# Do I subtract end-of-year deaths from total population? (prevalence and incidence)

For calculating prevalence and incidence rates... I am not sure if I should be subtracting "deaths at end of year" from the population total/at risk?

I know that incidence rate involves the the number of new cases/total AT RISK during x year (see question 1 example). Do I need to subtract the "disease deaths" from the "total at risk" population denominator if they occur by the end of year?

And for calculating prevalence at end of year (see example question #3), do I need to subtract "deaths from disease" from the total population denominator?

Example: During year 2010, 5000 newly diagnosed cases of a disease occurred in a city population of 100,000. At the beginning of the year, there were a total of 20,000 people with the disease in the city. By the end of the year 3000 died from the disease.

1. What is the incidence rate of the disease during 2010?

1. What is the prevalence of disease on Jan 1 2010?

3. What is the prevalence of disease on Dec 31 2010?

In your example, we do not have the exact about of person-time that each individual contributes, and we know that there are differences in the amount of person-time contributed, because some people are dying. We can instead calculate the cumulative incidence and convert that into an estimate of the incidence rate using the following formula: $CI = 1-e^{-I*t}$ where CI is the cumulative incidence, I is the incidence rate, and t is the time period. Cumulative incidence is calculated as the total number of new cases over the time period divided by the number of people at risk at baseline: here there were 5000 new cases and 80,000 (i.e. 100,000-20,000) people at risk on Jan 1. Your time period is 1 year. Note that this calculation assumes an exponential distribution of incidence times, but since you are given no information about failure times it is a reasonable assumption here.