Assumption: Treatment takes one day.
Objective: We would like to maximize the total number of patients treated.
Sets: Patient types are denoted with index $i \in I$, machines with $j \in J$, and days with $t \in T$.
Parameters: Let $d_{it}$ be the number of patients of type $i$ to start treatment on day $t$; and $s_{jt}$ denotes the number of available machines of type $j$ on day $t$. Each machine $j$ has a maximum efficiency $e_{ij}$ for patient type $i$.
Network flow: The underlying network is given in the figure. Green nodes are source $(0)$ and sink $(1)$ respectively. Light blue nodes denote patient type and start days $(i,t)$. Dark blue nodes denote machine type and treatment days $(j,t')$.
Light blue arcs are patient starts by type, so there is a capacity of $d_{it}$ for an arc between source node $(0)$ and light blue node $(i,t)$. Similarly, dark blue arcs indicate machine use, so there is a capacity of $s_{jt'}$ between dark blue node $(j,t')$ and sink node $(1)$. Red arcs don't have capacities. They exist only when $0 \leq t'-t \leq \left(10 e_{ij} - 8\right)^+$ due to the minimum efficiency constraint. Finally, the green arc from sink $(1)$ to source $(0)$ is also uncapacitated. We maximize the flow through this green arc.

Note: For notation brevity I did not use $p$, $q$ and $r$, but used the more general parameter $d$. Here, for instance, $d_{i,1} \equiv p_i$. I also defined the machine supply more generally. As it is a parameter, we can set $s_{it} = s_{i}$ for all $t$.