I am a medical student attempting to build mathematical models of disease, but am struggling with the calculus. In particular, I am very confused by this equation shown in attached photo, and feel worse because it is called "elementary". How is the second equation the integral of the first?

spasiocuo43

spasiocuo43

Answered question

2022-11-17

Basic calculus question about epidemic modeling
I am a medical student attempting to build mathematical models of disease, but am struggling with the calculus. In particular, I am very confused by this equation shown in attached photo, and feel worse because it is called "elementary". How is the second equation the integral of the first?
Feel free to ignore the first sentence, before "we consider."
The assumption (i) requires a fuller mathematical explanation, since the assumption of a recovery rate proportional to the number of infectives has no clearepidemiological meaning. We consider the "cohort" of members who were all infected at one time and let u(s) denote the number of these who are still infectivea time units after having been infected. If a fraction a of these leave the infective class in unit time then
u = α u ,
and the solution of this elementary differential equation is
u ( s ) = u ( 0 ) e α s

Answer & Explanation

Kennedy Evans

Kennedy Evans

Beginner2022-11-18Added 16 answers

"no slope" means a slope of zero
In this case the line is a horizontal line and since it passes through ( - 3 , 3 4 ) its equation is y = 3 4
Amy Bright

Amy Bright

Beginner2022-11-19Added 4 answers

Step 1
Your equation is equivalent to
d u d s = α u .
This is a separable differential equation. In particular,
d u d s = α u 1 u d u = α d s 1 u d u = α d s ln u = α s + c 1 e ln u = e α s + c 1 u ( s ) = c 2 e α s .
Step 2
Now using your initial condition, probably u ( 0 ) = 0, we arrive at c 2 = u ( 0 ), hence your solution,
u ( s ) = u ( 0 ) e α s .

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