- #1
caters
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- Homework Statement
- Given certain conditions, what percentage of the population is pregnant after 1 month of trying. Assume that no reproductive assistance therapy is done.
- Relevant Equations
- $$People_{Pregnant_n} = People_{Concieved} + People_{Pregnant_{n-1}} - Miscarriages_{Total}$$
$$People_{Concieved} = People_{Fertile} * \frac{2}{3}$$
$$Miscarriages_{Total} = (People_{Pregnant_3} * 30.9\%) + (People_{Pregnant_4} * (35.4\% * 60\%)) + (People_{Pregnant_5} * (26.9\% * 70\%))$$
So this question is a math question having to do with me calculating the rate of population growth starting from a population of 100,000. I have already gotten the first 3 steps done (sex ratio, ratio of cycle time, and pregnancy ratio) after a week among those in the fertile timeframe (calculating the ratio amongst the entire female population, which is what I'm after, should be relatively easy afterwards).
Monthly Cycle numbers
Here is the cycle ratio:
$$2_{early}:2_{fertile}:1_{late}$$
And the numbers:
$$20,000_{early}:20,000_{fertile}:10,000_{late}$$
Now, let's divide the early into 2 groups, pre-fertile, and safe and assume there is a 50/50 split between those 2 groups. Let's also assume that all the people in the fertile group are in the late group after a week, all those that are in the late group, are in the safe group after a week and so on. This suggests a cycle length of $4$, but let me verify it.
After a week:
$$10,000_{safe}:10,000_{pre-fertile}:10,000_{fertile}:20,000_{late}$$
After 2 weeks:
$$20,000_{safe}:10,000_{pre-fertile}:10,000_{fertile}: 10,000_{late}$$
After 3 weeks:
$$10,000_{safe}:20,000_{pre-fertile}:10,000_{fertile}:10,000_{late}$$
Yep, cycle length of $4$ is confirmed. To get the pregnancy ratio after a month of trying for pregnancy, I need the exact division which is a tad more complicated.
Figuring out pregnancy ratio
The ratio amongst the people in the fertile window of people who become pregnant is $2:3$ or $40\%$ Anti-miscarriage meds only work at or after 4 weeks has passed(more specifically, 4 weeks from the safe part of the cycle). Their effectiveness is $60\%$ at 4 weeks and $70\%$ at 5 weeks. It is 100% effective at 6 weeks. Here are the miscarriage rates:
- 3 weeks: 30.9%
- 4 weeks: 35.4%
- 5 weeks: 26.9%
So for the first week, $8,000$ become pregnant and the other $12,000$ in the fertile window go on to be in the late group. Ratio is $8,000_{pregnant}:42,000_{non-pregnant}$ which simplifies to $4:21$ or in terms of percents, $16\%$ of the female population.
After a week, another $4,000$ become pregnant. However, 30.9% of those from the starting week have a miscarriage. That is $2472$ people who miscarried, fewer than the number that became pregnant. Now the number is at $9,528$ pregnancies.
After another week, another $4,000$ become pregnant. 30.9% of those from the previous week miscarry. On top of that, 40% of the predicted 35.4% miscarry. So that is $1,236 + 783 = 2019$ miscarriages. This is fewer than those that become pregnant so there is an overall increase again. Now $11,509$ people are pregnant
After a third week, another $4,000$ become pregnant. 30.9% of those from the previous week, 40% of 35.4% of those that became pregnant the week before last, and 30% of 26.9% of those that became pregnant on the starting week miscarry. This adds up to $1,236 + 566 + 346 = 2,148$ miscarriages. Now $13,361$ people are pregnant.
After a fourth week, another $8,800$ become pregnant. At this point, there is no more push to become pregnant so the miscarriage calculations will get simpler from here. Here are the numbers for the miscarriages: $177 + 391 + 1236 = 1804$ miscarriages. Now, this is way less than the number that became pregnant so the total number of pregnancies now is $20,357$
Fifth week:
$2,719 + 391 + 192 = 3,302$ miscarriages
$17,055$ pregnancies
Sixth week:
$192 + 861 = 1,053$ miscarriages
$16,002$ pregnancies
Seventh week:
$421$ miscarriages
The final number of pregnancies is $15,581$ pregnancies
Percentage pregnant is $31.2\%$ which is approximately a $3:10$ ratio
Did I do my calculations correctly or did I make a mistake somewhere in the sea of multiplication, addition, and subtraction to calculate the miscarriage and pregnancy numbers week by week?
NOTE: My calculations assume no fertility after the miscarriage for several months, no other causes of fetal death besides spontaneous abortion(otherwise known as a miscarriage), and no maternal death while pregnant so despite matching the percent of the population that become pregnant after 1 try very closely, it might not be a realistic percentage for after 2-4 tries. Also, it is only this step, calculating the number of pregnancies that I need you to verify. Calculating how many are twins, etc. I should be able to do fine on my own.
Monthly Cycle numbers
Here is the cycle ratio:
$$2_{early}:2_{fertile}:1_{late}$$
And the numbers:
$$20,000_{early}:20,000_{fertile}:10,000_{late}$$
Now, let's divide the early into 2 groups, pre-fertile, and safe and assume there is a 50/50 split between those 2 groups. Let's also assume that all the people in the fertile group are in the late group after a week, all those that are in the late group, are in the safe group after a week and so on. This suggests a cycle length of $4$, but let me verify it.
After a week:
$$10,000_{safe}:10,000_{pre-fertile}:10,000_{fertile}:20,000_{late}$$
After 2 weeks:
$$20,000_{safe}:10,000_{pre-fertile}:10,000_{fertile}: 10,000_{late}$$
After 3 weeks:
$$10,000_{safe}:20,000_{pre-fertile}:10,000_{fertile}:10,000_{late}$$
Yep, cycle length of $4$ is confirmed. To get the pregnancy ratio after a month of trying for pregnancy, I need the exact division which is a tad more complicated.
Figuring out pregnancy ratio
The ratio amongst the people in the fertile window of people who become pregnant is $2:3$ or $40\%$ Anti-miscarriage meds only work at or after 4 weeks has passed(more specifically, 4 weeks from the safe part of the cycle). Their effectiveness is $60\%$ at 4 weeks and $70\%$ at 5 weeks. It is 100% effective at 6 weeks. Here are the miscarriage rates:
- 3 weeks: 30.9%
- 4 weeks: 35.4%
- 5 weeks: 26.9%
So for the first week, $8,000$ become pregnant and the other $12,000$ in the fertile window go on to be in the late group. Ratio is $8,000_{pregnant}:42,000_{non-pregnant}$ which simplifies to $4:21$ or in terms of percents, $16\%$ of the female population.
After a week, another $4,000$ become pregnant. However, 30.9% of those from the starting week have a miscarriage. That is $2472$ people who miscarried, fewer than the number that became pregnant. Now the number is at $9,528$ pregnancies.
After another week, another $4,000$ become pregnant. 30.9% of those from the previous week miscarry. On top of that, 40% of the predicted 35.4% miscarry. So that is $1,236 + 783 = 2019$ miscarriages. This is fewer than those that become pregnant so there is an overall increase again. Now $11,509$ people are pregnant
After a third week, another $4,000$ become pregnant. 30.9% of those from the previous week, 40% of 35.4% of those that became pregnant the week before last, and 30% of 26.9% of those that became pregnant on the starting week miscarry. This adds up to $1,236 + 566 + 346 = 2,148$ miscarriages. Now $13,361$ people are pregnant.
After a fourth week, another $8,800$ become pregnant. At this point, there is no more push to become pregnant so the miscarriage calculations will get simpler from here. Here are the numbers for the miscarriages: $177 + 391 + 1236 = 1804$ miscarriages. Now, this is way less than the number that became pregnant so the total number of pregnancies now is $20,357$
Fifth week:
$2,719 + 391 + 192 = 3,302$ miscarriages
$17,055$ pregnancies
Sixth week:
$192 + 861 = 1,053$ miscarriages
$16,002$ pregnancies
Seventh week:
$421$ miscarriages
The final number of pregnancies is $15,581$ pregnancies
Percentage pregnant is $31.2\%$ which is approximately a $3:10$ ratio
Did I do my calculations correctly or did I make a mistake somewhere in the sea of multiplication, addition, and subtraction to calculate the miscarriage and pregnancy numbers week by week?
NOTE: My calculations assume no fertility after the miscarriage for several months, no other causes of fetal death besides spontaneous abortion(otherwise known as a miscarriage), and no maternal death while pregnant so despite matching the percent of the population that become pregnant after 1 try very closely, it might not be a realistic percentage for after 2-4 tries. Also, it is only this step, calculating the number of pregnancies that I need you to verify. Calculating how many are twins, etc. I should be able to do fine on my own.