I am trying to model a human population. My simulation calculates at 1 day intervals. I want to track my population counts in cohorts of varying sizes (age 0, 1-14, 15-29, etc). The chance of death for a member of a cohort on a given day will be the average chance of death for the entire cohort for one day.
For simplicity, I am currently assuming a constant death rate (therefore an exponential distribution of ages). I plan to switch to a Weibull or mixed Weibull distribution after I am confident that my simulation is working.
From information I discovered here and here I am using the process below to determine the number of people in each cohort that die each day. Note that much of this simplifies for an exponential distribution, but I want to make sure I have the steps correct so that I can work things out in the more complicated case.
Assuming: $C_0\text{ is the initial age of the cohort in days}$; $C_1\text{ is the final age of the cohort in days} + 1$; $m\text{ is the mean time to death in days}$; $C_p\text{ is the number of people in the cohort}$ $$C_t = C_1 - C_0$$ $$\lambda = \frac1m$$ Reliability function: $$R(t) = e^{-\lambda t} = e^{\frac{-t}m}$$ The chance of surviving the cohort is the conditional probability of the surviving the length of the cohort age range given already having survived to the start of the cohort age: $$R(C_t|C_0) = \frac{R(C_0+C_t)}{R(C_0)} = e^{\frac{-C_t}m}$$ The chance of surviving a single day for a member of the cohort: $$r = {R(C_t|C_0)}^\frac1{C_t} = (e^{\frac{-C_t}m})^\frac1{C_t} = e^{\frac{-1}m}$$ The chance of dying each day for a member of the cohort: $$q = 1 - r = 1 - e^{\frac{-1}m}$$ The expected number of deaths in the cohort: $$d = C_pq = C_p(1 - e^{\frac{-1}m})$$
It is sufficient for my purposes to directly use the expected number of deaths. If some variance is needed, I can perturb the chance of dying by some random amount.
Updating to a Weibull distribution should be a matter of replacing the reliability function with the Weibull reliability function. $$R(t) = e^{-(\frac{t-\gamma}\eta)^\beta}$$
Do I have this correct?