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The national lockdown causes the “collateral deaths” of persons who would normally receive care but who remain untreated. The Dutch Volkskrant newspaper reports that 40% of normal hospital care has been cancelled.

  • People with a disease are vulnerable to Covid-19 and may fear a greater risk of infection within the health system itself.
  • Health care resources are reallocated from normal care to Covid-19 related cases. The latter can be rational, given that untreated infections are a risk for the whole population.

This shift in the burden of disease and death might be acceptable to a large extent. We must compare these “collateral deaths” to the “avoided deaths by treating those with infections“. However, at issue is whether some elements in this shift of the burden are dubious. We can understand these aspects a bit better when we have a better grasp of what is called “the value of life”.

Below, I will discuss a particular theoretical case of “collateral death”:

Consider a person of 20 years of age, with a potential gain of 60 additional years of living, who might not get proper treatment because the Intensive Care Units (ICU) are occupied with Covid-19 patients, who are elderly and might perhaps gain 10 more years of living. PM. Women in Holland at age 80 still have a life expectancy of a bit more than 9 years on average (statline). (It is not known whether this changes or whether they will have a full recovery from Covid-19.)

There is my 2003 working paper: On the value of life. This compares the lives-saved and life-years-gained measures, and develops a compromise: a “unit-square-root” measure. This regards each life as 100% and takes the square root of the relative gain – see below. The paper was intended for macro-economic issues and not for triage at the micro-level, but let us now investigate how it would work at the micro-level. PM. Said paper still suffers a lot from being a working paper, and requires much editing for readability.

Let us first look briefly at the flu and then proceed with above theoretical case at the ICU.

Relation to the flu

RIVM provides the following data for two flu-periods in Holland. Rather than the case fatality rate (CFR) (deaths per infected) we have the symptomatic fatality rate (SFR) (deaths per symptomatic). A well-known statement is that the flu is “an old man’s friend” – but this is disputed by some. I did not find the percentage at ICU. The Dutch health system is accustomed to the flu and some categories of (elderly) patients with the flu and comorbidity are no longer sent to the ICU.

Table 1. Flu incidence in Holland 2017-2019

RIVM data

Winter 2017 – 2018

Winter 2018 – 2019

Length of period (weeks)

18

14

Reported symptomatic people

900,000

400,000

Hospitalisation

16,000

10,000

Surplus deaths

9,500

2,900

Symptomatic Fatality Rate (SFR)

1.05%

0.73%

Covid-19 is “at least ten times deadlier than the regular flu” (quote) – a recent estimate is 7 times – and likely more contagious and more prone to mutation. At issue is whether the properties of Covid-19 warrant a different treatment at the ICU as is happening in these weeks. The Dutch health system is not accustomed to Covid-19 yet, and very likely they now admit Covid-19 patients who would not be admitted in the future. Let us consider the admission criterion.

The ICU admission criterion

Dutch ICU currently have the “incremental probability of survival” admission rule (document). This is the Δp = p b between the probability of survival of the treatment at the ICU (p) and the probability of surviving not at the ICU (b = background risk). Observe:

  • Patients with Covid-19 tend to have a worse Δp at the ICU than similarly diseased patients without Covid-19. Normally, we would not see many people with Covid-19 at the ICU (with comorbidity). Apparently, the health care system is not used to Covid-19 yet. They try to save patients who in retrospect wouldn’t have a chance at admission. (An aspect is that it is reported that 80% of the patients requiring breathing machines are overweight – an aspect of the obesitas health crisis.)
  • A comparable situation exists with influenza. I did not find a report about the use of ICU for flu patients. Quite likely, the availability of beds made it possible in the past that also categories of patients with influenza were admitted, also with comorbidity. The difference with Covid-19 is the increase in the case fatality rate – but this points to fewer admissions.
  • However, people have little reason to go to the hospital when they will not be treated. Untreated infections are a risk for others and thus it is better to present some scope for treatment. Keeping patients at the ICU is a form of quarantine, though an expensive one.

My 2003 paper On the value of life assumed a p = 100% chance of success of treatment (with a known subsequent life expectancy). Let us now discuss the case with a different value of p.

PM. The term “lives saved” might be emotionally biased. It seems better to say “lives extended”. Eventually everyone dies, and the term “lives extended” indicates that the important information of “how much” is not mentioned.

A treatment criterion in general

The general setup is: (i) maximise performance given costs, or (ii) minimise costs given a level of performance. Public Health budgets tend to be given, so we choose the first approach.

Let there be two types of treated patients in numbers n1 and n2. We select these treated numbers from the pools waiting for treatment n*1 and n*2. Costs of treatment per patient are c1 and c2, for example because of different lengths of stay at the ICU. Available resources are C, and these can be time of medical personel or plain costs. The probabilities of success are p1 and p2, with performance outcomes s1 and s2. All variables are nonnegative.

We can impose an additional (moral) condition that one type of patient gets a weight λ. The weight λ means that a person in that group in the new objective function becomes λ times as important as a person in that same group in the old situation. The effect depends upon the success criterion, see below. Sometimes this condition is imposed by making different beds for different types of patients. We might also manipulate the pool sizes. When a pool of cheap patients with a high treatment score is large, so that they get all the treatments, we may restrict the pool size in order to allow treatments for the other type.

The above gives the linear programming model for the variables n1 and n2:

Maximise    p1.s1.n1 + λ p2.s2.n2

Subject to  n1 ≤ n*1    and    n2 ≤ n*2    and    c1.n1 + c2.n2 ≤ C

The following diagram presents the C-simplex and the feasible region restricted to the n* pool-variables. Due to linearity, the objective function will tend to select one of the end-points. The shown optimal point is {n°1, n*2}. When the objective function and the feasible region have the same slope, then –c1/c2 = – p1.s1 / (λ* p2.s2), and λ* = s1/s2 (p1/p2) / (c1/c2). In this case we would select patients at random, but still keep representative numbers for the types of patients.

Figure. Linear programming for triage

Let us use this model for the theoretical example case from above, namely of the patient of 20 years without Covid-19 and the patient of 80 years with Covid-19. Let type 1 be the youngsters and type 2 be the elderly.

Simplification by using costed-patients

Each type of patient comes with a standard cost, and the linear programming problem becomes a bit more tractable by using this constancy.  Above problem can be simplified by looking at “costed-patients” qi = ci.ni. The conditions on the pool size become qi = ci.nici.n*i = q*i. Then we get:

Maximise (p1.s1/c1) q1 + λ (p2.s2/c2) q2

Subject to q1 ≤ q*1 and q2 ≤ q*2 and q1 + q2 ≤ C

For this formulation it may be easier to calculate the randomisation value of λ* = (p1.s1/c1) / (p2.s2/c2).

Since the maximand can be scaled arbitrarily, we may divide by the coefficient of the second group, and get this expression, so that it is fully clear that taking λ = λ* gives a maximand that is parallel to the cost condition. The shadow price of the cost condition will be max[λ , λ *].

Maximise    λ* q1 + λ q2

Subject to q1 ≤ q*1 and q2 ≤ q*2 and q1 + q2 ≤ C

Cohort size

The “direct gain ratio” is s1 / s2. For lives-extended, the direct gain ratio is 1 / 1 = 1, while for life-years gained above example gives 60 / 10 = 6, meaning that treating one younger person successfully gives the same effect as treating 6 elderly persons successfully.

By comparison λ* is the “effective gain ratio”, in which the direct gain ratio is corrected for the relative risk p1/p2 and the relative cost of treatment. The effective gain ratio means that treating 1 younger person (with the given chance of success) has the same effect as treating λ* elderly patients (with their chance of success) (and excluding the manipulation with λ).

Assuming that all costs are depleted, we can write c1.n1 + c2.n2 = as n2 (c1.n1/n2 + c2) = C. Taking μ = n1 / n2 as the “selected ratio”, then we can take a single cohort as consisting of μ youngsters + 1 elderly, or μ  + 1 persons. The cost per cohort is k = c1 μ + c2 = C / n2, using that there are n2 cohorts.

While above manipulation of choosing λ at λ* obviously can be done, it still leaves the problem of randomisation. The manipulation of the maximand is not so effective in this kind of problem. It is more effective to manipulate the pool of youngsters n*1. Above we tended to assume that this pool was exogenously given, but when this pool is so large that they claim all treatments, and we want to put a limit to this, then we effectively reduce the pool size. A relevant variable to consider is the selected ratio μ or the composition of the cohorts. If μ is chosen, then n2 = C / k, and then n*1 = n1 = μ n2 = μ C / k = μ C / (c1 μ + c2).

The choices of λ and μ are different since they apply at different aspects of the problem, i.e. the maximand or the boundary condition. Yet we can conceive of some combination. The value of λ does not always mean a representative cohort size. Only when the slopes happen to be equal and there is randomisation, then, in some cases, we might first treat μ = λ* youngsters before treating an elderly patient again (which single person also weighs as λ*).

Case 1. Using mortality only

The success criterion of lives-extended gives s1 = s2 = 1. The objective function is parallel to the feasible region when λ = λ*[lives] = (p1/p2) / (c1/c2), or p1 / p2 = λ c1 / c2.

The “incremental probability of survival” is a subcase that only compares p1 and p2.  This neglects costs, or sets c1 = c2, e.g. neglecting duration of treatment, or in fact sets λ = c2 / c1, i.e. compensating for duration of treatment (so that there is randomisation when the two probabilities are equal). The rule is also silent about the pool sizes. (In practice ICU have more criteria, with some implied λ.)

We would not discriminate patients when λ = 1, i.e. without additional moral judgement and when only the lives-extended criterion and the parameter values determine who is to live and who not. (We do not decide who dies: nature does.) The choice of  λ = 1 still allows for the happenstance that the slopes would be equal by chance. For example, when the younger person costs 2 financial units (or weeks) and the older person costs 3, then the slopes are the same when the survival probabilities have the same ratio: p1 = 2/3 p2. For other values of the survival probabilities, however, one type of patient is preferred to the other type.

E.g. when p1 = 0.80% and p2 = 0.51%, the maximand is 0.80 n1 + 0.51 n2 or 0.40 q1 + 0.17 q2, and then clearly the first type will be preferred. Only an additional quantitative restriction n*1 might prevent the neglect of all patients of type 2.

For another value than λ = 1, the objective function is 0.40 q1 + λ 0.17 q2. We randomise if λ = λ* = 40 / 17 = 2.35. This means that one old patient in the new maximand is valued as 2.35 old patients in the original maximand. If we take λ* as the cohort parameter μ, then after randomly choosing 2.35 youngsters, we would randomly choose 1 person from the elderly (who has weight 2.35 too).

The lives-extended measure neglects the “how much” of the life-expectancies.

Case 2. Using life-years-gained

The success criterion using life-years-gained gives s1 = 60 and s2 = 10. We find λ = λ[lys] = 60 / 10 (p1/p2) / (c1/c2) = 6 λ[lives].

We may also find the maximand of 24 q1 + λ 1.7 q2, and the objective function is parallel to the feasible region for λ*[lys] = 24 / 1.7 = 14.1. The latter means that 1 old person in the new maximand counts as 14.1 old persons in the original maximand. Perhaps with a bit more freedom, we might say that 1 year of life for an elderly person counts as 14.1 years for a younger person. For the cohort size, we might treat μ = λ* = 14 youngsters and then treat an elderly person (who counts as 14 too).

PM. The 10 life-years actually gained for an elderly person are regarded in the new maximand as 14.1 * 10 = 141 life-years gained. This reflects 141 / 60 = 2.35 youngsters. But the elderly are not really such youngsters, and we must also account for the the effect of the other parameters.

The life-years-saved criterion is biased in age and sex: it gives advantage to the young and women. Both criteria of lives-extended and life-years-gained have their drawbacks. There is scope for a compromise.

Case 3. Unit-square-root measure

The unit-square-root (UnitSqrt) measure uses:

a = age

d = life expectancy with disease, without treatment (for the ICU: d = 0)

x = expected life-years-gained when treatment is successful

patient score = Sqrt[x] / Sqrt[a + d + x] = Sqrt[x / (a + d + x)]

Substituting all parameter values gives the maximand 34.6 q1 + λ 5.7 q2.

For above young person the score is Sqrt[60 / (20 + 60)] = 0.866. Above Covid-19 patient has Sqrt[10 / (80 + 10)] = 0.333. We get λ = λ[sqrt] = 0.866 / 0.333  (p1/p2) / (c1/c2) = 2.6 λ[lives]. We thus have a compromise value between the ratio of only survival 1 / 1 = 1 and the ratio of life-years of 60 / 10 = 6, namely 0.866 / 0.333 = 2.6. For the example case, the “middle of the road” character of the UnitSqrt measure also shows from the randomisation value of λ* = 6.1: this lies between the earlier other two values of 2.4 and 14.1.

Collecting results

We can tabulate our findings for the discussed example. Obviously, this table applies to this particular example only. There can be quite some discussion about what kind of success measure, and possible “correction” or “discrimination” by means of the n* and λ and μ. In practice, an ICU may have quite different reasoning as well, like on the availability of medical personel for particular treatments.

While it remains possible to select one of these objective functions and be neutral with λ = 1, and allow an extreme outcome (except for the happenstance of parallel lines), it seems more likely, as said, that the more relevant choice concerns the cohort size μ + 1. Above discussion gives considerations for a reasoned choice of μ as one of the “gain ratio’s” mentioned in the table. Above discussion suggests the compromise value, for this particular example case, of μ = 6, i.e. treat first 6 youngsters of said type and then treat an elderly person of said other type.

Table 2. Comparing success measures for triage, a = {20, 80} and x = {60, 10}

p = {0.80, 0.51}
c = {2, 3}
Lives extended
Life-years gained
Unit square root
Success measure s = {1, 1} s = {60, 10} = x s = Sqrt[x / (a + d + x)]
Direct gain ratio (only s1/s2) 1 / 1 = 1 60 / 10 = 6 0.866 / 0.333 = 2.6
Maximand 80 n1 + λ 51 n2

40 q1 + λ 17 q2

48 n1 + λ 5.1 n2

24 q1 + λ 1.7 q2

69.3 n1 + λ 16.9 n2

34.6 q1 + λ 5.7 q2

Weight formula
λ[lives] λ[lys] = 6 λ [lives] λ[sqrt] = 2.6 λ[lives].
Randomisation,

Effective gain ratio (λ*)

λ* = 40 / 17 = 2.35 λ* = 24 / 1.7 = 14.1 λ* = 34.6 / 5.7 = 6.1

Queuing

The 2003 paper originated from a macro-economic context of the allocation of the budget over different types of treatment. At the micro-level, a hospital is faced with a queue of patients with all their own characteristics of age a, life expectation d, effect of treatment x, now extended with probabilities and costs c. We could order patients on their costed scores p / c Sqrt[/ (a + d + x)], and apply a λ per category. Obviously this is only a suggestion from theory.

Conclusions

Some conclusions are:

  1. The 2003 paper did not look at a probability of survival p other than 100%, and now we have found an useful adaptation, namely above maximand with the LP properties.
  2. The current admission criterion for the ICU of “incremental probability of survival” would tend to favour youngsters because of their better conditions and survival probabilities. The current admission of many elderly Covid-19 patients must derive from other considerations, like inexperience with Covid-19 and the effect on quarantine and containment. The creation of additional ICU beds might be seen as additional only.
  3. The admission criterion of the “incremental probability of survival” looks at “lives saved”, or rather “lives extended”. This causes questions about cost comparisons (e.g. length of stay at the ICU), pool sizes (n*) and moral values (λ). Switching to “life-years gained” comes with a larger information load and partly provides answers to those questions, but also causes more questions, since this criterion is biased on age and sex. The UnitSqrt measure takes each life as 100% and would remove the latter bias. It would eliminate this aspect in the choice of λ (except for the choice of other functions than the square root), so that its choice would be more dependent upon the parameters on costs and survival probabilities.
  4. While the original paper derived from the macro context, and this present weblog entry explored the micro context, the latter also highlights that the present public discussion about Covid-19 deaths does not yet consider the aspects on the life-years. The latter will be required for an evaluation of the wider consequences, like on the “collateral deaths” and on how to “prepare for the second wave”. However, the life-years are a biased criterion and it would seem to be advisable to see more statistics that use the unit square root too.

Disclaimer

(1) In 2002-2004 I collaborated at Erasmus Medical Center on the modeling of the Human Papilloma Virus (HPV) as the cause of cervical cancer. My background in modeling and also logistics was relevant because diseases may look like a Markov logistics process with stages and transition probabilities. There can be the same issues about test reliability, criteria of lives-extended and life-years-gained, and cost-effectiveness of screening and treatment. I also followed the discussion about the SARS epidemic of 2003. My period at Erasmus MC was too short to send in papers for peer review. (2) Now I am an elderly male and advantaged by the lives-extended and disadvantaged by the life-years criterion.

PM.

PM 1. We did not use “quality adjusted life-years” (qaly). The application with the UnitSqrt would be the same but the information load would be larger, and likely not available at an ICU.

PM 2. A bit more about the additive character in this weighted averaging: The current setup is that the patient with age a has a life expectancy of d without treatment, and supposedly also when the treatment would fail. If d = 0 then p is the acute probability of survival due to the treatment (relevant for an ICU). Some possible variants provide an indication that above linear weighted averaging is the relevant approach.

  • xeabt = p (xd) + (1 – pd = p x + d is the life expectation when admitted but before the treatment. This variable is less useful, see PM 3. Also Sqrt[p x / (a + d + p x)] for this state is not so useful.
  • scorexp = p Sqrt[x / (a + d + x)] + (1 – p) Sqrt[0 / (a + d + 0) is the expected score (relevant)

PM 3. Suppose that the outcome measures at failure are f1 and f2, and that s1 and s2 are measured incremental to such failure. Then we might consider maximising the total effect:

Maximise    {p1.(s1 + f1) + (1 – p1).f1}  n1 + {p2.(s2 + f2) + (1 – p2).f2}  n2

Which becomes:

Maximise    p1.s1.n1 + p2.s2.n2  + f1.n1 + f2.n2

With this maximand, there is a bias towards patients who would already have a good score when the treatment fails. This maximand is not the relevant one, because the failure outcome does not depend upon the treatment. Another maximand would include the background risk b, but this is also not affected by the treatment.