## Spread Of Infectious Diseases, Consumer Goods Adoption, And Technology S Curves

Rohit Tripathi

What do predicting the spread of an infectious disease 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.

Scientists predicting the spread of a new flu virus and consumer goods companies forecasting the adoption of new food item ask similar questions. Both know about the S curve, but how can they draw an accurate S curve for a new category?  A mathematical model and tools can help draw adoption curves with some precision beyond just a random sketch on a whiteboard.

### The Bass Model for predicting S curves

The Bass Model is a handy tool for predicting sales of new products, especially when no comparable sales history is 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 sales of a category, rather than an individual brand or solution, and has been widely embraced by academics and industry professionals.

While most Bass Model applications are for durable consumer goods, there is an opportunity to extend the model to predict adoption curves for new technologies and 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 on advertising, product reviews, etc., while the imitators depend on their interactions with those who have previously adopted the product (either innovators or earlier imitators).

### 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 the 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 can predict the sales or the adoption at a future time (t+1) with the equation:

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

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

(You aren’t required to understand all these equations to use the model; however, for inquiring minds, I’ll list them equations at the end of this article.)

### Getting started with 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 judgments; it’s harder to estimate the two coefficients p and q.

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

If there are no good analogous products available, 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, maintains a database of p, q, and N across different categories.

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

Product managers and marketers can use the Bass Model to make 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 that can be refined over time as we get 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. If you anticipate a strong innovator coefficient for your offering, i.e., you expect there will an immediate uptick in adoption, choose an appropriate value of p from the table of p and q values.

However, if you feel that the category needs a lot of education and convincing, you may not have that many imitators. Thus, you may choose a lower value of imitator coefficient q from analogous products to put some boundaries on adoption rates.

To illustrate this further, say the product manager uses the p and q values for calculators, which have a p value of 0.145 and a q value of 0.495.

Comparing this with the adoption curve for average p and q values for any generic product, we get some boundaries. This analysis now allows the solution manager or field personnel to plan the appropriate capacity, marketing, or lead-generation events. These are adjusted over time with actual sales data.

### Advantages and limitations of the model

The advantages of the Bass Model are its ease of implementation, as only simple Excel skills are required, and relatively easy interpretation of results. The model has a small number of input parameters, 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, making it relevant for global companies and a global user base.

The model has limitations, however. It does not predict the first purchase or sale at time 0; one could argue this is a function of the awareness and buzz created before the introduction. Second, the model tends to bucket consumers into two rigid categories.

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

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

### The math

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

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

q = propensity to imitate; the likelihood of 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

Once you’re done calculating and ready to start marketing, learn “How To Talk To Your Customers, Not At Them.”

This article originally appeared on The Future of Customer Engagement and Commerce.