IEEE Circuits and Systems Magazine - Q4 2021 - 9

The convergence result in Theorem 2 can be easily extended
to heterogeneous SIS models.
of the population is thus represented by the n-dimensional
vector () [, ]
x t 01 n
!
. Similar to the scalar model,
the contagion rate in node i has a nonlinear expression,
equal to the product of the infection rate
mi , the fraction
sti
of susceptible individuals in the ith group (), and the
strength of the interactions with other infected individuals
in the population (),mti
which depends on the network
structure and is equal to
()
mt = / ax t
!V
iij
j
j(),
(5)
as illustrated in the simple example in Fig. 5. For the sake
of simplicity, in this section, we will present our main results
under the assumption that the graph
GV E (, ,)A
=
is strongly connected. Results for directed networks that
are not strongly connected can be found in [27], [28].
Hence, the network SIS model is governed by a set of
n ODEs, one for each node of the network. The equation
that determines the evolution of (),xti
,
for a generic node
i V! is the following:
() (())
xto mn (6)
!V
ii 1 iij
j
=-xt / ax tx tji i(),
() -
where im and in are the infection and recovery rate of
node i, respectively. While in most of the literature it
is assumed that the epidemic parameters are homogeneous
in the population, that is,
,
mm i
all
= and
nni
= for
,
i V! the general formulation in Eq. (6) with heterogeneous
parameter has been adopted to model and
study different scenarios, for instance, to model diseases
that affect different age groups differently, which is
key to implement and study targeted interventions and
vaccination policies.
Note that, after introducing the n-dimensional vectors
m and n to gather all the im and in , i V!
tively, the dynamics can be written in a compact, vector
form as
xx At Mttt (),
.
()=-Kdiag(( )) ()where
K diag(),( ),diagMmn
1 xx
(7)
xm = 0.4
== and 1 is an n-dimensional
all-1 vector from which is straightforward to observe
that the domain [, ]01 n
is positively invariant. The
following result, initially presented in [26], extends the
results for the population SIS model in Theorem 1 to the
network SIS model. Different techniques have been used
to prove this extension, including Lyapunov arguments
[29] and positive systems theory [28].
FOURTH QUARTER 2021
the disease free equilibrium x 0= is globally asymptotically
stable. Otherwise, the SIS dynamics in Eq. (7)
converge to the unique (almost) globally asymptotically
stable endemic equilibrium x)
. Note that this expression
reduces to Eq. (8) in the homogeneous scenario. A technique
, respecxh
= 0
h
2
m
2 . 0.6
Figure 5. Example of the contagion rate in a node of the
network. The contagion rate in node i is equal to the sum of
the contributions coming from all its neighbors. Specifically,
nodes j and k contribute to the rate, proportionally to the fraction
of infected individuals in the nodes and the connection
strengths; nodes h and m do not contribute: in h there are no
infected individuals, while m is not a neighbor of i.
IEEE CIRCUITS AND SYSTEMS MAGAZINE
9
5. 0
5
i
2
k
xk = 0.6
4 . 0.2
4
j
xj = 0.2
Theorem 2. Consider the homogeneous network SIS
model in Eq. (6) with imm= and inn= , for all i V! on
,
a strongly connected network. If
()
n
m
where ()At
#
1
t A
,
(8)
is the spectral radius of the (weighted) adjacency
matrix A, then the disease free equilibrium x 0= is
globally asymptotically stable. Otherwise, if // (),A12mn t
the disease-free equilibrium is unstable and Eq. (7) has
a unique (almost) globally asymptotically stable endemic
equilibrium x)
.
Different from the population SIS model, where the
endemic equilibrium x) has a closed-form expression
depending on the model parameters, for the network
SIS model such an explicit formula cannot be derived in
general. However, iterative algorithms to compute such
an equilibrium have been proposed in the literature.
See, e.g., [12], [30].
The convergence result in Theorem 2 can be easily
extended to heterogeneous SIS models, where nodes
have different contagion rates im and/or recovery rates
n i
, as shown in [26], [28], [31]. In the heterogeneous case,
the behavior of the system is determined by the spectral
radius of the matrix AMK -
. Specifically, if
t () ,AM 1#
K -
(9)
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