IEEE Circuits and Systems Magazine - Q4 2021 - 6

is, () ()
st xt 0+=
oo
. Hence, the system of ODEs in Eq. (2)
can be reduced to the single nonlinear ODE:
() (()) () (),
xto =- -
mn1 xt xt xt
(3)
from which one can immediately observe that the domain
[, ]01 is positively invariant. From the analysis
of this equation and its equilibrium points, one can
conclude that, depending on the model parameters
m and
n , two outcomes may occur: either the origin
(disease-free equilibrium) is the unique equilibrium of
the system in [, ]01 and it is globally asymptotically
stable; or the origin becomes unstable, giving rise
to a (almost) globally asymptotically stable endemic
equilibrium
x (, ].01!)
These two behaviors are illustrated
in Fig. 2. The phase transition between these
two behaviors occurs when a certain relation between
S
µ
λ
(a)
I SRI
µ
λ
(b)
Figure 1. Schematics of the (a) SIS and (b) SIR models.
x(t)
0.2
0.4
0.6
02 4 68 t
(a)
x(t)
0.2
0.4
0.6
02 46 8
(b)
t
where the three equations are linearly dependent,
since () ()
st xt rt 0++ =
oo o
()
Figure 2. Two trajectories with different initial conditions of
the population SIS model in Eq. (3) with (a)
mn / = 0.5 (i.e.,
below the epidemic threshold) and (b) mn / = 2 (i.e., above
the epidemic threshold). Below the epidemic threshold, both
trajectories converge to the disease free equilibrium; above
the epidemic threshold, they both converge to the endemic
equilibrium x =)
0.5 denoted by the black dashed line.
6
IEEE CIRCUITS AND SYSTEMS MAGAZINE
. Hence, the system of ODEs
in Eq. (4) reduces to a planar system. The analysis of
such a planar system (see, e.g., [12]) shows that the ratio
/mn also plays an important role in the SIR model,
modulated by the initial fraction of susceptible individuals
s(0). If ()/
dividuals exponentially converges to 0. If ()/
mns 011 , then the fraction of infected inmns
012 ,
FOURTH QUARTER 2021
the model parameters is satisfied, which is called epidemic
threshold. In this survey, we opt for expressing
such a threshold in terms of the ratio between the
contagion rate m and the recovery rate .n The following
result from [5] fully characterizes the behavior of
the population SIS model.
Theorem 1. If the population SIS model in Eq. (3) satisfies
/1#mn , then the disease free equilibrium x = 0 is
globally asymptotically stable. Otherwise, if /12mn , the
disease-free equilibrium is unstable and Eq. (3) has a (almost)
globally asymptotically stable endemic equilibrium,
corresponding to
x 1 nm=-)
/.
The epidemic threshold is often associated with
the basic reproduction number R ,0
that is, the average
number of contagions that a single infected person will
cause in a population of susceptible individuals. The
concept of basic reproduction number, although already
touched upon in the original works on the SIS
model [5]-[7], was not formally introduced until the work
by George MacDonald in the early 1950s [10]. For the SIS
model in Eq. (3), in fact,
R0 mn= /. However, in more complex
models, e.g., those including additional contagion
mechanisms, stochasticity, heterogeneity, and dynamic
network structures, the basic reproduction number
may not be sufficient to characterize the behavior of the
system, which may depend on other factors such as its
variability [11]. For this reason, in this survey we prefer to
present the results in terms of the epidemic thresholds.
B. Extensions and Limitations
of Population Epidemic Models
In the population SIR model (see Fig. 1(b)), which was proposed
by W. O. Kermack and A. G. McKendrick in their
seminal works with the SIS model, a further compartment
named removed is introduced to account for individuals
that recover from the disease and become immune, or
die; () [, ]
rt 01!
(with () ()
* o
o
o
is the fraction of population in the removed
state. Hence, the heath state of the system is defined
by the three-dimensional state [( ), (),( )]
st xt rt 1++ = ), which evolves according to
st xt rt <
()
the following system of ODEs:
st
xt
rt
()
()
()
=
=
=
- m () ()
() () -
(),
mn
n
stxt
stxt xt
xt
()
(4)
IEEE Circuits and Systems Magazine - Q4 2021

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