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A new view on migration processes between SIR centra: an account of the different dynamics of host and guest

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A new view on migration processes between SIR centra:an account of the different dynamics of host and guest
Igor Sazonov
a
1
, Mark Kelbert
b
and Michael B. Gravenor
ca
College of Engineering, Swansea University, Singleton Park, SA2 8PP, U.K.
b
Department of Mathematics, Swansea University
b
Institute of Life Science, School of Medicine, Swansea University
Abstract
We study an epidemic propagation between
M
population centra. The novelty of the modelis in analyzing the migration of host (remaining in the same centre) and guest (migrated toanother centre) populations separately. Even in the simplest case
M
= 2
, this modiﬁcationis justiﬁed because it gives a more realistic description of migration processes. This becomesevident in a purely migration model with vanishing epidemic parameters. It is important toaccount for a certain number of guest susceptible present in non-host cenrta because thesesusceptiblemaybeinfectedandreturntothehostnodeasinfectives. Theﬂuxofsuchinfectivesis not negligible and is comparable with the ﬂux of host infectives migrated to other centra,because the return rate of a guest individual will, by nature, tend to be high. It is shown thattaking account of both ﬂuxes of infectives noticeably increases the speed of epidemic spreadin a 1D lattice of identical SIR centra.
Key words:
spatial epidemic models; migration dynamics; ourbreak time
AMS subject classiﬁcation:
92D30, 91D25
1 Introduction
The classical SIR model is one of the simplest models which describes qualitatively a typicaldirectly transmitted disease outbreak in a populated center, and remains the building block formany, more complicated applied epidemic models. The population is assumed to consist of threecomponents: susceptible (S), infected (I) and removed (R).Models of coupled epidemic centra are of particular interest because they describe epidemicspread through network of populated centra, and hence the overall population is not treated as ahomogenous system. This is a subject of intensive research, we mention here just a few recent
1
Corresponding author. E-mail: i.sazonov@swansea.ac.uk
1
publication [2, 3, 4, 5, 6] not trying to provide an extensive bibliography. The old scenario, knownfrom the middle ages, when the disease propagates locally from a village to the neighboring vil-lages is replaced now by almost instantaneous propagation around the globe. This phenomena wasanalyzed in a many papers (see, e.g., [7, 8]). In particular, it was observed that on heterogeneousnetworks an increase in the movement of population may decrease the size of the epidemic at thesteady state, although it increases the chances of outbreak. This motivates a detailed analysis of migration in inhomogeneous populations.The coupling between nodes of such a network is mainly caused by migration processes of infectives. There are several models describing such coupling (see [9, 1]), for example, in [10]the inﬂuence of various parameters on the spacial and temporal spread of the disease is studiednumerically, with particular focus on the role of quarantine in the form of travel restrictions. In[11, 12], the so-called diffusion like model is proposed and studied in the framework of a fastmigration time approximation. Note that the model in [10] is a particular case of the diffusionmodel when the migration time tends to inﬁnity but the coupling coefﬁcient introduced in [11]tends to zero.In all these models the guest population is completely mixed with the host one, so their dy-namics is indistinguishable. Nevertheless, a more detailed consideration suggests that while theepidemic dynamics is the same, the migration dynamics should be different, especially if consid-ered as part of a discrete randomized model approach (cf. [13, 14]).In the paper we start with consideration of the simplest network of only two interacting epi-demic SIR centra and study in detail the migration processes and their inﬂuence on the populationdynamics. Moreover, our interest in the model is motivated by the fact that it serves as a hydrody-namic approximation of a natural Markov process describing the stochastic dynamics of the system(cf. [15]). This topic will be explored more fully in a subsequent paper.To examine the migration model we ﬁrst consider here the case when epidemic parameters aretemporally switched off. The study of migration in isolation provides a simple tractable model andallows us to specify the parameters in a consistent way. Equally important, this analysis revealsthatmany models used in the literature (see eg [1, 16]) are unstable in the limit of vanishing infection.Other ones (see eg [17, 18]) remain stable but lead to non-realistic results.Note that even an isolated SIR model cannot be integrated explicitly, therefore a suitable ap-proximationisrequired toavoidnumerical integrationandto obtain practical formulasfor outbreak time, fade-out time and other parameters. In our previous works [11, 12, 14] the so-called smallinitial contagion (SIC) approximation was proposed, based on the assumption that an outbreak inevery population center is caused by relatively small number of initially infectives. This approxi-mation is appropriate when the model is applied to strongly populated centra like urban centra (i.e.in the situation when the reaction-diffusion model is not accurate).Inthepaperwealsoshowhowthemodelcanbegeneralizedonthegeneralnetworkofepidemiccentra (see Section 7). As an example a characteristic equation for the travelling wave in a chainof the population centers is derived and its numerical solution is plotted and analyzed.2
2 Governing equations
Consider two populated nodes, 1 and 2, with populations
N
1
and
N
2
, respectively. Let
S
n
(
t
)
,
I
n
(
t
)
,
R
n
(
t
)
be the numbers of host susceptibles, infectives and removed, respectively, in node
n
at time
t
. Let
S
mn
(
t
)
,
I
mn
(
t
)
,
R
mn
(
t
)
be numbers of guest susceptibles, infectives and removed,respectively, in node
n
migrated from node
m
at time
t
. Removed populations
R
n
,
R
nm
do notaffect dynamics of all others in the framework of the standard SIR model, and we omit them fromconsideration here. Then, two SIR centers (nodes) interacting due to the migration of individualsbetween them are described by the following model: the dynamics of hosts in node
n
obeys theODEs
˙
S
n
=
−
β
n
S
n
(
I
n
+
I
mn
)
−
˙
S
n
→
m
+ ˙
S
n
←
m
(1)
˙
I
n
=
β
n
S
n
(
I
n
+
I
mn
)
−
α
n
I
n
−
˙
I
n
→
m
+ ˙
I
n
←
m
(2)where
n
= 1
,
2
,
m
= 2
,
1
; and dot denotes the time derivative. Here the term
β
n
S
n
(
I
n
+
I
mn
)
appears due to infectives
I
mn
migrated from node
m
and contributing to the total disease transmis-sion process. Terms
˙
S
n
→
m
and
˙
I
n
→
m
describe migration ﬂuxes (rates) from node
n
to node
m
forsusceptibles and infectives, respectively. Terms
˙
S
n
←
m
and
˙
I
n
←
m
describe return migration ﬂuxes(rates) to node
n
for guest individuals in node
m
. We specify these below.The dynamics of guests in node
n
temporally arriving from node
m
can be described by anal-ogous ODEs
˙
S
mn
=
−
β
n
S
mn
(
I
n
+
I
mn
) + ˙
S
m
→
n
−
˙
S
m
←
n
(3)
˙
I
mn
=
β
n
S
nm
(
I
n
+
I
mn
)
−
α
n
I
mn
+ ˙
I
m
→
n
−
˙
I
m
←
n
(4)We assume the migration rate is proportional to the population size in the node from whichthey emigrate. So, we approximate the ﬂuxes as
˙
S
n
→
m
=
γ
S nm
S
n
,
˙
I
n
→
m
=
γ
I nm
I
n
,
˙
S
n
←
m
=
δ
S nm
S
nm
,
˙
I
n
←
m
=
δ
I nm
I
nm
(5)where
γ
’s and
δ
’s are the forward and backward migration coefﬁcients, respectively.Our interest in the dynamical equations presented above is motivated by the fact that theyserve as a hydrodynamic approximation of a Markov process model. In this context,
γ
’s can beassociated with the transition rate for a host individual to migrate to another node in a unit of time,and
δ
’s—with the transition rate for a guest individual to return to the host node.Clearly, average return rates should be higher:
γ
S nm
< δ
S nm
,γ
I nm
< δ
I nm
, otherwise an individualwould spend most of the time out of the home center.Substituting (5) into (1)–(2) and (3)–(4) yields a closed system of ODEs: for the hosts in node
n
˙
S
n
=
−
β
n
S
n
(
I
n
+
I
mn
)
−
γ
S nm
S
n
+
δ
S nm
S
nm
(6)
˙
I
n
=
β
n
S
n
(
I
n
+
I
mn
)
−
α
n
I
n
−
γ
I nm
I
n
+
δ
I nm
I
nm
(7)3
and for the guests migrated from node
m
into node
n
˙
S
mn
=
−
β
n
S
mn
(
I
n
+
I
mn
) +
γ
S mn
S
m
−
δ
S mn
S
mn
(8)
˙
I
mn
=
β
n
S
mn
(
I
n
+
I
mn
)
−
α
n
I
mn
+
γ
I mn
I
m
−
δ
I mn
I
mn
.
(9)Evidently, the dynamics of hosts and guests are different.Typical initial conditions for epidemiological problem describe a number of infectives, say
I
01
,
that appeared at
t
= 0
in node 1 only:
I
1
(0) =
I
01
, I
2
(0) = 0
,S
1
(0) =
N
1
−
S
12
(0)
−
I
01
, S
2
(0) =
N
2
−
S
21
(0)
,I
12
(0) = 0
, I
21
(0) = 0
,S
12
(0) =
γ
S
12
γ
S
12
+
δ
S
12
N
1
S
21
(0) =
γ
S
21
γ
S
21
+
δ
S
21
N
2
.
(10)The choice for values for
S
12
(0)
and
S
21
(0)
will be explained below in Section 3 by consideringthe migration processes before the epidemic outbreak starts.
3 Pure migration
Considermigrationofsusceptiblesbeforetheepidemicstartsinthenetwork. Setting
I
1
,I
12
,I
2
,I
21
=0
we obtain two decoupled systems of ODEs for
S
1
,S
12
and for
S
2
,S
21
describing the pure migra-tion processes in the absence of an outbreak. Say, for the pair
S
1
,S
12
we have
˙
S
1
=
−
γ
S
12
S
1
+
δ
S
12
S
12
(11)
˙
S
12
=
γ
S
12
S
1
−
δ
S
12
S
12
.
(12)Let migration start at
t
= 0
with the initial conditions
S
1
(0) =
N
1
,
S
12
(0) = 0
. The solution tosuch an initial value problem is
S
12
=
N
1
g
S
12
(
t
)
, S
1
=
N
1
−
S
12
(13)where
g
S
12
(
t
) =
γ
S
12
γ
S
12
+
δ
S
12
1
−
e
−
(
γ
S
12
+
δ
S
12
)
t
, t
≥
0
(14)is the response function (see below). Similar formulas are valid for the second pair:
S
2
,S
21
. Thus,the number of migrants exponentially tends to some limiting values
lim
t
→∞
S
12
=
γ
S
12
γ
S
12
+
δ
S
12
N
1
,
lim
t
→∞
S
21
=
γ
S
21
γ
S
21
+
δ
S
21
N
2
.
(15)These limits represent the dynamic equilibrium of migration processes in the absence of the out-break. At the equilibrium, the forward and backward migrations ﬂuxes compensate each other:
˙
S
1
→
2
= ˙
S
1
←
2
. So, in virtue of (5),
γ
S
12
S
1
=
δ
S
12
S
12
. Substituting
S
1
=
N
1
−
S
12
and resolving with4
respect to
S
12
yields (15). We take the equilibrium values from (15) as the initial conditions for theoutbreak problem, that is reﬂected in (10).The total populations in any node
S
Σ1
=
S
1
+
S
21
and
S
Σ2
=
S
2
+
S
12
are described as
S
Σ1
(
t
) =
N
1
−
N
1
g
12
(
t
) +
N
2
g
21
(
t
)
S
Σ2
(
t
) =
N
2
−
N
2
g
21
(
t
) +
N
1
g
12
(
t
)
.
They can be non-monotonic for some choice of parameters. Next,
S
Σ1
,
2
asymptotically convergesto
S
Σ1
(+
∞
) =
N
1
−
N
1
γ
S
12
γ
S
12
+
δ
S
12
+
N
2
γ
S
21
γ
S
21
+
δ
S
21
S
Σ2
(+
∞
) =
N
2
−
N
2
γ
S
21
γ
S
21
+
δ
S
21
+
N
1
γ
S
12
γ
S
12
+
δ
S
12
.
If both centra are identical then their total population remains constant.The migration dynamics described by this model seems reasonable. The migration processresembles a diffusion process in physics, in which the concentration tends monotonically to anequilibrium.Note that if
γ
S
12
≪
δ
S
12
then
S
12
(
t
)
≪
N
1
,
∀
t
, i.e. only a small share of the population fromof node 1 is currently in node 2 (and vice verse: if
γ
S
21
≪
δ
S
21
then
S
21
(
t
)
≪
N
2
,
∀
t
), which isappropriate for large population centra. So, in this approximation
S
mn
(
γ
S mn
/δ
S mn
)
S
m
.To understand why function (14) can be associated with a response function, consider a modelfor which the number of susceptibles can vary even in the absence of migration due to other reasons(e.g., birth and death). Let
˙
N
1
be the rate of incoming (
˙
N
1
>
0
) or outgoing (
˙
N
1
<
0
) individuals,i.e. the external source in the equations
˙
S
1
=
−
γ
S
12
S
1
+
δ
S
12
S
12
+ ˙
N
1
˙
S
12
=
γ
S
12
S
1
−
δ
S
12
S
12
(16)with initial conditions
S
1
(0) =
N
01
,
S
12
(0) = 0
. Integrating the second equation in view of relation
S
1
=
N
1
−
S
12
yields:
S
12
(
t
) =
∫
t
0
γ
S
12
exp
−
(
γ
S
12
+
δ
S
12
)(
t
−
t
′
)
N
1
(
t
′
)d
t
′
≡
˙
g
S
12
(
t
)
∗
N
1
(
t
)
where the asterisk denotes the convolution,
˙
g
S
12
(
t
)
is the derivative of function (14): recall that
˙
g
S
12
∗
H
(
t
) =
g
S
12
where
H
(
t
)
is the unit-step Heaviside function. So,
S
12
= ˙
g
S
12
(
t
)
∗
N
1
(
t
)
is theresponse in the number of guests in node 2 on the population variation in node 1.
4 Comparison with earlier models
The most epidemic network models deal with the total number of infectives:
I
Σ
n
=
I
n
+
I
mn
, andsusceptibles
S
Σ
n
=
S
n
+
S
mn
in node
n
. To compare these models take the sum of Eqs. (6) and (7)5

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