1. The diffusion and adoption of new ideas and new products often follows S-shaped growth patterns.
What are the positive feedbacks that generate the initial exponential growth of a successful innovation, and what are the negative feedbacks that limit its growth?
The spread of rumors and new ideas, the adoption of new technologies, and the
growth of new products can all be viewed as “epidemics spreading” by positive feedback as those who have adopted the innovation “infect” those who have not.

2. ..those who have already adopted the product, idea or technology come into contact with those who have not, exposing them to it and infecting some of them with the idea, or the desire to buy the new product and further increasing the population of adopters.
Any situation in which people imitate the behavior, beliefs, or purchases of others, any situation in which people jump on the bandwagon, describes a situation of positive feedback by social contagion.

3. Of course, once the population of potential adopters has been depleted, the adoption (infection) rate falls to zero.

4. People in the relevant community come into contact at a rate of c people per person per day.
The total rate at which contacts are generated by the potential adopter pool is then c P.

5. The proportion of adopters in the total population, A/N, gives the probability that any of these contacts is with an adopter who can provide word of mouth about the innovation.
Finally, the adoption fraction, i, is the probability of adoption given a contact with an adopter.
AR = cP A/N i

6. Adoption Rate_AR = cP A/N i
Adoption_Rate_AR = Contact_Rate_c*Potential_Adopters_P
*Adopters_A/Total_Population_N
*Adoption_Fraction_i

7. Logistic Model

10. One of the flaws in the logistic model of innovation diffusion is the startup problem.
The logistic model cannot explain the genesis of the initial adopters. Prior to the first sales of the VAX minicomputer, the installed base was zero. When growth processes begin, positive feedbacks depending on the installed base are absent or weak because there are no or only a few adopters.
Initial growth is driven by other feedbacks outside the boundary of the simple diffusion models. There are several channels of awareness that can stimulate early adoption of new innovations besides word of mouth and related feedback effects that depend on the size of the adopter population. These include advertising, media reports, and direct sales efforts.

11. Frank Bass (1969) developed a model for the diffusion of innovations that
overcomes the startup problem. The Bass diffusion model has become one of the most popular models for new product growth and is widely used in marketing, strategy, management of technology, and other fields.
Bass solved the startup problem by assuming that potential adopters become aware of the innovation through external information sources whose magnitude and persuasiveness are roughly constant over time.
The original Bass model was introduced primarily as a tool for forecasting
sales of new products, and Bass did not specify the nature of the feedbacks at the operational level. The positive feedback is usually interpreted as word of mouth (social exposure and imitation) and the external sources of awareness and adoption are usually interpreted as the effect of advertising.

12. Total Adoption Rate_AR = a P + c P A/N i

13. Bass Model

15. Total Adoption Rate_AR = a P + c P A/N i
Positive feedback dominates when d(AR)/dA > 0; negative feedback dominates when d(AR)/dA < 0
Positive feedback dominates if c*i > a.

16. Total Adoption Rate_AR = a P + c P A/N i
When A/N is small, advertising is the primary driver of adoption. As the adopter population grows, the impact of advertising and other external influences falls while word of mouth becomes more important. For most successful innovations,
a << c*i and word of mouth quickly becomes the most important source of new adopters.

17. The principal assumptions of the Bass model are listed below
The principal assumptions of the Bass model are listed below. Of course, no list of what is omitted from any model is ever complete.
· The total population (market size) is fixed. The model omits both demographic considerations (births, household formation, deaths, etc.) and endogenous changes in market size as attributes of the product or innovation change (e.g. lower price should increase total market size). Attributes of the innovation include price, features, functionality and performance, delivery time, reliability and quality, image, switching costs, the availability of complementary assets, etc.; all these are omitted.
· The population is homogeneous. All adopters and potential adopters are the same; no heterogeneity in the propensity to adopt or responsiveness to advertising and word of mouth is considered.
· Adoption is permanent. Product discards and replacement purchases, abandonment of the idea, or disadoption are not considered. The model cannot capture the decline in use as new, better generations of the product are introduced, as people grow tired of the product, or as units wear out and are discarded.

18. · Advertising and word of mouth are the only sources of adoption
· Advertising and word of mouth are the only sources of adoption. This implies that all other feedback effects are omitted; if the parameters of the model are estimated from time series data on actual adoption, the strength of the advertising and word of mouth loops will be overestimated since they will capture the impact of all omitted feedbacks.
· Advertising effectiveness (the parameter a) is constant, implying both constant advertising expenditures and that the ads have constant impact. Attributes of the product or innovation (price, features, functionality and performance, delivery time, reliability and quality, image, switching costs, the availability of complementary assets, etc.) have no impact on adoption from advertising.
· Adoption via external influences (advertising) and via internal influences (word of mouth) are independent (additive). Exposure to advertising does not make a word of mouth encounter more persuasive, nor does prior word of mouth enhance the impact of ads.

19. · Contacts between potential and actual adopters occur at a constant rate and do not depend on either external demographics or endogenous responses to the attributes of the product. The product has no impact on social networks and communication patterns in the population.
· The persuasiveness of word of mouth contacts between potential and actual adopters is constant, and does not depend on the attributes of the product (e.g. price, features, functionality and performance, delivery time, reliability and quality, image, switching costs, the availability of complementary assets, etc.).
· All adopters generate favorable word of mouth. There are no unhappy adopters or former adopters who generate unfavorable word of mouth or work against the adoption of the product or innovation.
· No competition is considered. The availability and attractiveness of competing products or ideas is assumed to be constant. The model cannot represent changing market share for the product or idea.

20. · There are no delays in the adoption decision either from advertising or from word of mouth. People exposed to advertising or word of mouth do not go through a process of deliberation or search for the best supplier, but instantly adopt the product.
· The model does not explicitly represent the stock of product in the hands of the adopter population (indeed, the model doesn’t distinguish between the adopters and the items they are adopting, implicitly assuming a constant number of items per adopter).
· The model assumes the product can be acquired instantly, and that demand always equals supply (the decision to buy the product is always matched by the ability of suppliers to provide it). There is no distinction between the decision to adopt, the search for the best vendor, the placing of orders, and the delivery of the product. These delays are often substantial, and capacity constraints (e.g. production capacity limitations) during the growth phase often constrain shipments. The model does not represent the capacity of the suppliers to deliver the product; implicitly the market clears instantly.

21. Without advertising, the tipping point defining an epidemic of adoption for the fad product, is defined by cid > 1, assuming everyone is a potential adopter (the product is new and no one is immune to the fad). Given the parameters, the threshold value of the adoption fraction i is i > 1/cd = 1/(100*1) = 0.01
Thus the fast case with i = is well above the tipping point, causing a highly successful fad.
The fad ends not when every potential adopter has purchased the product, but when the pool of potential adopters drops so low that the reproduction rate cid(P/N) falls below one.
With the contact number cid = 2.5 very few people are left without the product in their closet at the end of the fad, and even these few remaining non-adopters continue to adopt through advertising at the rate of 1%/year.