Today. Logistic equation in many contexts

Transcription

1 Today Logistic equation in many contexts Classic example of the power of mathematics - one unifying description for many apparently unrelated phenomena.

2 Rates of change that are proportional to two things

3 Rates of change that are proportional to two things A chemical reaction with only one reactant occurs at a rate proportional to the how much reactant is present (e.g. radioactive decay): dr = kr

4 Rates of change that are proportional to two things A chemical reaction with only one reactant occurs at a rate proportional to the how much reactant is present (e.g. radioactive decay): dr = kr A chemical reaction with two reactants occurs at a rate proportional to the how much of both reactants are present: dr 1 = kr 1 R 2

5 Rates of change that are proportional to two things A chemical reaction with only one reactant occurs at a rate proportional to the how much reactant is present (e.g. radioactive decay): dr = kr A chemical reaction with two reactants occurs at a rate proportional to the how much of both reactants are present: dr 1 = kr 1 R 2

6 Rates of change that are proportional to two things A chemical reaction with only one reactant occurs at a rate proportional to the how much reactant is present (e.g. radioactive decay): dr = kr A chemical reaction with two reactants occurs at a rate proportional to the how much of both reactants are present: dr 1 = kr 1 R 2

7 Rates of change that are proportional to two things Probability of hitting a blue ball: A chemical reaction with only one reactant occurs at a rate proportional to the how much reactant is present (e.g. radioactive decay): dr = kr A chemical reaction with two reactants occurs at a rate proportional to the how much of both reactants are present: dr 1 = kr 1 R 2

8 Rates of change that are proportional to two things Probability of A chemical reaction with only one reactant occurs at a rate proportional to the how much reactant is present (e.g. radioactive decay): dr = A chemical reaction with two reactants occurs at a rate proportional to the how much of both reactants are present: hitting a blue ball: blue area/total area = kr r 2 N A dr 1 = kr 1 R 2

9 Rates of change that are proportional to two things Probability of hitting a blue ball: A chemical reaction with only one reactant blue area/total area = r 2 N A occurs at a rate proportional to the how much reactant is present (e.g. radioactive decay): N dr A = = blue concentration kr A chemical reaction with two reactants occurs at a rate proportional to the how much of both reactants are present: dr 1 = kr 1 R 2

10 Rates of change that are proportional to two things Probability of hitting a blue ball: A chemical reaction with only one reactant blue area/total area = r 2 N A occurs at a rate proportional to the how much reactant is present (e.g. radioactive decay): N dr A = = blue concentration kr A chemical reaction with two reactants occurs at a rate proportional to the how much of both reactants are present: dr 1 = kr 1 R 2

11 Logistic equation in different contexts...

12 Rates of change that are proportional to two things

13 Rates of change that are proportional to two things Infectious disease: bsi (S=susceptible, I=infected)

14 Rates of change that are proportional to two things Infectious disease: bsi (S=susceptible, I=infected) Spread of rumour: bnh (N = not heard rumour, H = heard rumour)

15 Rates of change that are proportional to two things Infectious disease: bsi (S=susceptible, I=infected) Spread of rumour: bnh (N = not heard rumour, H = heard rumour) Spread of new words: bnu (use word or not).

16 Rates of change that are proportional to two things Infectious disease: bsi (S=susceptible, I=infected) Spread of rumour: bnh (N = not heard rumour, H = heard rumour) Spread of new words: bnu (use word or not). Spread of new technologies: bnu (use tech or not).

17 Rates of change that are proportional to two things Infectious disease: bsi (S=susceptible, I=infected) Spread of rumour: bnh (N = not heard rumour, H = heard rumour) Spread of new words: bnu (use word or not). Spread of new technologies: bnu (use tech or not). Active oil exploration sites: bud (undiscovered and discovered).

18 Rates of change that are proportional to two things Infectious disease: bsi (S=susceptible, I=infected) Spread of rumour: bnh (N = not heard rumour, H = heard rumour) Spread of new words: bnu (use word or not). Spread of new technologies: bnu (use tech or not). Active oil exploration sites: bud (undiscovered and discovered). Waterlillies in a pond: bsw (waterlillies and space for waterwillies).

19 ...two things that are just different forms of a single thing

20 ...two things that are just different forms of a single thing When X meets Y, there s a chance Y turns into X.

21 ...two things that are just different forms of a single thing When X meets Y, there s a chance Y turns into X. Lose Y: dy = bxy and gain X: dx = bxy

22 ...two things that are just different forms of a single thing When X meets Y, there s a chance Y turns into X. dy Lose Y: and gain X: = bxy dx = bxy X+Y= constant = C so Y=C-X.

23 ...two things that are just different forms of a single thing When X meets Y, there s a chance Y turns into X. dy Lose Y: and gain X: = bxy dx = bxy X+Y= constant = C so Y=C-X. dx = bx(c X)

24 Dr. Erin Mears: Once we know the R0, we'll be able to get a handle on the scale of the epidemic. Minnesota Health #4: So, it's an epidemic now. An epidemic of what? Dave: We sent samples to the CDC. Dr. Erin Mears: In seventy two hours, we'll know what it is, if we're lucky. Minnesota Health #4: Clearly, we're not lucky.

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26 N individuals, I of them have a flu, S=N-I do not.

27 N individuals, I of them have a flu, S=N-I do not. If everyone interacts, new cases appear at a rate proportional to SI.

28 N individuals, I of them have a flu, S=N-I do not. If everyone interacts, new cases appear at a rate proportional to SI. The DE describing the spread of disease:

29 N individuals, I of them have a flu, S=N-I do not. If everyone interacts, new cases appear at a rate proportional to SI. The DE describing the spread of disease: (A) = bi(n I) (C) ds = bsi (B) = bi(n I) (D) = bsi (assume b>0)

30 N individuals, I of them have a flu, S=N-I do not. If everyone interacts, new cases appear at a rate proportional to SI. The DE describing the spread of disease: (A) = bi(n I) (C) ds = (B) = bi(n I) (D) = bsi dp = rp P Compare this with 1. K bsi

31 What is the carrying capacity? = bi(n I) (A) b/n (C) I (B) N/b (D) N

32 What is the carrying capacity? = bi(n I) (A) b/n (C) I (B) N/b (D) N dp = rp 1 P K

33 What is the carrying capacity? = bi(n I) = bni 1 I N (A) b/n (C) I (B) N/b (D) N dp = rp 1 P K

34 What is the carrying capacity? = bi(n I) = bni { r 1 I N (A) b/n (C) I (B) N/b (D) N dp = rp 1 P K

35 What is the carrying capacity? = bi(n I) = bni 1 { (A) b/n (C) I r I N { K (B) N/b (D) N dp = rp 1 P K

36 What is the carrying capacity? = bi(n I) = bni 1 { (A) b/n (C) I r I N { K (B) N/b dp = rp 1 (D) N P K Everyone gets sick!

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38 Suppose infected people recover at a rate proportional to how many there are.

39 Suppose infected people recover at a rate proportional to how many there are. The DE describing the spread of disease with recovery: (A) (C) = bi(n I) µs = bi(n I)+µI (B) (D) = bi(n I) µi = bi(n I)+µI

40 = bi(n I) µi

41 = bi(n I) µi = bin bi 2 µi

42 = bi(n I) µi = bin bi 2 µi = bi N µ b I

43 = bi(n I) µi = bin bi 2 µi = bi N K µ b I

44 = bi(n I) µi = bin bi 2 µi If µ b >N then = bi N µ b I K = N µ b < 0 K

45 = bi(n I) µi = bin bi 2 µi If µ b >N then = bi N K µ b I K K = N µ b < 0 I 0 N I

46 = bi(n I) µi = bin bi 2 µi If µ b >N then = bi N K µ b I K K = N µ b < 0 I 0 N I and the disease dies out.

47 = bi(n I) µi = bin bi 2 µi = bi N K µ b I

48 = bi(n I) µi = bin bi 2 µi If µ then b <N K = N µ b > 0 = bi N µ b I K

49 = bi(n I) µi = bin bi 2 µi If µ then b <N K = N µ I 0 b > 0 = bi N µ b I K N I K

50 = bi(n I) µi = bin bi 2 µi If µ then b <N K = N µ I 0 b > 0 = bi N µ b I K N I K and the disease becomes an epidemic.

51 = bi(n I) µi = bin bi 2 µi If µ then b <N K = N µ I 0 b > 0 = bi N µ b I K N I K and the disease becomes an epidemic. R 0 = Nb µ > 1

52 = bi(n I) µi = bin bi 2 µi If µ then b <N K = N µ I 0 b > 0 = bi N µ b I K N I K and the disease becomes an epidemic. R 0 = Nb µ > 1

53 = bi(n I) µi = bin bi 2 µi If µ then b <N K = N µ I 0 b > 0 = bi N µ b I K N I K and the disease becomes an epidemic. R 0 = Nb µ > 1