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Spread of epidemics in time-dependent networks

Authors:

Abstract

We consider SIS models for the spread of epi-demics. In particular we consider the so called nonhomogeneous case, in which the probability of infection and recovery are not uniform but depend on a neighborhood graph which describes the possibility of infection between individuals. In addition it is assumed, that infection, recovery probabilities as well as the interconnection structure may change with time. Using the concept of the joint spectral radius of a family of matrices conditions are provided that guarantee robust extinction of the epidemics.
Spread of epidemics in time-dependent networks
V. S. Bokharaie, O. Mason and F. Wirth
Abstract We consider SIS models for the spread of epi-
demics. In particular we consider the so called nonhomogeneous
case, in which the probability of infection and recovery are not
uniform but depend on a neighborhood graph which describes
the possibility of infection between individuals. In addition it is
assumed, that infection, recovery probabilities as well as the
interconnection structure may change with time. Using the
concept of the joint spectral radius of a family of matrices
conditions are provided that guarantee robust extinction of the
epidemics.
I. INTRODUCTION
Mathematical modelling of the spread of diseases is a
classical topic in mathematical biology, [1], [2]. Interestingly,
in recent times models of disease spread have been applied
by the computer science community to model the spread of
malignant software in computer networks, [3], [4], [5], [6],
[7]. This has the benefit that in many cases the spread of
computer viruses is much better documented than the spread
of viruses in biological populations.
The modelling of epidemics distinguishes two cases of
fundamental difference. In one case the state of an individual
may change from susceptible to infected and after recovery to
susceptible again, the corresponding acronym is SIS models.
On the other hand the change of states may be evolve from
susceptible to infected to recovered with the assumption that
recovered individuals cannot become infected again (SIR
models). In this paper we concentrate on SIS models.
In general, there are several approaches to modelling
SIS epidemics. These range from discrete-time models and
ordinary differential equations to Markov chain models. Most
notably however the difference between homogeneous and
nonhomogeneous models has been widely discussed in recent
years. In homogeneous models a basic assumption is that
infection probabilities are uniform over the whole population
and the same assumption is made for recovery probabilities.
While this leads to appealingly simple models the drawback
to be expected is that too much is oversimplified. In non-
homogeneous models a graph is introduced that describes
the interaction between individuals and the possibilities of
one individual infecting another. Indeed the influence of the
graph structure on the dynamics of the infection is well
documented, [4], [5].
This work is supported by the Irish Higher Education Authority (HEA)
PRTLI Network Mathematics grant, by Science Foundation Ireland (SFI)
grant 08/RFP/ENE1417.
V. S. Bokharaie is with Hamilton Institute, National University of Ireland,
Maynooth, Ireland vahid.bokharaie@nuim.ie
O. Mason is with Hamilton Institute, National University of Ireland,
Maynooth, Ireland oliver.mason@nuim.ie
F. Wirth is with Institut f
¨
ur Mathematik, Universit
¨
at W
¨
urzburg, W
¨
urzburg,
Germany wirth@mathematik.uni-wuerzburg.de
The paper is organized as follows. In the next Section II
we provide a description of the problem being considered,
develop a discrete-time model for non-homogeneous disease
spread and present criteria for extinction and epidemic terms
of the joint spectral radius of a set of matrices. In Section
III, the corresponding analysis for continuous models is
described.
II. PROBLEM DESCRIPTION
We consider an undirected graph G = (V, E) of n = |V |
individuals. Each individual can either be healthy or infected.
Infections can spread in one time step among neighbors of
the graph. That is, if there is an edge (i, j) between two
vertices i and j and one of the vertices, say i, is infected,
then there is a certain probability β that in the next time
step j will be infected. Conversely, there is also a certain
probability δ that node i will be cured in the next time step.
In the following we develop a model for the evolution of
probabilities of infection in the graph. In contrast to results
in the literature we will not assume that the graph G, or δ
and β are fixed quantities but allow some time dependence.
We denote the set of neighbors of node i at time t by
N
i
(t). We assume that the process of infection between
neighbors is independent from the other probabilistic events.
The probability that node i is infected at time t is denoted
by p
i
(t). Following [3] the probability ζ
i
(t) that in the t-th
time interval node i is not infected by its neighbours is
ζ
i
(t) =
Y
jN
i
(t)
(p
j
(t)(1 β
j
(t)) + (1 p
j
(t))) (1)
=
Y
jN
i
(t)
(1 β
j
(t)p
j
(t)) , t N . (2)
Assuming independence of the probabilistic events that
can take place the time evolution of the health probability of
node i thus satisfies
1 p
i
(t + 1) = (3)
(1 p
i
(t))ζ
i
(t) + δp
i
(t)ζ
i
(t) +
1
2
δp
i
(t)(1 ζ
i
(t)) = (4)
1
2
δp
i
(t) + ζ
i
(t)
1 +
1
2
δ 1
p
i
(t)
, t N.
In (4) we have first a term that quantifies the probability
of not being infected at time t and not receiving infection
from t to t + 1, then we have a term characterising recovery
after being infected at time t and finally there is a term that
characterises recovery and reinfection from t to t + 1. In
this formula the factor 1/2 is somewhat arbitrary and would
depend on the particular sequence of events. We thus assume
Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems – MTNS 2010 • 5–9 July, 2010 • Budapest, Hungary
ISBN 978-963-311-370-7
1717
implicitly that in the time interval from t to t+1 the event of
recovery as well as the event of being infected by a neighbor
happens in a uniform manner.
To simplify the previous equations note that
ζ
i
(t)
1 +
1
2
δ 1
p
i
(t)
= (5)
1 +
1
2
δ 1
p
i
(t) +
X
jN
i
(t)
β
j
(t)p
j
(t) + o(kpk) .
Defining D(t) = diag
δ
1
(t) . . . δ
n
(t)
as the diagonal
matrix of recovery probabilities at time t and A(β) as the
weighted adjacency matrix of the graph of connections this
leads to the final formulation of the evolution of p(t) as
p(t+1) = (I D(t) + A(β(t))) p(t)+F
δ(t)(t)
(p(t)) , (6)
where the ith component of F
δ(t)(t)
(t) contains the higher
order products contained in (5) , i = 1, . . . , n and the index
denotes the dependence on the special choice of δ and A(β).
We note for later use, that the set of possible A(β) describing
the interconnection structure and infection probabilities is
bounded because it is contained in the set of nonnegative
matrices with entries in the interval [0, 1]. Thus for every
r > 0 there exists a uniform constant η (depending on r)
such that for all p, kpk r and all choices of δ, β we have
kF
δ,β
(p)k ηkpk
2
. (7)
Thus if we linearize the equation (3) in the equilibrium
of interest p
=
p
1
. . . p
n
T
= 0 the linear system
becomes
p(t + 1) = (I D(t))p(t) + A(β(t))p(t) . (8)
We now assume we have a set M R
n×n
+
of matrices
describing the possible matrices that can appear in (8). It is
reasonable to assume, that
M = I D + A ,
where D represents the set of possible recovery matrices
and A is the set of possible disease transmission matrices,
which are all the weighted adjacency matrices appearing. By
assumption the matrices in A have zero diagonal and entries
in [0, 1].
Associated to the set of matrices M is the joint spectral
radius, which is defined by
ρ(M) := lim
t→∞
sup{kM(t 1)M(t 2) . . . M(0)k
1/t
(9)
| M (s) M, s = 0, . . . , t 1} . (10)
Recall that ρ(M) < 1 is equivalent to the existence of
constants C 1, γ (0, 1) such that for all t N we
have
kM
t1
· · · M
1
M
0
k Cγ
t
for all M
s
M, s = 0, . . . , t .
(11)
Thus ρ(M) < 1 characterizes exponential stability of the
linear inclusion
p(t + 1) {Mp(t) | M M} .
It is known that this stability property is equivalent to the
existence of norms with respect to which all matrices M
M are contractions. This is the content of the following
theorem, [8].
Theorem 2.1: Let M R
n×n
be a nonempty compact set
of matrices, then ρ(M) < 1 is equivalent to the existence of
a norm v on R
n
and a constant α such that
v(M) α < 1 , for all M M .
This is also of relevance to the nonlinear system as the
next two results show.
Theorem 2.2: Assume that the joint spectral radius of M
satisfies
ρ(M) < 1 (12)
then p
= 0 is a locally exponentially stable fixed point of
(6).
Proof: The proof follows a classical approach of
linearization theory for stability. According to Theorem 2.1
we can pick a norm v on R
n
such that v(M) α < 1 for all
M M. Let C > 1 be a constant such that v(x) Ckxk
for all x R
n
. Using the norm v as a Lyapunov function
along the solutions of (6) we obtain for kp(t)k r and a
fixed r > 0 that
v(p(t + 1)) v(M
δ(t)(t)
p(t)) + v(F
δ(t)(t)
(p(t)))
αv(p(t)) + v(F
δ(t)(t)
(p(t)))
αv(p(t)) + Cηv(p(t))
If we choose r sufficiently small, so that α + Cη < 1, we
see that solutions are decaying no matter which choice of δ
and A(β) is applied. This shows the assertion.
In other words the condition ρ(M) < 1 guarantees that
infections will die out for all possible time evolutions of the
nonlinear system (3). Similarly, a criterion for epidemics is
Theorem 2.3: Assume that the joint spectral radius of M
satisfies
ρ(M) > 1 (13)
then p
= 0 is a locally exponentially unstable fixed point
of (3).
Proof: By the generalized spectral radius theorem, due
to Berger and Wang, [9], [8] if ρ(M) > 1 then there is a
finite sequence (M
0
, M
1
, M
2
, . . . , M
t1
) M
t
such that
the spectral radius of the product
r (M
t1
· · · M
1
M
0
) > 1 .
If we consider the corresponding sequence of choices for the
infection and recovery probabilities (δ
s
, A(β
s
)), s = 1, . . . , t
and consider the periodic nonlinear system (6) obtained by
periodically applying this sequence, then we obtain a system
which has the periodic linear system given by the matrices
M
0
, . . . , M
t1
as its linearization. As the linearization is
exponentially unstable, so is the fixed point p
= 0 for the
corresponding nonlinear system.
It should be noted that the content of the previous result
is that for certain choices of sequences (δ
s
, A(β
s
)), s N
the nonlinear system is unstable. This does not rule out the
possibility that for particular choices the fixed point is stable.
V. S. Bokharaie et al. • Spread of Epidemics in Time-Dependent Networks
1718
III. CONTINUOUS MODELS
The results that have been obtained in the previous section
translate readily to continuous time SIS models. Again p
i
(t)
denotes the probability that node i is infected at time t, but
now t R
+
. A model class of this type is given in [7] as
˙p
i
(t) = β(1 p
i
(1))
X
jN
i
p
j
(t) δ
i
p
i
(t) , (14)
where now the constant β may be interpreted as the infection
rate of the disease and δ
i
is the recovery rate of individual
i. As in the discrete time case we generalize this model to
the time-varying case by considering
˙p
i
(t) = (1 p
i
(1))
X
jN
i
(t)
β
ij
(t)p
j
(t) δ
i
(t)p
i
(t) , (15)
where we assume that the functions β
ij
, δ
i
, N
i
are suffi-
ciently regular to ensure existence of solutions. E.g. by
considering standard switching signals or by imposing a
measurability assumption. The linearization of (15) at p
= 0
is then given by
˙p
i
(t) =
X
jN
i
(t)
β
ij
(t)p
j
(t) δ
i
(t)p
i
(t) . (16)
which is easily seen to be the infinitesimal formulation of
the discrete-time systems described in (8). Formulated as a
system we obtain
˙p = (A(β(t)) D(t)) p(t) , (17)
where D(t) = diag
δ
1
(t) . . . δ
n
(t)
is the diagonal
matrix of recovery rates at time t and A(β(t)) as the
weighted adjacency matrix of the graph of connections.
We now assume we have a set M R
n×n
+
of matrices
that describe the possible matrices that can appear in (17).
It is reasonable to assume, that
M = A D ,
where D represents the set of possible recovery matrices and
A is the set of possible disease transmission matrices, which
are all the weighted adjacency matrices appearing.
As in the discrete time case, we can define the continuous
time-version of the joint spectral radius (also called maximal
Lyapunov exponent) ρ corresponding to the linear inclusion
˙p {Mp | M M} , (18)
see [10], [11]. The proof of the following two results is then
similar to that of the discrete time case. In particular, also in
the continuous time case it is known that exponential stability
of the inclusion (18) is equivalent to the existence of a norm
which serves as a Lyapunov function.
Theorem 3.1: Assume that the joint spectral radius of the
linear inclusion (18) satisfies
ρ(M) < 1 (19)
then p
= 0 is a locally exponentially stable fixed point of
the system given by (15).
Theorem 3.2: Assume that the joint spectral radius of the
linear inclusion (18) satisfies
ρ(M) > 1 (20)
then p
= 0 is a locally exponentially unstable fixed point
of the system given by (15).
IV. CONCLUSION AND FUTURE WORK
In this note we have presented a brief introduction into the
modelling of the spread of epidemics in time-varying situa-
tions. For certain cases the conditions on the joint spectral
radius can be significantly simplified. We will explore this in
future publications. Similarly, we plan to investigate related
Markov models.
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Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems – MTNS 2010 • 5–9 July, 2010 • Budapest, Hungary
1719
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