Phase-Type Survival Trees to Model a Delayed Discharge and Its Effect in a Stroke Care Unit

Abstract

The problem of hospital patients’ delayed discharge or ‘bed blocking’ has long been a challenge for healthcare managers and policymakers. It negatively affects the hospital performance metrics and has other severe consequences for the healthcare system, such as affecting patients’ health. In our previous work, we proposed the phase-type survival tree (PHTST)-based analysis to cluster patients into clinically meaningful patient groups and an extension of this approach to examine the relationship between the length of stay in hospitals and the destination on discharge. This paper describes how PHTST-based clustering can be used for modelling delayed discharge and its effects in a stroke care unit, especially the extra beds required, additional cost, and bed blocking. The PHTST length of stay distribution of each group of patients (each PHTST node) is modelled separately as a finite state continuous-time Markov chain using Coxian-phase-type distributions. Delayed discharge patients waiting for discharge are modelled as the Markov chain, called the ‘blocking state’ in a special state. We can use the model to recognise the association between demographic factors and discharge delays and their effects and identify groups of patients who require attention to resolve the most common delays and prevent them from happening again. The approach is illustrated using five years of retrospective data of patients admitted to the Belfast City Hospital with a stroke diagnosis.

1. Introduction

The problem of the delayed discharge or ‘bed blocking’ [1,2,3,4,5,6,7,8] of elderly patients in hospitals has long been a challenge for healthcare managers and policymakers [9]. For example, Ref. [10] reported that from 1988–1989, delayed discharge cost the NHS one million pounds per day. The literature considers it in epidemiology due to its detrimental effects on care systems [11]. Delayed discharge negatively affects the hospital performance metrics [5] and has other severe consequences for the healthcare system, such as bed blocking and adversely affecting patients’ health. For example, Ref. [9] found that in the district hospital, which they considered for their study, 36.7% of elderly patients had delayed discharge. Among these delayed discharge patients, a large number of patients (25.9%) experienced delayed discharge because private nursing homes or residential home beds were not available. In 2003, Ref. [12] reported that the single most significant reason for delayed discharge (26% of all delayed discharged patients) was awaiting the availability of a private nursing home or residential home beds. Other reasons for the delayed discharge were waiting for pre-discharge assessments for further therapy needs, additional NHS care, funding, domiciliary care, and other reasons, including transport, rehabilitation, or family reasons [9,12]. Ref. [9] also reported that 8.6% of delayed discharge patients died, and 12% developed a medical condition while awaiting discharge. Patients also have stress and anxiety about their future and feel unsupported when their discharge is delayed [13]. Ref. [14] analyzed the causes of discharge delays and mitigation measures. Many studies demonstrated a direct relationship between discharge delays and increased mortality rates in the UK [15] and EU countries [16,17,18,19]. Prompt recognition of the discharge delays and correctly estimating their impact is challenging [5].

Discharge delay is defined as the inappropriate use of hospital resources by patients medically fit for discharge or transfer [10,20,21,22,23]. The scoping review in [6], define discharge delay as “an instance where a medically-fit patient is needlessly kept in hospital due to internal organisational/operational factors or where a patient is flagged as in need of an alternate level of care and is delayed because of deferred transition of care and/or lack of external transfer-of-care arrangements”. In a survey carried out in six hospitals in the Netherlands, Ref. [24] found that in the average length of stay of 28 days, 36% of durations (10 days) were affected by discharge delays (the common reason was to wait for a nursing home bed). A study among patients from five brain injury treatment centres across England [23] found that one in three beds was occupied by patients whose discharge had been delayed by more than 30 days (the median delay was more than seven months). Among these patients, the discharge of 41% of patients was delayed due to the untimely availability of a suitable placement. For 31% of patients, it was due to the unavailability of post-discharge support. The average cost of a delayed discharge was £18,100 per patient, even after deducting post-discharge (Social services) costs.

Ref. [25] and, more recently, Ref. [26] emphasised determining the association between demographic factors and discharge delay to explain the reasons behind delayed discharge. In their study, Ref. [27] found that stroke patients’ discharge is more likely to get delayed, while other studies, such as [28], found that elderly (those over 75) patients are more likely to have delayed discharge. Elderly patients experience delayed discharge more as an ageing co-morbid population, so medical issues requiring longer in-hospital management will increasingly be an issue in the future. Ref. [29] presented a simple model of delayed discharge developed using the stock and flow structure in the ithink software. Ref. [30] used a logistic regression model to examine the relationship between demographic factors and reasons for delayed discharge. At the same time, Ref. [31] used the multivariate regression method to understand this association. Ref. [4] used a simulation model to understand the effect of delayed discharge in the Critical Care Unit of a large teaching hospital. Refs. [6,22,32,33] reviewed the literature to identify the causes of discharge delay. The reasons for discharge delays include lack of rehab/nursing home space [9,34,35], social isolation/lack of family support [13,17,36,37,38], inadequate social services [18,39], conflict of interest [40,41], procedural delays [41,42], confusion and redundancy [42,43], inadequate discharge planning [44], and patient’s age [1,8].

The impact of discharge delays includes bed blocking [5,35,45,46], overcrowding [1,5,8], and increased costs [42,46,47,48,49].

2. Materials and Methods

The discharge destinations play a crucial role as a factor in discharge delays. Therefore, we also model the discharge delays based on discharge destinations and their impact in terms of the extra beds required, additional cost, and bed blocking (overcrowding).

In our previous work [50,51,52,53,54,55,56,57,58], we proposed and validated the phase-type survival tree (PHTST)-based analysis to cluster patients into clinically meaningful patient groups and, in an extension of this approach, to examine the relationship between the length of stay in hospital and destination on discharge. This paper describes how PHTST-based clustering can be used for modelling delayed discharge and its effects in a stroke care unit. The PHTST length of stay distribution of each patient group (each PHTST node) stratified by demographic factors and their conditions diagnosed is separately modelled as a finite state continuous-time Markov chain using Coxian-phase-type distributions (C-PHDs). Delayed discharge patients waiting for discharge are modelled as a ‘blocking state’, an additional state of the Markov chain. For PHTST construction, we used splitting criteria based on the weighted-average information criterion [53].

We can use this model to recognise the association between demographic and other factors and discharge delays and their effects on the average length of stay, cost of care, etc. We can also utilise the model to identify groups of patients who require attention to resolve the most common delays and prevent them from happening again. Each patient cluster can reveal unique issues. The approach is illustrated using five years of retrospective data [59,60] for 1985 stroke patients (hemorrhagic stroke, cerebral infarction, transient ischaemic attack TIA, and unspecified stroke). The patients were admitted to the Belfast City Hospital between January 2003 and December 2007 and discharged between 9 January 2003 and 11 March 2008. No information that identified individual patients was supplied. Patients were aged between 24 years and 101 years. The length of stay (LOS) range was 0 to 1425 days, the mean LOS was 29.01 days, and the standard deviation was 52.84 days [59,60]. The underpinning phase-type model-based clusters were previously validated for these data using statistical log-rank tests for equality between survival distributions for different discharge clusters [54].

We have used our model to understand the effect of delays in discharge from the stroke unit of the Belfast City Hospital. Patients can be discharged to either private nursing homes or other destinations such as their usual residences. In addition, some patients can eventually die during their treatment. The estimated daily cost of care is calculated using estimates from [61], adjusted from 2005. We attach unit costs of £164.80 per day for stay-in acute care (phase 1) and £114.80 per day for stay-in rehabilitative care or long-stay care (phases 2–4).

The following section describes the methodology used, including the description of the phase-type distributions, phase-type survival trees, and their mathematical foundation. The third section elaborates on the results, followed by a discussion and the conclusion sections.

2.1. Phase Type Survival Tree (PHTST)

Phase-type distributions are a preferred choice to fit the spell length of stay data [62,63], thanks to their memoryless property and ability to provide an intuitive description of the patient flow in a care system. A PHTST is constructed by recursively partitioning patient length of stay data into subgroups (or clusters) based on covariates maximising within-node homogeneity based on splitting criteria [53]. These covariates represent patient’’ characteristics such as gender, age at admission, and disease (diagnosed) available in the dataset used and have previously been identified in the literature and anecdotally as useful [52,54,60,64]. Each cluster is separately fitted to the C-PHDs [62,63]. C-PHDs model a patient’s journey through the care system as an n-state continuous time-absorbing Markov chain (Figure 1) with a single absorbing state. These partially observable states represent stages through the time spent in the hospital before discharge or death. This single absorbing state might cover both death and discharge, making the model focussed on the institution (e.g., capacity, costs) as patient-centred. Therefore, the community care-centred perspectives would consider multiple absorbing states in Section 2.3 (extended PHTST) and Section 2.4.2 (multi-absorbing state). A patient can be admitted to the hospital in the first state and then sequentially move to the next state with a transition rate λ_i. (i = 1,2,…n), constituting movement from any state i to the absorbing state (representing death or discharge from hospital) n + 1 with transition rate μ_i. The transition matrix Q for the absorbing Markov chain can be defined as follows:

$$\mathit{Q}=\left(\begin{array}{ccccccc}-\left({\lambda }_1+{\mu }_1\right)& {\lambda }_1& 0& \cdots & 0& 0& {\mu }_1\\ 0& -\left({\lambda }_2+{\mu }_2\right)& {\lambda }_2& \cdots & 0& 0& {\mu }_2\\ ⋮& ⋮& ⋮& \cdots & ⋮& ⋮& ⋮\\ 0& 0& 0& \cdots & -\left({\lambda }_{\left(n-1\right)}+{\mu }_{\left(n-1\right)}\right)& {\lambda }_{n-1}& {\mu }_{\left(n-1\right)}\\ 0& 0& 0& \cdots & 0& -{\mu }_n& {\mu }_n\\ 0& 0& 0& \cdots & 0& 0& 0\end{array}\right)$$

(1)

The time spent in the hospital before discharge or death has the probability density function:

$$f\left(t\right)=\mathit{p}\left(\exp \left(\mathit{Q}t\right)\right)\mathit{q}$$

(2)

where the row vector p of size (n + 1) is:

$$\mathit{p}=\left(\begin{array}{cccc}1& 0& 0& \dots \end{array} \begin{array}{cc}0& 0\end{array}\right)$$

(3)

Q represents the initial patient” distribution, and column vector q, representing absorption rates, is defined as follows:

$$\mathit{q}={\left(\begin{array}{cccc}\begin{array}{cc}{\mu }_1& {\mu }_2\end{array}& \dots & {\mu }_n& {\mu }_{n+1}\end{array}\right)}^T.$$

(4)

The log-likelihood function is defined as follows [63]:

$$\mathrm{Log} \mathrm{likelihood}={\displaystyle \sum }_{i=1}^N\left(\log \left(\mathit{p}\exp \left\{\mathit{Q}{t}_i\right\}\mathit{q}\right)\right) .$$

(5)

where N is the total number of patients.

Figure 1. A patient pathway as an n-state absorbing Markov chain.

Figure 1. A patient pathway as an n-state absorbing Markov chain.

Splitting criteria

A partition maximising within-node homogeneity at each node is selected based on a splitting criterion. There are several splitting criteria available to use [53], such as the criteria based on the maximum likelihood ratio, the criteria based on Akaike Information Criterion [65], the criteria based on Schwartz Bayesian Information Criterion [66], the criteria based on Corrected AIC [67], the criteria based on Corrected BIC [68], and the criteria based on the weighted-average information criterion [69]. Ref. [53] critically assessed each of these splitting criteria. It found that the splitting criteria based on the Weighted-Average Information Criterion (WIC) outperforms other splitting criteria listed here with both small and large sample sizes and also in situations where the sample size is unknown [69,70]. Therefore, we will use WIC-based splitting criteria to construct PHTST. The WIC for the fit of a node can be defined as follows [53,69]:

$$\begin{gathered} \mathrm{WIC}=-2\mathrm{Log} \mathrm{likelihood}+df \\ +\left(\frac{df\left(\left(\left(\log (N\right)-1\right)\log (N\right))\left(N-\left(df+1\right){)}^2+2N\left(N+\left(df+1\right)\right)\right)}{\left(2N+\left(\log (N\right)\left(N-\left(df+1\right)\right)\right))\left(N-\left(df+1\right)\right)}\right) \end{gathered}$$

(6)

where df is the degree of freedom (or the number of free parameters required to be estimated), and df = 2n − 1, with n being the number of phases in the C-PHD model (we have used phase and states interchangeably in the paper). Here, we model the discharge delay unrelated to the patient’s discharge destination, i.e., mainly due to the hospital environment rather than the proposed destination. N is the total number of patients, and Log-likelihood is calculated in (5).

2.2. PHTST Construction

PHTST is constructed by recursively partitioning nodes into more homogeneous subgroups based on covariates representing patient characteristics such as age, gender, and disease diagnosed. A split, minimising WIC is selected to partition the node into child nodes at each branch. If no split improves WIC, the node is designated as a terminal node. Figure 2 is the schematic representation of the phase-type survival tree constructed using the WIC-based splitting criteria for the length of stay data on stroke patients from the Belfast City Hospital.

Table 1. Phase-type survival tree construction using the WIC-based splitting criteria (nodes and possible splits).

Node	Covariate	Covariate Value	Number of Patients	Mean LOS	Standard Deviation (LOS)	Coefficient of Variation	WIC	Number of Phases	Total WIC	Gain in WIC
All	Complete dataset	Root node	1985	29.01	52.84	1.82	16,847.73	3	16,847.73	-
1 (Root node)	Gender	Male	933	26.59	44.06	1.66	7736.85	2	16,845.79	1.94
		Female	1052	31.15	59.47	1.91	9108.94	3
	Age	Young	624	19.26	39.15	2.03	4650.09	2	16,729	118.73
		Old	1361	33.48	57.49	1.72	12,078.9	3
	Diagnosis	Hemorrhagic	154	33.6	56.45	1.68	1338.96	3	16,567.05	280.68
		Cerebral	655	36.66	47.68	1.3	5978.78	2
		TIA	425	9.31	19.95	2.14	2612.38	2
		Other	751	32.54	65.05	2	6636.92	2
2 Hemorrhagic	Gender	Male	80	28.2	52.1	1.85	660.85	4	1332.41	6.55
		Female	74	39.45	60.25	1.53	671.56	2
	Age	Young	50	24.56	55.12	2.24	370.13	4	1324.75	14.21
		Old	104	37.95	56.56	1.49	954.63	2
3 Cerebral	Gender	Male	302	33.71	49.88	1.48	2694.19	2	5981.96	−3.18
		Female	353	39.19	45.55	1.16	3287.78	2
	Age	Young	194	24.07	42.45	1.76	1586.02	2	5949.18	29.61
		Old	461	41.96	48.79	1.16	4363.16	2
4 TIA	Gender	Male	207	8.7	22.68	2.61	1229.17	2	2619.64	−7.26
		Female	218	9.89	16.94	1.71	1390.47	2
	Age	Young	176	5.84	11.16	1.91	924.58	2	2593.06	19.32
		Old	249	11.77	24.02	2.04	1668.48	2
5 Other strokes	Gender	Male	344	30.74	43.41	1.41	3014.02	2	6645.27	−8.35
		Female	407	34.07	78.8	2.31	3631.25	2
	Age	Young	204	24.96	43.76	1.75	1662.38	2	6611.95	24.97
		Old	547	35.37	71.17	2.01	4949.57	2
6 Hemorrhagic Young	Gender	Male	29	30.52	69.11	2.26	226.49	2	375.19	−5.07
		Female	21	16.33	22.81	1.4	148.71	2
7 Hemorrhagic Old	Gender	Male	51	26.88	39.2	1.46	437.87	2	954.43	0.19
		Female	53	48.6	67.58	1.39	516.56	2
8 Cerebral Young	Gender	Male	104	24.67	49.27	2	853.33	2	1591.24	−5.22
		Female	90	23.37	32.94	1.41	737.91	2
9 Cerebral Old	Gender	Male	198	38.45	49.67	1.29	1836.51	2	4369.07	−5.92
		Female	263	44.6	47.94	1.07	2532.56	2
10 TIA Young	Gender	Male	88	5.74	11.33	1.97	460.65	2	933.56	−8.98
		Female	88	5.93	11	1.85	472.91	2
11 TIA Old	Gender	Male	119	10.89	28.08	2.58	767.35	2	1674.19	−5.71
		Female	130	12.58	19.53	1.55	906.85	2
12 Other strokes Young	Gender	Male	119	30.11	52.77	1.75	1006.47	3	1665.64	−3.26
		Female	85	17.75	24.66	1.75	659.17	2
13 Other strokes Old	Gender	Male	225	31.08	37.52	1.21	2000.22	2	4955.54	−5.97
		Female	322	38.37	87.17	2.27	2955.32	2

lists nodes of the tree and possible splits of these nodes. The total value of WIC for all the terminal nodes of the survival tree is 16,478.75, and the total improvement in the WIC is 368.98. For the root node, WIC is 16,847.73.

2.3. Extended PHTST

To construct extended PHTST (ePHTST), we further grow the PHTST by partitioning the terminal nodes into subgroups with more homogeneous patient pathways based on covariates representing outcome measures such as the discharge destination. Although the discharge destination information is unavailable at the time of admission, we can assign the probability to each discharge destination using cohort analysis. Figure 3 is the schematic representation of the ePHTST for the length of stay data constructed using the WIC-based splitting criteria. The resulting tree now has 26 terminal nodes.

Table 2. Tree extension using the WIC-based splitting criteria (nodes and possible splits by the covariate discharge destination).

Node	Destination	Number of Patients	Mean LOS	Standard Deviation (LOS)	Coefficient of Variation	WIC	Number of Phases	Degrees of Freedom (${\mathit{df}}_{\mathbf{max}}$)	Total WIC	Gain in WIC
6 Hemorrhagic Young	All	50	24.56	55.12	2.24	370.13	4	7	-	-
	Death	17	16.41	40.62	2.48	98.52	2	10	369.19	0.94
	Other	33	28.76	60.83	2.12	270.67	4
14 Hemorrhagic Old Male	All	51	26.88	39.2	1.46	437.87	2	3	-	-
	Death	21	10.9	14.73	1.35	141.1	2	7	395.23	42.64
	Other	27	31.89	38.85	1.22	243.81	1
	PNH	3	93.67	67.48	0.72	10.32	2
15 Hemorrhagic Female	All	53	48.6	67.58	1.39	516.56	2	3	-	-
	Death	27	24.15	35.47	1.47	223.82	2	7	490.63	25.93
	Other	24	74.17	85.36	1.15	257.48	1
	PNH	2	74.17	85.36	1.15	9.33	2
8 Cerebral Young	All	194	24.07	42.45	1.76	1586.02	2	3	-	-
	Death	14	21.29	35.22	1.65	112.65	2	7	1585.46	0.56
	Other	174	22.28	40.5	1.82	1405.34	2
	PNH	6	82.33	63.22	0.77	67.48	1
9 Cerebral Old	All	461	41.96	48.79	1.16	4363.16	2	3	-	-
	Death	112	35.66	46.4	1.3	1028.5	1	5	4339.19	23.97
	Other	296	36.94	43.09	1.17	2732.54	2
	PNH	53	83.34	62.02	0.74	578.15	1
10 TIA Young	All	176	5.84	11.16	1.91	924.58	2	3	-	-
	Death	2	57.5	12.5	0.22	8.47	2	7	890.83	33.75
	Other	173	4.8	7.77	1.62	873.57	2
	PNH	1	81	0	0	8.79	1
11 TIA Old	All	249	11.77	24.02	2.04	1668.48	2	3	-	-
	Death	11	33.27	30.31	0.91	101.53	1	5	1655.42	13.06
	Other	231	10.58	23.32	2.2	1497.54	2
	PNH	7	17.29	18.01	1.04	56.35	1
12 Other strokes Young	All	204	24.96	43.76	1.75	1662.38	2	3	-	-
	Death	22	20.27	27.84	1.37	179.13	1	7	1638.66	23.72
	Other	179	25.13	45.48	1.81	1454.59	2
	PNH	3	49.33	19.48	0.39	4.94	2
13 Other strokes Old	All	547	35.37	71.17	2.01	4949.57	2	3	-	-
	Death	142	41.82	123.73	2.96	1295.56	2	11	4895.2	54.37
	Other	358	28.27	33.35	1.18	3104.34	3
	PNH	47	70.02	50.18	0.72	495.31	2

lists the original terminal nodes of the tree and possible splits of these nodes by the covariate discharge destination. The total improvement in the WIC is 218.943. The total value of the WIC of the extended tree in Figure 3 is 16,259.8, and the total value of the WIC of the survival tree in Figure 2 is 16,478.75.

2.4. Modelling Blocking State

Delayed discharge patients waiting for discharge can be modelled as being in a special state of the finite state continuous time absorbing Markov chain modelling the length of stay of the group of patients. This special state is called the ‘blocking stat’, which represents the clinical state of “awaiting” or “ready” for “discharge”. A cluster represented by a terminal node of ePHTST generated as a result of a split of a terminal node of PHTST by the covariate discharge destination is considered a single ‘absorbing state cluster’, as all patients in the cluster are expected to discharge to the same destination. Suppose a cluster represented by a terminal node of ePHTST or a terminal node of PHTST is considered a multi-absorbing state. In that case, cluster patients may expect to be discharged to different destinations. All terminal nodes in the ePHTST of Figure 3 represent single-absorbing state clusters, and no cluster in the ePHTST has multi-absorbing states. In the following section, we will first model the blocking state for single-absorbing state clusters of ePHTST and then extend our model for ePHTST clusters with multi-absorbing states.

2.4.1. Single-Absorbing State

A patient pathway represented by a cluster of the ePHTST having a blocking state with a single-absorbing state can be modelled as an n + 1 state absorbing Markov chain (See Figure 4). There will be no blocking state for the cluster representing the discharge destination ‘death’. The transition matrix Q for the absorbing Markov chain can be defined as follows:

$$\mathit{Q}=\left(\begin{array}{cccccccc}-\left({\lambda }_1+{\mu }_1\right)& {\lambda }_1& 0& \cdots & 0& 0& {\mu }_1& 0\\ 0& -\left({\lambda }_2+{\mu }_2\right)& {\lambda }_2& \cdots & 0& 0& {\mu }_2& 0\\ ⋮& ⋮& ⋮& \cdots & ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& \cdots & \left({\lambda }_{\left(n-1\right)}+{\mu }_{\left(n-1\right)}\right)& {\lambda }_{n-1}& {\mu }_{\left(n-1\right)}& 0\\ 0& 0& 0& \cdots & 0& -{\mu }_n& {\mu }_n& 0\\ 0& 0& 0& \cdots & 0& 0& -\frac{1}{\tau }& \frac{1}{\tau }\\ 0& 0& 0& \cdots & 0& 0& 0& 0\end{array}\right)$$

(7)

Here, ${\mu }_i$ is the transition rate from the transient state i (i = 1,2,……, n) to the blocking state, λ_i represents the transition rate from the transient state i to the next transient state i + 1, and τ is the waiting time in the blocking state.

Figure 4. Schematic representation of single absorbing state process with the blocking state.

Figure 4. Schematic representation of single absorbing state process with the blocking state.

The initial distribution (at time t = 0) of patients in each state is represented by vector s₀ and is defined as follows:

$${\mathit{s}}_0=\left\{s_{0,1}, s_{0,2}, s_{0,3},\dots , s_{0,n}, s_{0,n+1}, s_{0,n+2}\right\} .$$

(8)

where s_0,i is the initial (at t = 0) number of patients in state i. s_0,n+2 represents the initial number of patients in the absorbing state.

2.4.2. Multi-Absorbing State

A cluster representing pathways with more than one absorbing state can have one blocking state if all patients delay by the same duration, and the transition matrix Q for the absorbing Markov chain for such a cluster can then be represented as follows:

$$\mathit{Q}=\left(\begin{array}{ccccccccccc}-\left({\lambda }_1+{\displaystyle \sum _{i=1}^m}{\mu }_{1,i}\right)& {\lambda }_1& 0& \cdots & 0& 0& {\displaystyle \sum _{i=1}^{m-1}}{\mu }_{1,i}& 0& 0& \cdots & {\mu }_{1,m}\\ 0& -\left({\lambda }_2+{\displaystyle \sum _{i=1}^m}{\mu }_{2,i}\right)& {\lambda }_2& \cdots & 0& 0& {\displaystyle \sum _{i=1}^{m-1}}{\mu }_{2,i}& 0& 0& \cdots & {\mu }_{2,m}\\ ⋮& ⋮& ⋮& \cdots & ⋮& ⋮& ⋮& ⋮& ⋮& \cdots & ⋮\\ 0& 0& 0& \cdots & \left({\lambda }_{\left(n-1\right)}+{\displaystyle \sum _{i=1}^m}{\mu }_{\left(n-1\right),i}\right)& {\lambda }_{n-1}& {\displaystyle \sum _{i=1}^{m-1}}{\mu }_{\left(n-1\right),i}& 0& 0& \cdots & {\mu }_{\left(n-1\right),m}\\ 0& 0& 0& \cdots & 0& -\left({\displaystyle \sum _{i=1}^m}{\mu }_{n,i}\right)& {\displaystyle \sum _{i=1}^{m-1}}{\mu }_{n,i}& 0& 0& \cdots & {\mu }_{\mathrm{n},m}\\ 0& 0& 0& \cdots & 0& 0& -\frac{1}{\tau }& \frac{{{\displaystyle \sum }}_{j=1}^n{\mu }_{j,1}}{\tau \ast {{\displaystyle \sum }}_{j=1}^n\left({{\displaystyle \sum }}_{i=1}^{m-1}{\mu }_{j,i}\right)}& \frac{{{\displaystyle \sum }}_{j=1}^n{\mu }_{j,2}}{\tau \ast {{\displaystyle \sum }}_{j=1}^n\left({{\displaystyle \sum }}_{i=1}^{m-1}{\mu }_{j,i}\right)}& \cdots & 0\\ 0& 0& 0& \cdots & 0& 0& & 0& 0& \cdots & 0\\ 0& 0& 0& \cdots & 0& 0& & 0& 0& \cdots & 0\\ ⋮& ⋮& ⋮& \cdots & ⋮& ⋮& & ⋮& ⋮& \cdots & ⋮\\ 0& 0& 0& \cdots & 0& 0& & 0& 0& 0& 0\end{array}\right)$$

(9)

where ${\mu }_{i,j}$ is the rate of transition (or absorption) from the transient state i (i = 1,2,……, n) to the absorbing state j (j = 1,2,……, m), assuming there is no blocking state, λ_i represents the rate of transition from the transient state i to the next transient state i + 1, and τ is the waiting time in the blocking state. A cluster representing pathways with m absorbing states can have m − 1 blocking states if patient delay depends on the discharge destination. Here, we assume that the absorbing state m represents death, and there is no delay and thus no blocking state for patients who die in the hospital. The transition matrix Q for the absorbing Markov chain for such a cluster can then be represented as follows:

$$\mathit{Q}=\left(\begin{array}{ccccccccccccccc}-\left({\lambda }_1+{\displaystyle \sum _{i=1}^m}{\mu }_{1,i}\right)& {\lambda }_1& 0& \cdots & 0& 0& {\mu }_{1,1}& {\mu }_{1,2}& \cdots & {\mu }_{1,\left(m-1\right)}& 0& 0& \cdots & 0& {\mu }_{1,m}\\ 0& -\left({\lambda }_2+{\displaystyle \sum _{i=1}^m}{\mu }_{2,i}\right)& {\lambda }_2& \cdots & 0& 0& {\mu }_{2,1}& {\mu }_{2,2}& \cdots & {\mu }_{2,\left(m-1\right)}& 0& 0& \cdots & 0& {\mu }_{2,m}\\ ⋮& ⋮& ⋮& \cdots & ⋮& ⋮& ⋮& ⋮& \cdots & ⋮& ⋮& ⋮& \cdots & ⋮& ⋮\\ 0& 0& 0& \cdots & \left({\lambda }_{\left(n-1\right)}+{\displaystyle \sum _{i=1}^m}{\mu }_{\left(n-1\right),i}\right)& {\lambda }_{n-1}& {\mu }_{\left(n-1\right),1}& {\mu }_{\left(n-1\right),2}& \cdots & {\mu }_{\left(n-1\right),\left(m-1\right)}& 0& 0& \cdots & 0& {\mu }_{\left(n-1\right),m}\\ 0& 0& 0& \cdots & 0& -\left({\displaystyle \sum _{i=1}^m}{\mu }_{n,i}\right)& {\mu }_{n,1}& {\mu }_{n,2}& \cdots & {\mu }_{n,\left(m-1\right)}& 0& 0& \cdots & 0& {\mu }_{n,m}\\ 0& 0& 0& \cdots & 0& 0& -\frac{1}{{\tau }_1}& 0& \cdots & 0& \frac{1}{{\tau }_1}& 0& \cdots & 0& 0\\ 0& 0& 0& \cdots & 0& 0& 0& -\frac{1}{{\tau }_2}& \cdots & 0& 0& \frac{1}{{\tau }_2}& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \cdots & ⋮& ⋮& ⋮& ⋮& \cdots & ⋮& ⋮& ⋮& \cdots & ⋮& ⋮\\ 0& 0& 0& \cdots & 0& 0& 0& 0& \cdots & -\frac{1}{{\tau }_{\left(m-1\right)}}& 0& 0& \cdots & \frac{1}{{\tau }_{\left(m-1\right)}}& 0\\ 0& 0& 0& \cdots & 0& 0& 0& 0& 0& 0& 0& 0& \cdots & 0& 0\\ 0& 0& 0& \cdots & 0& 0& 0& 0& 0& 0& 0& 0& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \cdots & ⋮& ⋮& 0& 0& 0& 0& ⋮& ⋮& \cdots & ⋮& ⋮\\ 0& 0& 0& \cdots & 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right)$$

(10)

where ${\mu }_{i,j}$ is the rate of transition (or absorption) from the transient state i (i = 1,2,……, n) to the absorbing state j (j = 1,2,……, m), assuming there is no blocking state, which is essentially equal to the transition rate to the blocking state j (j = 1,2,……,m − 1). λ_i represents the transition rate from the transient state i to the next transient state i + 1. τ_j is the waiting time in the blocking state j. The absorbing state m represents death; thus, there is no blocking state corresponding to the absorbing state m.

3. Results

To evaluate our proposed approach, we applied it to the data collected from the stroke unit of the Belfast City Hospital.

3.1. The Stroke Unit of the Belfast City Hospital

As discussed in the introduction, the primary reasons for discharge delays are related to discharge destinations, such as a lack of rehab/nursing home space, social isolation/lack of family support, and inadequate social services. Therefore, we also model the discharge delays based on the discharge destination. Consequently, we will first model without clustering based on discharge destinations. Then we will model patient clusters based on their discharge destinations: (i) the cluster of patients discharged to the nursing homes and (ii) the cluster of patients discharged to other destinations such as the patient’s usual or family residence. The impact of discharge delays includes bed blocking, overcrowding, and increased costs. Here, the discharge destinations play a crucial role as a factor in discharge delays. Therefore, the impact is measured in terms of extra beds required, additional cost, and bed blocking (overcrowding).

3.1.1. Discharge Delay Distribution without Clustering Based on Discharge Destinations

Here, we model the discharge delay unrelated to the patient’s discharge destination, i.e., mainly due to the hospital environment rather than the proposed destination. In Figure 5, we can see that the estimated blocking state size increases linearly with the increase in the discharge delay if the mean discharge delay is the same for all patients (expected to be discharged to either private nursing homes or other destinations). In addition, the estimated blocking state size grows exponentially with an increasing duration (number of days) since the first patient gets delayed and fast approaches an asymptotic value. Blocking states for both types of patients, expecting to be discharged to a private nursing home or other destinations, follow the same pattern (See Figure 6). Figure 6 shows the proportion of patients in the blocking state expected to be discharged to each destination for the mean discharge delay (τ) of 10 days. This proportion remains constant irrespective of the mean discharge delay. Figure 7 shows the proportion of patients in the blocking state expected to be discharged from each phase of the stroke care unit for the mean discharge delay of 10 days. Again, the proportion remains constant irrespective of the mean discharge delay. The estimated number of additional beds required by the delayed discharge patients increases linearly with the mean discharge delay (See Figure 8) irrespective of the duration (number of days) since the first patient was delayed.

The growth rate of the blocking state size and, therefore, the expected number of additional beds (Figure 9) and expected daily cost required (Figure 10), are different for patients discharged to different destinations. The estimated proportion of beds occupied by delayed discharge patients (with respect to the total bed requirement of the stroke care unit) is shown in Figure 11. The estimated proportion of delayed discharge patient’’ costs follows almost the same pattern as shown in Figure 12. On the other hand, the estimated number of additional beds required by patients discharged from different phases varies (see Figure 13). Therefore, the additional cost incurred by patients discharged from different phases is different (see Figure 14). Figure 15 shows that the estimated daily cost of care of patients in the stroke care unit expected to be discharged to each destination grows in a negative exponential fashion with the increasing number of days since the first patient gets delayed. The estimated cost of care for patients expected to be discharged from each phase also grows exponentially with the increasing number of days since the first patient gets delayed (see Figure 16). The proportion of the estimated cost of care of patients expected to be discharged from each blocking state phase is the same as that of patients in each stroke care unit phase (see Figure 15 and Figure 16).

3.1.2. Discharge Delay for the Patients Expected to Be Discharged to Private Nursing Homes

Figure 17 and Figure 18 show that the estimated number of patients in the blocking state and, therefore, the daily cost of care of patients in the blocking state expected to be discharged to private nursing homes grows exponentially with time. Figure 18 also shows how this cost will be distributed among patients expected to be discharged from each phase of the stroke care unit. Figure 17 and Figure 19 show an almost linear growth in the estimated daily care cost of patients expected to be discharged to private nursing homes, with the increase in the mean discharge delay and the cost distribution among patients expected to be discharged from each phase of the stroke care unit.

3.1.3. Discharge Delay with the Patients Expected to Be Discharged to Other Destinations

Figure 20 and Figure 21 show that the estimated number of patients in the blocking state and, therefore, the daily cost of care of patients in the blocking state expected to be discharged to other destinations grows exponentially with time. Figure 21 shows how this cost will be distributed among patients expected to be discharged from each phase of the stroke care unit. Figure 20, Figure 21 and Figure 22 show that the estimated daily cost of care for patients expected to be discharged to other destinations grows almost linearly with the increase in the mean discharge delay. Figure 21 and Figure 22 also show the cost distribution among patients expected to be discharged from each phase of the stroke care unit.

4. Discussion

The first contribution of this paper is to illustrate a stochastic model of blocking states and delayed discharge and its effects in a care unit. It models the process dynamics of a care system more realistically by representing the patient journey through the care system as a finite state continuous-time Markov chain. It has better explainability to healthcare professionals as the blocking state is modelled as a partially observable state of a continuous Markov chain exhibiting the stochastic nature of discharge delay. Another contribution is the PHTST-based model for analysing the effects of individual clusters and their interactions with the whole stroke care unit. We can also use the model to determine the association between demographic factors and the impact of a discharge delay. This information can identify the reasons for discharge delay and develop better solutions to minimise discharge delay. We can also use it to estimate the effects of demographic change in the patient population on discharge delay and its impact and continually review the trends in discharge delay to resolve these. The model helps recognise groups of patients who require attention by determining the most common delays, identifying the groups with high discharge delays or highly impacted by them and preventing such delays from happening again. The information about the cost of discharge provided by our model helps health managers and policymakers to plan and allocate budgets for alternate solutions. These novel models can estimate discharge delays and their impact, as seen in the results section. They can also help to monitor the healthcare service delivery continuously and promptly recognise discharge delays or changes in the healthcare process dynamics by using the models to identify when such problems are developing. For example, the higher actual length of stay distribution than the distribution estimated by our models indicates a higher discharge delay. However, there is a lack of comparable models in the literature. Most of the delayed discharge analysis is based on the identified delays. It needs a rigorous analysis of medical records. In contrast, we here demonstrate how the model projections, such as those provided in the previous section, can identify when such problems develop and facilitate corrective intervention.

More sophisticated models are needed to analyse more realistic scenarios. For example, if the discharge delay is due to constrained bed availability in private nursing homes and only a fixed number of beds are available in a given duration. Another scenario is constrained resource availability for assessing patients for their suitability for discharge (irrespective of their expected discharge destinations), and a fixed number of patients can be assessed in a given duration. Another possible scenario is constrained resource availability for assessment of care support (i.e., care package) requirements for patients expected to be discharged to their usual homes or other destinations.

These models consider long-term averages (expected values) and incorporate the variability only based on known covariates and representing a realistic scenario. However, in a more sophisticated model, capturing all possible variabilities, estimating their effects, and planning resource availability incorporating them is necessary. More complex analytical models (formulating the variances to estimate the variability and its impact) or simulation models, such as discrete event simulation models proposed by [52,53,54,71], can be used to meet these objectives. Another limitation of these models is that they characterise expected discharge delays. We are presently analysing the medical records of a large cohort of patients to extract the actual delays in each patient care activity and their reason. It will help us to develop more accurate and advanced models. We can forecast the impact of a discharge delay after a given period, such as after 10 or 20 years or as the result of adding more resources, such as extra beds. Furthermore, we can use the Internet of Things (IoT) [72] and blockchain technology [73] for secure health information (medical records) collection and exchange and the tracking of each stage of patient progress through the healthcare system (hospital).

5. Conclusions

In this paper, we demonstrated the use of phase-type survival-tree-based analysis for modelling blocking states and the effect of delayed discharge in a hospital’s stroke unit. Using PHTST-based analysis, we can have more realistic deterministic models for blocking states and better estimate the effect of delayed discharge in a stroke unit by analysing the impact of individual clusters and their interactions with the whole stroke care unit. We can also use our model to monitor the trends in discharge delays continually and promptly resolve these issues. More sophisticated models are required to analyse more realistic scenarios, such as if the discharge delay is due to constrained bed availability in private nursing homes or constrained resource availability for assessing patients for their suitability for discharge. We can forecast the impact of discharge delay after a given period, such as after 10 or 20 years or as the result of adding more resources, such as extra beds. In addition, the effects of variability of expected values can be considered using more complex models (analytically formulating the variances to estimate the variability) or discrete event simulation (DES) models, such as those proposed by [52,54,71].