Application of a simple analytical model of capacity requirements Martin Utley Steve Gallivan, Mark Jit Clinical Operational Research Unit University College London

Outline A simple model for estimating capacity requirements for hospital environments Applications Evaluation of Treatment Centres Paediatric intensive care unit

Variable length of stay A hospital environment with unlimited capacity Variable admissions Variable demand for beds ? ? Variable length of stay ?

Variable length of stay A hospital environment with unlimited capacity Variable admissions Variable demand for beds ? ? Variable length of stay ?

Contribution to bed demand from admissions on an earlier day Probability Number of admissions Still in hospital? 1 2 3 4 5 Contribution to demand N Y Start Calculate demand for beds on day of interest

Contribution to bed demand from admissions on an earlier day Probability 1 2 3 4 5 N Y Number of admissions Still in hospital? Contribution to demand Booked admissions (incorporating non-attendance) + emergencies... ...may follow weekly cycle, seasonal trends etc.

Contribution to bed demand from admissions on a previous day Probability 1 2 3 4 5 N Y Number of admissions Still in hospital? Contribution to demand Reflects variability in length of stay No need to parameterise length of stay distribution

Contribution to bed demand from admissions on an earlier day Probability Number of admissions Still in hospital? 1 2 3 4 5 Contribution to demand N Y Start Calculating the probability distribution of this contribution is relatively straightforward

Contributions to bed demand from admissions on a sequence of days

Contributions to bed demand from admissions on a sequence of days Probability Number of admissions Still in hospital? Y Contribution to demand

Contribution to bed demand from a sequence of days TOTAL DEMAND

Capacity required to meet demand on 95% of days Having calculated distribution of demand, one can estimate capacity required to meet demand on a given % of days. This depends on both average demand and the degree of variability Capacity required to meet demand on 95% of days

The impact of variability Average occupancy 10 patients Capacity required to meet 95% of demand 14 beds* 95% Demand

The impact of variability Average occupancy 10 patients Capacity required to meet 95% of demand 15 beds* 95% Demand

The impact of variability Average occupancy 10 patients Capacity required to meet 95% of demand 17 beds* 95% Demand

Application: UK Treatment Centres Intention is to separate routine elective cases from complex & emergency cases Hospital 1 Hospital 1 TC Hospital 2 Hospital 2

Application: UK Treatment Centres In Prague we presented plans for a 3 yr project to evaluate UK NHS Treatment Centres in terms of: Organisational Development; Knowledge Management; Throughput and the efficient use of capacity. Here, I’ll briefly present our methods and findings.

How could a TC improve efficiency? Genuinely reducing length of stay Management of patients Genuinely reducing variability in length of stay Gains in efficiency for whole system? Economies of Scale organisation of service Structure and Reducing variability in length of stay through patient selection Note: our work was limited to study effects associated with organisation of services

Compare capacity requirements We evaluated a large number of hypothetical scenarios... Hospital 1 TC Hospital 2 ...to identify circumstances in which a TC might be an efficient use of capacity

Compare capacity requirements  = Capacity requirements* of Hospital 1 Hospital 2 TC  < 1 The system is more efficient with a TC * In order to meet demand on 95% of days

What factors to consider? Number of elective admissions Level of emergency admissions Impact of TC () Number of Non-TC hospitals Success of patient selection

How to quantify success of patient selection? Number of elective admissions Level of emergency admissions Impact of TC () Success of patient selection Number of Non-TC hospitals

Worst Case Refer to TC Refer to hospital Frequency Frequency Length of stay (days) Frequency Refer to hospital Length of stay (days) Length of stay (days)

Best Case Refer to TC Refer to hospital Frequency Frequency Length of stay (days) Frequency Refer to hospital Length of stay (days) Length of stay (days)

Modelling patient selection Use mathematical function with parameter to separate length of stay distribution a

Modelling patient selection

Modelling patient selection

Modelling patient selection

Modelling patient selection

RESULTS

Example patient population Probability

Efficiency of capacity use N (number booked admissions per day) Efficiency of capacity use if a TC is introduced 20 18 Worse r > 1 16 14 Marginally better 0.975 < r ≤ 1 12 Better 0.95 < r ≤ 0.975 10 8 Much better r < 0.95 6 4 2 1 3 5 9 11 13 15 17 19 21 23 a (degree of success in identifying shorter stay patients)

Urology patients Probability Data: Elective urology inpatients to an English NHS Trust Jan 01 – Oct 04

Level of emergencies 0% 10% 20% Worse Marginally better Better Much better

Findings In many circumstances, there is no theoretical benefit associated with treatment centres*. Theoretical benefits exist if there is successful identification of shorter-stay patients and a number of participating non-TC hospitals. Benefits rely on cooperation between providers *in terms of the efficient use of capacity

Is cooperation between providers likely? Payment by Results It has come to this... Fixed Tariffs Patient Choice

George Box said... “All models are wrong... ...some are useful”

All models are wrong ...some are useful Simple model of capacity requirements All models are wrong ...some are useful

Summary Simple analytical model very useful for examining the interplay of key factors in a large number of hypothetical scenarios (1800 overall). Also has application in practical planning problems: admissions to paediatric intensive care restructuring services for anxiety and depression?

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Modelling patient selection % referred to hospital 100% We used the Michaelis-Menten function to represent the degree to which intelligent selection of patients can be achieved. 50% Refer to TC Refer to hospital Frequency Length of stay Length of stay

Reminder: generating functions Let y be a positive integer valued random variable where Prob(y = i) = ri. The generating function that describes the probability distribution of y, Y(s), is defined as 0 < s  1 Expectation and variance of y, E(y) and Var(y) given by and The parameter s is a dummy variable used only to define the generating function and has no physical significance.

Xb(s) A single hospital environment with unlimited capacity Ad(s) Based on Ad(s), the generating function for the number of patients entering on day d, and Xb(s) , that for the number of beds occupied by a single patient b days after entering, Xb(s) Ad(s) the distribution of ud, the occupancy on day d, is given by the generating function

Xb(s) A single hospital environment with unlimited capacity Ad(s) Based on Ad(s), the generating function for the number of patients entering on day d, and Xb(s) , that for the number of beds occupied by a single patient b days after entering, Xb(s) Ad(s) (1) the contribution from admissions on day (d-b)

Xb(s) A single hospital environment with unlimited capacity Ad(s) Based on Ad(s), the generating function for the number of patients entering on day d, and Xb(s) , that for the number of beds occupied by a single patient b days after entering, Xb(s) Ad(s) (2) summed over all days from d backwards

Example 1 Ad(s) = sN Xb(s) = (1-pb) + pbs Gallivan et al, BMJ, 2002 N booked admissions per day that all attend Xb(s) = (1-pb) + pbs bed occupied b days after admission with probability pb, maximum value of b = M Gallivan et al, BMJ, 2002

Example 2 Xb(s) = (1-pb) + pbs Utley et al, HCMS, 2003 N booked admissions that attend with probability (1-n) and a prob qi of i emergencies Xb(s) = (1-pb) + pbs bed occupied b days after admission with probability pb, maximum value of b = M Utley et al, HCMS, 2003

1 Non-TC 1 Non-TC + 1 TC TC Non-TC