Original article Biomath 1 (2012), 1209255, 1–7

B f

Volume ░, Number ░, 20░░ 

BIOMATH

 ISSN 1314-684X

Editor–in–Chief: Roumen Anguelov  

B f

BIOMATH
h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / Biomath Forum

Multiregional SIR Model with Infection
during Transportation

Diána H. Knipl∗ and Gergely Röst†
∗Analysis and Stochastics Research Group – Hungarian Academy of Sciences,

Bolyai Institute, University of Szeged, Szeged, Hungary
Email: knipl@math.u-szeged.hu

† Bolyai Institute, University of Szeged, Szeged, Hungary
Email: rost@math.u-szeged.hu

Received: 12 July 2012, accepted: 25 September 2012, published: 18 October 2012

Abstract—We present a general epidemic model to
describe the spread of an infectious disease in several
regions connected by transportation. We take into account
that infected individuals not only carry the disease to a
new place while traveling from one region to another, but
transmit the disease during travel as well. We obtain that
a model structured by travel time is equivalent to a large
system of differential equations with multiple delays. By
showing the local Lipschitz property of the dynamically
defined delayed feedback function, we obtain existence and
uniqueness of solutions of the system.

Keywords-epidemic spread; transportation model; dy-
namically defined delay; Lipschitz continuity

I. INTRODUCTION

We consider an arbitrary n number of regions which
are connected by transportation, and present an SIR
based model which describes the spread of infection
in the regions and also during travel between them.
We show that our model is equivalent to a system of
functional differential equations

x′(t) = F(xt), (1)

where t ∈ R, t ≥ 0 and x : R � R3n. We use the nota-
tion xt ∈ C, xt(θ) = x(t + θ) for θ ∈ [−σ, 0], where for
σ > 0, we define our phase space C = C([−σ, 0], R3n) as
the Banach space of continuous functions from [−σ, 0]
to R3n, equipped with the usual supremum norm || · ||.
In the sequel we use the notation |v| for the Euclidean

TABLE I
VARIABLES AND KEY MODEL PARAMETERS (j, k ∈ {1, . . . n})

Sj , Ij , susceptible, infected, recovered, all

Rj , Nj individuals in region j

sk,j , ik,j , susceptible, infected, recovered, all

rk,j , nk,j individuals during travel from region k to j

Λj incidence in region j

λk,j incidence during travel from region k to j

αj recovery rate of infected individuals in region j

αk,j recovery rate during travel from region k to j

µj,k travel rate from region j to region k

τk,j duration of travel from region k to j

norm of any vector v ∈ Rm for m ∈ Z+. In order to
obtain the general existence and uniqueness result for the
system, we prove that F : C � R3n satisfies the local
Lipschitz condition on each bounded subset of C, that
is, for every M > 0 there exists a constant K = K(M)
such that the inequality |F(φ) − F(ψ)| ≤ K||φ − ψ||
holds for every φ,ψ ∈ C with ||φ||, ||ψ|| ≤ M.

The paper is organised as follows. In Section 2 we
introduce our model, then we obtain the compact form
of the system in Section 3. Section 4 concerns with the
proof of the local Lipschitz condition for various types
of incidence (new cases per unit of time).

Citation: D. Knipl, G. Röst, Multiregional SIR Model with Infection during Transportation, Biomath 1 (2012),
1209255, http://dx.doi.org/10.11145/j.biomath.2012.09.255

Page 1 of 7

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D. Knipl et al., Multiregional SIR Model with Infection during Transportation

II. MODEL DESCRIPTION

We formulate a dynamical model describing the
spread of an infectious disease in n regions and also
during travel from one region to another. Divide the
entire populations of the n regions into the disjoint
classes Sj, Ij, Rj, j ∈ {1, . . .n}, where Sj(t) Ij(t),
Rj(t), j ∈ {1, . . .n} denote the number of susceptible,
infected and recovered individuals at time t in region j.
For the total population in region j at time t, we use the
notation

Nj(t) = Sj(t) + Ij(t) + Rj(t).

The incidence in region j is denoted by
Λj
(
Sj(t),Ij(t),Rj(t)

)
, model parameter αj represents

the recovery rate of infected individuals in region j. We
denote the travel rate from region j to region k by µj,k
and we set µj,j = 0 for j,k ∈ {1, . . .n}.
Let sk,j, ik,j, rk,j denote susceptible, infected and
recovered travelers, where lower index-pair {k,j},
j,k ∈ {1, . . .n} indicates that individuals are traveling
from region k to region j. Let τk,j > 0 denote the
time required to complete the travel from region k to
region j, which is assumed to be fixed. To describe the
disease dynamics during travel, for each t∗ we define
sk,j(u; t∗), ik,j(u; t∗), rk,j(u; t∗), j,k ∈ {1, . . .n} as the
densities of individuals with respect to u who started
travel at time t∗ and belong to classes sk,j, ik,j, rk,j,
where u ∈ [0,τk,j] denotes the time elapsed since
the beginning of the travel. Then sk,j(τk,j; t − τk,j),
ik,j(τk,j; t − τk,j), rk,j(τk,j; t − τk,j) express the inflow
of individuals arriving from region k to compartments
Sj, Ij, Rj at time t, respectively. Let

nk,j(u; t∗) = sk,j(u; t∗) + ik,j(u; t∗) + rk,j(u; t∗)

denote the total density of individuals during travel from
region k to j, where j,k ∈ {1, . . .n}. The total density
is constant during travel, i.e. nk,j(u; t∗) = nk,j(0, t∗) for
all u ∈ [0,τk,j]. During the course of travel from region
k to j, λk,j

(
sk,j(u; t∗), ik,j(u; t∗),rk,j(u; t∗)

)
represents

the incidence, and let αk,j denote the recovery rate.
All variables and model parameters are listed in Table I.
Based on the assumptions formulated above, we obtain
the following system of differential equations for the

disease transmission in region j, j ∈ {1, . . .n}:


Ṡj(t) = −Λj(·) −

(
n∑
k=1

µj,k

)
Sj(t)

+
n∑
k=1

sk,j(τk,j; t − τk,j),

İj(t) = Λj(·) −

(
n∑
k=1

µj,k

)
Ij(t)

− αjIj(t) +
n∑
k=1

ik,j(τk,j; t − τk,j),

Ṙj(t) = αjIj(t) −

(
n∑
k=1

µj,k

)
Rj(t)

+
n∑
k=1

rk,j(τk,j; t − τk,j).

(Lj)

For each j,k ∈ {1, . . .n} and for each t∗, the following
system (Tk,j) describes the evolution of the densities
during the travel from region k to region j which started
at time t∗:




d
du
sk,j(u; t∗) = −λk,j(·),

d
du
ik,j(u; t∗) = λk,j(·) − αk,jik,j(u; t∗),

d
du
rk,j(u; t∗) = αk,jik,j(u; t∗).

(Tk,j)

For sake of simplicity, in systems (Lj) and (Tk,j) we
use the notations Λj(·) and λk,j(·) for the incidences,
where these functions are meant to be evaluated at
the appropriate points. For u = 0, the densities are
determined by the rates individuals start their travels
from region k to region j at time t∗. Hence, the initial
values for system (Tk,j) at u = 0 are given by


sk,j(0; t∗) = µk,jSk(t∗),

ik,j(0; t∗) = µk,jIk(t∗),

rk,j(0; t∗) = µk,jRk(t∗).
(IV Tk,j)

Notice that µj,j = 0 for j ∈ {1, . . .n} implies that
for each t∗, it holds that sj,j(u; t∗) = ij,j(u; t∗) =
rj,j(u; t∗) ≡ 0, as there is no travel from region j to
itself. Since travel from region k to region j takes τk,j
units of time to complete, we need to assure that there
exists a unique solution of system (Tk,j) on [0,τk,j] (see
Proposition IV.1).
Now we turn our attention to the terms sk,j(τk,j; t−τk,j),
ik,j(τk,j; t−τk,j), rk,j(τk,j; t−τk,j), j,k ∈ {1, . . .n} in

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D. Knipl et al., Multiregional SIR Model with Infection during Transportation

system (Lj), which give the inflow of individuals arriving
to classes Sj, Ij, Rj at time t, upon completing a travel
from region k. At time t, these terms are determined
by the solution of system (Tk,j) at u = τk,j with initial
values (IV Tk,j) for t∗ = t − τk,j, since individuals who
left region k with rate µk,j at time t − τk,j will enter
region j at time t.
Next we specify initial values for system (Lj) at t = 0.
Since for k ∈ {1, . . .n}, travel from region k to region
j takes τk,j units of time to complete, arrivals to region
j at time t are determined by the state of the classes of
region k at t−τk,j, via the solution of system (Tk,j) and
initial values (IV Tk,j). Thus, we set up initial values as
follows: 


Sj(θ) = ϕS,j(θ),

Ij(θ) = ϕI,j(θ),

Rj(θ) = ϕR,j(θ),
(IV Lj)

where θ ∈ [−τ, 0] for τ := maxj,k∈{1,...n} τk,j, moreover
ϕS,j, ϕI,j and ϕR,j are continuous functions for j ∈
{1, . . .n}.

III. THE COMPACT FORM OF THE SYSTEM

For each j,k ∈ {1, . . .n} and t∗ ≥ 0, we define
y(u) = yt∗k,j(u) = (sk,j(u; t∗), ik,j(u; t∗), rk,j(u; t∗))

T

and g = gk,j = (gS,gI,gR)T , where y : [0,τk,j] � R3,
g : R3 � R3 and

gS(y) = −λk,j(y1,y2,y3),
gI(y) = λk,j(y1,y2,y3) − αk,jy2,
gR(y) = αk,jy2.

Then for each j,k and t∗, system{
y′(u) = g(y(u)),

y(0) = y0
(2)

is a compact form of system (Tk,j) with initial
values (IV Tk,j) for y0 = (µk,jSk(t∗), µk,jIk(t∗),
µk,jRk(t∗))T . Let y(u, 0; y0) denote the solution of
system (2) at time u with initial value y0.
The feasible phase space is the nonnegative cone C+ =
C([−τ, 0], R3n+ ) of the Banach space of continuous func-
tions from [−τ, 0] to R3n with the sup norm. For every
j,k ∈ {1, . . .n}, let hk,j : R3n � R3 be defined by
hk,j = (hS,k,j, hI,k,j, hR,k,j)T , where

hS,k,j(v) = µk,jv3k−2,

hI,k,j(v) = µk,jv3k−1,

hR,k,j(v) = µk,jv3k.

For φ ∈ C+, we use the notation ŷφ(−τk,j )(u) =
y(u, 0; hk,j(φ(−τk,j))). Furthermore we define Wk :
C+ � R3n as

Wk(φ) =
(
ŷφ(−τk,1)(τk,1), . . . ŷφ(−τk,n)(τk,n)

)T
.

Let x(t) = (S1(t), I1(t), R1(t), . . .Sn(t),In(t),Rn(t))T

for t ≥ 0, and f = (fS,1,fI,1,fR,1, . . .fS,n,fI,n,fR,n)T ,
where for j ∈ {1, . . .n},

fS,j(x) = −Λj(x3j−2,x3j−1,x3j) −

(
n∑
k=1

µj,k

)
x3j−2,

fI,j(x) = Λj(x3j−2,x3j−1,x3j) − αjx3j−1

−

(
n∑
k=1

µj,k

)
x3j−1,

fR,j(x) = αjx3j−1 −

(
n∑
k=1

µj,k

)
x3j.

Clearly the union of systems (Lj) with initial conditions
(IV Lj), j ∈ {1, . . .n} can be written in a closed form
as 


x′(t) = f(x(t)) +

n∑
k=1

Wk(xt) =: F(xt),

x0 = ϕ,

(3)

where F : C+ � R3n and for ϕ ∈ C+, ϕ :=
(ϕS,1,ϕI,1,ϕR,1, . . . ,ϕS,n,ϕI,n,ϕR,n)T .

IV. THE LOCAL LIPSCHITZ PROPERTY

This section is devoted to the proof of the general
existence and uniqueness result of system (3). First, we
obtain the following simple result.

Proposition IV.1. Assume that λk,j possesses the lo-
cal Lipschitz property on each bounded subset of
R3. Moreover, assume that λk,j(q1,q2,q3) ≥ 0 and
λk,j(0,q2,q3) = 0 hold for q1,q2,q3 ≥ 0. Then there
exists a unique solution of system (2) which continuously
depends on the initial data, and for u ∈ [0,τ] and y0 ≥ 0
the following inequality holds componentwise:

0 ≤ y(u, 0; y0) ≤
√

3 |y0|.

Proof: The local Lipschitz condition guarantees
the existence of a unique solution which continuously
depends on the initial data (see Picard-Lindelöf theorem
in Chapter II, Theorem 1.1 and Chapter V, Theorem 2.1
in [1]). We have also seen that nk,j(u; t∗) is constant
for all u in the maximal interval of existence, moreover
from the nonnegativity condition of λk,j it follows that

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D. Knipl et al., Multiregional SIR Model with Infection during Transportation

nonnegative initial data give rise to nonnegative solution.
Hence we obtain

0 ≤ nk,j(0; t∗) = nk,j(u; t∗),
0 ≤ sk,j(0; t∗) + ik,j(0; t∗) + rk,j(0; t∗)

= sk,j(u; t∗) + ik,j(u; t∗) + rk,j(u; t∗)

= µk,j(Sk(t∗) + Ik(t∗) + Rk(t∗)),

0 ≤ sk,j(u; t∗), ik,j(u; t∗),rk,j(u; t∗)
≤ µk,j(Sk(t∗) + Ik(t∗) + Rk(t∗)),

(4)

where we used (IV Tk,j). Using the definition of y, (4)
implies that the inequality

0 ≤
(
y(u, 0; y0)

)
1
,
(
y(u, 0; y0)

)
2
,
(
y(u, 0; y0)

)
3

≤ (y0)1 + (y0)2 + (y0)3

≤
√

3
√

((y0)1)
2 + ((y0)2)

2 + ((y0)3)
2

holds for u ∈ [0,∞), where y0 =
(
(y0)1, (y0)2, (y0)3

)T
is the initial value and we used the arithmetic-quadratic
mean inequality. We conclude that the solution exists on
[0,τ] and is bounded.

Now we prove that if we assume that Λj and λk,j
possess the local Lipschitz property, then F is also
locally Lipschitz continuous.

Lemma IV.2. Let us suppose that for all j,k ∈
{1, . . .n}, Λj and λk,j possess the local Lipschitz prop-
erty, λk,j(q1,q2,q3) ≥ 0 and λk,j(0,q2,q3) = 0 hold
for q1,q2,q3 ≥ 0. Then F satisfies the local Lipschitz
condition on each bounded subset of C+.

Proof: We claim that for every M > 0 there
exists a constant K = K(M) such that the inequality
|F(φ) −F(ψ)| ≤ K||φ−ψ|| holds for every φ,ψ ∈ C+
with ||φ||, ||ψ|| ≤ M.
Fix indices j,k ∈ {1, . . .n}. For ||ψ|| ≤ M it holds
componentwise that 0 ≤ ψ(−τk,j) ≤ M, so due to the
continuity of hk,j, there exists a constant L

h
k,j(M) such

that 0 ≤ hk,j(ψ(−τk,j)) ≤ Lhk,j is satisfied component-
wise. For y0 = hk,j(ψ(−τk,j)) Proposition IV.1 implies
that there exists a Jk,j = Jk,j(Lhk,j) = Jk,j(M) such that
the inequality |ŷψ(−τk,j )(u)| ≤ Jk,j holds for u ∈ [0,τ]
(for instance we can choose Jk,j =

√
3Lhk,j).

The local Lipschitz property of hk,j follows from its
definition. We assumed that λk,j is Lipschitz contin-
uous, this implies the Lipschitz continuity of g. Let
Khk,j = K

h
k,j(M) be the Lipschitz constant of hk,j on

the set {v ∈ R3n : |v| ≤ M}, we denote the Lipschitz
constant of g = gk,j on the set {v ∈ R3 : |v| ≤ Jk,j} by
K
g
k,j = K

g
k,j(J) = K

g
k,j(M). For any ||φ||, ||ψ|| ≤ M,

it holds that |φ(−τk,j)|, |ψ(−τk,j)| ≤ M. Since solu-
tions of (2) can be expressed as y(u, 0; y0) = y0 +∫u
0
g(y(r, 0; y0)) dr, we have∣∣ŷφ(−τk,j )(u) − ŷψ(−τk,j )(u)∣∣
=
∣∣∣∣hk,j(φ(−τk,j)) +

∫ u
0
g(ŷφ(−τk,j )(r)) dr

−
(
hk,j(ψ(−τk,j)) +

∫ u
0
g(ŷψ(−τk,j )(r)) dr

)∣∣∣∣
≤ |hk,j(φ(−τk,j)) − hk,j(ψ(−τk,j))|

+
∫ u

0

∣∣g(ŷφ(−τk,j )(r)) − g(ŷψ(−τk,j )(r))∣∣ dr
≤ Khk,j||φ − ψ||

+
∫ u

0
K
g
k,j

∣∣ŷφ(−τk,j )(r) − ŷψ(−τk,j )(r)∣∣ dr

(5)

for u ∈ [0,τ]. Define

Γ(u) =
∣∣ŷφ(−τk,j )(u) − ŷψ(−τk,j )(u)∣∣

for u ∈ [0,τ]. Then (5) gives

Γ(u) ≤ Khk,j||φ − ψ|| + K
g
k,j

∫ u
0

Γ(r) dr,

and from Gronwall’s inequality we have

Γ(u) ≤ Khk,j||φ − ψ||e
K

g
k,ju. (6)

Applying the definition of Wk, we arrive to the inequal-
ity

|(Wk(φ))j − (Wk(ψ))j|
=
∣∣ŷφ(−τk,j )(τk,j) − ŷψ(−τk,j )(τk,j)∣∣

≤ Khk,je
K

g
k,jτk,j||φ − ψ||,

where we used (6) at u = τk,j.
It is straightforward that Wk has the Lipschitz con-
dition for any k ∈ {1, . . .n}, KWk = KWk (M) =√∑n

j=1

(
Khk,je

K
g
k,jτk,j

)2
is a suitable choice for the

Lipschitz constant.
Finally, the assumption that Λj is Lipschitz continuous
for any j ∈ {1, . . .n} implies the Lipschitz continuity
of f, so let Kf = Kf (M) be the Lipschitz constant
of f on the set {v ∈ R3n : |v| ≤ M}. Then for any
||φ||, ||ψ|| ≤ M, |φ(0)|, |ψ(0)|, |φ(−τ)|, |ψ(−τ)| ≤ M
hold and thus

|F(φ) − F(ψ)| ≤ |f(φ(0)) − f(ψ(0))|

+
n∑
k=1

|Wk(φ) − Wk(ψ)|

≤ Kf||φ − ψ|| +
n∑
k=1

KWk||φ − ψ||.

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D. Knipl et al., Multiregional SIR Model with Infection during Transportation

|Λ1(p) − Λ1(q)| = β1
∣∣∣∣ q1q2q1 + q2 + q3 − p1p2p1 + p2 + p3

∣∣∣∣
= β1

∣∣∣∣ q1q2q1 + q2 + q3 − q1p2q1 + q2 + q3 + q1p2q1 + q2 + q3
−

q1p2
q1 + p2 + q3

+
q1p2

q1 + p2 + q3
−

q1p2
q1 + p2 + p3

+
q1p2

q1 + p2 + p3
−

q1p2
p1 + p2 + p3

+
q1p2

p1 + p2 + p3
−

p1p2
p1 + p2 + p3

∣∣∣∣
≤ β1

(∣∣∣∣ q1q2q1 + q2 + q3 − q1p2q1 + q2 + q3
∣∣∣∣

+
∣∣∣∣ q1p2q1 + q2 + q3 − q1p2q1 + p2 + q3

∣∣∣∣ +
∣∣∣∣ q1p2q1 + p2 + q3 − q1p2q1 + p2 + p3

∣∣∣∣
+
∣∣∣∣ q1p2q1 + p2 + p3 − q1p2p1 + p2 + p3

∣∣∣∣ +
∣∣∣∣ q1p2p1 + p2 + p3 − p1p2p1 + p2 + p3

∣∣∣∣
)

= β1

(
|q2 − p2|

∣∣∣∣ q1q1 + q2 + q3
∣∣∣∣ + |p2 − q2|

∣∣∣∣ q1p2(q1 + q2 + q3)(q1 + p2 + q3)
∣∣∣∣

+|p3 − q3|
∣∣∣∣ q1p2(q1 + p2 + q3)(q1 + p2 + p3)

∣∣∣∣
+|p1 − q1|

∣∣∣∣ q1p2(q1 + p2 + p3)(p1 + p2 + p3)
∣∣∣∣ + |q1 − p1|

∣∣∣∣ p2p1 + p2 + p3
∣∣∣∣
)

≤ β1 (2|q2 − p2| + |p3 − q3| + 2|p1 − q1|)
≤ 5β1|q − p|

(7)

Hence Kf +
∑n

k=1

√∑n
j=1

(
Khk,je

K
g
k,jτk,j

)2
is a suit-

able choice for K, the Lipschitz constant of F for the
set {ψ ∈ C+ : ||ψ|| ≤ M}.

The assumptions of Lemma IV.2 on
the incidences Λj(Sj(t),Ij(t),Rj(t)) and
λk,j(sk,j(u; t∗), ik,j(u; t∗),rk,j(u; t∗)) can be fulfilled
by several choices on the type of disease transmission.
For instance, let βj > 0 be the transmission rate in
region j and let βTk,j > 0 denote the transmission
rates during travel. For j,k ∈ {1, . . .n} and for
q = (q1,q2,q3) ∈ R3, define

Λj(q) = −βj
q1

q1 + q2 + q3
q2,

λk,j(q) = −βTk,j
q1

q1 + q2 + q3
q2.

This implies that Λj and λk,j have the form

Λj(Sj,Ij,Rj) = −βj
Sj
Nj

Ij,

λk,j(sk,j, ik,j,rk,j) = −βTk,j
sk,j
nk,j

ik,j,
(8)

which is called standard incidence. Now we prove the
following existence-uniqueness theorem.

Theorem IV.3. With the incidences Λj and λk,j defined
in (8), there exists a unique solution of system (3).

Proof: Recall Theorem 3.7 from [2]:

Suppose that F satisfies the local Lipschitz property
on each bounded subset of C+ = C+([−τ, 0], R3n+ ),
moreover let M > 0. There exists A > 0, depending
only on M such that if φ ∈ C+ satisfies ||φ|| ≤ M,
then there exists a unique solution x(t) = x̂(t, 0; φ) of
(3), defined on [−τ,A]. In addition, if K is the Lipschitz
constant for F corresponding to M, then

max
−τ≤η≤A

|x(η, 0; φ) − x(η, 0; ψ)| ≤ ||φ − ψ||eKA

holds for ||φ||, ||ψ|| ≤ M.

We showed in Lemma IV.2 that the local Lipschitz
continuity of F follows from the local Lipschitz property
of the incidences and the nonnegativity condition of λk,j.
The latter condition clearly holds, hence it is sufficient
to prove that the incidences defined in (8) possess the
local Lipschitz property.
As one may observe, the definition of the Λj-s and λk,j-

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s (j,k ∈ {1, . . .n}) only differ in a constant multiplier,
hence it is sufficent to prove the local Lipschitz property
only for one of them, i.e. for Λ1. Moreover, we prove
this property only on the nonnegative cone R3+, which
is invariant under systems (3) and (2). For p, q ∈ R3+,
by (7) we obtain the Lipschitz constant K = 5β1, where
we used that for any a,b,c > 0, it holds that a

a+b+c
< 1.

Remark 1. It follows from the proof of Theorem IV.3
that the incidences Λj and λk,j defined in (8) also satisfy
the global Lipschitz property, meaning that the Lipschitz
constant K arises independently of M. In this case, the
solution of system (3) exists on [0,∞).

Another natural choice for the incidences can be the
following: for q = (q1,q2,q3) ∈ R3 and for j,k ∈
{1, . . .n}, let

Λj(q) = −βjq1q2,
λk,j(q) = −βTk,jq1q2,

which leads to the mass action-type disease transmission,
therefore Λj and λk,j have the form

Λj(Sj,Ij,Rj) = −βjSjIj,
λk,j(sk,j, ik,j,rk,j) = −βTk,jsk,jik,j.

(9)

Theorem IV.4. With incidences Λj and λk,j defined in
(9), there exists a unique solution of system (3).

Proof: Similarly as by Theorem IV.3, it is enough
to show that Λj and λk,j satisfy the local Lipschitz
property, we detail the proof only for Λ1 and consider
the nonnegative subspace R3+. For any M > 0, for any
p, q ∈ R3+ such that |p|, |q| ≤ M, we obtain

|Λ1(p) − Λ1(q)| = | − β1p1p2 + β1q1q2|
≤ β1|p1p2 − q1q2|
≤ β1|p1p2 − p1q2 + p1q2 − q1q2|
≤ β1(|p1p2 − p1q2| + |p1q2 − q1q2|)
≤ β1(p1|p2 − q2| + q2|p1 − q1|)
≤ 2Mβ1|q − p|,

so we can choose K(M) = 2Mβ1.

Remark 2. Although the global Lipschitz property does
not hold for Λj and λk,j defined in (9), it is possible to
show that the solution of (3) is bounded and hence exists
on [0,∞).

V. CONCLUSION

The topic of epidemic spread of infectious diseases via
transportation networks has recently been examined in
several studies (see [3], [4], [5], [6], [7]), although these
works mostly consider only two connected regions. We
introduced a dynamic model which describes the spread
of an infectious disease in and between n regions which
are connected by transportation. We used the commonly
applied SIR model as a basic epidemic building block
in the regions and also during the travel. The model
formulation led to a system structured by travel time,
which turned out to be equivalent to a system of differen-
tial equations with multiple dynamically defined delays.
We showed that under local Lipschitz conditions on the
infection terms within the regions and during travel, the
usual existence and uniqueness results hold.
Recent epidemics like the 2002-2003 SARS outbreak
and the 2009 pandemic influenza A(H1N1) highlighted
the importance of the global air travel network in the
study of epidemic spread. During long distance travel
such as intercontinental flights, a single infected indi-
vidual may infect several other passengers during the
flight, and since the progress of these diseases is fast,
even a short delay (a fraction of a day) arising due to
transportation may play a significant role in the disease
dynamics. In this paper we illustrated by proving an
existence and uniqueness result that such epidemiologi-
cal situations can be studied in the framework of delay
differential equations.

ACKNOWLEDGMENT

DHK was partially supported by the Hungarian Re-
search Fund grant OTKA K75517 and the TÁMOP-
4.2.2/B-10/1-2010-0012 program of the Hungarian Na-
tional Development Agency. RG was supported by Eu-
ropean Research Council StG Nr. 259559, Hungarian
Research Fund grant OTKA K75517 and the Bolyai Re-
search Scholarship of Hungarian Academy of Sciences.

REFERENCES

[1] P. Hartman, “Ordinary differential equations”, Classics in Ap-
plied Mathematics, vol. 38, SIAM, 2002.

[2] H. L. Smith, An Introduction to Delay Differential Equations
with Applications to the Life Sciences, Springer, 2010.

[3] D. H. Knipl, “Fundamental properties of differential equation
with dynamically defined delayed feedback”, preprint, 2012.

[4] D. H. Knipl, G. Röst, J. Wu,“Epidemic spread of infectious
diseases on long distance travel networks”, preprint, 2012.

[5] J. Liu, J. Wu, Y. Zhou, “Modeling disease spread via transport-
related infection by a delay differential equation”, Rocky Moun-
tain J. Math., vol. 38 No 5, pp. 1225–1541, 2008.

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D. Knipl et al., Multiregional SIR Model with Infection during Transportation

[6] Y. Nakata, “On the global stability of a delayed epidemic model
with transport-related infection”, Nonlinear Analysis Series B:
Real World Applications, vol. 12 No 6, pp. 3028–3034 2011.
http://dx.doi.org/10.1016/j.nonrwa.2011.05.004

[7] Y. Nakata, G. Röst, “Global analysis for spread of an infectious
disease via global transportation”, submitted, 2012.

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http://dx.doi.org/10.11145/j.biomath.2012.09.255

	Introduction
	Model Description
	The Compact Form of the System
	The Local Lipschitz Property
	Conclusion
	References