Signature
AR = A / N
| Inputs | Definition | Unit |
|---|---|---|
A | Number of new cases meeting the case definition during the period | count of people |
N | Number of people at risk at the start of the period, excluding people known to be immune, greater than zero | count of people |
AR | Proportion of the people at risk who became cases during the stated period | proportion with no time unit, often reported as a percentage |
|---|
Function
Attack rate, secondary attack rate and final epidemic size function
Maps counts of new cases and of people at risk over a stated period to an attack rate, a risk expressed as a proportion rather than a rate per unit of time. The same structure gives the secondary attack rate among contacts of primary cases, and comparing attack rates between groups gives the attack rate ratio and vaccine efficacy or effectiveness. Under the assumptions of the simple susceptible, infectious and recovered model, the final attack rate of a completed epidemic is linked to the basic reproduction number R_0 through the final size relation. In an economic evaluation the attack rate is either a fixed input probability of infection, as in a static model, or an output that changes with the intervention, as in a dynamic transmission model.
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Implementations
Excel
Attack rate from named case and population counts
Excel divides the named cell holding the new cases by the named cell holding the population at risk, and returns #N/A when that population is not positive.
=IF(AtRisk>0,NewCases/AtRisk,NA())
Assumptions
Attack rate denominator limited to the population at risk
N counts only people who could become a case, so for an infection people known to be immune are removed. Everyone counted in A is also counted in N, which is why the attack rate is a proportion and cannot exceed 1.
Attack rate tied to a stated period and case definition
AR applies only to the period over which A was counted, such as the duration of an outbreak, and to one case definition. An attack rate of 20% over a two-week outbreak and one of 20% over a winter are different quantities.
Worked examples
Overall attack rate in the CDC shigellosis outbreak
In the CDC teaching example, 18 people in a community of 1,000 developed shigellosis, an overall attack rate of 0.018, or 1.8%.
A = 18; N = 1000; AR = 0.018
Attack rate with no vaccination programme in the illustrative epidemic
In the article's illustrative epidemic with R_0 of 2, 79,700 of 100,000 fully susceptible people are infected, an attack rate of 0.797.
A = 79700; N = 100000; AR = 0.797
Attack rate among the unvaccinated at 30% coverage
With 30% of the population vaccinated before the illustrative epidemic, the 35,800 infections fall among the 70,000 unvaccinated people, an attack rate of about 0.511 among them. The denominator leaves out the vaccinated, who are assumed fully protected.
A = 35800; N = 70000; AR = 0.511
Common errors
Treating an attack rate as a rate per unit of time
An attack rate has no time unit and carries time only through its stated period. Comparing an outbreak attack rate directly with an incidence rate in cases per 1,000 person-years, or reading it as a weekly or annual rate, mixes a risk with a rate.
Leaving immune people in the attack rate denominator
Illustrative figures: 18 cases in a population of 1,000 of whom 400 are known to be immune give an attack rate of 0.03 among the 600 at risk, not 0.018. Leaving immune people in N makes the infection look less transmissible among susceptible people than it is.
Spreading a seasonal attack rate evenly across model cycles
Rescaling a seasonal attack rate to equal monthly probabilities with the standard probability to rate conversion assumes a constant underlying rate. Outbreak risk rises and falls with the epidemic curve, so the model places cases and costs in the wrong cycles.
Sources
CDC definitions of overall and food-specific attack rates
Centers for Disease Control and Prevention. Principles of Epidemiology in Public Health Practice. 3rd ed. Atlanta: CDC; 2006, page last reviewed 2012. Lesson 3: Measures of Risk, section 2: Morbidity frequency measures. Attack rate as a synonym for risk over a specified period such as an outbreak, the overall and food-specific attack rates, the population at risk as the denominator, and the shigellosis example.
Canonical Identity
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