so that at the last time-step, no members are vulnerable or compromised, while minimizing the maximum number of treatments that occur at each time-step. This minimum is the treatment number, and it depends on the choice of a pre-determined length of time $r$ that a vertex can remain in a treated state and length of time $s$ that a vertex can remain in a vulnerable state without being treated again. We denote the pathwidth of graph $H$ by $pw(H)$ and prove that the treatment number of $H$ is bounded above by $\lceil \frac{1+pw(H)}{r+s}\rceil$. This equals the best possible lower bound for a cautious treatment plan, defined as one in which each vertex, after being treated for the first time, is treated again within every consecutive $r+s$ time-steps until its last treatment. However, many graphs admit a plan that is not cautious. When $r=s=1$, we find a useful tool for proving lower bounds, show that the treatment number of an $n\times n$ grid equals $\lceil\frac{1+n}{2}\rceil$, characterize graphs that require only one treatment per time-step, and prove that subdividing one edge can reduce the treatment number. It is known that there are trees with arbitrarily large pathwidth; surprisingly, we prove that for any tree $T$, there is a subdivision of $T$ that requires at most two treatments per time-step.