Before everyone is infected, the epidemic will then fall and come to an end
without any change in the infection or recovery parameters c and r.
Not everyone will catch the disease. Some people escape the disease
completely because they do not have an effective encounter with an infective.
The environment gradually becomes safer and safer for them because the
number of infectives decreases as they get over the disease and become immune
to it. Thus there are not enough new encounters to generate sufficient new
infectives to keep the disease on the growth path. Eventually, the infectives
almost disappear, and the population consists almost entirely of susceptible and
recovered. Applying this model to narratives: because not everyone is infected,
some people will say after an economic narrative epidemic that they never even
heard of the narrative, and they will be skeptical of its influence on the economy
even if the narrative is indeed very important to economic activity.
Which factors combine to spread a major disease that ultimately reaches a lot
of people (the total fraction of the population ever infected and recovered)? The
disease’s reach is determined by the ratio c/r. As time goes to infinity, the
fraction of people who have ever had the disease goes to a limit R∞ (called the
size of the epidemic) strictly less than 1. It follows directly from the first and
third equations that 
 Given the initial condition on the fraction of the
population initially infected I0 that 
, and because I∞ = 0, 1 = S∞ +
R∞, we have:
which provides the relationship between the ultimate number ever infected by
the disease and c/r. If we could choose c and r, we could make the size of the
epidemic R∞ anything we want between I0 and 100%. If we define “going viral”
as 
, then we see a viral event happening from I0 close to zero when 
.
If we multiply both parameters, c and r, by any positive constant a, then the
same three equations are satisfied by S(at), I(at), R(at).
Higher c/r corresponds to higher size of epidemic R∞, regardless of the level
of c or r, while higher c itself, holding c/r constant, yields a faster epidemic. For
an epidemic to get started from very small beginnings, when S is close to 1, c/r
must be greater than 1. Depending on the two parameters c and r, there can be
both fast and slow epidemics that look identical if the plot is rescaled. If we also

vary the ratio c/r, we can have epidemics that play out over days and reach 95%
of the population, or epidemics that play out over decades and reach 95% of the
population, or epidemics that play out over days and reach only 5% of the
population, or epidemics that play out over decades and reach 5% of the
population. But in each case, we can have hump-shaped patterns of infected that
on rescaling look something like the heavy line in Figure A.1.

Variations on the SIR Model
The Kermack-McKendrick SIR model is the starting point for mathematical
models of epidemics that have, over the better part of a century since, produced a
huge literature. Among the different versions, the basic compartmental model
has been modified to allow for gradual loss of immunity, so that recovereds are
gradually transformed into susceptibles again (the SIRS model).4 The SIR model
can also be modified so that an encounter between a susceptible and an infected
leads to an increase in exposed E, a fourth compartment who become infected
later (the SEIR model). The model has also been modified to incorporate partial
immunity after cure, birth of new susceptibles, the presence of superspreaders
with very high contagiousness, and geographical patterns of spread.
These models, with modifications appropriate for the disease studied, have
been useful for predicting the course of epidemics. For example, the SEIR model
has been modified to explain the spread of influenza geographically with the
assumption that the exposed but still asymptomatic are capable of long-distance
travel. Applying the model to influenza data and data on intercity volume of air
transportation, R. F. Grais and her coauthors found that their model helps explain
intracity and intercity time patterns of influenza outbreak.5
Another compartmental model example is a stochastic extension of an
SEIHFR model, where S is susceptible, E is exposed, I is infected, H is
hospitalized, F is dead but not buried, and R is recovered or buried. This model
has been fitted to data on African Ebola epidemics,6 and it takes into account
public efforts to stem contagion of the disease through hospitalization and proper
disposal of bodies.
The SEIHFR compartmental model has six compartments, but future models
of economic narratives might well benefit from even more compartments. For
example, a model for the spread of the technological unemployment narrative
(see chapter 13) might include separate compartments for unemployed and
infected and unemployed and uninfected, employed and infected and employed
and uninfected, as well as extra equations that come from conventional
economic models.
Economic models might also take inspiration from the medical literature on
co-epidemics to incorporate contagious economic narratives into economic
models. In a medical setting, a co-epidemic occurs when the progress of one