Signature
P = I * D / (1 + I * D); P_approx = I * D
| Inputs | Definition | Unit |
|---|---|---|
I | New cases per person-year among people free of the condition | cases per person-year |
D | Mean time from diagnosis to recovery or death | years |
P | Share of the population with the condition on the survey date | proportion |
|---|---|---|
P_approx | Product of incidence rate and mean duration, close to P when prevalence is small | proportion |
Function
Prevalence estimation from a cross-sectional survey and its relation to incidence and duration
Estimates the prevalence of a condition from a single survey round, weighting respondents for how the sample was drawn, and links point prevalence to incidence and mean duration in a steady state. The same link shows why averages taken over prevalent cases weight each form of a condition by incidence times duration, not by incidence alone. Notation follows the Cross-Sectional Design article and its prevalent-case utility example.
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Implementations
Excel
Steady-state prevalence and its approximation from named cells
With IncRate and MeanDur named, the formulas return the exact steady-state prevalence and the approximation, held in PrevExact and PrevApprox.
=IncRate*MeanDur/(1+IncRate*MeanDur); =IncRate*MeanDur
Assumptions
Steady-state population for the prevalence relation
Incidence, mean duration and the size of the population are stable over time and there is no net migration of cases; Reichenheim and Coutinho list stationarity among the conditions for relating cross-sectional measures to incidence.
Consistent time units for incidence and duration
I and D use the same time unit, such as per person-year and years; a rate per year with a duration in months overstates I D twelvefold.
Worked examples
Condition with an incidence of 2 per 1,000 and a mean duration of six years
Half of new cases last 10 years and half 2, so the mean duration is six years; I times D is 0.012, the article's total prevalence of 1.2 per cent, and the exact steady-state value is 0.012 / 1.012, about 0.011858 (exact value computed here for illustration).
I = 0.002; D = 6; P = 0.011858; P_approx = 0.012
Mild form with an incidence of 1 per 1,000 and ten years' duration
The approximation gives the article's 0.010; the exact steady-state value is 0.01 / 1.01, about 0.009901 (exact value computed here for illustration).
I = 0.001; D = 10; P = 0.009901; P_approx = 0.01
Common condition where the approximation fails
With an incidence of 0.05 per person-year and five years' duration, I times D is 0.25 but the steady-state prevalence is 0.20, so the approximation overstates it by a quarter (computed here for illustration).
I = 0.05; D = 5; P = 0.2; P_approx = 0.25
Common errors
Using P equal to I times D for a common condition
HealthKnowledge states the approximation for prevalence below about 0.11; at I times D of 0.25 (illustrative) it overstates prevalence by a quarter, so for common conditions the odds form should be used.
Back-calculating incidence when a factor changes duration
Dividing prevalence by a common duration assumes every group has the same mean duration; Hill and colleagues show that an association seen among prevalent cases can be spurious when a risk factor influences mortality from the disease studied, and so the duration of prevalent cases.
Sources
Prevalence as a function of incidence density and mean duration
Reichenheim ME, Coutinho ESF. Measures and models for causal inference in cross-sectional studies: arguments for the appropriateness of the prevalence odds ratio and related logistic regression. BMC Medical Research Methodology. 2010;10:66. doi:10.1186/1471-2288-10-66 (full text read). Formal relations between measures, equation (3): P_i = ID_i T_i / (ID_i T_i + 1), with ID_i the incidence density and T_i the mean duration of the outcome, equivalent to P / (1 minus P) = ID times T (equation 8 rearranged); Structuring conditions: the population must be in steady state over the study period, with no selective survival and equal mean duration across exposure groups.
Approximation of prevalence by incidence times duration below about 0.11
Faculty of Public Health. Measures of disease frequency and disease burden. HealthKnowledge epidemiology resource, section 5. Accessed 3 October 2026 (full text read). A population in which the numbers with and without the disease remain stable is a steady-state population; there the point prevalence is approximately equal to the product of the incidence rate and the mean duration of disease (from diagnosis to recovery or death), providing that prevalence is less than about 0.11.
Interrelation of prevalence, incidence and duration in a steady state
Freeman J, Hutchison GB. Prevalence, incidence and duration. American Journal of Epidemiology. 1980;112(5):707-723. doi:10.1093/oxfordjournals.aje.a113043 (abstract read). Abstract: prevalence, incidence and duration of a condition in a steady-state population are interrelated so that two of these quantities may be used to obtain the third.
Neyman's bias when a risk factor affects survival with the disease
Hill G, Connelly J, Hébert R, Lindsay J, Millar W. Neyman's bias re-visited. Journal of Clinical Epidemiology. 2003;56(4):293-296. doi:10.1016/s0895-4356(02)00571-1 (abstract read). Abstract: in case-control studies using prevalent cases an apparent association may be spurious if the risk factor affects survival; a compartment model shows it can explain the association only if the risk factor influences mortality from the disease being studied.
Canonical Identity
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