Page 216 - IJOCTA-15-4

H. Kravitz et al. / IJOCTA, Vol.15, No.4, pp.750-778 (2025)
spread for years; this behavior is characteristic of v 20 (Radom, Kielce, and Rybnik), relative to the
a very low η. A high β may occur in communities data.
with high contact rates and/or when a disease is
highly contagious. For example, the probability 3.3. Application: flattening the curves
of gonorrhea transmission is about 50% per sex
act; 99 paired with lack of access to treatment, 100 During the early phases of COVID-19 spread,
there was worldwide discourse 101,102 around what
this disease fits the profile. The low initial in-
people can do to “flatten the curve,” lowering
fected population occurs when the population is
the peak infection rate while causing the peak
taken to be the entire population of a city (a nat-
to occur later in time. 103–105 The intention was
ural choice for geographic studies), rather than a
to lower the number of people needing care at a
smaller group.
given time so the healthcare system would not
be overwhelmed. Some non-pharmaceutical in-
3.2. Results of parameter estimation terventions (NPIs) aimed at reducing the spread
of COVID include mask-wearing, social distanc-
Because our Rosenbrock objective function is cor- ing, hand washing, and quarantine after contact
rugated and has many local near-minima, we fit with an infected individual. Due to the severity of
to just one plausible combination, which can be the pandemic, Poland, like many other countries,
found in Appendix D. Without knowing the pre- implemented temporary stay-at-home orders or
cise parameter values, it is hard to tell which pa-
lockdowns beginning in the spring of 2020 to min-
rameter value is “right.” We find good agree-
imize non-essential human contact. These mea-
ment between the model and the data in terms
sures restricted public gatherings, closed schools
of both the 2-norm and the shape of the curves.
and businesses, and significantly altered patterns
The fine-tuning of the parameters to ensure that 106
of human mobility.
the shape of the curves match was guided by our
As an application of our model, we explore
global sensitivity analysis, which can be found in
how reductions in travel between cities can further
Appendix C. For example, we know the time of
reduce or delay the epidemic peak. We note that
peak infection at a particular vertex is highly sen- the data to which we have fit the model already
sitive to both β and η with some dependence on includes the flattening effect of the NPIs imple-
α, d, and λ. Figure 5 shows a subset of highly mented in Poland (lockdowns, social distancing,
populated vertices, with the rest in Figure A15 in masking, etc.), though these interventions were
Appendix D. The non-normalized data is found not always adhered to uniformly. 41 Our analysis,
in Figure 6. As expected, the amplitudes are off, therefore, explores the effect of further limiting
but the shapes generally match. We note that we
inter-city travel while still allowing for some es-
do not claim that the amplitudes we achieve are
sential traffic flow. Keeping the same parameters
fully representative of the actual COVID num-
found in Section 3, we uniformly reduce the α
bers in Poland, just that we find a possible set of
terms over the whole network by 50%, 80%, and
parameters under which to explore the plausible
90%, respectively, and compare the results to the
geographic progression of the epidemic.
original model, with the results shown in Figure 7.
The model captures the timing of peak infec- To quantify how well the reductions flatten
tion to within at most 4 days in all cases except the curve, we use a Kolmogorov-Smirnov (K-S)
´
v 21 (Swinouj´scie), where highly oscillatory time- methodology 107 to explore whether a particular
series data made fitting difficult – see Figure A15 reduction in α will affect the overall distribution
in Appendix D (though v 21 has the lowest popu- of the infection in a statistically significant way.
lation and was, therefore, assigned the lowest pri- We find that a traffic reduction of 90% or more
ority in our fitting procedure). Notably, we did statistically changes the distribution at a signif-
not fit the amplitudes of our curves; neverthe- icance level of 5% - the probability distributions
less, the relative amplitudes align fairly well. The and the empirical cumulative distribution func-
most populated vertex, v 8 (Warszawa) exhibits tions are shown in Figure 8.
the highest peak by far, as expected, and the
groups {v 1 , v 2 , v 3 , v 4 } ({Pozna´n, Wroc law, Ka-
3.4. Intra- versus inter-city infections
towice, Krak´ow}), {v 7 , v 9 } ({ L´od´z, Gda´nsk }),
and {v 13 , v 16 } ({Szczecin, Bydgoszcz}) form clus- The cumulative infected population on each
R R
ters with similar dynamics in both the model and edge ( I e (x, t) dx dt, the integral of the non-
the data. The model underestimated the relative normalized left panel of Figure A17 in the Appen-
contribution of v 5 and v 22 (Rzesz´ow and Toru´n) dix) is plotted on top of the network structure in
while overestimating the infection at v 6 , v 14 , and Figure 10. The infected population on each edge
758