11. Broadcast, Diffusion, and Contagion
As contagion of sickness makes sickness, contagion of trust can make trust.
—Marianne Moore
In this chapter, we model the spread of information, technologies,
behaviors, beliefs, and diseases throughout a population using models of
broadcast, diffusion, and contagion. These models play central roles in
communication, marketing, and epidemiology. All three models partition
the population into people who know or have some thing and those who do
not. Over time, people move between those two groups. Someone moves
from being susceptible to a disease to being infected, or from being
uninformed about a new product or idea to being informed.
Empirical plots of the number of people who over time catch a disease, buy
a product, or know a piece of information (the adoption curve) tend to be
either concave or S-shaped. How people learn the information or catch the
disease—that is, whether it spreads by broadcast or diffusion—determines
the shape of that graph. One contribution of this chapter will be to link the
micro-level processes of how ideas and diseases spread to the shape of
these adoption curves. The chapter begins with an analysis of the broadcast
model, which applies when people hear of an idea or catch a disease from a
single source. This model produces plots with an r-shape. We then cover
the diffusion model, in which spread occurs from contact, as when a
disease spreads from person to person. This model produces an S-shaped
curve.
Many products, programs, ideas, and pieces of information spread by both
broadcast and word of mouth. We can model these environments by
allowing for both broadcast and diffusion. The resulting model, known as
the Bass model, plays a central role in marketing. Whether it produces
more of an r-shape or S-shape depends on the strengths of the two
processes. The last model we cover, the SIR model of contagion from
epidemiology, includes a rate of recovery. This assumption could capture
an immune system fighting off a disease, behaviors or styles dropping out
of fashion, or information becoming less worthy of passing on to others.
The SIR model produces a tipping point, where small changes in the
attributes of the product or a disease spell the difference between failure
and success. A slight reduction in virulence can transform a mass infection
into a minor outbreak. A small increase in the probability of spreading
word of a hot new band can be the difference between the Beatles and a
band that played pubs in Liverpool for a few months in the 1960s.
The Broadcast Model
All of the models we cover in this chapter assume a relevant population,
denoted by NPOP. This consists of those people who could potentially catch
the disease, learn the piece of information, or adopt the product. The
relevant population is not the entire population of, say, a city or country. If
we are modeling the spread of a continuous aortic suture method, the
relevant population is heart surgeons, not everyone in the city of
Philadelphia.
At any moment in time, some people have the disease, know the
information, or adopt the behavior. We refer to these people as either the
infected or the informed (denoted by It). The remaining members of the
relevant population are susceptible (denoted by St). These people could
catch the disease or learn the information or behavior.1 The relevant
population equals the sum of the number of people infected (or informed)
plus the number of susceptible people: NPOP = It + St.
Broadcast Model
It+1 = It + Pbroad · St
where Pbroad denotes the broadcast probability, and It and St equal the
number informed and susceptible at time t.
Initially, I0 = 0 and S0 = NPOP.
The broadcast model captures the spread of ideas, rumors, information, or
technologies through media like television, radio, or the internet.
Knowledge of most current events spreads through broadcast. The model
captures processes in which a source, which could be the government, a
corporation, or a newspaper, spreads information. It could also capture
contaminations that spread through a water supply. The model does not
apply to diseases or ideas that spread from person to person. As the
broadcast model better fits the spread of ideas and information than disease,
we refer to the number of people informed, as opposed to the number
infected.
The number of informed people in a given time period equals the number
informed in the previous period plus the probability that a susceptible
person hears of the information multiplied by the number of susceptible
people (see box). By convention, the initial population contains only
susceptible people. Calculating the number of informed people in all future
periods involves plugging the number of informed and susceptible people
into the difference equation. The result will be an r-shaped adoption curve.
Imagine that the mayor of a city with 1 million residents announces a new
tax policy. Prior to the announcement, no one could have known about the
policy. If we assume the probability that someone hears the news on any
given day equals 30% (Pbroad = 0.3), then 300,000 people hear about it the
first day. On the second day, 30% of the remaining 700,000 people, or
210,000, hear about it. In each period, the number of informed people
increases and does so at a decreasing rate, as shown in figure 11.1.
image
Figure 11.1: The r-Shaped Adoption Curve Produced by the Broadcast
Model
In the broadcast model, everyone in the relevant population learns the
information or buys the product. Using initial sales data, we can therefore
estimate the relevant population size. Suppose that a company introduces a
new line of shoes for people who practice tai chi, and in the first week, it
receives orders for 20,000 pairs of shoes. If in the second week it receives
orders for 16,000 pairs, we can make a crude estimate of eventual total
sales, the size of the relevant population, to be 100,000.
Fitting the Broadcast Model to Data
Period 1: I1 = 20, 000 = Pbroad · NPOP
Period 2: I2 = 36, 000 = 20, 000 + Pbroad · (NPOP − 20, 000)
Solution:2 Pbroad = 0.2 and NPOP = 100, 000
We should not have a great deal of confidence in any estimate based on two
data points. The model leaves out any number of real-world features.
People might be hearing by word of mouth as well as through media, some
people may have bought multiple pairs, or advertising may have targeted
likely buyers. Including these features would change the estimates. Caveats
aside, the model provides a rough estimate. The firm should not expect to
sell exactly 2 million pairs, but it should be confident that they will sell
more than 100,000 pairs. As more data arrives, the estimate can be
improved. If week three’s sales equals 13,000 pairs (the amount the model
predicts), then the firm can place more confidence in the initial prediction.
The Diffusion Model
Most diseases as well as information about many products, ideas, and
breakthroughs spread by word of mouth. The diffusion model captures such
processes. It assumes that when one person adopts a technology or catches
a disease, that person has some probability of passing it on to those with
whom she comes in contact. In the case of a disease, choice pays no role.
The probability a person catches the disease depends on factors such as
genetics, the virulence of the disease, and even the temperature. Malaria
will spread much faster during a hot, wet season than during a cold, dry
one.
The spread of a technology involves a choice on the part of the adopters, so
technologies that are more useful will be adopted with a higher probability.
We do not explicitly consider choice in the model, however. Therefore, the
hipness of the Apple Watch plays the same role as the virulence of a flu
strain.
We again emphasize the spread of information, so we will refer to people as
informed or uninformed. New people become informed if they meet an
informed person and the information spreads between them. These are two
distinct events that vary by context. People in cities may have higher
contact probabilities than rural people, and information with high salience
—say, news that aliens have landed—has a higher sharing probability than
news of the reintroduction of pretzel M&M’s. Thus, we write the diffusion
probability as the product of a contact probability and a sharing
probability. We can write the model in terms of the diffusion probability,
but when we estimate or apply the model, we must keep track of the two
probabilities independently.
The diffusion model assumes random mixing, that is, that any two people in
the relevant population are equally likely to make contact. This assumption
should raise a red flag. It may be an accurate assumption for a model of
disease spread in a preschool where children interact with high frequency.
It is problematic to apply it to a city-level population. People do not
randomly mix. People live and work in neighborhoods; they belong to work
teams, families, and social groups. Their interactions are primarily within
those groups. Remember, though, an assumption need not be accurate to be
part of a useful model. We proceed with the assumption but keep an open
mind toward changing it.
Diffusion Model
image
where Pdiffuse = Pspread · Pcontact.
In this model as well, in the long run everyone in the relevant population
learns the information. However, in this model, the adoption curve for the
diffusion model has an S-shape. Initially, few people are informed; I0 is
small. It follows that the number of susceptible people who meet an
informed person must also be small. As the number of informed people
grows larger, the number of contacts between informed and uninformed
people increases, producing larger increases in the number of informed
people. When nearly everyone in the relevant population is informed, the
number of newly informed people decreases, forming the top of the Sshape. Technological adoption curves often have this shape. For example,
adoption curves for hybrid seed in the last century vary by state (Iowans
adopted hybrid seeds faster than Alabamans), but all of the curves have an
S-shape.3
In the broadcast model, estimating the relevant population size from data is
straightforward. The initial numbers of adopters correlates strongly with the
relevant population. In contrast, estimating the size of the relevant
population using data from a diffusion model can be difficult. The same
increases in product sales could result from a large diffusion probability
among a small relevant population or a small diffusion probability among a
large population. Figure 11.2 shows data for two hypothetical smartphone
applications. On the first day, one hundred people bought each application.
For each of the next five days, the first application realizes both higher total
sales and larger increases in sales. Absent a model, we would probably
predict the first application to have the larger market. Fitting the model to
the two data streams shows the opposite to be true.
image
Figure 11.2: Two Adoption Curves for Sales of an Application
The first application fits a diffusion probability of 40% and a relevant
population of 1,000 people, while the second application corresponds to a
diffusion probability of 30% and a relevant population of 1 million people.4
Within a few days, we would come to realize that there is a larger relevant
population for the second application. Nevertheless, absent the model, we
would make the incorrect inference about total sales if we based it on just
five days of data.
When using the diffusion model to guide action, we unpack the probability
of diffusion as the product of the probability of sharing and the probability
of making contact. To increase the speed of an application’s sales, its
developer could increase the rate at which people meet or increase the
probability that they share information about the application. Changing the
first probability would be difficult. To increase the second probability, the
developer could provide incentives for signing up new buyers, which many
developers do. A game developer might give points within the game to
users who sign up new buyers. Though this would increase the speed of
diffusion, it would not affect total sales, at least not according to the model.
As mentioned, total sales equals the relevant population size, regardless of
the probability of sharing. Increasing the rate of sales produces no longterm effect.
Most consumer goods and information spread through both diffusion and
broadcast. Our next model, the Bass model, combines the two processes in
a single model.5 The difference equation in the Bass model equals the sum
of the difference equations from the broadcast model and the diffusion
model. The adoption curve of the Bass model will be more S-shaped the
larger the probability of diffusion. The adoption curves for televisions,
radios, cars, computers, telephones, and cell phones all are combinations of
r-shapes and S-shapes.
Bass Model
image
where Pbroad = probability of broadcast and Pdiffuse = probability of
diffusion.
The SIR Model
In the models that we have covered so far, a person who adopts a
technology never abandons it. That makes sense for the adoption of
technologies like electricity, the dishwasher, and television; we never
reverse our adoptions. That assumption does not hold for all things that
spread by diffusion. After we catch a disease, we recover. When we adopt a
fashion or fad, such as a particular style of dress or a dance step, we can
abandon it. Following convention, we refer to people who drop an adoption
as recovered. The resulting model, the SIR model (susceptible, infected,
recovered), occupies a central position in epidemiology.
Given the model’s origins and that recovery occurs more naturally in
diseases, we describe the model using the spread of a disease, such as
COVID. To avoid overcomplicating the mathematics, we assume that
people who recover reenter the susceptible pool, that being cured of the
disease does not create future immunity.
SIR Model
image
where Pspread, Pcontact, and Precover equal the probability of spreading the
disease, the probability of contact, and the probability of recovery.
Epidemiologists keep separate track of the probability of contact and the
probability of spreading, so we will as well. Contact rates depend on how
the disease passes from one person to another. HIV spreads through sexual
contact. Diphtheria spreads through saliva. Flu viruses spread through the
air. Thus flu has a higher contact probability than diphtheria, which has a
higher contact probability than HIV. Once contact occurs, the probability of
spread also varies. Pertussis (whooping cough) transfers to another person
more readily than COVID.
The SIR model produces a tipping point at what is known as the basic
reproduction number (R0), the ratio of the probability of contact times the
probability of spread to the probability of recovery. A disease with an R0
greater than 1 can spread through the population. Diseases with R0’s less
than 1 dissipate. In this model, the information, or in this case the disease,
need not spread to the entire relevant population. Whether or not it does
depends on the value of R0. Hence, government agencies like the Centers
for Disease Control rely on estimates of R0 to guide policy.6
R0: The Basic Reproduction Number
image
As shown in the table below, measles, which can spread through the air, has
a higher reproduction number than HIV, which spreads through sexual
contact and needle sharing.
image
Estimates of R0 do not assume that people change their behavior in
response to a disease. People responded to the COVID pandemic by
wearing masks, keeping distances from others, and avoiding large crowds.
If these actions lower the probabilities of contact and spread sufficiently,
they create an effective reproduction number which will be less than the
basic reproduction number. If the effective reproduction number, denoted
by Rt, falls below one, the virus will cease to spread. Quarantines offer a
sure fire way reduce Rt below one, but they are costly.7
Vaccines also stop the spread of disease. a vaccine exists, then vaccination
can prevent disease spread. Disease spread can be prevented even without
vaccinating everyone. The proportion of people who must be vaccinated,
the vaccination threshold, is given by the formula image which we can
derive from the model.8
The vaccination threshold increases with R0. To prevent the spread of polio,
which has an R0 of 6, the vaccine must cover image of the population. To
stop the spread of measles, which has an R0 of 15, the vaccine must cover
image of the population. The mathematical derivation of the vaccination
threshold provides guidance to policymakers. If too few people are
vaccinated, then the disease will spread, so governments vaccinate more
than the threshold amount estimated by the model. For diseases with high
basic reproduction numbers, such as measles and polio, governments try to
vaccinate everyone.
Some people worry about side effects of vaccines and choose not to
participate in vaccination programs. If these people constitute a small
percentage of the population, the vaccination of others prevents them from
catching the disease. Epidemiologists call this phenomenon herd immunity.
The people who choose not to get the vaccine free ride off the vaccinations
of others. We study free riding in greater detail later in the book.9
R0, Superspreaders, and Degree Squaring
The derivation R0, the basic reproduction number, assumes random mixing:
in each time step, individuals in the population randomly meet one another.
As noted above, the random mixing assumption may approximate airborne
diseases or diseases spread by touch, but it makes less sense for diseases
that spread through sexual contact.
If we embed the SIR model on a network, we see the importance of the
degree distribution to disease spread. Here, we compare a rectangular grid
network (a checkerboard)—where each node is connected to the nodes to
the north, south, east, and west—to a hub-and-spoke network where a hub
node connects to all other nodes.
Assume that a disease randomly occurs at a node. We set Pcontact = 1
within the network so that each person makes contact with everyone to
whom he is connected. In the next period, the disease potentially spreads to
each neighbor independently with a given probability corresponding to the
virulence of the disease.
First consider the rectangular grid network. In each period, the disease can
spread to any of the four nodes to the north, south, east, and west. If the
probability of the disease spreading exceeds image, we would expect the
disease to spread. If we look ahead one period, we see that if one new node
caught the disease, then that node has three possible neighbors who could
catch the disease. If two neighbors, those to the north and east of the
original node, caught the disease, then there exist five new nodes to which
the disease could spread. This network, in this case, does not have much of
an effect on the likelihood of disease spread.
image
Next, consider the hub-and-spoke network. The first node to get the disease
could be the hub or a spoke. If the hub catches the disease, then it could
spread the disease to any one of the spokes. We would expect the disease to
spread, even for a low probability of spreading. If a spoke caught the
disease, then the only possible node that could catch the disease is the hub.
And as we just learned, if the hub catches the disease, the disease will
spread even for low probabilities of spreading.
For the hub-and-spoke network, R0 is less informative because if the hub
catches the disease, the disease will spread. Epidemiologists refer to highdegree hub people as superspreaders. Superspreaders contributed to the
early spread of both HIV and SARS.10 A superspreader need not be
extremely social or well connected. A superspreader may have an
occupation—tollbooth operator, bank teller, dental hygienist—that puts him
in contact with people who belong to distinct social networks. Mary Mallon
(Typhoid Mary) worked as a cook in New York at the turn of the twentieth
century. She moved from family to family infecting each with typhoid
fever. Once discovered as the source, Mary was quarantined against her
will.
To derive the effect of high-degree nodes, we note first that a high-degree
node is both better able to spread the disease and more likely to catch it. A
person with three times as many friends as another will be approximately
three times as likely to catch the disease and able to spread it three times as
widely. His total contribution to the spread of the disease will therefore be
nine times that of the other. Thus, a node’s contribution to the spread of a
disease (or an idea) correlates with the square of the node’s degree. If node
A has a degree K times larger than node B, then node A will be K times as
likely to spread the disease and spread it to K times as many others as B. Its
total effect will be K2 times larger than B’s, a phenomenon known as
degree squaring.
One-to-Many
Though the SIR model was designed to examine the spread of diseases, we
can apply it to social phenomena that spread by diffusion and then fade:
books, songs, dance steps, phrases, websites, diets, and exercise regimens.
We can estimate probabilities of contact, spread, and recovery and basic
reproduction numbers in these contexts as well. The model implies that
small changes in these probabilities could spell the difference between
success and failure by moving the basic reproduction number above zero.
Success can hinge on what John Updike, in describing Ted Williams’s last
at-bat, called the “tissue-thin difference between a thing done well and a
thing done ill.”11 Suppose that you think up a new joke. Making the joke a
little bit funnier might push the basic reproduction number above 1 and
cause the joke to spread. The same logic applies to the stickiness of ideas.
If an idea sticks in people’s minds a little longer, the recovery rate will be
lower, increasing the basic reproduction number.
Not all cases lie on the threshold. The Beatles had enormous talent. Their
reproduction number surely exceeded 1 by a large amount. That is of
course conjecture. For current pop stars, we can use internet downloads to
estimate basic reproduction numbers. Pop star Justin Bieber had an
estimated R0 of 24, making him more virulent than the measles.12
In the SIR model, we derived two critical thresholds, R0 and the
vaccination threshold. These thresholds are contextual tipping points, at
which small changes in the environment (the context) have large effects on
the outcomes. These differ from direct tipping points, where small actions
at a particular moment in time forever alter the path of a system. Direct tips
occur at unstable points, such as when a ball is perched atop a hill. A small
push in either direction sends the ball down one side of the hill or the other.
That small push is a direct tip.13
At a contextual tipping point, a change in a parameter changes how the
system behaves. At a direct tipping point, the trajectory of future outcomes
takes a sharp turn. A kink, such as the first bend in the S-shaped adoption
curve produced by the diffusion model, satisfies neither definition of
tipping point. The kink in the adoption curve corresponds to the point
where the slope has maximal increase. At that point, the diffusion is well
under way. No tip occurs.
Figure 11.3 shows the number of users of Google+ in its first two weeks.14
A kink in the graph occurs six days after the release. By that time, the
process of diffusion was well under way. It is not the case that Google+
struggled early and that a direct tip occurred on day six, with the result that
within two weeks Google+ had over 16 million users. This conflation of
tips with sharp upturns leads to an overuse of the term tipping point.
Moments identified in the news media and on the internet as tipping points
rarely satisfy the formal definition.
image
Figure 11.3: A Kink (Not a Tip) in the Number of Google+ Users
We can also think of obesity as an epidemic. Though people cannot catch
obesity the way they might catch a cold, they can be socially influenced to
adopt behaviors that contribute to obesity.15 To reverse the obesity
epidemic we must lower its basic reproduction number, which can be
accomplished by decreasing the probabilities of contact or sharing or
increasing the probability of recovery. The SIR model applied to obesity,
school dropout rates, or crime is not better than economic or sociological
models. It is a different model, so it produces different explanations and
predictions. It also possibly points to different actions or policies. It
contributes to an ensemble of models that help us make sense of the world.
It is not a golden arrow that will solve the problem.
In applying models of broadcast, diffusion, and contagion to social
phenomena, we may find that some assumptions hold and others do not. In
the spread of a disease, each contact has an independent probability of
spreading the disease. In social domains, contagion may become more
likely with more exposure because adoption is a choice. We do not choose
the flu. We catch it. We do choose to buy tight-fitting jeans. As more
people wear tight jeans, we may become more likely to as well. Similar
logic applies to the choice to become involved in a social movement, to
adopt a new technology, or to get a tattoo. In these cases, as well as in the
contagion of beliefs or of trusting behavior, we may have to emend the
model to allow for the possibility that the probability of adoption per
exposure increases with more exposures.16 Such modifications are often
necessary when broadening the set of applications of a model.