Why calculate vaccine benefit and risk on a common scale?
A vaccine can reduce the probability of illness while also carrying a small risk of an adverse event. The balance is not determined by comparing the number of events alone: outcomes differ in severity, duration, and consequences, and their probabilities can vary by age and by the disease exposure expected under a coverage scenario. Expected health utility makes those quantities comparable by weighting each outcome by its probability and health consequence.
The calculations below use hypothetical data for illustration, not estimates for any real vaccine or population. The time horizon is one year. Utility is expressed in quality-adjusted life-years (QALYs): remaining in full health for one year is assigned 1 QALY, and an illness or adverse event is represented by its QALY loss over that year. This is a deliberately transparent model; its conclusions are conditional on the probabilities, utility losses, and assumptions supplied.
Model and assumptions
For each age group and coverage scenario, let \(r\) be the probability of symptomatic infection during the one-year horizon without vaccination. Conditional on infection, let \(q\) be the probability of severe illness; otherwise infection is mild. The unvaccinated probabilities are therefore \(rq\) for severe illness, \(r(1-q)\) for mild illness, and \(1-r\) for no symptomatic infection.
Let \(VE\) be vaccine efficacy against symptomatic infection, represented here as a proportional reduction in infection probability. Thus the vaccinated infection probability is \(r(1-VE)\). This example assumes that vaccination does not change the severe fraction among breakthrough infections. Let \(a\) be the probability of the specified vaccine adverse event during the same horizon. Let \(L_m\), \(L_s\), and \(L_a\) denote the QALY losses from mild infection, severe illness, and the adverse event, respectively.
The model assumes that the probabilities apply to the stated age group and scenario, and that the event probabilities are suitable for the one-year horizon. Infection outcomes are mutually exclusive. An adverse event may occur in a person who also experiences infection; for this calculation, event losses are additive. This permits use of marginal event probabilities without an independence assumption, but it does assume that the combined utility loss is the sum of the separate losses. If joint events have a different utility consequence, that joint outcome needs its own utility value.
Here \(EU_0\) is expected utility without vaccination, and 1 is the one-year reference utility in QALYs. With vaccination, the expected utility is
The incremental expected utility of vaccination is \(\Delta EU=EU_1-EU_0\). A positive value favors vaccination on this modeled health-utility scale; a negative value favors no vaccination. These statements do not, by themselves, account for costs, distributional preferences, or other outcomes not included in the model.
Why the probability-weighted calculation works
For a single event with probability \(p\), suppose utility is \(1-L\) if the event occurs and 1 if it does not. Its expected utility is \(p(1-L)+(1-p)1=1-pL\). For multiple events with additive losses, the utility in each possible joint state is the reference utility minus the losses for events in that state. Taking expectations gives the reference utility minus the sum of each marginal event probability times its loss. This follows from linearity of expectation; event independence is not required for additive losses.
For infection, the probability of mild illness is \(r(1-q)\) and that of severe illness is \(rq\). Applying the single-event calculation to these mutually exclusive outcomes gives the unvaccinated expression above. Replacing \(r\) by \(r(1-VE)\) and including the adverse-event loss gives the vaccinated expression. Subtracting yields a useful expression for incremental utility:
The first term is expected QALYs preserved through fewer infections; the second is expected QALYs lost through the adverse event. The expression makes the trade-off explicit, while the full expected-utility calculations remain useful for reporting outcome probabilities and utilities in context.
Worked Example: A Younger Age Group at Lower Coverage
Suppose a hypothetical younger group has a 4% one-year symptomatic infection probability without vaccination. Among infections, 1% are severe. Vaccine efficacy against symptomatic infection is 75%. A mild infection costs 0.002 QALY, a severe illness costs 0.20 QALY, and a vaccine adverse event occurs with probability 0.00001 and costs 0.50 QALY.
Check the outcome probabilities. Without vaccination, mild infection has probability \(0.04(1-0.01)=0.0396\), severe illness has probability \(0.04(0.01)=0.0004\), and no symptomatic infection has probability \(1-0.04=0.96\). These sum to 1. With vaccination, infection probability is \(0.04(1-0.75)=0.01\), so mild and severe probabilities are 0.0099 and 0.0001, respectively, with probability 0.99 of no symptomatic infection.
Calculate expected utility without vaccination. Expected QALY loss is \(0.0396(0.002)+0.0004(0.20)=0.0000792+0.0000800=0.0001592\). Thus \(EU_0=1-0.0001592=0.9998408\) QALY per person over one year.
Calculate expected utility with vaccination. Expected loss is \(0.0099(0.002)+0.0001(0.20)+0.00001(0.50)=0.0000198+0.0000200+0.0000050=0.0000448\). Thus \(EU_1=1-0.0000448=0.9999552\) QALY per person. The incremental utility is \(0.9999552-0.9998408=0.0001144\) QALY per person. Under these assumptions, vaccination has positive expected health utility in this group.
Worked Example: Comparing Age Groups
Now consider a hypothetical older group in the same low-coverage scenario. Its one-year infection probability without vaccination is 0.20, with 5% of infections severe. Retain 75% vaccine efficacy. In this group, a mild infection costs 0.003 QALY, severe illness costs 0.25 QALY, and the adverse-event probability is 0.00002 with a loss of 0.50 QALY.
Without vaccination, mild and severe illness probabilities are \(0.20(0.95)=0.19\) and \(0.20(0.05)=0.01\). Expected loss is \(0.19(0.003)+0.01(0.25)=0.00057+0.00250=0.00307\) QALY, so \(EU_0=0.99693\) QALY.
With vaccination, infection probability is \(0.20(0.25)=0.05\). Mild and severe probabilities are \(0.05(0.95)=0.0475\) and \(0.05(0.05)=0.0025\). The total expected loss is \(0.0475(0.003)+0.0025(0.25)+0.00002(0.50)=0.0001425+0.000625+0.000010=0.0007775\) QALY. Therefore \(EU_1=0.9992225\), and \(\Delta EU=0.9992225-0.99693=0.0022925\) QALY per person.
The modeled gain is larger for the older group, despite its slightly higher adverse-event probability, because its baseline infection risk and severe-outcome burden are higher. This comparison is conditional on the stated age-specific risks and losses; it is not a general claim that age alone determines vaccine benefit.
Coverage scenarios and population aggregation
Coverage can affect the infection risk expected in a population. Rather than derive transmission dynamics here, this worked assessment treats infection risks under low and high coverage as scenario inputs from a hypothetical forecast. The risks apply to both vaccinated and unvaccinated people within each scenario. This makes the calculation conditional on each scenario and prevents a direct-effect comparison from being mistaken for a full transmission model.
Worked Example: Aggregating Two Coverage Scenarios
Suppose a hypothetical population has 10,000 younger and 10,000 older people. Coverage is the same in both age groups. The younger and older infection risks without vaccination are, respectively, 0.04 and 0.20 under low coverage, and 0.02 and 0.08 under high coverage. Use the age-specific severity fractions, utility losses, vaccine efficacy, and adverse-event risks from the previous examples. Calculate mean expected utility per person under 30% and 80% coverage.
For low coverage, the individual expected losses already calculated are 0.0001592 QALY for an unvaccinated younger person and 0.0000448 for a vaccinated younger person; for older people they are 0.00307 and 0.0007775. At 30% coverage, average loss in the younger group is \(0.70(0.0001592)+0.30(0.0000448)=0.00012488\) QALY per person. In the older group it is \(0.70(0.00307)+0.30(0.0007775)=0.00238225\). With equal group sizes, the population mean loss is \((0.00012488+0.00238225)/2=0.001253565\) QALY per person, giving mean utility \(1-0.001253565=0.998746435\) QALY.
For high coverage, first recalculate using its lower infection risks. In the younger group, unvaccinated expected loss is \(0.0198(0.002)+0.0002(0.20)=0.0000796\). Vaccinated infection risk is \(0.02(0.25)=0.005\), giving loss \(0.00495(0.002)+0.00005(0.20)+0.00001(0.50)=0.0000249\). At 80% coverage, the group-average loss is \(0.20(0.0000796)+0.80(0.0000249)=0.00003584\).
For the older group, unvaccinated loss is \(0.076(0.003)+0.004(0.25)=0.001228\). Vaccinated infection risk is 0.02, with mild and severe probabilities 0.019 and 0.001. Including the adverse event, its loss is \(0.019(0.003)+0.001(0.25)+0.00002(0.50)=0.000317\). The group-average loss is \(0.20(0.001228)+0.80(0.000317)=0.0004992\). The population mean loss is \((0.00003584+0.0004992)/2=0.00026752\), so mean utility is \(0.99973248\) QALY per person.
Under these specified scenarios, mean modeled utility is higher at 80% than at 30% coverage. The comparison combines different coverage levels and different assumed infection risks; it should not be interpreted as an estimate of vaccine effect alone or as proof that the assumed risks will occur.
Diagnostics, alternatives, and limitations
Before interpreting a result, check the inputs and the model structure. Infection probabilities must be between 0 and 1; mild and severe probabilities must sum to the total infection probability; and the vaccinated infection probability must reflect the stated efficacy definition. Confirm that utilities use a consistent time horizon and scale. For population aggregation, check that subgroup sizes and coverage weights sum to the population total. A calculation that produces negative event probabilities, subgroup weights that do not sum to one, or an implausibly large utility gain signals an input or implementation error.
A useful diagnostic is a component breakdown: report infection-related QALYs preserved and adverse-event QALYs lost separately before reporting the net difference. Also test plausible alternative values for baseline risk, efficacy, severity mix, and utility losses. When a rare adverse event has a very small probability but a large loss, small changes in its estimated risk can matter; report the event probability and resulting expected loss rather than hiding them in a net score.
This expected-utility approach is one alternative to comparing event counts or presenting outcomes separately. Event counts are easy to communicate but treat outcomes of very different severity as equivalent. Separate outcome reporting preserves detail but does not by itself yield a single overall ranking. A utility-weighted analysis provides a common summary, but its result depends on the specified utility losses and may conceal meaningful differences between outcomes. For consequential decisions, present the component outcomes alongside the summary and examine how the ranking changes under plausible inputs or preferences.
The model is limited by its short, fixed horizon; assumed efficacy against infection; unchanged severity profile among breakthrough cases; and the chosen health losses. It does not include costs, indirect protection beyond the scenario-specific infection risks, waning, repeated doses, uncertainty in parameters, or differences in how people value the outcomes. These omissions do not automatically invalidate the calculation, but they restrict what its conclusion can support. In particular, a positive expected utility is a conditional comparison, not a guarantee of benefit for every individual.
Common mistake
Check Your Understanding
Work each problem using the one-year expected-utility framework above. Solutions are included after the questions.
- A group has infection risk 0.10 without vaccination, with 2% of infections severe. Vaccine efficacy is 60%. Mild and severe illness losses are 0.003 and 0.20 QALY; adverse-event risk is 0.00002 and its loss is 0.40 QALY. What is the incremental expected utility of vaccination?
- Why can the expected losses from infection and adverse events be added without assuming the events are independent, under the model used here?
- If coverage is 50%, unvaccinated expected loss is 0.0010 QALY and vaccinated expected loss is 0.0004 QALY in a group, with no coverage-related change in infection risk. What is the group’s average expected loss and utility?
- What conclusion is and is not supported by finding higher mean utility in a high-coverage scenario that also assumes lower infection risk?
Solutions. (1) Without vaccination, mild and severe probabilities are 0.098 and 0.002, so loss is \(0.098(0.003)+0.002(0.20)=0.000294+0.000400=0.000694\) QALY. With vaccination, infection risk is 0.04; mild and severe probabilities are 0.0392 and 0.0008. Vaccinated loss is \(0.0392(0.003)+0.0008(0.20)+0.00002(0.40)=0.0001176+0.0001600+0.0000080=0.0002856\). Thus incremental utility is \(0.000694-0.0002856=0.0004084\) QALY per person. (2) Additivity of event losses and linearity of expectation allow the marginal expected losses to be summed; independence is not required. (3) Average loss is \(0.50(0.0010)+0.50(0.0004)=0.0007\) QALY, so mean utility is \(1-0.0007=0.9993\) QALY per person. (4) The result supports higher modeled utility under the stated high-coverage scenario. It does not isolate the causal effect of coverage or establish that the assumed infection risks will occur.