Modelling Ward Transitions

Given the potential complexity and uncertainty associated with assigning patients to wards (Section 4.3.5), an algorithm which accounts for all the factors considered by hospital staff cannot be obtained based on the information commonly contained within PA databases. However, a number of methods exist in the literature for approximating patient routing behaviour stochastically,

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based on data obtained from the hospital being modelled. For example, Chow et al. (2011) employ a so-called “trace-driven” approach, in which entire patient pathways (including lengths-of-stay) are sampled for each simulated patient from a database of observed hospital stays. Gallivan and Utley (2005) and Helm and Van Oyen (2014) derive “persistence matrices” from available data, which return the probability of being on a particular ward given the amount of time the patient has already spent in hospital. Günal (2008) computes “transition matrices” which contain the estimated probability of transitioning between any two wards, based patient transitions observed in the hospital data.

While persistence matrices treat patient routing as a function of time, and transition matrices treat patient routing as a function of location, it is not difficult to think of other factors which might influence routing decisions in a real hospital. For example, it is expected that transferring a patient to a ward with no available beds should be less likely than transferring the patient to a ward on which beds are available. For this reason, a preferred ward routing policy is one which can be generalised to respond to other factors which might influence patient transitions.

In this chapter, fixed or “static” transition matrices (STMs) will be used in the simulation model under consideration. This method captures not only the potential uncertainty associated with a given patient’s path through the hospital network, but also the average rate of transition between any two wards. The term “static” is used here to indicate that the probability of transitioning between any two wards is estimated independently of time or any other variable which might influence the likelihood of transitioning between wards. However,

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transition matrices are expected to be able to be formulated as functions of

other factors which influence patient routing, if necessary. This is because a snapshot of the hospital can be taken from the PA data at the time of any recorded transition. Persistence matrices on the other hand, only incorporate ward location information at pre-determined times after admission, meaning any information relating to what might have caused a particular transition is lost. Patient routes generated by a trace-driven approach, clearly cannot respond to the state of the modelled system, because the entire ward-stay trajectory is sampled when the simulation entity is created.

As shown in Figure 4.2, the modelled wards form a complete graph, meaning each ward is connected to every other ward once implemented in Micro Saint Sharp. The STMs govern the likelihood of a patient transitioning to other wards once the LOS on their current ward has ended. For newly arriving emergency patients, a set of entry transition probabilities govern which ward a patient is admitted to on arrival. For the elective patients, the ward of admission and weekday of arrival are considered to be part of the elective schedule; therefore, it is not necessary to estimate the probability of arriving on a particular ward for this admission type.

While all the modelled wards are connected to one another in Micro Saint Sharp, the probability is allowed to be zero if there is no evidence of it occurring in the PA data; effectively disconnecting the two wards. The transition probabilities are estimated from the PA data in the following way:

𝜋̂

=

∑𝑤+1𝑘=1𝑛𝑖,𝑘

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𝜋̂_𝑖,𝑗 is the estimated probability of transitioning from ward 𝑖 to 𝑗, while 𝑤 is the number of modelled wards. The (𝑤 + 1)th ward corresponds to being discharged from the hospital, rather than being a modelled ward in its own right. The 𝑛_𝑖,𝑗 represent the number of observed transitions from 𝑖 to 𝑗 in the PA data. Bed transfers within the same ward are modelled as a single LOS period, therefore 𝜋̂𝑖,𝑖 = 0 for all wards except Other, where reflexive transfers represent transfers between the smaller wards of which Other is composed. The 𝜋̂_𝑖,𝑗 are calculated for each of the two admission types (emergency and elective), and these form two separate transition matrices to account for the likely differences in pathways through the hospital for the two patient groups.

The STMs shown in Tables 4.2 and 4.3 are used in the offline model, and the outcome of patient transfers from “row” wards to “column” wards are drawn from these distributions. Since reflexive transfers are not considered to be ward transfers, entries where the row and column wards match are zeroed. Entry and Exit are dummy wards which cannot be revisited, therefore they only occur as row and column wards respectively. It is also worth noting that the entry row of the transition matrix does not apply to elective patients, since their first ward is decided by the elective admissions schedule and is therefore deterministic.

While STMs of the type shown in Tables 4.2 and 4.3 are fixed with respect to other factors which might influence patient placement decisions, it has already been mentioned that matrices of constant probabilities such as these might be generalisable to matrices of functions of the state of the hospital. This is made possible because draws from transition matrices (static or otherwise) occur at the same time as the simulated transition occurs, meaning it is possible for the

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state of the hospital to influence the probability of transition at that time. This concept is explored further in Chapter 5, in which relationships between ward level occupancy and transition probability are sought from the PA data to emulate the effect of outliers in the simulation. The outputs generated by this simulation are compared against the outputs generated by the simulation using STMs in this chapter to assess the relative effect of modelling transition probability as a function of occupancy.

Table 4.2: The Static Transition Matrix estimated for the emergency patients.

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