Page 230 - IJOCTA-15-4

H. Kravitz et al. / IJOCTA, Vol.15, No.4, pp.750-778 (2025)
• Time t of maximum infection at vertex v. Zabrze, & Bytom), two highly populated adja-
Z T
• Total cumulative infection, I v (t) dt. cent vertices with a short edge in between. No-
0 table effects also arise from changes in β at v 7
( L´od´z), v 8 (Warszawa), v 9 (Gda´nsk & Gdynia),
Six input parameters were varied: c β ∈
and v 2 (Wroc law). Increasing the transmission
[1, 6], c η ∈ [1, 3.5], α ∈ [0.3, 0.95], c λ ∈
rate at v 2 (Wroc law), and to a lesser extent at
[0.1, 0.95], c v ∈ [0.1, 0.5], d ∈ [0.07, 1.2]. We per-
v 8 (Warszawa), has the greatest effect in reduc-
formed n = 100 runs sampled using Latin hyper- ing the time to peak infection (Figure A14b). A
cube sampling 117 to ensure an even distribution weaker but similar effect occurs at v 9 (Gda´nsk &
across the parameter space. 117,120
Gdynia), where higher β shifts the peak earlier.
Figure A13 shows the absolute mean and stan- The effects in Figure A14c (cumulative infections)
dard deviation of the elementary effects (EE) of largely mirror the population distribution across
each variable divided up by indicator: peak in- the vertices.
fection rate, time of peak infection, and cumu- A more in-depth location-based sensitivity
lative infections for Poland (the entire network), study is planned, as some spatial patterns are be-
Warszawa (the most populated city), and Pozna´n ginning to emerge that could potentially open the
(a city of medium population). Somewhat un- door to richer analysis. Examining the EE under
surprisingly, the amplitude of the peak infection changes to location-specific parameters might of-
and the time of peak infection at individual ver- fer hints or perspectives that could be relevant for
tices depend highly on the transmission rate β informing the control of the infection. 110
and removal rate η, with a small influence from
the diffusion coefficient d. The time of the peak Appendix D. Selected additional
infection in the entire network, however, depends results
fairly heavily on the diffusion coefficient d, with
an increase in diffusion rate causing the peak to Agreement between the normalized model results
occur earlier. This suggests a strong influence and the data at the most populated vertices was
from the edges in the metric graph - a rich area presented in Figure 5. Figure A15 shows the rest
for future exploration. Interestingly, the cumula- of the vertices.
tive infections are influenced by many of the pa- The susceptible population at each vertex,
rameters. The high standard deviations indicate S v (t) is plotted as a function of time in Fig-
possible nonlinear effects, which could also be ex- ure A16. The reduction in population from S v (0)
plored in the future. Table 4 shows the signed is the portion of “native” infections. The plot
means of the EEs to give an indication of the di- shows that the susceptible population is not de-
rection of influence, i.e., increasing the recovery pleted, despite large spikes in the infected popu-
rate η (roughly equivalent to reducing the dura- lation.
tion of infection) leads to a decrease in the peak
infection rate. D.1. Edge population

Having a metric graph model allows us to gen-
erate granular model data that is not typically
After global optimization, our method in- available, namely the infected population on each
volves adjusting the parameters at each vertex. edge. In Figure A17, some population dynamics
Therefore, we briefly consider the effect of per- are shown for the nine edges with the highest in-
turbing β at each vertex. In this example, we fected populations. The left panel shows the edge
R
vary each β v by a constant amount while keeping infected population over time ( I e (x, t) dx) as a
the rest of the parameters fixed at the values se- percent of the total population on all edges at
lected for the model (Table 3). Unsurprisingly, time t, while the right panel shows the contribu-
µ(EE) of β v at vertex v is several magnitudes tion of each edge to its incident vertices over time
P
I
larger than for any other vertex. The interesting (α v e|e∼v e (v, t)) as a percent of the total con-
scenario in this case is to look at the effect of vary- tributions at time t (note that the other 39 edges
ing β v on the entire network. Figure A14 contains are not pictured).
the signed means of the EE for the peak infection We can see some interesting behavior in this
rate, time of peak infection, and cumulative infec- figure. The edge connecting v 1 with v 16 (Pozna´n
tions for the entire network. Figure A14a shows with Bydgoszcz) initially contributes the highest
that the strongest effects on peak infection rate percent to its incident vertices, but over time its
occur when changing β at v 4 (Krak´ow) and v 3 influence decreases slightly. This behavior is seen
(the metropolis made up of Katowice, Sosnowiec, in the infection curves as well (both the data
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