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Data-driven optimization and parameter estimation for an epidemic model
mechanism for the spread of vector-borne dis- along a network, 44 have been shown to outper-
ease, transporting mosquitoes further in a sin- form models that assume a homogeneous struc-
gle direction than spatial diffusion alone would ture in capturing the directional and constrained
dictate. 5–8 Regional transportation infrastructure nature of the spread of disease.
also contributes to the spread of disease through
the movement of infected humans - the surfaces of 1.2. Parameter selection
vehicles like buses and ambulances can facilitate
Unfortunately, even the most basic SIR model has
surface-to-hand contact of Methicillin-resistant parameters that are not easily observable from
Staphylococcus aureus (MRSA), 9–12 while indi- 45
data, as complex interactions between the pa-
viduals in close quarters may spread respiratory rameters change the results of the model in un-
infections like COVID-19. 13–15 The movement of
predictable ways. This problem is further com-
infected individuals is also a well-documented
plicated by unreliable or inconsistent data that
pathway of geographic spread. For example, the do not correspond directly to the equations mod-
recent COVID-19 pandemic spread first between eled in SIR systems, e.g., reporting typically con-
countries through the air transportation network sists of new cases, while the model’s I(t) covers
and then along local highway systems. 16–21
both new and ongoing cases. Fitting SIR-type
models to the COVID-19 pandemic has proven
especially challenging. The first challenge is the
vast scale of the pandemic - one study estimated
1.1. SIR-type models that by November 2021, 43.9% of the global pop-
ulation had been infected with SARS-CoV-2 at
The spatial spread of infections is often stud- least once. 46 For a pandemic on such an enor-
ied using compartmental epidemiological mod- mous scale, both the reliability of the data and
els, typically using variations of the Susceptible- the selection of model parameters are further
Infected-Removed (SIR) system of equations 22-28 . complicated by inconsistent adherence to non-
One common way to incorporate geographic pharmaceutical intervention (NPI) guidelines (so-
structure in this type of model is the inclusion of cial distancing, masking, hand-washing, etc.), 41
corridors of fast diffusion, one-dimensional lines under-reporting of cases (further obscured by the
that facilitate faster travel than spatial diffu- presence of asymptomatic cases), 47–49 and het-
sion alone would allow. 29–31 This framework has erogeneity in disease dynamics across population
been used to model cholera transmission along groups. 41–43 Despite these challenges, SIR-type
river systems, 32,33 the movement of disease vec- models remain valuable for gaining mathematical
6
tors along highways, and the large-scale spread of insight into epidemic dynamics. They clarify how
disease along major transportation corridors. 34,35 key parameters influence outbreak thresholds and
A recent metric graph-based model intro- long-term behavior and help identify which mech-
duced by Besse and Faye 31 consists of a network of anisms most affect disease spread, especially when
cities connected by one-dimensional edges, adding data are unreliable.
a “traveling infection” component to the stan- Even though parameter estimation in SIR
dard SIR model. This model is able to incorpo- models is difficult, several strategies have been
rate both geographic network structure and spa- developed to address this challenge. One com-
tial heterogeneity in the form of different param- mon technique is nondimensionalization, in which
eters for individual vertices and edges. Compart- the system is simplified using variable substitu-
mental epidemiological models incorporate re- tion to reduce the number of parameters. 50,51
gional variation in many different ways, includ- While useful for theoretical analysis (e.g., bifur-
ing heterogeneity in initial conditions 36,37 (e.g., cation identification), nondimensionalization re-
seeding infections in major cities) and spatially- moves the physical meaning of the parameters
varying uniform parameters 38–40 (e.g., population and makes fitting the model to data more dif-
density, diffusion coefficient, transmission rate). ficult. More advanced fitting often includes
The ability to represent regional disparities is es- statistical techniques like maximum likelihood
pecially important for studying large-scale epi- estimates, 26,52 machine learning algorithms, 53–55
demics such as COVID-19, where disease bur- and Bayesian approaches. 56,57
den and response strategies varied significantly A more direct approach, which we adopt here,
across jurisdictions, especially during the begin- involves first approximating the differential equa-
ning of the pandemic. 41–43 Compartmental mod- tions using finite difference methods over short
els that incorporate geographic structure, such time intervals. 53,58–61 This allows key parame-
as location-specific parameters 38,44 or diffusion ters like transmission and recovery rates to be
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