Incidence, Prevalence and the Right Denominator: The Epidemiology Beneath Cost-of-Illness and Markov Models

อ่านฉบับภาษาไทย (Thai version)

Abstract

Incidence, prevalence, risk, rate and odds describe different quantities, even when they concern the same disease. Each needs a denominator and a time window before entering an economic model. Prevalence counts the existing pool; incidence adds new cases, while recovery and death remove them. Longer survival can enlarge that pool without faster onset. A prevalent sample also favours people with longer disease episodes. Our illustrative example separates annual care costs from discounted lifetime costs assigned to new cases. It makes explicit the approximation used when total population replaces disease-free exposure. Prevention changes future costs before the existing caseload disappears, so avoided lifetime costs are not immediate cash savings. Converting a longer-term risk into an annual probability requires an assumption about the hazard over time. Death before disease requires a competing-risk calculation. These distinctions help clinicians check registry counts, choose model inputs, and explain which policy question a cost total answers.

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Name the denominator before the cost

Start with the event, eligible people, and observation window. Risk is the probability of a new event within a stated interval: $R(t)=\Pr(T\leq t)$. Here, $T$ is time until the event among people initially event-free.

An incidence rate divides new events by disease-free person-time. Person-time is the time people accumulate while observed and eligible to develop disease. For first onset, accumulation stops at disease, death, or observation ending.

Odds compare event probability with its complement: $O=R/(1-R)$. They are not the proportion who develop disease. Point prevalence divides existing cases by the population alive at the same instant, including people diagnosed earlier [1].

The pool changes with entry and duration

Prevalence is a stock: the pool present at an instant. Incidence supplies a flow of new cases; recovery and death empty the pool. In a stationary population with stable incidence and disease duration, $P/(1-P)=I\bar D$ [2]. Here, $I$ is incidence per disease-free person-time and $\bar D$ is mean disease duration.

The approximation $P\approx I\bar D$ works when prevalence is small. Changing diagnosis, migration, or survival can invalidate the steady-state relationship. Longer survival can raise prevalence without any increase in onset.

Prevalent samples favour long disease episodes because those people remain available for sampling longer [3]. This is length bias. Such samples can underrepresent early deaths and diagnosis costs; use incident cohorts, enrolled at onset, to study those pathways.

Annual care or the cost of new cases?

Prevalence-based cost-of-illness counts care delivered during a calendar period to people with disease. It supports annual burden estimates and service planning. Multiplying point prevalence by an annual cost is a simple proxy; actual expenditure also depends on entries and exits.

Incidence-based cost-of-illness assigns future disease costs to newly arising cases [5]. It helps value prevention because preventing onset avoids a future care pathway. A model starting with prevalent cases can also evaluate prevention if it projects changes in incidence.

Our illustrative population contains 1,000,000 people. Incidence is 2 per 1,000 disease-free person-years; mean duration is 5 years. Annual care costs 20,000 THB per case, with a 3.00% annual discount rate. These are teaching inputs, not observed Thai costs or a policy recommendation.

Discounting gives future payments less weight than current payments. Assume each new case incurs equal year-end payments for 5 years. Mean duration alone does not justify that schedule; it is an additional illustrative assumption.

# Illustrative inputs. Writes: stdout only.
population <- 1000000; incidence <- 2 / 1000
duration <- 5; annual_cost <- 20000; discount <- 0.03
prevalence <- incidence * duration / (1 + incidence * duration)
new_cases <- population * incidence
cohort_cost <- new_cases * annual_cost * sum((1 + discount)^-(1:duration))
cat("Prevalence (%):", formatC(100 * prevalence, format = "f", digits = 2), "\n")
cat("Existing cases (rounded):", round(population * prevalence), "\n")
cat("Annual care (THB m):", formatC(population*prevalence*annual_cost/1e6, 1, format="f"), "\n")
cat("New cases per year (approximate):", new_cases, "\n")
cat("Cohort lifetime cost (THB m):", formatC(cohort_cost / 1e6, 1, format="f"), "\n")
cat("Halving onset avoids (THB m):", formatC(cohort_cost / 2e6, 1, format="f"), "\n")

Prevalence (%): 0.99
Existing cases (rounded): 9901
Annual care (THB m): 198.0
New cases per year (approximate): 2000
Cohort lifetime cost (THB m): 183.2
Halving onset avoids (THB m): 91.6

Convert risk to the model cycle

A hazard is an instantaneous event rate among people still event-free. With constant hazard $r$ and no competing event, the cycle probability is $p=1-e^{-rt}$ [4]. The cycle length $t$ must use the same time unit as the hazard.

Take an illustrative 5-year risk of 30.00%. Its odds are 0.429. Preserving survival gives an annual risk of 6.89%; dividing the risk by years instead gives 6.00%. That division ignores the shrinking population still free of the event.

Use the unrounded converted probability in the model. If hazards change with age or time since onset, use a model that allows that change.

# Illustrative inputs; constant hazard, no competing event. Writes: stdout only.
risk <- 0.30
years <- 5
odds <- risk / (1 - risk)
hazard <- -log(1 - risk) / years
annual_risk <- 1 - exp(-hazard)
divided_risk <- risk / years
cat("Odds:", formatC(odds, format = "f", digits = 3), "\n")
cat("Annual risk (%):", formatC(100 * annual_risk, format = "f", digits = 2), "\n")
cat("Risk divided by years (%):", formatC(100 * divided_risk, format = "f", digits = 2), "\n")

Odds: 0.429
Annual risk (%): 6.89
Risk divided by years (%): 6.00

Prevalence cannot replace a transition

A Markov model moves people between states using probabilities conditional on their current state. It can start with an incident cohort or a prevalent population. Prevalence sets initial occupancy; onset, recovery, and death govern subsequent movement.

Death before disease is a competing risk: it prevents later disease onset. Treating those deaths as ordinary censoring makes $1-\mathrm{KM}$ overstate cumulative incidence [6]. Here, KM denotes Kaplan-Meier survival. Use Aalen-Johansen, which combines event hazards with survival free of all competing events.

Convert competing exits jointly, since they share the same population at risk. Include remaining in the current state so each transition row sums to unity. Keep uncertainty estimates with the input rates.

Check counts against observable follow-up

For Thai registry or claims data, define first onset and remove repeated visits before counting cases. Match the numerator to people whose eligible follow-up is observable. A count without disease-free person-time is not an incidence rate.

Document entry, prior disease, death, and observation end. For costs, state whose resources count and the price year. Read more about rates [[incidence-rate-vs-cumulative-incidence]] and competing events [[cumulative-incidence-competing-risks-kaplan-meier]].

The series, in learning order

1. Health Economics Series Guide: Read the Evidence Before the Costs
2. Incidence, Prevalence and the Right Denominator: The Epidemiology Beneath Cost-of-Illness and Markov Models (this article)
3. Design, Model and Estimand: What Each Study Design Can Actually Estimate
4. Risk Ratio, Odds Ratio, Risk Difference and NNT: One 2×2 Table, Four Answers
5. The Hazard Ratio Is Not a Risk Ratio: Proportional Hazards, Restricted Mean Survival Time and Model Inputs
6. Chance, Bias and Confounding: Why an Association Is Not Yet a Treatment Effect
7. Beyond the Evidence Pyramid: GRADE Certainty and Choosing Effectiveness Inputs for Economic Models
8. From Sensitivity to Cost per Case Detected: Predictive Values, Prevalence and the Economics of Screening
9. Whose Cost, in Which Year? Perspective, Opportunity Cost, Inflation and Discounting
10. QALY, DALY and the Utilities Behind Them: What Is Actually Being Measured
11. ICER, Dominance and the Cost-Effectiveness Plane: Reading the Thai Threshold of 160,000 Baht per QALY
12. Decision Trees, Markov Cohorts and Probabilistic Sensitivity Analysis: How a Model Turns Evidence into a Decision
13. Cost-Effective Is Not Affordable, and Efficient Is Not Fair: Budget Impact and Equity

Related in the wiki: Rate-Based Measures in Clinical Research: Cumulative Incidence vs Incidence Rate · Why 1 Minus Kaplan-Meier Overstates Risk When Patients Can Die First