# Neural Network Augmented Compartmental Pandemic Models

Lorenz Kummer, Kevin Sidak

(July 2021)

###### Abstract

Compartmental models are a tool commonly used in epidemiology for the
mathematical modelling of the spread of infectious diseases, with their most
popular representative being the Susceptible-Infected-Removed (SIR) model and
its derivatives. However, current SIR models are bounded in their capabilities
to model government policies in the form of non-pharmaceutical interventions
(NPIs) and weather effects and offer limited predictive power. More capable
alternatives such as agent based models (ABMs) are computationally expensive
and require specialized hardware. We introduce a neural network augmented SIR
model that can be run on commodity hardware, takes NPIs and weather effects
into account and offers improved predictive power as well as counterfactual
analysis capabilities. We demonstrate our models improvement of the state-of-
the-art modeling COVID-19 in Austria during the 03.2020 to 03.2021 period and
provide an outlook for the future up to 01.2024.

##  1 Introduction

In the period between March 15th 2020 and April 4th, Austria enacted it’s
first series of COVID-19 NPIs [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], resulting
in a country-wide lockdown and restrictions of public life, followed by a
series of repeated loosening and tightening regulations that have not been
completely taken back until today [12, 13, 14, 15, 16, 17, 18, 19].
Nonetheless, as of March 1st 2021, 8574 people were registered as deceased
with or by COVID-19 [20, 21] in Austria. Although a highly socio-politically
relevant question, it remains unclear how effective these NPIs were, what
would have happened without any NPIs or only a subset of the historical NPIs
and whether a less intrusive set of NPIs exists which provides an outcome
similar to the historical one. These are all counterfactual questions which
cannot be empirically answered per definition but only through simulation.
Given the correlation between weather and pandemic spread [22, 23, 24, 25], we
recognize that any such simulation must not only model the pandemic given a
certain set of NPIs, but also account for weather effects.

###  1.1 Related Work

ABMs, used by Austrian government agencies as decision support tool [26], are
powerful instruments for simulating disease spread under NPIs (e.g. contact
tracing [27]) as they simulate individual agents and their movement, behaviour
and interactions but are also capable of evaluating immunization levels [28]
and vaccination strategies [29] as well as estimating undetected cases [30].
However, they are computationally complex [28] which limits their use to
institutions with access to sufficient computing power or specialized hardware
[31].

A computationally less expensive model is the SIR model. Since its inception
[32], the SIR model has been extended numerous times [33, 34, 35] and it
recently found an application in the modelling of COVID-19 [36, 37, 38, 26,
39, 40]. The predictive power of the SIR model in the context of COVID-19 and
been confirmed by [41]. Currently though, SIR models are limited in their
capability to simulate NPIs or weather effects, with proposed solutions
ranging from segmenting and picking epidemiological parameters for various
phases of the pandemic manually or automatic change point detection and
inference of epidemiological parameters for different phases of the pandemic
[42] to training simple exponential regression models on predicting
epidemiological parameters [40]. Other publications use projections of the
effects of specific NPIs on reproduction number based on empirical studies of
that particular NPI and incorporate them in the SIR-Model, but completely
disregard weather effects [37]. In any case, the mix of varying weather
conditions and large combinations of different NPIs can not be modelled by SIR
models currently.

###  1.2 Motivation and Contribution

We propose separating the problem of estimating the epidemiological parameters
as function of weather conditions and NPIs, which we do in an inductively
learnt pre-prepossessing step, from the computation of the SIR model and to
apply an inductively learnt post-processing step to the SIR-models result to
compensate for the SIR-models intrinsic lack of representative power. Using
our technique, we achieve improved predictive power at slightly higher
computational cost than running non-augmented SIR models and are able to
provide an efficient and accurate tool for backwards analysis of NPIs.

##  2 Model Pipeline

As mentioned in sec. 1.2, we separated the modelling of the pandemic in three
separate tasks. Our pipeline, as illustrated in fig. 1, consists modelling
reproduction number as function of weather and NPIs, which we accomplish using
a neural network, modelling the spread of COVID-19 based on the modelled
reproduction number using a SEIR-FV-Model and subsequently using the SEIR-FV-
Model output as input for a group of neural networks correcting SEIR-FVs
intrinsic error and inferring hospital resource utilization.

Figure 1: Model Pipeline

###  2.1 Reproduction Number

For modelling reproduction number $R_{eff}$ as function of weather and NPIs,
we used numerically encoded NPI data from Oxford COVID-19 Government Response
Tracker (OxGRT) [43] which is already provided in machine-learnable format at
daily resolution, monthly weather data from 1955 till today provided by Hohe
Warte weather station near Austria’s capital Vienna [44] and daily
reproduction number estimates provided by AGES [45]. Zentralanstalt für
Meteorologie und Geodynamik (ZAMG) [46], which could provide more fine
granular weather data not only for Vienna but all of Austria, was contacted
but unfortunately did not wish to cooperate. In order to be suitable input for
a neural network, weather data is first up sampled to match OxGRT datas
resolution using Fouriers method and most significant features are selected
using linear regression-analysis. Based on the selected features, additional
features are engineered via power and log transforms to allow more room for
choice in the forthcoming step. Next OxGRT data and weather data are merged
and scaled and linear regression-analysis is employed again to eliminate
independent variables not affecting the dependent variable $R_{eff}$ for a
significance level of $p=0.05$ with exception to NPIs. For NPIs, we manually
selected which to keep and which to remove in order to be able to keep the
model from picking measures not directly relevant for pandemic spread such as
”Fiscal Measures”, ”Debt Contract Relief” or ”Income Support” but nonetheless
part of the OxGRT dataset and because we wanted measures with particularly
severe potential human collateral damage (e.g. ”School Closures”, ”Workplace
Closing”) (CITE) to be kept even if not chosen for the given significance
level. Finally, after the preprocessing of the training data is finished, we
train a neural network on each of the 3 $R_{e}ff$ estimates (lower, upper,
mean) provided by AGES. We chose a neural network with non-linear activation
functions over a linear regression model because despite achieving a good fit
(R2=0.99) during regression analysis in previous steps, the linear regression
model performed very poorly for OOS data (we found setting multiple NPIs in
the dataset to 0.0 could lead to the linear regression model predicting
negative reproduction numbers), a deficit we did not find using a non-linear
neural network.

Figure 2: Predicting $R_{eff}$

Dependent on application, for backwards-analysis we apply a post-processing
step that introduces historical $R_{eff}$ estimates as lower bound for
scenarios where a specific NPI is completely removed s.t. e.g. in a scenario
without mandatory face masks, $R_{eff}$ can never be lower than in a
historical case where face masks were mandatory. For forward-analysis the
post-processing step is naturally not applied.

|  Low $R_{eff}$ | Mean $R_{eff}$ | High $R_{eff}$
---|---|---|---
R2 | 0.94 | 0.93 | 0.94
Table 1:  R2 scores for predicting $R_{eff}$ on 30% validation set via a
neural network

###  2.2 Pandemic Model

The predicted lower, mid and upper vectors containing daily $R_{eff}$ values
for a given time interval is then handed to three SEIR-FV models (one for each
estimate of lower, mid, upper $R_{eff}$) which, for each age group, tracks
Susceptible, Exposed, Infectious, Recovered, Fatal, Vaccinated population
groups, see fig. 3 for a simplified illustration.

Figure 3: Susceptible, Exposed, Infectious, Recovered, Fatal, Vaccinated

The initial age group distribution is obtained from [47], Hospitalization,
Intensive Care Unit (ICU), Infection Fatality (IFR) (fig. 4) ratios per age
group is obtained from [48].

Figure 4: Per Age Group IFR

The age groups data is homogenized to groups from $<5$ to $>84$ in 5-year
intervals. The rate at which people are vaccinated with any vaccine is
obtained from [49] and for ease of use, an average daily vaccine doses
available per day is computed by fitting a linear function through the data
(fig. 5) and using its slope.

Figure 5: Vaccination Rate in Austria

The SEIR-FV models follow an old-first vaccination strategy, i.e. available
vaccine doses are distributed among age groups, preferring older age groups
over younger age groups. For duration of vaccine and recovery immunity and
efficiency, we use estimates based on [50]. Incorporating ICU and
Hospitalization rates into the model is done statistically: based on the
a-priory known hospitalization and ICU rates for each age group as well as the
current number of infected computed by the SEIR-FV model, we infer the number
of persons of an age group requiring a hospital bed or an ICU bed. For cases
where hospital requirements exceed hospital resources, we made the simplifying
assumption that all patients requiring hospital treatment but not receiving it
become a fatality.

###  2.3 Error Correction Networks

The pandemic models provided by SEIR-FV even for the time dependent $R_{eff}$
computed from weather and NPI data by the NN regressor is, while qualitatively
making accurate predictions where the peaks in disease spread are,
quantitatively off by orders of magnitude (fig. 6). We solve this by training
so-called error correction neural networks (see sec. 2, fig. 1 for the step in
the modelling pipeline) on correcting the SEIR-FVs outputs error (fig. 7) and
furthermore predict the amount of hospital resources used (fig. 8).

Figure 6: SEIR-FV, R obtained from NN Regressor

As illustrated by the figures, our error correction neural networks approach
greatly improves the SEIR-FVs outputs quantitatively, with the historical
numbers of infected and fatal cases being inside the given 95% confidence
interval for the whole 03.2020 to 03.2021 time period.

Figure 7: NN Error Correction Augmented SEIR-FV, R obtained from NN Regressor,
03.2020 to 03.2021 time frame

For hospital resources (regular and ICU beds), the augmented model produces a
quantitatively and qualitatively equally accurate output for a 85% confidence
interval. Regarding network architecture, we use simple 4-Layer ReLU activated
MLPs. To account for the uncertainty that comes with using such networks that
always incorporate randomness due to their random initial state, we train
groups of 10 networks for each of the target variables (infected, fatal,
hospital bes, ICU) and provide the results in the form of confidence
intervals. We do this for each of the three SEIR-FV models computed for the
three estimates for $R_{eff}$.

Figure 8: Hospital Resource Requirements computed by NN, 03.2020 to 03.2021
time frame

###  2.4 NPI Collateral Damage Model

Secondary to modelling pandemic spread, we decided to model human collateral
damage to NPIs as well. Our simple collateral damage model exploits the known
correlation between life expectancy and health and economic development [51,
52, 53] and the observed economic depression [54, 55] during the pandemic,
which we conjecture to be caused by business and border closures. Other
potential causes of collateral damage such as mental health [56, 57, 58, 59,
60, 61, 62, 63, 64, 65, 66, 67], disrupted medical supply chains [68],
skipping of early medical examinations [69] and problems for people with
disabilities [70] were initially considered, but we decided to disregard them
due to a lack of useful coherent data. For our economy-based collateral damage
model, we first confirmed that a statistically significant (p=0.05) relation
between economic factors and life expectancy existed in pre-corona times, a
purpose for which we used OECDs GDP and life-years-lost (LYL) data sets [71],
and that the same holds for NPIs and GDP. Next, as depicted in fig. 9 we
trained a NN to predict GDP dependent on NPIs and another NN to predict LYL
dependent on GDP to finally obtain a pipeline modelling LYL dependent on NPIs.
Admittedly, this approach suffers from not taking the collateral damage
produced by corona itself into account (which would certainly be strong in the
case of e.g. an unrestricted spread of the pandemic) but the lack of a
baseline for such scenarios made it impossible to model them.

Figure 9: NPI Collateral Damage Model

##  3 Results

We applied our approach in two different tasks. First, we applied it to the
backwards analysis task, i.e. asking the counterfactual question ”What would
have happened if NPI $<x>$ had not been in place?” to find out which NPIs were
most crucial in the past. Second, we apply our approach to predicting future
developments. In this case, we will make a simple statistical weather
prediction for future weather based on the historical data provided by Hohe
Warte weather station to use as input for our NN regressors trained to predict
$R_{eff}$.

###  3.1 Backwards Analysis

For backwards analysis of NPI effectiveness, we follow a ”take one out”
approach. We compute each scenario under the assumption of historical weather
and NPIs but with a particular NPI not implemented, i.e. it’s zeroed in the
input data matrix.

Figure 10: No Workplace Closing, Infected and Fatalities, 03.2020 to 03.2021
time frame Figure 11: No Workplace Closing, Hospital Ressource Requirements
computed by NN, 03.2020 to 03.2021 time frame

Fig. 10, 11 display infections, fatalities and hospital resource usage for the
hypothetical scenario where no workplaces closures were implemented. The model
predicts a slightly higher peak during the winter months and accordingly a
higher hospital resource consumption. Particularly critical is the elongated
period ICUs are over capacity compared to the historical scenario, leading to
slightly higher number of fatalities in vulnerable age groups. Another
potentially interesting scenario is no obligatory facial mask usage (fig. 12,
13). Infections and fatalities nearly triple for this scenario, hospital
resources are catastrophically overburdened for a period of over six months
which contributes massively to fatalities in older population groups likelier
to require ICU care in case of an infection.

Figure 12: No Facial Masks, Infected and Fatalities, 03.2020 to 03.2021 time
frame Figure 13: No Facial Masks, Hospital Ressource Requirements computed by
NN, 03.2020 to 03.2021 time frame

Finally, we want to explore the case where no mitigation measures were
implemented at all and COVID-19s spread is only controlled by weather dynamics
and population composition (fig. 14, 15). In this scenario, we see a massive
peak for infections in September, with multiple million people suffering
COVID-19 infections simultaneously. The massive and constant operation of
hospitals over capacity over the whole evaluated time frame leads to up to a
quarter million fatalities within the first 12 months of the pandemic.

Figure 14: No Mitigation Measures, Infected and Fatalities, 03.2020 to 03.2021
time frame Figure 15: No Mitigation Measures, Hospital Resource Requirements
computed by NN, 03.2020 to 03.2021 time frame

####  3.1.1 NPI Collateral Damage

Using the collateral damage model described in sec. 9, computed collateral
damage for two scenarios to illustrate the functionality of our model. Fig. 16
illustrates the scenario where historical NPIs were in place, except
”Workplace Closing”. As shown by the visualization, removing the ”Workplace
Closing” NPI dramatically reduces collateral damage, with the prediction of
LYL for the next decade being not worse than what would have been the case
without any NPI induces collateral damage.

Figure 16: No Workplace Closing, Collateral Damage, 03.2020 to 03.2021 time
frame

A scenario with historical NPIs but no facial masks is shown by fig. 16. As
you can see, our collateral damage model predicts only a slight improvement
compared to historical measures.

Figure 17: No Facial Masks, Collateral Damage, 03.2020 to 03.2021 time frame

###  3.2 Predictive Power

For long-term predictions illustrated in fig. 18 and 19 the model pipeline is
fitted on historical data of the 03.2020 to 03.2021. Future weather data for
the 03.2021 to 01.2024 period is statistically generated from historical
weather data and government measures are assumed to be repeated on a yearly
basis, but of course scenarios for alternative sets of future government
measures could be computed as well. As you can see, the model predicts with a
high probability a potential future peak in infections as well as ICU
requirements exceeding capacity in December 2021 and January 2022. After the
winter 2021/2022 peak, the number of infected slowly declines and fatalities
have reached their maximum due to vulnerable age groups having been immunized
via recovery or vaccination.

The predictive power of the model pipeline is to be taken with a grain of
salt, though. The current prediction algorithm is only capable to predict
future developments from the very beginning of the pandemic due to the way the
SEIR-FV model is initialized. Correcting this deficit will be subject to
future work, but it offsets the predictions quantitatively.

Figure 18: Long Term Prediction computed by NN, 03.2020 to 03.2024 time frame
Figure 19: Long Term Hospital Requirements computed by NN, 03.2020 to 03.2024
time frame

##  4 Conclusion

We present a new approach of pandemic modelling through augmenting a regular
SEIR-FV compartmental pandemic model with pre- and post-processing neural
networks prediction reproduction number dependent on weather conditions and
NPIs and demonstrate its usefulness for backwards analysis tasks as well its
potential for predicting future developments w.r.t. COVID-19. Admittedly, the
predictive power of our model leaves room for improvements, but we think that
despite our approaches deficits, it still advances the state-of-the-art in
cheap but accurate compartmental pandemic models due to its ability to
incorporate weather data and fine granular NPI data from OxGRT. We also
provide a simple NPI collateral damage model capable of predicting the
development of future life-years-lost dependent on NPIs.

##  5 Future Work

We intend to fix the pandemic model’s current issues when making long-term
predictions and release it in the form of a web-based interface to allow the
public to play out hypothetical backwards analysis scenarios for various sets
of NPIs as well as to make future predictions dependent on user chosen NPI
sets and weather conditions. Furthermore, we intend to use this report as a
first draft for a publication in a journal or a conference focusing on
pandemic modelling. The web interface and the potential journal or conference
paper will not contain the collateral damage model because we consider it too
weak to support the potentially strong political implications that could be
derived from it.

### 5.1

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