What do predicting the spread of an infectious diseases or a consumer good or a technology adoption like growth of mobile phones have in common? All of them follow the classical S curve model of adoption.

Everyone, especially the software product managers know about the S curve, the question is, how do does one draw an accurate S curve for their product category? How should product managers to the best of their ability draw the S curve that best fits their product category?

Similar questions are faced by scientists who are predicting the adoption of the new flu virus or the consumer goods companies predicting the adoption of new food category item. In this article I want to highlight a mathematical model and related tools that exist that help product managers draw these adoption curves with some precision beyond just a random sketch on a whiteboard.

### Bass Model for predicting S curves

The Bass Model is a handy tool for predicting sales for new products, especially when past comparable sales history is not available. The Bass Model was developed by Frank M. Bass in 1969 to study the diffusion of innovation in consumer products. This model is useful for predicting sale of a category and not that of an individual brand or solution, and has been widely embraced by both academics and industry professionals.

While most applications of Bass Model are for durable consumer goods, there is an opportunity to extend the model in predicting the adoption curves for new technologies and also new technology appliance categories.

The model assumes that the market consists of two types of consumers, innovators, and imitators. The innovators are intrinsic adopters and depend upon advertising; product reviews etc., while imitators depend upon their interactions with those who have previously adopted the product (can be both innovators and imitators from previous periods).

### Conceptual setup and mathematics behind the model

In a market consisting of N total consumers, who will ultimately adopt a product (or technology), in any given period there are:

- innovators (coefficient of innovators, p) who adopt technology independent of decisions of others, this usually decreases over time
- imitators (coefficient of innovators, q) who are influenced in their decision by other members, this usually increases with time
- For each period the total adoption is given by sum of innovators and imitators

Thus the three input parameters for the model are market size N, coefficient of innovators p, and coefficient of imitators q. Using these three inputs, we are now able to predict the sales or the adoption at a future time (t+1) given by the equation:

Sales or adoption at time (t+1) = p x (N-Q(t)) + q/N x Q(t) x (N-Q(t))

Plot this equation for a given value of N, p, q and you will get an S curve.

*(It is not required to understand all of these equations to use the model, however for the inquiring minds I do list these equations at the end of this article.)*

**Getting Started with Using the Model**

In order to start using the model for predicting sales over time, we need to know the three input variables, N, p, and q.

While, the total potential market N is often relatively easy to estimate from various sizing estimates, triangulations and management judgment; the harder to estimate inputs are the two coefficients p and q.

In case of product categories where there is no previous sales data available, the most popular approach of estimating coefficients p and q is via analogous products already in the market. Management judgment is needed is picking the right analogous products and often using a set of analogous products to check for sensitivities.

If however, there are no good analogous products available then we can use average values of p and q, based on values across a range of several categories. Professor Christophe Van Den Bulte, from The Warton School of Management maintain a database of p, q, and N across different categories, here is a sample set of such values

If sales data exists for reasonable time period for the product or technology category then p and q can be estimated using simple regression analysis.

Product managers and marketers can use the Bass Model to make some reasonable predictions about the growth potential for new innovations as they come to market. These predictions instead of being simple guesses or a gut feel can be based on quantitative inputs which can be refined over time as we get to know the actual sales results.

I assume as a product manager or marketer you have done some market sizing and have an idea of N, the total market segment for your product category, now if you anticipate a strong innovator coefficient for your offering, i.e. you expect there will an immediate uptick in adoption, thus choose an appropriate value of p from the table of p and q values.

However, if as the product manager you feel that the category needs a lot of education and convincing then you may not have that many imitators and thus choose a lower value of imitator coefficient q from analogous products. Thus, getting some boundaries on the adoption rates for the appliance.

To illustrate this further, say the product manager uses the p, q values for calculators. Calculators as seen from table on previous page had p value of 0.145 and q value of 0.495.

Comparing this with the adoption curve for average p, q values for any generic product (also from the table) we get some boundaries. This analysis now allows the solution manager or field personnel to plan the appropriate capacity or marketing or lead generation events. These are then adjusted as we go along with actual sales data.

### Advantages and limitations of the model

The advantages of Bass Model are its ease of implementation, simple Excel skills are required and relatively easy interpretation of results. The model has small number of input parameters thus making it easier for the product manager or marketer to start some initial analysis.

Finally, the model parameters can be compared across products and geographic markets, thus making it relevant for a global companies and global user base.

The model has its limitations, it does not predict the first purchase or sale at time 0, one could argue this is a function of the awareness and buzz that is created before the introduction and second the model tends to bucket the consumers in two rigid categories.

However, in absence of other meaningful data, this tool is better than just simple assumption on growth rates which may or may not be grounded in reality or analogous comparisons. Further, the model does not include disruptive or new technologies being introduced, however this can be adjusted based on resets in the modeling assumptions based on business judgment.

So there you go, now you have a tool to draw meaningful adoption curves and makes some real business decisions. If nothing else at least now you know what predicting the spread of infectious diseases and software product management has in common.

**Appendix:**

**Mathematical Setup of the Bass Model**

Total market potential is N, this is total across all time because there is a set population of customers for the category

p = propensity to innovate, likelihood someone adopts new technology based on external factors

q = propensity to imitate, likelihood adoption due to “word-of-mouth” or pressure from existing users

Q(t) = cumulative sales till time t

Remaining market potential at t = (N-Q(t))

Total number of innovators at time (t+1) = p x (N-Q(t))

Also, the existing total adopters, Q(t), will interact with the remaining (N-Q(t)), leading to

total of Q(t) x (N-Q(t)) interactions, Of these interactions, q/N result in imitations

Total number of imitators at time (t+1) = q/N x Q(t) x (N-Q(t))

Total sales at time (t+1), S(t+1) = New innovators + New imitators

= p x (N-Q(t)) + q/N x Q(t) x (N-Q(t))

% Adoption can be estimated by dividing the total sales by the total market size