Modelling Patient Flow in an Emergency Department

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1 Modelling Patient Flow in an Emergency Department Mark Fackrell Department of Mathematics and Statistics The University of Melbourne

2 Motivation Eastern Health closes more beds Bed closures to cause ambulance delays Hospital targets cost lives Girl waited for hours for emergency care ACT meets elective surgery targets but fails on emergency waits

cost lives Girl waited for hours for emergency care

3 Motivation Public hospitals will be required by 2015 to ensure that 90% of all patients spend no more than 4 hours in the emergency department. Currently (2012) 54% major metropolitan hospitals 63% major regional hospitals 67% large metropolitan hospitals 78% large regional hospitals

4 The Australian Triage Scale ATS Category Description Treatment Acuity % Adherence 1 Resuscitate Immediate 100% 2 Emergency 10 minutes 80% 3 Urgent 30 minutes 75% 4 Semi-urgent 60 minutes 70% 5 Non-urgent 120 minutes 70%

100% 2 Emergency 10 minutes 80% 3 Urgent 30 minutes

5 The Emergency Department Ward 1 x x Triage Desk Treatment Queue Assessment and Treatment Bed Queue Ward 2 Ward 3 Ward 4 Do Not Wait Discharge

6 Modelling the Bed Queue For each six hour block (00-06, 06-12, 12-18, 18-24), and each day of the week, we assumed that the arrival rate to the bed queue is constant, ie. λ, and departure rate from the bed queue depends on the number of patientsnwaiting in the bed queue, ie. µ(n). Data available from 1 January 2001 to 18 April 2005 (200,000 entries). Parameters estimated using spline regression.

λ, and departure rate from the bed queue depends on the number of patientsnwaiting in the bed

7 Modelling the Bed Queue We model the bed queue (service centre) with a continuous-time Markov chain. C beds (servers) State spaces = {0,1,2,...,C} λ Arrival rate n current number of beds occupied (number of customers being served) µ(n) Departure rate No queueing (ie. if the bed queue becomes full the emergency department goes on bypass.)

..,C} λ Arrival rate n current number of beds occupied (number of customers being

8 Transition rates The transition rates for the Markov chain are α nn+1 = α nn 1 = λ, 0 n C 1 0, n = C 0, n = 0 µ(n), 1 n C

9 Transition Probabilities The Markov chain stays in statenfor a random timet n and then moves to either staten+1orn 1with probabilities P(n n+1)= λ λ+µ(n), 0 n C 1 P(n n 1)= µ(n) λ+µ(n), 1 n C

10 Time Until an Arrival or Departure In statenlet T + n T n be the time until the next arrival, and be the time until the next departure. Note thatt + n exp(λ) andt n exp(µ(n)). We have that T n = T 0 +, n = 0 min(t n +, Tn ), 1 n C 1 T C, n = C

next departure. Note thatt + n exp(λ) andt n exp(µ(n)).

11 Density Function The density function oft n is f n (t) = λe λt, n = 0 (λ+µ(n))e (λ+µ(n))t, 1 n C 1 µ(c)e µ(c)t, n = C

12 Probability of Reaching Capacity p n (t) probability of moving fromntoc patients in the time interval[0, t] p n (t x) probability of moving fromntoc in[0,t] given that the first transition fromnoccurs at timex p n (t x) = 0, n < C,x > t λ λ+µ(n) p n+1(t x) + µ(n), n < C,x t λ+µ(n) p n 1(t x) 1, n = C

fromntoc in[0,t] given that the first transition fromnoccurs at timex p n

13 Probability of Reaching Capacity For0 n C p n (t) = 0 p n (t x)f n (x)dx. Thus, p 0 (t)= p n (t)= t 0 t 0 p 1 (t x)λe λx dx [λp n+1 (t x)+µ(n)p n 1 (t x)]e (λ+µ(n))t dx p C (t)=1

14 Laplace Transforms Taking Laplace transforms gives P 0 (s)= λ s+λ P 1 (s) P n (s)= λ s+λ+µ(n) P n+1 (s)+ µ(n) s+λ+µ(n) P n 1 (s) P C (s)= 1 s To findp n (t) invert P n (s) numerically Euler method, Abate and Whitt, 1995.

15 Probability of Ambulance Bypass Probability of Bypass on Monday (capacity of 20 cubicles) Probability Hours

(capacity of 20 cubicles) Probability 0.

16 The ED Capacity Prediction Tool

17 Model Validation To validate the model we used data observed from 19 April 2005 to 26 September 2006 (48,000 entries). Choose a thresholdc. At the beginning of each 6 hour block calculate the probability of reaching capacity C using the model. Group the blocks into 10 bins of equal sizen i according to the probabilities. For each bini, calculate the mean probability p i, and the variance of the probabilitiesv i.

At the beginning of each 6 hour block calculate the probability of reaching capacity C using the model.

18 Model Validation Fori = 1,2,...,10, record the number of times capacityc is reached,o i. Calculate the expected number of times capacity C is reached, E i = N i p i. The goodness-of-fit statistic is G 2 = 10 i=1 (O i E i ) 2 var(o i ) whereg 2 χ 2 10 (Hosmer and Lemeshow, 1980) and var(o i ) N i p i (1 p i ) N i V i.

Calculate the expected number of times capacity C is reached, E i = N i p i.

19 Model Validation Bin N i O i E i (O i E i ) 2 var(o i ) p-value = 0.16

91 5 182 4 2.36 1.16 6 182 8 4.92 1.98 7 182 13 9.77 1.

20 Future Work Apply and validate the model with more recent data. Analyse the data to see if the mandated government targets were met. Model patient flow through the entire emergency department. Determine strategies to improve the running of the emergency department based on stochastic models.

Model patient flow through the entire emergency department.

21 Headlines You ll Never See More hospital beds to open at Lyell McEwin ABC News Website 5 July, 2013

Existing Barriers to Patient Flow from ED to the ward the ED experience

Existing Barriers to Patient Flow from ED to the ward the ED experience Dr. de Villiers Smit Acting Director Emergency and Trauma Centre The Alfred Hospital Definitions: Emergency: a sudden, urgent, usually