Classroom Companion: Business

Box 18.4 The SIR Model
The simplest model of interactive 
games is similar to the model used in 
biological sciences to describe how epi-
demic diseases spread in a population, 
the SIR model (Murray, 
2002
).
The SIR model consists of three 
groups of individuals: Susceptible (S), 
Infected (I) and Recovered (R). This 
corresponds to B, P, and Q, respec-
tively, in the BPQ model. The flow of 
new infected individuals in the SIR 
model depends only on network 
effects—someone must infect you; the 
flow of recovered individuals does not 
depend on network effects—you 
recover independently of how anybody 
else recovers. The parameter β
−
1
denotes the time between contacts 
which is required for the transmission 
of the disease. The parameter γ
−
1
is the 
time it takes to recover from the dis-
ease. 
.
Figure 
18.8
 shows the SIR 
model. The resulting differential equa-
tions for the SIR model are:
dS
dt
SI


,
dI
dt
SI
I




dI
dt
I


The set of differential equations is non-
linear and does not have a closed-form 
solution; however, the solution is easily 
found by numerical integration. The 
most important conclusion is that, ini-
tially, the number of infected increases 
very slowly (as in the Bass model with 
only imitators) and then to increase 
very rapidly.
The spread of the COVID-19 pan-
demic follows the simple SIR model. 
Countries have implemented several 
countermeasures to reduce the spread 
of the disease. The differential equation 
shows that this is achieved by reducing 
the term βSI. Examples of countermea-
sures that reduce this product are:
5
Increased social distance and hand 
washing reduces β.
5
Isolation of particularly vulnerable 
people, curfews, and prohibiting 
many people to assemble in places 
where social distance cannot be 
upheld reduces S.
5
Isolating infected and possibly 
infected people reduces I.
The SIR model was published by A. G. 
McKendrick and W. O. Kermack in a 
series of papers in the period from 1927 
to 1933. The SIR model is the basis for 
more advanced compartmental models 
in epidemiology, such as the SIS model, 
MSIR model, and the SEIR model. 
The major differences between these 
models are the number of compart-
ments (user groups) and the interaction 
between them.
Compartmental models have 
inspired academics to develop similar 
models for the evolution of digital 
goods and services in the digital econ-
omy—the Bass model, the model with 
competition and churning, and the