Introduction

In epidemiology, the “Measurement of Disease” is the foundation of quantitative analysis. We cannot compare the health status of populations, investigate outbreaks, or evaluate health interventions without quantifying disease occurrence.

By the end of this module, you should be able to: 1. Distinguish between ratios, proportions, and rates. 2. Calculate and interpret Prevalence and Incidence. 3. Understand the mathematical relationship between Prevalence and Incidence. 4. Calculate basic measures of mortality.

1. Basic Mathematical Tools

Before measuring disease, we must define our mathematical tools.

1.1 Ratio

One number divided by another. The two numbers are not necessarily related. \[ Ratio = \frac{A}{B} \] * Example: The sex ratio (Males / Females). * Real-life: In a classroom, there are 20 males and 30 females. The ratio is \(20:30\) or \(0.67\).

1.2 Proportion

A ratio where the numerator is included in the denominator. Usually expressed as a percentage. \[ Proportion = \frac{A}{A + B} \times 100 \] * Example: The proportion of students who are male. * Calculation: \(20 / (20 + 30) = 0.40\) or \(40\%\).

1.3 Rate

A ratio that involves time. It measures the speed of occurrence of an event. \[ Rate = \frac{\text{Number of events}}{\text{Person-Time at risk}} \]

2. Measures of Morbidity

Morbidity refers to the state of being diseased or unhealthy. We use two primary measures: Prevalence and Incidence.

2.1 Prevalence (Status)

Prevalence measures the proportion of people who have the disease at a specific time (existing cases). It is a “snapshot” of the population.

\[ \text{Prevalence} = \frac{\text{Number of existing cases}}{\text{Total Population}} \times 100 \]

Types of Prevalence:

  1. Point Prevalence: Do you have the disease right now?
  2. Period Prevalence: Did you have the disease at any point during a specific time period?

R Example: Calculating Prevalence

Imagine a village of 1,000 people. We survey them and find 150 people currently have Hypertension.

total_pop <- 1000
existing_cases <- 150

prevalence <- (existing_cases / total_pop) * 100

paste("The Point Prevalence of Hypertension is:", prevalence, "%")
## [1] "The Point Prevalence of Hypertension is: 15 %"

2.2 Incidence (New Events)

Incidence measures the occurrence of new cases of disease in a population at risk over a period of time.

A. Cumulative Incidence (Risk)

The proportion of a fixed population that develops the disease over a specified period.

\[ CI = \frac{\text{Number of NEW cases}}{\text{Population at risk at start}} \]

Important: The denominator must exclude people who already have the disease or are immune.

Real-Life Example (Food Poisoning): At a wedding with 200 guests, 100 people ate the chicken salad. Within 24 hours, 25 of those who ate the salad became ill. \[ Attack Rate = \frac{25}{100} = 25\% \]

B. Incidence Density (Incidence Rate)

Used when the population is dynamic (people enter and leave) or observation times differ. The denominator is Person-Time.

\[ IR = \frac{\text{Number of NEW cases}}{\text{Total Person-Time of observation}} \]

R Example: Incidence Rate Visualization

Let’s simulate a cohort study of 5 patients followed for 5 years.

  • Patient A: Followed for 5 years, healthy.
  • Patient B: Followed for 2 years, then got disease (Event).
  • Patient C: Followed for 5 years, healthy.
  • Patient D: Followed for 3 years, then moved away (Censored).
  • Patient E: Followed for 4 years, then got disease (Event).
# Create data
cohort_data <- data.frame(
  id = c("A", "B", "C", "D", "E"),
  years_followed = c(5, 2, 5, 3, 4),
  status = c("Healthy", "Disease", "Healthy", "Lost", "Disease")
)

# Calculate Person-Years
total_person_years <- sum(cohort_data$years_followed)
new_cases <- sum(cohort_data$status == "Disease")

inc_rate <- new_cases / total_person_years

paste("Total Person-Years:", total_person_years)
## [1] "Total Person-Years: 19"
paste("New Cases:", new_cases)
## [1] "New Cases: 2"
paste("Incidence Rate:", round(inc_rate, 3), "cases per person-year")
## [1] "Incidence Rate: 0.105 cases per person-year"

3. Relationship between Prevalence and Incidence

The relationship is often described using the “Bathtub Analogy”: * Faucet (Incidence): New water flowing in. * Tub Level (Prevalence): The amount of water currently in the tub. * Drain (Death/Cure): Water leaving the tub.

If the disease is stable (steady state), the relationship is:

\[ P \approx I \times D \] Where: * \(P\) = Prevalence * \(I\) = Incidence * \(D\) = Average Duration of disease

Visualizing the Relationship

  • Scenario A (Common Cold): High Incidence, Short Duration \(\rightarrow\) Low Prevalence.
  • Scenario B (Diabetes): Low Incidence, Long Duration \(\rightarrow\) High Prevalence.

4. Measures of Mortality

While morbidity measures illness, mortality measures death.

4.1 Crude Death Rate (CDR)

The actual observed mortality in a population.

\[ CDR = \frac{\text{Total Deaths}}{\text{Mid-year Population}} \times 100,000 \]

4.2 Case Fatality Rate (CFR)

A measure of the severity or virulence of a disease. It answers: “If I get this disease, how likely am I to die?”

\[ CFR = \frac{\text{Deaths from Disease X}}{\text{Total Cases of Disease X}} \times 100 \]

Real-Life Comparison: Rabies vs. COVID-19 vs. Seasonal Flu

  • Rabies (untreated): CFR \(\approx 100\%\) (Very virulent).
  • Seasonal Flu: CFR \(\approx 0.1\%\) (Low virulence, though high morbidity).

4.3 Proportionate Mortality Ratio (PMR)

Of all the deaths that happened, what proportion was caused by a specific disease?

\[ PMR = \frac{\text{Deaths from Specific Cause}}{\text{Total Deaths}} \times 100 \]

R Example: Proportionate Mortality

Imagine a town had 1,000 deaths in 2023. * Heart Disease: 300 deaths * Cancer: 250 deaths * Accidents: 100 deaths * Other: 350 deaths

deaths <- data.frame(
  Cause = c("Heart Disease", "Cancer", "Accidents", "Other"),
  Count = c(300, 250, 100, 350)
)

# Calculate PMR
deaths$PMR <- (deaths$Count / sum(deaths$Count)) * 100

# Pie Chart
ggplot(deaths, aes(x="", y=PMR, fill=Cause)) +
  geom_bar(stat="identity", width=1) +
  coord_polar("y", start=0) +
  theme_void() +
  geom_text(aes(label = paste0(round(PMR), "%")), 
            position = position_stack(vjust = 0.5)) +
  labs(title = "Figure 3: Proportionate Mortality Ratio (PMR)")

Summary Cheat Sheet

Measure Formula Key Concept
Prevalence Cases / Total Pop Burden of disease
Cumulative Incidence New Cases / Pop at Risk Individual Risk
Incidence Rate New Cases / Person-Time Speed of spread
Case Fatality Rate Deaths / Cases Virulence/Severity

Practice Problem

Scenario: On January 1st, a population of 100 people is screened for Disease X. * 10 people already have the disease. * The remaining 90 healthy people are followed for 1 year. * During that year, 9 new cases occur.

Task: 1. Calculate Point Prevalence on Jan 1st. 2. Calculate Cumulative Incidence (Risk) over the year.

Solution Code:

# 1. Point Prevalence
# 10 cases out of 100 total people
prev <- 10 / 100
print(paste("Prevalence:", prev))
## [1] "Prevalence: 0.1"
# 2. Cumulative Incidence
# 9 NEW cases.
# Denominator is Population AT RISK (Total - Existing Cases)
# Pop at risk = 100 - 10 = 90
ci <- 9 / 90
print(paste("Cumulative Incidence:", ci))
## [1] "Cumulative Incidence: 0.1"

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  3. Paste the code above into the file.
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