IEEE Circuits and Systems Magazine - Q4 2021 - 5

variolation, helping its increasing adoption until the development
of the smallpox vaccine in 1796. A latter milestone
of mathematical modeling of epidemics is materialized in
the studies of the 1849 and 1854 Cholera outbreaks in London
by William Farr [2] and John Snow [3], respectively.
However, it is not until the beginning of the 20th century
that differential and difference equations started
being adopted as tools to model and analyze the spread
of epidemic diseases. The very origin of this approach
can be found in a paper by William Heaton Hamer [4],
in which the British epidemiologist pioneered the use
of a nonlinear formula to model the rate of the contagion
process, proportional to the product between the
number of susceptible individuals and the number of
infectious individuals in the population. Such a modeling
approach has been formalized and popularized by
William Ogilvy Kermack and Anderson Gray McKendmodeling,
namely the susceptible-infected-susceptible
(SIS) and the susceptible-infected-removed (SIR) models
and the concept of epidemic threshold (and consequently,
phase transition).
A. Population SIS Model
In the SIS model, it is assumed that infected individuals
do not acquire immunity after recovery, and thus
become again susceptible to the disease. This model
became very popular to study recurrent and endemic
diseases, including sexually transmitted diseases, such
as Gonorrhea [8]. Here, we present the deterministic SIS
model in a continuous-time framework, i.e., employing
differential equations (as in the original contribution by
Kermack and McKendrick [5]). Note that, discrete-time
implementations of the model using difference equations
have also been extensively studied. More details
on discrete-time models can be found, in, e.g., [9].
Formally, a unit mass population is split between two
compartments: susceptible and infected. Let () [, ]
and () [, ] be the fraction of susceptible and infected
individuals in the population at time t 0$ rest
01!
xt 01!
,
spectively. The health state of the population evolves
according to two mechanisms: contagion and recovery,
illustrated in Fig. 1(a). Susceptible individuals may become
infected if they interact with infected individuals,
while infected individuals spontaneously recover, becoming
again susceptible to the disease. According to
the contagion mechanism, the total number of newly
infected individuals is a nonlinear expression proportional
to the number of susceptible individuals and to
the number of infectious individuals in the population
(as proposed in [4]), i.e., equal to
mstxt
() (),
(1)
where m 2 0 is a positive parameter termed contagion
rate. The total number of recoveries in the population
is proportional to the number of infected individuals,
yielding the term ()
nxt , where n 2 0 is the recovery rate.
Hence, the heath state of the population is described
st xt <
by the two-dimensional state variable [( ), ()], which
evolves according to the following system of ordinary
differential equations (ODEs):
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'
rick. In their seminal paper [5], and in two subsequent
works [6], [7], the two Scottish researchers laid out the
basis for the current mathematical theory of epidemic
o
st
xt
o
()
()
=
=
mn
mn
() ()
-+
-
stxt xt
stxt xt
() () ()
().
(2)
Note that the two equations are linearly dependent,
since the total mass of the population is preserved, that
Lorenzo Zino and Ming Cao are with the Faculty of Science and Engineering, University of Groningen, 9747 AG Groningen, The Netherlands
(e-mails: {lorenzo.zino, m.cao}@rug.nl).
FOURTH QUARTER 2021
IEEE CIRCUITS AND SYSTEMS MAGAZINE
5
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IEEE Circuits and Systems Magazine - Q4 2021

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