Do the maths…

Just like football... training primary care NHS

By far, the most elegant of the disciplines is mathematics. Nothing much works without numbers, chugging away in the background.

Look what a zero and a one can achieve, in a binary format. It’s the foundation of all computing.

Look at the elegant calculations of the parabola that enabled NASA to take a man to the moon and bring him home safely.

What did Archimedes do for us? Everything!

Perhaps, lesser known is Agner Erlang. The Danish mathematician, who died far too early, aged 51.

In 1909 his work was focussed on ‘the probabilities of telephone conversations’ and what the capacity of a telephone exchange might need to be.

Bell Telephone and the British Post Office used his work and since then Erlang formulae have been used in calculating call-centre capacity and a lot more.

The so-called Erlang equation is used as a measure of ‘busyness’. Expressed in healthcare it helps us with the number of beds we need, the average occupancy and the turn-away-rate.

The turn-away-rate is the vital measure for patients. Particularly in psychiatric facilities.

We are told there are 10 Trusts unable to find psychiatric beds, and resorting to using out-of-area-placements and in consequence are being threatened with cuts to funding and targets, if they don’t pull their socks up.

Either the managers of these hospitals are complete idiots or there is something structurally wrong with the infrastructure provision and I’ll wager it is the latter.

In a fascinating paper from Dr Rodney Jones we learn, using the Erlang formula,

occupancy levels above 80% lead to stressful working environments and a figure of around 3%-5% turn-away is suggested as the near optimum balance between access and the cost of capital.

Psychiatric hospitals with fewer than 100 beds should be operating below 85% average occupancy and larger establishments should be limited to a maximum of 85%.

This is to protect staff and patients from untoward incidents. Hence 85% is often quoted as good practice.

I wonder, by what margin, that figure will be broken today?

Jones tells us;

‘… A recent study into the adverse effects of high occupancy has demonstrated that in a large acute teaching hospital with 650 beds the probability of severe adverse events, i.e. an in-hospital fall resulting in a fracture or an overdose of medication requiring treatment or a longer stay to rectify, increased significantly above 75% occupancy (actual occupancy on the day rather than an average occupancy figure).

Half of all falls resulting in a fracture occurred above 99% occupancy and half of medication events occurred above 98% occupancy.’

Occupancy rates, the cause of the problems of the unfortunate 10, in psychiatric settings, can reach 100% and more if weekend trial leave is factored in.

Erlang’s queuing-theory is the only way to calculate the link between the size of the bed pool and the likely turn-away at different average occupancies. (See Fig1)

Turn-away is the percentage time that a bed will be unavailable for the next arriving patient.

In both maternity and psychiatry immediate access is required and a turn away around 0.1% requires an average occupancy of below 85%.

• In England, 45% of Mental Health Trusts are operating in the >1% to <5% turn-away band.

• 27% operate above 20% turn-away and this reflects capital constraints and pressure to reduce bed numbers.

Hence Simon Stevens’ recent Damascus Road conversion to Erlang and ‘bring me more beds’. NHS hospitals are using temporary ‘escalation beds’ all year round.

Queuing for a bed is very efficient as it means the bed is idle for the minimum period, but it is useless if there are people in the queue whose need is urgent.

Hence any-bed, anywhere, is the only solution available to the Luckless 10 and patients get ship across the country.

The management solution is not to be found in bellicose memos, threats of reducing funding or management by target. Nor, even, throwing money at the problem

The answer is in mathematics. We can know precisely what our bed demand is, what is safe, and how to manage the flow. From that, it is possible to commission it or to create it.

There are too few specialty inpatient beds and too many managers unwilling to do the numbers and face the facts.

It’s not about the memos. It’s all about the maths.

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Reproduced at by kind permission of Roy Lilley.