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Problem 69 Hard Difficulty

In a study of the spread of illicit drug use from an enthusiastic user to a population of $ N $ users, the authors model the number of expected new users by the equation
$$ \gamma = \int_0^\infty \frac{cN (1 - e^{-kt})}{k} e^{-\lambda t}\ dt $$
where $ c $, $ k $ and $ \lambda $ are positive constants. Evaluate this integral to express $ \gamma $ in terms of $ c $, $ N $, $ k $, and $ \lambda $.

Source: F. Hoppensteadt et al., "Threshold Analysis of a Drug Use Epidemic Model," Mathematical Biosciences 53 (1981): 79-87.

Answer

$\frac{c N}{k}\left(\frac{1}{\lambda}-\frac{1}{k+\lambda}\right)=\frac{c N}{k}\left(\frac{k+\lambda-\lambda}{\lambda(k+\lambda)}\right)=\frac{c N}{\lambda(k+\lambda)}$

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Video Transcript

this lesson in this lesson, we have evaluated the integral. Yeah, So the best thing to do is to let all the constant that C N k know that that is no related to t come out of the integral science. Or, for example, we can have the c n nick a out. Then we have the Integra. Oh, oh, yeah Mhm Oh, yeah, The next thing to do Oh, it's gonna stink too is to multiply the e to the power Negative lambda t by all that we have in a bracket. So here we have right t outside. Negative t outside. Then we have key Mhm. Last lambda, then the t Okay, so we can write it. Us what? Mhm limits us t approaches infinity of c n. Okay, then the integral from zero to t. Okay, well, okay. Yeah, Okay. Okay. Yeah. Uh huh. Yeah. So now we look at the evaluation of data ground. Uh huh. Yeah. When you integrate this Mhm, you have one over negative lambda. Yeah. Okay. Uh, that is a t a negative t. And that's when you have a positive, because that's a negative. Already. So one over. Okay, Last Lambda negative. T Cape last lambda. Okay, so the whole thing as from zero to t okay? Yeah. No, the ones that are involved in the zero can be replaced. And the T s Well, okay. Yeah. No. So, c m okay. When you put you there, you have t all around. So that is one over lambda. I'm putting t in place of itself, so nothing really much okay. Yeah, going on. Just putting t back. The last step is yeah. Mm. Yeah. You put zero there. Okay, so here, you know? Mm hmm. Okay, let me write it. At this point, you put zero into them, so the whole thing becomes 11 But this negative one over Lambda Less one. Yeah. Over key last London. Oh, okay. Yeah. Problem. Okay. Uh huh. Yeah, yeah. Right now, unless you followed them using the Lambda. Now the limits okay. T approaches infinity of anything that is related to then I get to of t mhm. As is there. Okay. Mm. So this is zero means that this and that would become zero as t approach zero. And it means that there were no matter so it would would have a negative, multiplying all these. So we find the lambda becomes so one over best negative here and multiplies all of them. Okay, Because as he becomes very large, the whole of this goes to zero. So this becomes there as well. So it goes to zero, then this becomes zero. Okay, so that's why we would ignore it entirely. Okay, so here is what we have. We can write in another way that is having a base for them. Oh, one thing that Okay, Okay. This isn't affected by the limits, because there's a constant and nothing like the at T. Okay, so here we can have a base that is Lambda K plus Lambda. And here we have Lambda. Okay, Minus lambda. Okay, so we end up having okay, C n then this one is that we have key. Okay, hold on. All right. So the whole thing now becomes this cake cause us out of C n. Then Lavenda key, less lumber. So this is the solution. Okay. Well, yeah, so thanks for your time. This is the end of the lesson. Yeah,