April 29 2013 I Divorce a Diffusion helps us know how information attitudes behavioral choices can spread through a population b Can continue this sort of analysis to the next generation c After 5 generations get table p 357 i Divorce rate drops for each generation to 10 in Generation 5 ii This result depends of course on the made up numbers for divorce rates contagion etc d Algebra supplied table p 358 i Algebra simplified because there was 100 rate for one and 0 rate for one ii Di divorce rate in generation 1 iii a divorce rate for couples having one divorced parent e Steady State Equilibrium i same divorce rate from one generation to another ii Algebra p 359 leads to these possibilities 1 D rate is zero 2 D rate is 100 3 A is 50 iii Furthermore 1 2 If a 50 D rate approaches 0 If a 50 D rate approaches 100 Epidemiology how disease bad habits spread through the population a Two key questions II i What is the rate of diffusion ii What is the pattern of diffusion b Mostly just have models of the rate though c Example With swine flu epidemiologists noticed that they should give immunizations to college campuses because college students are very mobile with good immune systems who feel fine but will spread the flu d Rate i Can get overall rate by just dividing distance etc by time ii But more often have different rates at different points during a trip Graph p 362 Note different slopes iii Delta difference iv Rate distance etc time v If you graphed change in distance over change in time you would probably not get a straight line like you did in fourth grade vi You have different rates at different points in time of a diffusion process different rates at beginning middle and time vii Delta stands for change III Model of Information Explosion Birth Model a Problem is diffusion of info about new boss b Assumptions i Each hour every person who has the info tells 3 people who don t have it c First Results ii Everyone who hears it believes it i At first Carter knows ii Hour 1 Carter tells A B C total 4 iii Hour 2 each of the 4 tells 3 people total 4 4x3 16 d N as a function of T i Graph p 367 n vertical t horizontal axes ii Slow horizontal at first then moves sharply upward iii Rate gets higher and higher faster and faster Snowballs iv P 368 roughly same shape graph if do n t vertical v t horizontal v n t 3n 1 Where does the 3 come from vi More generally n t an 1 a depends on transmission speed and fraction believing the information vii All diffusion curves for this model have roughly the same shape e Limits i The analysis so far birth model would lead to more people knowing the information than there are people ii Various sorts of things limit the diffusion process iii For ex as diffusion continues it becomes harder to find someone who doesn t already know f Birth Model with Limits into account i Taking the growth limit eg number of people in the organization ii N growth limit n number who have the info at the start of the time period t time period a diffusion rate 1 n t a n N n iii NB really only n is changing through the diffusion process g More Limits i n t a n N n ii Common sense observations 1 N will be larger than n until they are equal 2 The a n term will be small at first 3 The N n term will be large at first 4 Equation is a times number who know it times the number who don t iii A worked ex p 373 know how to do this example iv The graph p 374 Called an ogive oh jive it looks like a flat S number of people who know is vertical time is horizontal axis 1 Growth in number who know accelerates with the fastest growth where half of them know 2 Flattens out at end when targets are rare
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