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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

j∈N

i

(t)

(p

j

(t)(1 − β

j

(t)) + (1 − p

j

(t))) (1)

=

Y

j∈N

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

j∈N

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),β(t)

(p(t)) , (6)

where the ith component of F

δ(t),β(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

t−1

· · · 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),β(t)

p(t)) + v(F

δ(t),β(t)

(p(t)))

≤ αv(p(t)) + v(F

δ(t),β(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

t−1

) ∈ M

t

such that

the spectral radius of the product

r (M

t−1

· · · 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

t−1

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

j∈N

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

j∈N

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

j∈N

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.

REFERENCES

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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

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