# Machine learning spatio-temporal epidemiological model to evaluate Germany-
county-level COVID-19 risk

Lingxiao Wang  Tian Xu  Till Hannes Stoecker  Horst Stoecker  Yin Jiang  Kai
Zhou  Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1, 60438
Frankfurt am Main, Germany  Department of Physics, Beihang University,
Beijing, 100191, China  Black Hole KG, 61440 Oberursel (Taunus), Germany
Institute for Theoretical Physics, Goethe University Frankfurt, 60438
Frankfurt am Main, Germany  Beijing Advanced Innovation Center for Big Data-
Based Precision Medicine, School of Medicine and Engineering, Beihang
University, 100191, China

###### Abstract

As the COVID-19 pandemic continues to ravage the world, it is of critical
significance to provide a timely risk prediction of the COVID-19 in multi-
level. To implement it and evaluate the public health policies, we develop a
framework with machine learning assisted to extract epidemic dynamics from the
infection data, in which contains a county-level spatiotemporal
epidemiological model that combines a spatial Cellular Automaton (CA) with a
temporal Susceptible-Undiagnosed-Infected-Removed (SUIR) model. Compared with
the existing time risk prediction models, the proposed CA-SUIR model shows the
multi-level risk of the county to the government and coronavirus transmission
patterns under different policies. This new toolbox is first utilized to the
projection of the multi-level COVID-19 prevalence over 412 Landkreis
(counties) in Germany, including t-day-ahead risk forecast and the risk
assessment to the travel restriction policy. As a practical illustration, we
predict the situation at Christmas where the worst fatalities are 34.5
thousand, effective policies could contain it to below 21 thousand. Such
intervenable evaluation system could help decide on economic restarting and
public health policies making in pandemic.

###### keywords:

SUIR Model, COVID-19 Pandemic , Deep Learning, Cellular Automata

††journal: XXX

[figure]style=plain,subcapbesideposition=top

##  1 Introduction

Public health security is the cornerstone of economic and social stability.
CoronaVirus Disease 2019 (COVID-19), an infectious disease caused by the
Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2), became a global
pandemic that spread out across the world from the beginning of 2020. Since
middle of March, the number of infected cases in Germany experienced a rapid
growth. Although the severe restriction of travel eases the situation in the
summer, the number of infected cases have been increasing again with a more
critical trend since mid-September, named routinely as the second-wave
pandemic worldwide. By November 25, 2020, the pandemic has caused 983588
confirmed cases and 15160 fatalities in Germany. Many states are facing
dilemma challenges of public heath and economics in curbing the spread of the
COVID-19, e.g., Berlin, Bavaria and Hessen. The medical resources of Germany
(ICU beds and doctors) have not reached their ceiling, but the current phase
of the pandemic is not encouraging. Being a high-infectious disease, COVID-19
is expected to continue spreading or be even worse in the coming winter, which
will cause a higher number of infections and mortality in the next few months.
With no perfect medical treatments nor enough vaccines available until now,
implementation of relatively effective public health policies such as social
distancing and travel restriction become the primary measures of the states to
mitigate the spread of epidemic, which has been proven to be an effective
containment in East Asia, e.g. China [1] and South Korea, also in Europe at
early stage [2]. In this worsening health crisis, a dire need emerges to
establish a multi-level (Nation-State-County) COVID-19 risk prediction model,
which can guide the evaluation and improvement on the public health policies
of governments (states) at multi-level. Such intervenable risk evaluation
model is beneficial to policymakers in decision-making on economics restarting
and public health policies at different scales during epidemics. The multi-
level risk information can help governments to improve the public restriction
strategies such as cross-state travel restriction, home office rules, severe
home isolating regulations and vaccination resources allocation. Residents,
who are concerning the present situation, can also benefit from the multi-
level risk information, to make effective access of medical resources and help
implementing or following the public health policies.

Evaluation to the risk of the on-going COVID-19 pandemic across the
geographical-dependent administrative divisions is challenging because of the
substantial heterogeneity rooted in economic composition, industrial
diversity, traffic conditions and political views. Nonetheless, it is
inevitably an emergent demand for governments at all levels. Although a lot of
studies have been devoted to the dynamics of the epidemics based on
mathematical models [3, 4], (see also reviews111Since the number of COVID-19
papers are too many to list them all, the following references are highly
relevant to the topic we discussed in this paper. [5, 6, 7]), they mainly
focus on time-domain analysis and ignore the influence of geographic
distribution and population migration [2] which palpably presents as cross-
county transfers at daily-scale. A model which can predict the spatio-temporal
evolution of the COVID-19 cases is urgently needed to assess the local risk at
multi-level regions. Besides works which concern spatial distribution or local
evolution based on the historical statistical data [8, 9, 10] , recently there
are several studies sharing similar goals in different countries [11, 12, 13,
14], which is merited to be one of the most urgent needs across the world. In
a practical perspective, a reasonable and efficient risk prediction model
needs to be performed comprehensively with all Landkreis 222It is the primary
administrative subdivision higher than a city in Germany, also could be named
as county in English., since people is highly mobile and connected by
highways, railways and airways in Germany. Conceivably, a county-level risk
evaluation to the public and local governments is informative and ponderable.

Machine learning (ML), a branch of artificial intelligence, efficiently
integrates statistical and inference algorithms and thus offer the opportunity
to uncover hidden structure of evolution in complex coronavirus data and to
describe it with finite dynamical parameters. Therefore, combinatory usage of
ML algorithms and spatio-temporal epidemiological model for risk prediction in
the context of COVID-19 pandemic are worth exploring. Although the state-of-
the-art ML techniques have been deployed into myriad fields of COVID-19
pandemic [15, 16, 17, 18], for the governments and residents, the highest
concern is how fast the COVID-19 diffuses. Available ML models that focus on
this exhibit promising implications in individual or national scale, but are
still impeded by the paucity of validation information and limited privacy
rules in Germany, thus lack the capability of predicting multi-scale
evolution.

In this study, we aim to establish a new evaluation paradigm by introducing a
spatio-temporal epidemiological model with machine learning assisted. The
dynamical model of the epidemic contains a Cellular Automaton (CA) [19, 20]
and a modified Susceptible-Undiagnosed-Infected-Removed (SUIR) model [13, 21].
The SUIR model is more inclusive than the previous Susceptible-Infectious-
Recovery(SIR)-type models since consideration of the Undetected or
Asymptomatic populations [22, 23] hidden under the lack of detection. As an
important extension of the SIR model, we divide the infectious population into
two compartments: one is the undiagnosed who manifest as normal susceptible
individuals but with possible infectious and no timely detection; the other
one is the infected (diagnosed) who are confirmed by detection and have been
admitted to hospital or isolated at home. In this new temporal model, we
constrain the mobility of the people in the susceptible and undiagnosed
populations, in which the understanding to the latter will help address the
underestimation issue concerning infection cases in the public databases. As a
matter of fact, the movement of the undiagnosed people across the counties
will contribute to the deteriorating epidemic situation, thus we introduce the
CA model to capture the influences of the population mobility. Being assisted
with ML to extract the dynamical parameters, this new evaluation system can
inform the public and governments in different regions with effective warning,
which should be considered into adjusting the restriction rules.

The article is organized as follows. In Section. 2, we propose a full
framework of risk prediction to COVID-19 pandemic in Germnay with introducing
the spatio-temporal CA-SUIR model. The machine learning techniques are applied
to extract the dynamical parameters from reported data of Robert Koch-Institut
(RKI). Section. 3 reproduces the evolution of infection maps from 25 February
to 28 March 2020, and extracts the infectious parameters from RKI data in
September 2020 with a well-trained machine which was training on the data set
from CA-SUIR model simulations. The well-learned machine predicts county-level
infection maps from September to November 2020, which is matched remarkably
with the reported situation. With regard to the up-coming Christmas holiday,
the different public health policies are evaluated in the end of the section,
in which the entire, partial and none lock-down strategies are compared. In
the last section, some concluding remarks are presented.

##  2 Machine Learning Spatio-temporal Epidemiological Model

Figure 1: A framework to evaluate and predict the COVID-19 risk in Germany.
The left panel is the geographic distribution of the infection cases in
Germany, which is named as the real map at nation level. The central panel is
the topological equivalent map of the infection cases, in which the main
cities are mapping onto the sites on square lattice at state level. The right
panel is the microscopic dynamics on each site, which is spatially depicted by
Cellular Automata and temporally evolved through SUIR model at county level.
The color of the characters in SUIR corresponds to the subsequent Fig. 3.

In this section, we propose a spatio-temporal model to predict the multi-level
COVID-19 risk in Germany, named as CA-SUIR model in the following parts.
Machine learning techniques are introduced to learn the dynamical parameters
of the CA-SUIR model given a ’movie’ of infection maps, which is sequentially
utilized to evaluate the COVID-19 risk from RKI infection map timely in
Germany. The full flow chart is assembled in Fig.1 to present the multi-level
spatio-temporal simulation of epidemics, where as the first step real maps are
projected onto the topological equivalent maps as explained in Appendix. A.
The cumulative infected cases in the real profiles333see the map at COVID-19
pandemic in Germany are reassembled into a topological equivalent map with
$25\times 25$ sites in Germany, which contains 412 counties (see details in
Appendix. A). The micro-states of each county, its number of infected cases,
are driven by the SUIR and CA simultaneously in the model, which construct the
topological infection map. With simulated data from the CA-SUIR model, the
Conv2LSTM neural networks [24] consist of the Time-distributed Convolution
Neural Networks(CNNs) and the Recurrent Neural Networks(RNNs) are designed to
learn to extract dynamical parameters of CA-SUIR model. Subsequently, we
transfer the well-trained neural networks to make timely prediction and risk
evaluation on RKI infection maps.

###  2.1 Deep learning dynamical parameters

Figure 2: Machine learning the dynamical parameters of CA-SUIR model from the
evolution of infection maps. The Time-Distributed Convolutional Neural
Networks(CNNs) layers and Long Short-Term Memory(LSTM) units are combined to
extract the information encoded in the input short movie (one image per
frame). The undiagnosed population, infectious rate, removed rate, diagnosed
rate and transfer rate are decoded from the Conv2LSTM neural networks.

CNN model and Long Short-Term Memory (LSTM) model have been popularly adopted
and show remarkable performance in many researches like in computer vision or
Natural Language Processing tasks. While the CNN model can do efficient
pattern recognition with convolutional filters on image-like tensor data, the
LSTM model is capable of capturing dependencies along time direction between
sequential tallies. In this work we combine CNN and LSTM model into Conv2LSTM
for analysing the infection dynamical evolution.

The Conv2LSTM neural networks we built are demonstrated in Fig. 2, in which
the Time Distributed Convolutional Neural Networks(CNNs) layers and LSTM
layers are combined to decode the information encoded in the short movie
constructed by the infection maps. The details of the structure are arranged
in Appendix. C. The time distributed CNNs could extract features (with
temporal structure preserved) from sequential image inputs and then embed them
into the following LSTM layers, which is able to efficiently drill evolution
rules from the short movies.

###  2.2 Temporal SUIR model

We propose a proposed modified Susceptible-Undiagnosed-Infected-Removed (SUIR)
model and demonstrate it in Fig 3, where the infectious rate $\lambda$
controls the rate of spread which represents the probability of transmitting
disease between a susceptible and an infectious individual. The diagnosed rate
$\sigma$ dictate the probability of latent infectious individuals becoming
confirmed by testing. Removed rate $\gamma$ contains the rate of mortality and
recovery from the infectious. Besides the classical definition of SIR model,
considering the special features of COVID-19, we introduce the modified SUIR
model: there is a considerable population that could be infected but without
any symptoms, or, they just have no access to perform a detection(with
possibility $p$), which is concluded in a new compartment, Undiagnosed(U). In
contrast to the Infected who is usually subjected to be isolated in medical
facilities or quarantine at home, the Undiagnosed population is with normal
mobility, which leads to the high infection; the infectivity could decrease
after $d$ days [25], but there is no clear evidence that they will get special
immunity, which means immunity loss.

Figure 3: The SUIR diagram shows how individuals are infected and transferred
among each compartments in a close area.

In a closed population with no births or mortalitys per day,the modified SUIR
model becomes,

$\displaystyle\frac{dS}{dt}$ $\displaystyle=-\frac{S}{N}\lambda
U+\frac{\sigma^{\prime}}{\text{days}}U$ (1) $\displaystyle\frac{dU}{dt}$
$\displaystyle=\frac{S}{N}\lambda U-\sigma
U-\frac{\sigma^{\prime}}{\text{days}}U$ (2) $\displaystyle\frac{dI}{dt}$
$\displaystyle=\sigma U-\gamma I$ (3) $\displaystyle\frac{dR}{dt}$
$\displaystyle=\gamma I$ (4)

where $N=S+U+I+R$ is the total population. The redefinition parameters in
Fig.3 are $\sigma\equiv\sigma p,\sigma^{\prime}\equiv\sigma(1-p)$. The detail
explanation of the parameters are in Appendix. B.

###  2.3 Spatio-temporal CA-SUIR model

We construct a cellular automaton model for tracking the epidemics spatial
evolution in a closed system (Nation with fixed boundary as Fig. 1 shown,
which matches the extreme restriction adopt by the German government from
March 16. to May). The underlying structure is a $L\times L$ cell grid, where
$L$ is the system size, and the number is from the topological map. The color
of cell characterizes the population in minimum geographical unit, which is
county in our case. It’s an instructive example to set the cell size to a
typical area of city, such as $10^{2}\sim 10^{3}\text{km}^{2}$, which can
simulate the epidemics in the corresponding resolution approximately. The
resolution could be naturally increased into community level, but it needs
more detailed data which is laboriously accessible, thus the following
discussion focuses on county level and corresponding resolution. This model
adopts the Moore neighbor, and cells update their states by transition tensor
$P_{m,n}^{i}(t)$ with $4\times L\times L$ shape, which means the possibility
of transfer from city $(m,n)$ to direction $i$ at $t$ time-step. In Fig. 4,
the $P^{i}(t)$ shows as 4 matrices
$(P^{\text{u}}(t),P^{\text{d}}(t),P^{\text{l}}(t),P^{\text{r}}(t))$ with same
lattice size, in which different colors labels the direction of transfer. In
the following simulation, we adopt the mean field approximation [26], which
leads to the uniform transition possibility
$P^{\text{u}}(t)=P^{\text{d}}(t)=P^{\text{l}}(t)=P^{\text{r}}(t)=\epsilon$.
The transfer rate $\epsilon$ is introduced to describe the average movement of
the residents on each site. To mimic the border control induced closed
boundary, a transfer rate $\epsilon=0$ has been used in the primary epidemic
stage [27].

Figure 4: Transition matrices in Cellular Automaton model. The size of the
grid is $L=25$ in our case.

The evolution of the CA-SUIR model is as follows,

Step.1 Initialization. Set the position of infection burst $(x,y)$ and
generate population distribution $S(t)$ at the $L\times L$ lattice. At $t=0$
time-step, epidemic turns on and population begins to migrate;

Step.2 SUIR dynamics. At $t$ time-step, evolve SUIR model $t^{\prime}$ times
on each site;

Step.3 Movement. The parts of susceptible and undiagnosed people (S and U) on
site $(m,n)$ migrate into the neighbor site with transition matrices
$P^{k}(t,m,n)$ at $t+1$ time-step. Update all sites synchronously;

Step.4 Update. Turn to Step.2 until $T$ time-step is reached.

##  3 Results

###  3.1 Pandemic dynamics modeling, data set generation and network capacity

Figure 5: The comparison of RKI data and simulated cumulative infection cases.
The data is from 25 Feb.2020 to 28 Mar.2020, in which the shadow bars are from
uncertainty estimation. The orange panel is the variation space of SUIR model
parameters, which shows a reliable area under $10\%$ changes. The green panel
is from the $10\%$ uncertainty of the undiagnosed population(U), which is
actually from the stability issues of RT-PCR testing [28] and unreported cases
in practical situation.

With the daily number of cumulative cases in Germany (Data is from the open
database of Robert Koch-Institut (RKI) 444 www.rki.de/covid-19 and COVID-19
GeoHub Deutschland), we fix the SUIR parameters to be
$\lambda^{\prime}\equiv\lambda-\frac{N}{S}\sigma^{\prime}/\text{days}=0.293,\sigma=0.129,\gamma=0.628$
by fitting from redefined exponential behavior parameters(see Appendix. B).
This parameter set can fit well the first-wave burst out in March especially
the rapidly increasing stage (from 25 Feb.2020 to 28 Mar.2020), as presented
in Fig. 5. The number of initial undiagnosed cases is determined in the model
as $U_{0}\simeq 700$ which distributes as the infection cases. Fig.5 suggests
that there is an overestimation of the simulation at early stage(first 20
days), which might be naturally understood to be related with the
underestimation in reality at that time since the limited detection ability
and the supply of test reagents were insufficient [29]. Conceivably, by
introducing more practical factors from clinical diagnosis or management, the
SUIR model can be predictably improved to match with the record data, which
will be discussed in our further works but is not the focus in the current
one. According to the initial infection cases distribution and the locations
of main cities of Germany, we set the initial undiagnosed population
distribution $U_{0}(m,n)$ to be proportional to the population distribution of
the cities $N(m,n)$. In the simulations for March we choose the transfer rate
$\epsilon=0.08$ as a constant, the population of transfer on site is
proportional to the rate in each time-step. With the fitted parameters from
above SUIR model, we generate the simulated evolution of the cumulative
infection cases map in Fig.6, which demonstrates the high consistency with the
RKI data map (named as real profiles, see the full evolution sample in
Appendix. A).

[] []

Figure 6: Simulated and data evolution of the infection distribution in
Germany. The infection maps are evaluated through CA-SUIR model, from the same
initial distribution at 25 Feb.2020. The color bar is set in an uniform orange
color scale, the darkest color is 2000 cases with a logarithmic re-scale. The
horizontal and vertical coordinates correspond to the real geographical
latitude and longitude, which is consistent in the following figures.

To prepare training data for Conv2LSTM, we generate 30-day simulation maps as
short movies by associating the SUIR model parameters a $10\%$ uncertainty
range. The total number of the data set is 10000, which contains 9000
infection movies as training data-set and the rest 1000 as testing data-set.
The 30-day movie contains 30 frames, which is fed to the machine with
continuous 7-day segments as slip windows [30]. The structure of the Conv2LSTM
neural networks are exhibited in Appendix. C. With the short movies to be the
input, the Conv2LSTM is trained to predict the involved epidemiological model
simulation parameters. The learning curve is shown in Appendix. C. The four
sub-figures in Fig. 7 demonstrate the comparison between the prediction from
the trained Conv2LSTM and ground-truth for testing data, they are the transfer
rate $\epsilon$, the infectious rate $\lambda$, the diagnosed rate $\sigma$
and undiagnosed population $U$ respectively. The well-trained machine extracts
parameters from the RKI infection map of March 2020 as
$\lambda^{\prime}=0.26,\sigma=0.09,U_{0}=1352,\epsilon=0.068$. The first and
last parameters are numerically matched with the fitting results, and the
diagnosed rate and initial undiagnosed population maintain the consistency
with fitting parameters in the relationship $\sigma U_{0}$ (see Appendix. B).

[] [] [] []

Figure 7: The performance of Conv2LSTM neural networks to extract dynamical
parameters from infection maps. (a) The transfer rate $\epsilon$ is tested on
the testing data-set with the coefficient of determination $R^{2}=0.996$. (b)
The infectious rate $\lambda$ is tested on the testing data-set with the
coefficient of determination $R^{2}=0.998$. (c) The diagnosed rate $\sigma$ is
tested on the testing data-set with the coefficient of determination
$R^{2}=0.987$. (d) The undiagnosed population $U$ is tested on the testing
data-set with the coefficient of determination $R^{2}=0.993$.

###  3.2 Machine Learning-assisted Risk Prediction

To assess the COVID-19 risk in realistic situation timely, we deploy the well-
trained neural network into the current cases with transfer learning method.
Transfer learning is a machine learning method where a model developed for a
task is generally reused as the starting point for a model on a second task.
In our case, transferring the pre-trained models into the on-going second-wave
COVID-19 epidemics in Germany, we extract the dynamical parameters of the
spatio-temporal pandemics from the 30-day RKI infection maps with beginning
from September 8, 2020, which yields the dynamical parameters as
$\lambda^{\prime}=0.21,\sigma=0.167,U_{0}=7426,\epsilon=0.09$.

[] []

Figure 8: Comparison of simulation and RKI infection maps from September to
November 2020 in Germany. The color bar is set into an uniform red color
scale, the darkest color is 20000 cases with a logarithmic re-scale. (a) As of
the end of November, the cumulative infection cases are presented in dynamical
simulation with parameters learned by machine learning at 2020-09-25,
2020-10-25 and 2020-11-25 respectively. (b) As of the end of November, the
cumulative infection cases are presented in real profiles at 2020-09-25,
2020-10-25 and 2020-11-25 respectively. The top 5 biggest cities in Germany,
Berlin (13°23E, 52°31N), Hamburg (10°00E, 53°33N), München (11°35E, 48°03N),
Köln (6°58E, 50°56N) and Frankfurt (8°41E, 50°07N) shows a worse situation
than the other sites.

Fig. 8 describes the homogeneity between simulation and RKI infection maps
from September to November 2020 in Germany. The former cumulative infection
maps shown in Fig. 8 are generated from the SUIR model with the machine
learning parameters during 3 months. The latter maps shown in Fig. 8 are
projected from the RKI data, in which we present the cumulative infection
cases in real profiles until the end of November. The slight mismatching on
the map of November is due to the actual soft lock-down policy has been
executed from 2 November. The functional regulations will contain the pandemic
moderately, which will be examined in the following section under different
policies.

###  3.3 Public Policies Evaluation

Table 1: Public Policies for Different Spots. This table lists specific rules which lead to three scenarios: a) unlocked, b) soft-locked, c) locked. The last one is the well-know lock-down policy which shows most powerful restriction to the mobility of human. | Unlocked | Soft-locked | Locked
---|---|---|---
Office | Office work in schedule | Home office for large group | Home office
Campus | Open | Close | Close
School | Open | Close | Close
Supermarket | Open | Open with distance constrains | Close
Restaurant | Customer number constrain | Only Take-out | Close
Travel | Free movement | Long distance travel is prohibited | Home isolation

Three different degrees of restriction rules, named as the unlocked, soft-
locked and locked strategies in public policy, are evaluated in our COVID-19
evaluation framework. The examples of the corresponding policies are listed in
Table. 1, in which we list some representative government policies in
different spots. It is of crucial concern for both governments and residents
who are enduring the social-economical pressure in pandemic [11, 31, 32, 33].
The hot spots include but are not limited to office, campus, school,
restaurant and cross-county travel. It should be mentioned that although we
introduce the heterogeneous rules in different spots the comprehensive
influences at county-level are averaged, which means the effect of policies
are coarse graining to the county-level. Reasonably, it could be improved if
we collect more precise data below county-level, e.g., individual or
community-level [34].

[] [] []

Figure 9: Predicted cumulative infection maps at Christmas under three
different public policies. The color bar is set into an uniform red color
scale, the darkest color is 20000 cases with a logarithmic re-scale. The
dynamical parameters are extracted from RKI infection maps by the well-trained
machine. (a) The prediction is based on the existing public policies. (b) The
soft lockdown rules contains the relative loose restrictions to some hot
spots, thus the diffusion of the epidemic is alleviated partially. (c)Locked:
These strictest rules prohibit any form of mobility, which yield the least
infections.

The differences among them in simulation are the transfer rate and susceptible
population: the most strict locked rules has transfer rate $\epsilon=0$, the
soft-locked with half transfer $\epsilon=0.045$, and the rule with no
additional restriction has $\epsilon=0.09$ as default; meanwhile, the
containment policies could result in a consequence of infectious population
decreasing that effectively deplete the susceptible population [1]. It leads
to the reduction of the susceptible population, reaching to $1/2,1/8$ for the
locked and soft-locked rules respectively, in which the $1/8$ is from the
rough estimation to the necessary out time. These three strategies are
implemented from the same time point 25 November 2020, and their 30-day
simulation results are shown in Fig. 9. In our prediction assisted by the
transfer learning method, 2.3 million people will be infected before Christmas
in the worst situation, where daily increase is approximately 80 thousand
(estimate with mortality rate $\sim 1.5\%$ [35]). Compared with the prediction
of the Institute for Health Metrics and Evaluation
(IHME)555http://www.healthdata.org/covid, they achieve the consistency in
order of magnitude, which means the total mortality will reach 34.5 thousand
approximately. With regard to the soft-locked policy, the infection number
reduces to 1.4 million and the corresponding daily increasing cases are 20
thousand. In the most strict rules, although it restricts the mobility, the
number of infection cases will be 1.07 million and the daily increase is 4
thousand.

##  4 Discussion

In this paper, we construct a multi-level spatio-temporal epidemiological
model that combines a spatial cellular Automaton (CA) with a temporal hybrid
Susceptible-Undiagnosed-Infected-Removed (SUIR) model which contains the human
mobility across the counties. This new toolbox enables the projection of the
state-county-level COVID-19 prevalence over 412 Landkreise in Germany,
including t-day-ahead risk forecast and the risk evaluation related to a
public policy. The modular design in our survey can reach a more accurate
estimation of the COVID-19 risk with replacing the modules by the other
elaborate spatial or temporal models conveniently. With the help of machine
learning methods, we extract the dynamical parameters directly from the
infection maps, which are topologically equivalent to the real geographic
maps. After training the Conv2LSTM neural networks composed of time
distributed CNNs and LSTM on data set generated from CA-SUIR model, we
transfer the well-trained neural networks into the data set collected from the
open database of RKI.

The CA-SUIR model reproduces the evolution of the infection maps in March and
September acknowledged as the first- and second-wave of COVID-19 pandemic.
Even though the prediction focuses on the infectious dynamics in the current
paper, it is conveniently feasible to derive recovery and mortality rate from
the model under the guarantee of accurate data sources confronted. The
simulations perform properly to describe and predict the data in the initial
30-day phase, while it shows a tendency towards a faster smoothing-out of the
pronounced local fine structure and persistent hot spots, as compared to the
data. It could be understood that the observed infection cases concentrated in
the big cities, such as Berlin, Hamburg, München, Köln and Frankfurt than the
other sites in the simulation, which reflects high population density enhances
the diffusion of the COVID-19. The prediction could be evidently improved by
introducing more abundant interactions across and within the counties. As some
practical improvements, e.g. replacing the geographic lattice to traffic
networks [5, 34, 36, 37, 38], introducing the higher resolution lattices
beyond the county-level which also means higher-order interactions among the
residents [12, 20, 34, 39], can be implemented and devoted in further future
works. With the help of the deep learning, the machine learns the dynamical
parameters matched with the data of March, which is transferred into the
prediction to second-wave sequentially. Starting in September, the simulation
maps are remarkably consistent with the data. Regarding the slight uncertainty
in extracting parameters for the dynamics in September, which might has root
in the difference of development trend between the first- and second-wave, it
could be avoided by training the neural networks on more heterogeneous data
set. We will present the further improvement in our future papers.

Compared to the prevailing COVID-19 risk prediction models, the machine
learning-assisted CA-SUIR model here indicates that knowledge of risk
evaluation in the county-level can be displayed directly to governments and
residents. The prediction to the effects of different public health policies
suggests that the transmission modes of coronavirus could be shaped by the
efficient non-pharmaceutical interventions. Besides, the 7-day windows could
give a robust prediction in long-term, which is advantageous to timely assess
the current situation. The other valuable application of our evaluation system
is to improve allocation of medical resources which are routinely unequal in
different counties, some of them are suffering from the epidemic that far
exceeds their medical capacity. It is of essential significance to implement
the intervenable risk evaluation model for decision-making on restarting
economics and public health policies in the COVID-19 pandemic.

##  5 Acknowledgments

The authors thank Esteban Vargas and Maria Babarossa for useful discussions
and comments. The work on this research is supported by the BMBF through the
ErUM-Data funding and the Samson AG AI grant (L.W, K.Z), by the National
Natural Science Foundation of China, Grant No. 11875002(T.X, Y.J.), by the
Zhuobai Program of Beihang University(Y.J.). The authors also thank the
donation of NVIDIA GPUs from NVIDIA Corporation.

##  Appendix A Infection Maps

Figure 10: Mapping from real infection map of Germany onto topological
equivalent lattice. The left map is divided into $25\times 25$ sites as the
size of the right part. The geometric shape of the suqare lattice is
irrelevant to the real geographic topography. The site on lattice depict a
typical county-level unit of the real map.

In order to simulate the population evolution on lattice, the topological
mapping is adopted in our CA computation. We firstly embed the map in a
rectangle area whose length and width are chosen as the maximum value of the
Germany map, as in Fig. 10(left). Then the area is segmented into uniform
sites of $L\times L$. And each site is labelled by its row and column numbers
as $B_{mn}$, where $1\leq m,n\leq L$. If a county occupies $N$ sites, all
kinds of the population, such as susceptible, undiagnosed, infected and
removed, are assigned equally into each site. And if there is more than one
county or county part on a site, the populations of the site is set as the sum
of all these counties or county parts. Because the geographic distance is not
important in our simulation, which focus on the population distributions, we
set each site as a square when plotting the distribution in Fig. 10(right). In
the current simulation, we confine all the population in Germany and thus set
the out-of-Germany transfer possibility at the border as zero.

[] []

Figure 11: Comparison of simulated and RKI infection maps during March 2020 in
Germany. The color bar is set in an uniform orange color scale, the darkest
color is 2000 cases with a logarithic re-scale. (a) As of the end of March,
the cumulative infection cases are presented in dynamical simulation with
fitting parameters at 2020-03-05, 2020-03-15 and 2020-03-25 respectively. (b)
As of the end of March, the cumulative infection cases are presented in real
profiles at 2020-03-05, 2020-03-15 and 2020-03-25 respectively.

##  Appendix B SUIR model parameters explanation

In this part, we discuss the fitting details based on the mathematical
simplification to the SUIR model. In the large susceptible population limit
and ignoring the recovery and mortality cases for the uncertainty at early
stage, there are only two equations are relevant,

$\displaystyle\frac{dU}{dt}$
$\displaystyle=(\lambda-\sigma-\frac{\sigma^{\prime}}{\text{days}})U$ (5)
$\displaystyle\frac{d(I+R)}{dt}$ $\displaystyle=\sigma U$ (6)

The solutions are clearly derived as,

$\displaystyle U(t)$
$\displaystyle=U_{0}Exp[(\lambda-\sigma-\frac{\sigma^{\prime}}{\text{days}})t]$
(7) $\displaystyle I+R$ $\displaystyle=I_{0}+R_{0}-\frac{\sigma
U_{0}}{\lambda-\sigma-\frac{\sigma^{\prime}}{\text{days}}}+\frac{\sigma
U(t)}{\lambda-\sigma-\frac{\sigma^{\prime}}{\text{days}}}$ (8)

The $I+R$ fitting is easy now with $I+R=a_{*}+b_{*}Exp[c_{*}t]$. This means we
can not obtain all the independent parameters in the solution, but the
following redefinition could be helpful,

$\displaystyle a_{*}$ $\displaystyle=I_{0}+R_{0}-\frac{\sigma
U_{0}}{\lambda-\sigma-\frac{\sigma^{\prime}}{\text{days}}}$ (9) $\displaystyle
b_{*}$ $\displaystyle=\frac{\sigma
U_{0}}{\lambda-\sigma-\frac{\sigma^{\prime}}{\text{days}}}$ (10)
$\displaystyle c_{*}$
$\displaystyle=\lambda-\sigma-\frac{\sigma^{\prime}}{\text{days}}$ (11)

or equivalently only the following combinations could be obtained

$\displaystyle I_{0}+R_{0}=a_{*}+b_{*}$ (12) $\displaystyle\sigma
U_{0}=b_{*}c_{*}$ (13)
$\displaystyle\lambda-\sigma-\frac{\sigma^{\prime}}{\text{days}}=c_{*}$ (14)

As for the product $\sigma U_{0}$, which means it is difficult to determine
the $\sigma$ and $U_{0}$ individually. It will introduce the inevitable
uncertainty to the two parameters, while the confidence to the multiplicative
variable could be reserved.

##  Appendix C Conv2LSTM Neural Networks

Figure 12: The Conv2LSTM neural networks. The nuts-and-bolts of the neural
network are: Convolution layers within TimeDistributed wrapper, from
Convolution 2D (64, kernel size=3x3, ‘ReLU’, ‘same’ padding) to Convolution 2D
(64, kernel size=3x3, strides=2, ‘ReLU’, ‘same’ padding) to Convolution 2D(64,
kernel size=3x3, ‘ReLU’, ‘same’ padding) to Convolution 2D(64, kernel
size=3x3, ‘ReLU’, ‘same’ padding) and Flatten; for the LSTM part shown in Fig.
12, 256 LSTM cells with ‘sequence=False’ and activation=‘tanh’ are used; dense
layers contains 256 neurons with ‘ReLU’ active function, before the output is
the ‘sigmoid’ function to fit the normalized targets. Figure 13: Learning
curve of the Conv2LSTM neural networks. The red line is the learning curve, in
which the Mmean-Square Error (MSE) loss is reducing with the epoch increasing.
The blue line is located in validation data set, which shows the similar
results in learning data set. The increasing of the R square with training is
shown in subfigure, which indicates the correlation between the prediction
from neural networks and truth tends to be close.

## References

  * Maier and Brockmann [2020] B. F. Maier, D. Brockmann,  Effective containment explains subexponential growth in recent confirmed COVID-19 cases in China,  Science 368 (2020) 742–746. doi:10.1126/science.abb4557.
  * Walensky and del Rio [2020] R. P. Walensky, C. del Rio,  From Mitigation to Containment of the COVID-19 Pandemic: Putting the SARS-CoV-2 Genie Back in the Bottle,  JAMA 323 (2020) 1889\. doi:10.1001/jama.2020.6572.
  * Barbarossa et al. [2020a] M. V. Barbarossa, J. Fuhrmann, J. H. Meinke, S. Krieg, H. V. Varma, N. Castelletti, T. Lippert,  Modeling the spread of COVID-19 in Germany: Early assessment and possible scenarios,  PLOS ONE 15 (2020a) e0238559. doi:10.1371/journal.pone.0238559.
  * Barbarossa et al. [2020b] M. V. Barbarossa, J. Fuhrmann, J. Heidecke, H. V. Varma, N. Castelletti, J. H. Meinke, S. Krieg, T. Lippert,  A first study on the impact of current and future control measures on the spread of COVID-19 in Germany,  medRxiv (2020b) 2020.04.08.20056630. doi:10.1101/2020.04.08.20056630.
  * Estrada [2020] E. Estrada,  COVID-19 and SARS-CoV-2. Modeling the present, looking at the future,  Physics Reports 869 (2020) 1–51. doi:10.1016/j.physrep.2020.07.005.
  * Vespignani et al. [2020] A. Vespignani, H. Tian, C. Dye, J. O. Lloyd-Smith, R. M. Eggo, M. Shrestha, S. V. Scarpino, B. Gutierrez, M. U. G. Kraemer, J. Wu, K. Leung, G. M. Leung,  Modelling COVID-19,  Nat. Rev. Phys. 2 (2020) 279–281. doi:10.1038/s42254-020-0178-4.
  * Wang et al. [2020] L. Wang, Y. Wang, D. Ye, Q. Liu,  Review of the 2019 novel coronavirus (SARS-CoV-2) based on current evidence,  International Journal of Antimicrobial Agents 55 (2020) 105948. doi:10.1016/j.ijantimicag.2020.105948.
  * Kergassner et al. [2020] A. Kergassner, C. Burkhardt, D. Lippold, S. Nistler, M. Kergassner, P. Steinmann, D. Budday, S. Budday,  Meso-scale modeling of COVID-19 spatio-temporal outbreak dynamics in Germany,  medRxiv (2020) 2020.06.10.20126771. doi:10.1101/2020.06.10.20126771.
  * Linden et al. [2020] M. Linden, J. Dehning, S. B. Mohr, J. Mohring, M. Meyer-Hermann, I. Pigeot, A. Schöbel, V. Priesemann,  The foreshadow of a second wave: An analysis of current COVID-19 fatalities in Germany,  ArXiv201005850 Phys. Q-Bio (2020). arXiv:2010.05850.
  * Long et al. [2020] Y. Long, Y. Chen, Y. Li,  Multifractal scaling analyses of the spatial diffusion pattern of COVID-19 pandemic in Chinese mainland,  ArXiv201002747 Phys. (2020). arXiv:2010.02747.
  * Chande et al. [2020] A. Chande, S. Lee, M. Harris, Q. Nguyen, S. J. Beckett, T. Hilley, C. Andris, J. S. Weitz,  Real-time, interactive website for US-county-level COVID-19 event risk assessment,  Nat. Hum. Behav. (2020) 1–7. doi:10.1038/s41562-020-01000-9.
  * Rader et al. [2020] B. Rader, S. V. Scarpino, A. Nande, A. L. Hill, B. Adlam, R. C. Reiner, D. M. Pigott, B. Gutierrez, A. E. Zarebski, M. Shrestha, J. S. Brownstein, M. C. Castro, C. Dye, H. Tian, O. G. Pybus, M. U. G. Kraemer,  Crowding and the shape of COVID-19 epidemics,  Nat. Med. (2020) 1–6. doi:10.1038/s41591-020-1104-0.
  * Wang et al. [2020] J. Wang, X. Lin, Y. Liu, Qilegeri, H. Lin, M. Wu,  Global dynamics of a SUIR model with predicting COVID-19,  ArXiv200412433 Phys. Q-Bio (2020). arXiv:2004.12433.
  * Zhou et al. [2020] Y. Zhou, L. Wang, L. Zhang, L. Shi, K. Yang, J. He, B. Zhao, W. Overton, S. Purkayastha, P. Song,  A Spatiotemporal Epidemiological Prediction Model to Inform County-Level COVID-19 Risk in the United States,  Harv. Data Sci. Rev. (2020). doi:10.1162/99608f92.79e1f45e.
  * Gao et al. [2020] Y. Gao, G.-Y. Cai, W. Fang, H.-Y. Li, S.-Y. Wang, L. Chen, Y. Yu, D. Liu, S. Xu, P.-F. Cui, S.-Q. Zeng, X.-X. Feng, R.-D. Yu, Y. Wang, Y. Yuan, X.-F. Jiao, J.-H. Chi, J.-H. Liu, R.-Y. Li, X. Zheng, C.-Y. Song, N. Jin, W.-J. Gong, X.-Y. Liu, L. Huang, X. Tian, L. Li, H. Xing, D. Ma, C.-R. Li, F. Ye, Q.-L. Gao,  Machine learning based early warning system enables accurate mortality risk prediction for COVID-19,  Nat. Commun. 11 (2020) 5033. doi:10.1038/s41467-020-18684-2.
  * Shen et al. [2020] C. Shen, A. Chen, C. Luo, J. Zhang, B. Feng, W. Liao,  Using Reports of Symptoms and Diagnoses on Social Media to Predict COVID-19 Case Counts in Mainland China: Observational Infoveillance Study,  J. Med. Internet Res. 22 (2020) e19421. doi:10.2196/19421. arXiv:2004.06169.
  * Yeşilkanat [2020] C. M. Yeşilkanat,  Spatio-temporal estimation of the daily cases of COVID-19 in worldwide using random forest machine learning algorithm,  Chaos, Solitons & Fractals 140 (2020) 110210. doi:10.1016/j.chaos.2020.110210.
  * Zou et al. [2020] D. Zou, L. Wang, P. Xu, J. Chen, W. Zhang, Q. Gu,  Epidemic Model Guided Machine Learning for COVID-19 Forecasts in the United States,  medRxiv (2020) 2020.05.24.20111989. doi:10.1101/2020.05.24.20111989.
  * White et al. [2007] S. H. White, A. M. del Rey, G. R. Sánchez,  Modeling epidemics using cellular automata,  Applied Mathematics and Computation 186 (2007) 193–202. doi:10.1016/j.amc.2006.06.126.
  * Schneckenreither et al. [2008] G. Schneckenreither, N. Popper, G. Zauner, F. Breitenecker,  Modelling SIR-type epidemics by ODEs, PDEs, difference equations and cellular automata – A comparative study,  Simulation Modelling Practice and Theory 16 (2008) 1014–1023. doi:10.1016/j.simpat.2008.05.015.
  * Jun-feng [2020] L. Jun-feng,  Assessment and Prediction of COVID-19 Based on SEIR Model with Undiscovered People,  J. Univ. Electron. Sci. Technol. China 49 (2020) 1–9.
  * Chen et al. [2020] Y.-C. Chen, P.-E. Lu, C.-S. Chang, T.-H. Liu,  A Time-dependent SIR model for COVID-19 with Undetectable Infected Persons,  IEEE Trans. Netw. Sci. Eng. (2020) 1–1.
  * Ivorra et al. [2020] B. Ivorra, M. R. Ferrández, M. Vela-Pérez, A. M. Ramos,  Mathematical modeling of the spread of the coronavirus disease 2019 (COVID-19) taking into account the undetected infections. The case of China,  Communications in Nonlinear Science and Numerical Simulation 88 (2020) 105303\. doi:10.1016/j.cnsns.2020.105303.
  * Shi et al. [2015] X. Shi, Z. Chen, H. Wang, D.-Y. Yeung, W.-k. Wong, W.-c. Woo,  Convolutional LSTM Network: A machine learning approach for precipitation nowcasting,  in: Proceedings of the 28th International Conference on Neural Information Processing Systems - Volume 1, NIPS’15, MIT Press, Cambridge, MA, USA, 2015, pp. 802–810.
  * Hernandez-Vargas and Velasco-Hernandez [2020] E. A. Hernandez-Vargas, J. X. Velasco-Hernandez,  In-host Mathematical Modelling of COVID-19 in Humans,  Annual Reviews in Control (2020). doi:10.1016/j.arcontrol.2020.09.006.
  * Wang and Jiang [2019] L. Wang, Y. Jiang,  Escape dynamics based on bounded rationality,  Physica A 531 (2019) 121777. doi:10.1016/j.physa.2019.121777.
  * Eckardt et al. [2020] M. Eckardt, K. Kappner, N. Wolf, Covid-19 Across European Regions: The Role of Border Controls, SSRN Scholarly Paper ID 3688126, Social Science Research Network, Rochester, NY, 2020.
  * Li et al. [2020] Y. Li, L. Yao, J. Li, L. Chen, Y. Song, Z. Cai, C. Yang,  Stability issues of RT-PCR testing of SARS-CoV-2 for hospitalized patients clinically diagnosed with COVID-19,  J Med Virol 92 (2020) 903–908. doi:10.1002/jmv.25786.
  * Vandenberg et al. [2020] O. Vandenberg, D. Martiny, O. Rochas, A. van Belkum, Z. Kozlakidis,  Considerations for diagnostic COVID-19 tests,  Nat. Rev. Microbiol. (2020) 1–13. doi:10.1038/s41579-020-00461-z.
  * Ramos et al. [2017] A. A. Ramos, I. S. Requerey, N. Vitas,  DeepVel: Deep learning for the estimation of horizontal velocities at the solar surface,  A&A 604 (2017) A11. doi:10.1051/0004-6361/201730783.
  * Bavel et al. [2020] J. J. V. Bavel, K. Baicker, P. S. Boggio, V. Capraro, A. Cichocka, M. Cikara, M. J. Crockett, A. J. Crum, K. M. Douglas, J. N. Druckman, J. Drury, O. Dube, N. Ellemers, E. J. Finkel, J. H. Fowler, M. Gelfand, S. Han, S. A. Haslam, J. Jetten, S. Kitayama, D. Mobbs, L. E. Napper, D. J. Packer, G. Pennycook, E. Peters, R. E. Petty, D. G. Rand, S. D. Reicher, S. Schnall, A. Shariff, L. J. Skitka, S. S. Smith, C. R. Sunstein, N. Tabri, J. A. Tucker, S. van der Linden, P. van Lange, K. A. Weeden, M. J. A. Wohl, J. Zaki, S. R. Zion, R. Willer,  Using social and behavioural science to support COVID-19 pandemic response,  Nat. Hum. Behav. (2020) 1–12. doi:10.1038/s41562-020-0884-z.
  * Hsiang et al. [2020] S. Hsiang, D. Allen, S. Annan-Phan, K. Bell, I. Bolliger, T. Chong, H. Druckenmiller, L. Y. Huang, A. Hultgren, E. Krasovich, P. Lau, J. Lee, E. Rolf, J. Tseng, T. Wu,  The effect of large-scale anti-contagion policies on the COVID-19 pandemic,  Nature 584 (2020) 262–267. doi:10.1038/s41586-020-2404-8.
  * Wong et al. [2020] G. N. Wong, Z. J. Weiner, A. V. Tkachenko, A. Elbanna, S. Maslov, N. Goldenfeld,  Modeling COVID-19 Dynamics in Illinois under Nonpharmaceutical Interventions,  Phys. Rev. X 10 (2020) 041033. doi:10.1103/PhysRevX.10.041033.
  * Chang et al. [2020] S. Chang, E. Pierson, P. W. Koh, J. Gerardin, B. Redbird, D. Grusky, J. Leskovec,  Mobility network models of COVID-19 explain inequities and inform reopening,  Nature (2020) 1–8. doi:10.1038/s41586-020-2923-3.
  * Stafford [2020] N. Stafford,  Covid-19: Why Germany’s case fatality rate seems so low,  BMJ 369 (2020). doi:10.1136/bmj.m1395.
  * Topirceanu et al. [2020] A. Topirceanu, M. Udrescu, R. Marculescu,  Centralized and decentralized isolation strategies and their impact on the COVID-19 pandemic dynamics,  ArXiv200404222 Phys. Q-Bio (2020). arXiv:2004.04222.
  * Valba et al. [2020] O. Valba, V. Avetisov, A. Gorsky, S. Nechaev,  Self-isolation or borders closing: What prevents the spread of the epidemic better?,  Phys. Rev. E 102 (2020) 010401. doi:10.1103/PhysRevE.102.010401. arXiv:2003.12290.
  * Ye et al. [2020] Y. Ye, Q. Zhang, Z. Ruan, Z. Cao, Q. Xuan, D. D. Zeng,  Modeling the heterogeneous disease-behavior-information dynamics during epidemics,  ArXiv200507012 Phys. (2020). arXiv:2005.07012.
  * Charoenwong et al. [2020] B. Charoenwong, A. Kwan, V. Pursiainen,  Social connections with COVID-19–affected areas increase compliance with mobility restrictions,  Sci. Adv. 6 (2020) eabc3054. doi:10.1126/sciadv.abc3054.