Page 212 - IJOCTA-15-4

H. Kravitz et al. / IJOCTA, Vol.15, No.4, pp.750-778 (2025)
Table 2. Description of model parameters.

Parameter Description Range Number of parameters per graph
β v transmission rate β v > 0 |V |
at vertex v
removal rate at η v > 0 |V |
η v
vertex v
d e diffusion coeffi- No restriction |E|
cient on edge e
α v rate of exchange (0, 1) |V |
from incident
edges to vertex v
v
λ v e rate of exchange (0, 1) and P e λ < 1 P v deg(v) = 2|E|
e
from vertex v to
edge e
[0, 1) and
rate of travel from X v
v < P 2
edge e m to e m,e n deg(v) − deg(v)
v v m̸=n vP
2
e m,e n = deg(v) − 2|E|
adjacent edge e n α v + P v
without stopping at v e n n̸=m
v v
e n,e m
Junction Conditions vertex v and the outflux through the same edges.
∂ The rate at which the exchanges occur is governed
D v I e + K v I e = Λ v I v . (4) v
∂x by parameters α v and λ , which can be specified
e
The following matrices appear in the junction for each vertex-edge pair. Equation 3 describes
condition equations: the propagation of the epidemic along the edge e
of the metric graph.
• D v = diag (d e ) is a real, square matrix of
size deg(v) × deg(v). The Robin junction conditions in Equation 4,
• K v = A v + N v is a real, square matrix of applied at the end of the metric graph line seg-
size deg(v) × deg(v), where: ments (vertices), have been employed in several
– A v = diag (α v ) is a real, square ma- metric graph-based models 6,30,31 and encode a
∂
trix of size deg(v) × deg(v). balance of fluxes at the vertices. ∂x is the out-
– N v is a real, square matrix of size ward normal derivative (taken away from the ver-
deg(v) × deg(v) such that: tex). I e is a vector of length deg(v) with elements
∗ For n ̸= m (non-diagonal I e (x 0 , t), where x 0 is the end of the edge. I v is a
terms), (N v ) n,m = −v v . vector of length deg(v) with all elements equal
e m,e n
∗ Diagonal terms: (N v ) n,n = to the infected population at vertex v. These
P v junction conditions allow for a non-uniform prob-
v (sum of the abso-
m̸=n e n,e m
lute values of everything else in ability of travel from a vertex to its adjacent
edges; they allow for fluxes to go in different di-
the column).
rections, as well as acting as a source for the ver-
• Λ v = diag λ v .
e e∼v
tex. We note that the boundary conditions in
Equation 1 represents the susceptible popula-
Equation 4 do not enforce continuity across the
tion in a city. Following the framework of Besse graph, as is typical in metric graph problems. 78
and Faye, 31 Berestycki, Roquejoffre, and Rossi, 29 Though they occupy the same point in space,
and others, 30,32,71,72,77 the susceptible popula- (x 0 , t) ̸= I v (t). As discussed in
I e n (x 0 , t) ̸= I e m
tion is assumed to be ambient, i.e., any movement our previous work, 30 the structure may be con-
of susceptible individuals does not affect their ceived as two coupled graphs: a metric graph with
population density. The term β v S v I v in Equa- discontinuous junction conditions coupled with a
tions 1 and 2 captures the transition from suscep- combinatorial graph with populations at the ver-
tible to infected, while the term η v I v represents tices.
the removal of the infected population. The re-
P The existence and uniqueness of positive clas-
I
maining terms in Equation 2, α v e|e∼v e (v, t) −
sical solutions was established by Besse and

P v
λ I v (t) describe the influx of the in- 31
e|e∼v e Faye, who also characterized the long-term be-
fected population from all edges incident to the havior and established the conservation of total
754