Suppose there are three complimentary events well, sick, dead. At anytime, if you are not sick nor dead, you are well.
From literature we know the rate of sick is 123 / 100,000 per year, and rate of dead is 456 / 100,000 per year.
sick and dead once in 5 years time respectively?dead rate is 1 / 100,000 per month.Let be $r_1 = 0.00123$ and $r_2 = 0.00456$ the yearly rate of being sick and death respectively.
We have that the probability to survive one year is $$S(1) = 1-(r_1+r_2) = 0.99421.$$
In general the probability to survive until year $t$ is $$S(t) = S(1)^t,$$ so the probability to survive until year 5 is $S(5) = 0.9713833$.
Let's now calculate the probability of being sick. This probability is the sum of being sick during the first year plus being sick during the second year when surviving the first year plus being sick during the third year when surviving the first two years plus... In other words,
\begin{align}P(\text{"sick during 5 years"}) = P(\text{"sick during one year"}) + S(1) P(\text{"sick during one year"}) + \dots + S(4) P(\text{"sick during one year"})\end{align}
Therefore,
$$P(\text{"sick during 5 years"}) = r_1 + S(1) r_1 + S(1)^2 r_1 + S(1)^3 r_1 + S(1)^4 r_1 = 0.006079194.$$
In a similar way we can calculate the probability of dying during the five years
$$P(\text{"dying during 5 years"}) = r_2 + S(1) r_2 + S(1)^2 r_2 + S(1)^3 r_2 + S(1)^4 r_2 = 0.0225375$$
Notice that $0.9713833+0.006079194+0.0225375 = 1.$
If you have the rates in months you can apply the same reasoning but in months. To calculate "dying during 5 years" (5x12=60) instead of 5 terms you will have 60 terms. In months, you will get a more accurate result.
Here, the proposed solution is an adaptation of what is called the Aalen-Johansen estimator, which calculates the probability of event from observed data when the appearence of one event in one person provoques that the other event cannot appear or is not considered on that person (it is said that events are competing, competing risk survival analysis).
sick is in yearly rate but dead is in monthly rate? It is not unusual because 30-day mortality rates might be used for critical illness survivors. (Again, we assume sick is sick with minor ailment and is independent from dead.)
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Commented
Apr 5, 2017 at 2:10
Let $s_i$, $d_i$, and $w_i$ be the events that this person is sick, dead, and well in the $i$-th year, respectively. We know that: $$\text{Pr}(s)=0.00123$$ $$\text{Pr}(d)=0.00456$$ $$\text{Pr}(\bar{d})=1-0.00456=0.99544$$ $$\text{Pr}(w)=0.99421$$ As you mentioned in the comments, you want the probability that a person is sick in exactly one year in 5 years. But to shorten the answer, assume that we are considering three years instead of five. The event of being sick in exactly one year is equal to occurring one of the following disjoint events:
$$s_1,w_2,w_3$$ $$s_1,w_2,d_3$$ $$s_1,d_2$$ $$w_1,s_2,w_3$$ $$w_1,s_2,d_3$$ $$w_1,w_2,s_3$$ To compute the probability of these events, you need more subtle information. For example, to compute the probability of the first event, you need $\text{Pr}(w_2|s_1)$. But with an independence assumption, we can simply compute the probability of each of the above events. For example, $$\text{Pr}(w_1,s_2,d_3)=0.99421\times 0.00123 \times 0.00456$$ Other probabilities can also be simply computed.
The event of dying in one year is equal to occurring one of the following events: $$d_1$$ $$\bar{d_1},d_2$$ $$\bar{d_1},\bar{d_2},d_3$$ These probabilities are also simply computed. For example, $$\text{Pr}(\bar{d_1},d_2)=0.99544\times 0.00456$$
Note that the made assumption is necessary to obtain the above results, although it is not a reasonable assumption. For example, when a person is not sick in 4 years, she/he is less likely to be sick in the fifth year.
