Model Description
Models inspired by epidemiology and informed by the sociology of the spread dynamics, are applied to the diffusion of a specific scientific idea. Although the example will clearly not cover every
class of ideas, it will point to features of epidemic models that apply to idea diffusion. It will also
reveal features of these models that require modification, thereby producing more realistic candidate models.
At the onset of the spread of the idea, most of the population will be in the susceptible class (S), with a few individuals in the incubator class (E)—having been in contact with the idea—and a small number of adopters (I) manifesting it. These are the principal classes in the models below. In addition, inspired by the approaches of Daley and Kendall , also explored are models in which there may be competing and mutually exclusive ideas (e.g. where susceptibles are turned off from the idea and become skeptics or idea stiflers, represented by the class Z).
Furthermore, individuals may recover or become immune (R), and not manifest the idea again.
The total population is denoted by N(t), where N = S + E + I + Z + R. In the epidemic
models used in this study, the demographic dynamics are modeled by dN/dt = B(N) - µN, where B(N) is
referred to as the recruitment function. In our case, this denotes the arrival rate of new individuals susceptible to the idea, such as new graduate students starting in the field as well as other scientists who find the idea relevant for their research. The parameter µ>0 denotes the rate at which physicists stop using Feynman diagrams . Thus, the maximum value that 1/µcan take is the average lifespan of the idea within a generation of researchers in the relevant community.
Whenever B(N)>0 and µ>0, then the system in Fig. is said to have vital dynamics. If B(N) ≡ Λ>0, then
N(t) varies over time and approaches a stable fixed point, Λ/ µ, as t→∞, in other words, the community approaches its ‘‘carrying’’ capacity. The basic reproductive number, R0, is a measure of effectiveness of adoption.
*********** IGNORE THE .ODE FILE***********
Parameters:
Λ – Recruitment Rate
µ - Average lifetime of Idea
ε – Average idea incubation time
γ – Average recovery time
β- Per-capita S-I contact rate
δ- Per-capita E-I contact rate
b- Per-capita S-Z contact rate
l- S to Z transition probability given contact with skeptics
1-l – S to E transition probability given contact with skeptics
p- S to I transition probability given contact with adopters
$$\dot{S} = \Lambda - \beta S(I/N) - bs(Z/N) - \mu S \\$$ |
| (S)Population that is susceptible to the idea. |
$$\dot{E} = (1-p)\beta S(I/N) + (1-l) bS(Z/N) - \rho E(I/N) - \epsilon E - \mu E\\$$ |
| (E) Incubators of idea |
$$\dot{I} = p\beta S(I/N) + \rho E(I/N) + \epsilon E - \mu I\\$$ |
| (I) Idea adopters |
$$\dot{Z} = lbS(Z/N) - \mu Z$$ |
| (Z) Population of skeptics |
# way to fool XPP into cobwebbing # first I define a function that every other step evaluates # the map -- in the alternate steps, it just keeps the same # value so that it alternates between horizontal and vertical # jumps # # here is your map f(x)=r*x*(1-x) # # # the parameter scale sets the maximum value of x # for the logistic it is 1 # par scale=1 # g(x,y)=if(mod(t,2)<.5)then(f(x))else(y) # note that 't' is the iteration number 0,1,2,... # if t is even evaluate f otherwise keep the old y y(t+1)=g(x,y) x(t+1)=if(t==0)then(x)else(y) # note that x(t+2)=f(x(t)) so every other point is the map! par r=3.95 init y=0,x=.01 # always start y=0 # # add this plot to the graph to see the actual function # this assumes that you have chosen at least 100 iterates aux map=f(scale*t/100) aux st=scale*t/100 # some convenient settings for the graphics @ xlo=0,ylo=0,xhi=1,yhi=1 @ xp=x,yp=y # tell xpp that it is discrete and iterate 100 times @ meth=discrete,total=100 done
Biswas S, Arizona State University.
