A Discrete SIR Model with Spatial Distribution on a Torus for COVID-19

Analysis using Local Neighborhood Properties

Reinhard Schuster

1

, Klaus-Peter Thiele

2

, Thomas Ostermann

3

and Martin Schuster

4

1

Chair of Department of Health Economics, Epidemiology and Medical Informatics, Medical Advisory Board of Statutory

Health Insurance in Northern Germany (MDK Nord), 23554 L

¨

ubeck, Germany

2

Medical Director of the Medical Advisory Service Institution of the Statutory Health Insurance in North Rhine (MDK

Nordrhein), 40212 D

¨

usseldorf, Germany

3

Chair of Research Methodology and Statistics in Psychology, Witten/Herdecke University, Alfred-Herrhausen-Straße 50,

58448 Witten, Germany

4

Faculty of Epidemiology, Christian-Albrechts University Kiel, 24105 Kiel, Germany

martin.schuster@epi.uni-kiel.de

Keywords:

COVID-19, SIR Model, Torus, Differential Equation, Laplace Operator, Mean Value Operator.

Abstract:

The ongoing COVID-19 pandemic threatens the health of humans, causes great economic losses and may

disturb the stability of the societies. Mathematical models can be used to understand aspects of the dynamics of

epidemics and to increase the chances of control strategies. We propose a SIR graph network model, in which

each node represents an individual and the edges represent contacts between individuals. For this purpose, we

use the healthy S (susceptible) population without immune behavior, two I-compartments (infectious) and two

R-compartments (recovered) as a SIR model. The time steps can be interpreted as days and the spatial spread

(limited in distance for a singe step) shell take place on a 200 by 200 torus, which should simulate 40 thousand

individuals. The disease propagation form S to the I-compartment should be possible on a k by k square (k=5

in order to be in small world network) with different time periods and strength of propagation probability in

the two I compartments. After the infection, an immunity of different lengths is to be modeled in the two R

compartments. The incoming constants should be chosen so that realistic scenarios can arise. With a random

distribution and a low case number of diseases at the beginning of the simulation, almost periodic patterns

similar to diffusion processes can arise over the years. Mean value operators and the Laplace operator on the

torus and its eigenfunctions and eigenvalues are relevant reference points. The torus with ﬁve compartments is

well suited for visualization. Realistic neighborhood relationships can be viewed with a inhomogeneous graph

theoretic approach, but they are more difﬁcult to visualize. Superspreaders naturally arise in inhomogeneous

graphs: there are different numbers of edges adjacent to the nodes and should therefore be examined in an

inhomogeneous graph theoretical model. The expected effect of corona control strategies can be evaluated by

comparing the results with various constants used in simulations. The decisive beneﬁt of the models results

from the long-term observation of the consequences of the assumptions made, which can differ signiﬁcantly

from the primarily expected effects, as is already known from classic predator-prey models.

1 INTRODUCTION

The Covid-19 pandemic is a major challenge for phy-

sicians, politicians, scientists and much other groups.

Models are useful to discuss possible scenarios and

implications of interventions to expand background

knowledge and to design policy impact research, cf.

(Chinazzi et al., 2020), (Rosenbaum, 2020), (Pan

et al., 2020), (Behrens et al., 2020), (Tang et al.,

2020).

The contextual selection of adequate models

should be able to generate all known actual and hi-

storic observations by choosing suitable model clas-

ses and parameters, cf. (Bailey, 1975), (Keener and

Sneyd, 1998), (Xue et al., 2020), (Kucharski et al.,

2020). As an intermediate step, the advantages and

disadvantages of known mathematical models in me-

dicine and biology and their implementations in infor-

matics using different classes of parameters should be

analyzed, cf. (Kermack and McKendrick, 1933).

The biomathematical analysis of epidemiological

systems has a long history, cf. (Arnautu et al., 1989),

Schuster, R., Thiele, K., Ostermann, T. and Schuster, M.

A Discrete SIR Model with Spatial Distribution on a Torus for COVID-19 Analysis using Local Neighborhood Properties.

DOI: 10.5220/0010252504750482

In Proceedings of the 14th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2021) - Volume 5: HEALTHINF, pages 475-482

ISBN: 978-989-758-490-9

Copyright

c

2021 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

475

(Aronson, 1985), (Beretta and Capasso, 1986). The

family of SIR models with various generalizations of

the status elements S (susceptible), I (infected or in-

fectious) and R (removed or recovered) with respect

to other status elements as E (exposed) and subpopu-

lations use ordinary differential equations, cf. (Ahmed

and Agiza, 1998), (Andersen and May, 1979), (Capas-

so, 1993), (Redheffer and Walter, 1984).

The variables are fractions of the population mo-

delled as positive real numbers as in the mass acti-

on law supposing homogeneous distributions. Peri-

odic solutions of ﬁrst order ordinary differential equa-

tions are unstable with respect to system parameters.

A large number of SIR models use bilinear differenti-

al equations, cf.(R

¨

udiger et al., 2020).

We propose a model which could be presented as

a cellular automata. In order to use methods of par-

tial differential equations and mean value operators

in differential geometry we use differential manifolds

as a general modelling background and its discreti-

zation which generalizes the Euclidean structure of

cellular automata, cf. (Mikeler et al., 2005), (G

¨

unther

and Pr

¨

ufer, 1999), (Schuster, 1988), (Schuster, 1997),

(Wang et al., 2015), (Welch et al., 2011).

An essential feature will be the visualization of de-

velopment of the pandemic with special structures, cf.

(Murray, 2002), (Murray, 2003), (Barrat and Vespig-

nani, 2008), (Andersen and May, 1979). This will be

realized with Mathematica from Wolfram Research

which also allows to use a syntax without coordinates.

A visualization of the pandemic status of 40 thousand

individual can be reached with some hours of calcula-

tion time. The results to be considered in the followi-

ng have proven to be the optimal solution within the

used simulations with regard to the target variable that

is still to be discussed.

If one would change the discretization of the torus

structure of the used differential manifold by a graph

structure (cf. (Solimano and Bretta, 1982), (R

¨

udiger

et al., 2020), (Welch et al., 2011), (Boguna et al.,

2003)) much less individuals could be visualized.

Cyclical structures in the sense of a boundary cy-

cle or a cyclical band were discussed in chemical

and biological examples, where there are fundamen-

tal theoretical differences, but also commonalities bet-

ween discrete and continuous systems, cf. (Capas-

so and Maddalena, 1983), (Field and Burger, 1985).

When modeling exponential growth, there are not on-

ly differences between continuous and discrete sy-

stems, but also between spatially bounded and un-

bounded systems (compact and non-compact mani-

folds in differential geometry), cf. (Murray, 2002),

(Murray, 2003), (Barrat and Vespignani, 2008), (Be-

retta and Capasso, 1986), (Boguna et al., 2003), (So-

limano and Bretta, 1982). In this context, the r-value

plays a role in the current Covid-19 discussion that

has already been analyzed earlier in various contexts,

cf. (Dieckman et al., 1990). The r-value should de-

scribe the prevailing value of an exponential spread.

The exponential mapping provides the crucial relati-

onship between Lie algebras and Lie groups in diffe-

rential geometry. This goes far beyond the exponen-

tial function on real numbers, which already shows a

completely different qualitative behavior on complex

numbers. Using the exponential function to describe

the spread of Covid-19, which could practically ne-

ver be sensibly modeled with an exponential spread,

leads to theoretical and practical problems. The expo-

nential map on a discretized torus leads to a growth in

which the saturation value has to be estimated, which

is in principle impossible from short initial data. The

actual paper is a ﬁrst step in the stated context.

2 MATERIAL AND METHODS

Individuals are the elements of an n by m array (in the

examples we use the special values n = m = 200). In-

dividuals can be in the status S (susceptible: healthy

without immune behavior), I

1

or I

2

(infected or infec-

tious after incubation period) and R

1

or R

2

(recovered

with immune behavior) for each day after initializa-

tion. Further on for every individual in the I

1

, I

2

, R

1

or R

2

status the number of days being in that status is

considered, cf. Figure 1. It is also possible to model

different immune behavior speciﬁcally related to the

infectious disease in different S compartments, here

this is be done in the R compartments.

The right and left hand as well as the upper and

lower elements of the array resulting from the dis-

cretization of the torus are connected. If one would

change the upper and lower side of the right and left

sides of the array and respectively for the upper and

lower side we would get the discretization of double

M

¨

obius strip which can’t be realized in three dimen-

sional geometry. With an other boundary identiﬁcati-

on one also could reach the topology of a sphere.

As a local neighborhood of the array element (i, j)

we consider all elements (i + k, j + k) mod n with

−m ≤ k ≤ m (m = 5 in the examples). An S indivi-

dul in the local neighborhood of a I individual is in-

fected randomly with equal probability with the pro-

perty that during the infectious period r individuals (r

as the theoretical reproduction number) are infected.

This theoretical reproduction number supposes that

all other individuals in the local neighborhood are in

the S status.

The individuals of the I

2

compartment are less in-

HEALTHINF 2021 - 14th International Conference on Health Informatics

476

S

I

2

I

1

R

1

R

2

p

1

p

2

p

5

p

6

p

7

p

8

p

3

p

4

p

9

p

10

2

Figure 1: SIR compartments and transitions p.

fectious than those of the I

1

compartment but over a

longer period of time, both lead to the same resulting

reproduction number in the used simulation. This as-

sumption can be modiﬁed.

The individuals of the R

2

compartment should be

immune longer than those individual of the R

1

com-

partment. At the beginning of the simulation, all indi-

viduals should belong to the S compartment with the

exception of i

1

and i

2

individuals, respectively, ran-

domly determined individuals of the I

1

and I

2

com-

partments. The previous time in the I

1

or I

2

compart-

ment should also be chosen randomly. This means

that at the beginning of the observation of the spread

there is no immune behavior. With regard to measures

to be taken to limit the spread of the infection, this is

the most unfavorable option. Alternatively, one could

start with an initial distribution of individuals in R

1

and R

2

compartments. Since simulations have shown

that the same dynamic behavior will occur in the me-

dium term, it was easier to choose comparable initial

conditions.

After the infectious period of individuals of I

1

or

I

2

they change to R

1

or R

2

individuals by transiti-

on parameters. After immunity, the individuals switch

back to the S compartment. There also may be a

switch between I

1

and I

2

individuals. The transitions

p

i

(i=1,...10) between the compartments are shown

in Figure 1. In the considered context the p

i

are sto-

chastic processes depending on the previous length of

stay of the individuals of the compartments. In an ana-

log Markov model, these are only transition probabi-

lities.

The daily steps are simultaneously using the pre-

vious day’s conditions for all transitions. This inclu-

des that an individual in an R-compartment which

changes to the S-compartment can not change from

another individual to an S-compartment through in-

fection during this step. In this way, an overall effect

of all other individuals in the local contagious neigh-

borhood is achieved.

S individuals may be infected by all I

1

or I

2

indi-

viduals in the related local neighborhood with resul-

ting increasing probability with the number of these

individuals, similar to a bilinear infection in SIR mo-

dels without spacial structure. Thereby the new I

1

or

I

2

individuals are determined. After that step R

1

or R

2

individuals may switch to S individuals and I

1

or I

2

individuals may switch to R

1

or R

2

individuals. The

parameters of the simulation should be determined in

such a way that

1. the I

1

compartment is as small as possible

2. the transition probability from the S compartment

to the I

2

compartment (long infectious) is smal-

ler than that into the I

1

compartment (short infec-

tious) and

3. a large proportion of the population reaches the R

1

or R

2

compartment.

Small I compartments and large R compartments are

opposing objectives, since only I individuals can be-

come R individuals. I

1

individuals can be interpreted

as symptomatic cases or as individuals with stronger

disease symptoms and I

2

individuals as asymptoma-

tic cases or with lower disease symptoms. We want

to reach large immunity with low treatment cases as a

target. The system should be able to generate appro-

ximately stable cyclical behavior (including a cyclic

band with chaotic behavior in the inner band part),

because states of equilibrium can be unstable possib-

ly solely through external inﬂuences.

Of central interest is how the system settles. If the

ﬁrst or second oscillation (ﬁrst or second wave) is si-

gniﬁcantly higher than later oscillations of small am-

plitude for the diseases, a new infection from outside

is a high risk as immunity declines. A small regular

rate of contamination from outside should realistical-

ly be included in the model.

The approach considered is a discretization of a

partial integro-differential equation with an additional

time delay. The effect of the local environment can be

viewed as a mean value operator. It is known from

differential geometry that the eigenvalues of the La-

place operator are also eigenvalues of the mean value

operators and that eigenfunctions of the mean value

operators and can be calculated from those of the La-

place operator, with no time delays being included in

the considerations of differential geometry. Instead of

the time delays, as is usual in SIR models, transition

probabilities can be replaced, which can be modeled

as reciprocal values of the expected value of remai-

ning time in the compartment.

A Discrete SIR Model with Spatial Distribution on a Torus for COVID-19 Analysis using Local Neighborhood Properties

477

3 RESULTS

We start with 10 randomly infected individuals of the

I population, all other individuals should initially be

in the S compartment. This number is important with

regard to the number of initial clusters of the infection

and should initially model a low external inﬂuence. A

small number extends the initial phase, a high number

induces a dynamic that has to be synchronized with an

approximately periodically stable internal system dy-

namic. Half of the individuals in the I compartment

should initially be in the I

1

compartment, the other in

the I

2

compartment. An incubation period is not inclu-

ded in the results considered here, but in the previous

simulations they have not led to any other qualitative

results. These only show changes in the initial phase

of the development of the infection.

The results of this section are calculated with the

optimal parameters with respect to the mentioned goal

to reach low infections over the time including the in-

itial phase and a large number of immune individuals

over the time out of over 100 long time simulations.

In addition to the spatial distributions and progress ﬁ-

gures in the individual compartments, we will provi-

de qualitative comparisons with the other simulations

carried out.

Individuals in the I

1

compartment should be con-

tagious for two weeks, and individuals in the I

2

com-

partment for 10 weeks, this is a ratio of 5. Other ratios

have shown less stable results or worse results in re-

lation to the main target.

Due to the large number of incoming parameters,

the search for an optimum was only able to ﬁnd a lo-

cally optimal version. Further theoretical analyzes are

necessary at this point.

The time unit primarily used is also to be regarded

as provisional. The need for an adjustment may arise

from the cycle durations resulting from the long-term

behavior.

The r-value of the RKI (Robert Koch institute) of-

ten used in Germany results from ratios of two latest

available consecutive three day incidence numbers. If

the used time step is shorter than a day the r-value

becomes more stable.

Every individual of the I

1

population shall have

a theoretical (initial) r-value of 3. This shall men,

that this individual would infect 3 other individuals

throughout his infection period, if all individuals in

the considered local neighborhood under considerati-

on belonged to the S compartment. Half of the newly

infected individuals should get into the I

1

and I

2

com-

partments.

One could assume that smaller theoretical initial

r-values would be desirable. As a result of the mo-

re slowly developing immunity, however, they lead to

more diseases in the initial phase and to higher long-

term stability with repeated high numbers of diseases.

High initial r-values lead to high values in the I

1

com-

partment, with an interpretation that leads to high ca-

se numbers in hospital care and should therefore be

avoided as a priority. The proportion of hospital cases

within the I

1

compartment is not considered here, as it

has at most a marginal effect on the spread dynamics.

Individuals in the I

2

compartment should only be

5% as contagious as individuals in the I

1

compart-

ment, with half of the new infections reaching the I

1

and I

2

compartments. But this I

2

compartment has a

major inﬂuence on the dynamics of the spread, as the

longer residence time in this compartment has a sta-

bilizing effect on the spatial distribution of the com-

partments. If dynamic differences between the I

1

and

I

2

compartments are too small, there is no spatial sta-

bilization effect. If the differences are too large, the

overall effect of the I

2

compartment is too small. In

between there is at least a local optimum with regard

to the central target variable with regard to the ratio

considered here.

In addition, there should be an external possibili-

ty of infection. Complete isolation of a region is an

illusion, especially with regard to asymptotic cases,

and would only lead back to the unfavorable settling

phase.

Every ﬁfth day a randomly selected individual is

infected if this is in the S compartment. This results

in a slight acceleration in the initial phase of the dy-

namics of the epidemic and has almost no effect if the

immune behavior is widespread. This approach pre-

vents the infection from escalating under the illusion

that a region can be completely isolated from the out-

side world.

R-value = 1.10, day = 91

0 → 38 899

1 → 204

2 → 808

3 → 613

4 → 78

R-value = 0.99, day = 182

0 → 32 755

1 → 580

2 → 2545

3 → 4347

4 → 375

R-value = 0.97, day = 273

0 → 25 510

1 → 349

2 → 2497

3 → 11 392

4 → 854

R-value = 0.98, day = 365

0 → 22 634

1 → 119

2 → 918

3 → 15 851

4 → 1080

Figure 2: S-I-R compartments at day 91, 182, 273 and 365

(from top left to bottom right).

HEALTHINF 2021 - 14th International Conference on Health Informatics

478

After the contagion phase, 90% of the individuals

in the I

1

compartment should be randomly selected to

move to the R

1

compartment, the rest to the R

2

com-

partment. Based on the I

2

compartment, it should be

80% respectively. A thoroughly interesting interacti-

on between the I

1

and I

2

compartments is initially not

considered here, since this results in a strong interac-

tion with regard to a meaningful determination of the

parameters to be used and signiﬁcantly extends the

search process for meaningful combinations. The in-

dividuals in the R

1

compartment should remain there

for 180 days (about half a year), those in the R

2

com-

partment for three times that time. With these parame-

ters, after one quarter (91 days), the division into the

S-I-R compartments is shown in Figure 2 top left. The

colors are those used in Figure 1. The S compartment

with value 1 is shown in yellow. The I

1

compartment

is shown in lighter red (value 2) and I

2

in darker red

(value 3). Intuitively, red should act as a warning color

according to the trafﬁc light symbols. The temporari-

ly immune compartment is shown in green, the lighter

green (value 4) with shorter immunity than the darker

green (value 5).

The speciﬁed r-value in the ﬁgure is the value

used by the RKI (Robert-Koch Institute in Germany),

which describes the ratio of new infections from two

consecutive 3-day periods. If you follow the results

in detail, you can quickly see that this value descri-

bes the development of the infection very poorly and

misleadingly in the used simulation.

The epidemiological situation of the next quarter

(day 182) is described in Figure 2 top right.

In this simulation, over 75% of the population is

not yet affected by the epidemic. It can be seen to

a large extent through the strong creative power of

random processes caused the emergence of distribu-

tion patterns. Pattern creation has been discussed in

mathematical modeling in chemical, biological and

medical applications for many decades, especially in

the case of diffusion effects, and is relevant to the

approach considered here, cf. (Murray, 2002), (Mur-

ray, 2003), (Aronson, 1985), (Capasso and Madda-

lena, 1983). The Laplace operator already discussed

plays a decisive role in diffusion processes. Spectral

results (eigenvalues and eigenvectors) of the Laplace

operator are relevant for pattern formation.

Regarding the strong design ability of random

processes, it should be noted that with random primes

it was possible to depict properties of prime numbers

as motivated, the proof of which has not been suc-

cessful for hundreds of years. Why this is so is largely

unclear and is a challenge for future research. These

considerations are not based on simulations, but on

calculations using the stochastic approach. Such an

approach is also useful for the present topic.

The epidemiological situation of the next quarter

(day 273) is described in Figure 2 bottom left.

About half of the population is affected by the epi-

demic. The pattern emergence is largely shaped by the

previous quarter. One year after the start of the simu-

lation, a high level of immunity is achieved, although

this is associated with few new infections.

Before details of the next few years are presented,

the time series in the compartments should be consi-

dered in Figure 3. After a large initial swing, the S-

compartment stabilizes at approx. 75% of the popula-

tion with a variation of approx. 25% of the population

in the considered simulation.

As shown in Figure 3 the I

2

compartment shows

signiﬁcantly lower amplitudes than the I

1

compart-

ment.

500 1000 1500 2000 2500 3000 3500

25 000

30 000

35 000

40 000

S compartment

500 1000 1500 2000 2500 3000 3500

500

1000

1500

2000

2500

3000

I2 compartment

500 1000 1500 2000 2500 3000 3500

5000

10 000

15 000

R1 compartment

500 1000 1500 2000 2500 3000 3500

500

1000

1500

R2 compartment

Figure 3: Time series of the S compartment (top left), I

2

compartment (top right), R

1

compartment (bottom left) and

R

2

compartment (bottom right).

500 1000 1500 2000 2500 3000 3500

100

200

300

400

500

I1 compartment

Figure 4: Time series of the I

1

compartment.

While the I

1

and I

2

compartments show a high de-

gree of correspondence over time, there are signiﬁ-

cant qualitative differences for the R

1

and R

2

com-

partments.

After some years, the epidemic system is still in

the settling phase.

In the years nine and ten after the start of the simu-

A Discrete SIR Model with Spatial Distribution on a Torus for COVID-19 Analysis using Local Neighborhood Properties

479

R-value = 0.80, day = 730

0 → 32 374

1 → 47

2 → 266

3 → 6713

4 → 1202

R-value = 1.03, day = 1095

0 → 28 773

1 → 171

2 → 919

3 → 9379

4 → 1360

R-value = 1.01, day = 3285

0 → 32 635

1 → 233

2 → 1001

3 → 5762

4 → 971

R-value = 1.04, day = 3650

0 → 29 073

1 → 32

2 → 231

3 → 10 233

4 → 1033

Figure 5: S-I-R compartments after 2, 3 9 and 10 years

(from top left to bottom right).

lation, the epidemic essentially shows the long-term

dynamics.

If the time step is really shorter than one day, then

the number of years for the settling process is corre-

spondingly shorter. For long-term observations, annu-

al inﬂuences on the assumed constants of infection are

relevant. This process of synchronization between the

internal clock and the external clock has long been

discussed in the biomathematical literature and will

be taken up in a later paper.

4 DISCUSSION AND

CONCLUSIONS

The dynamics of the development of the epidemic is

inﬂuenced by the interaction of all parameters used.

Only then does it become apparent whether the time

unit used is compatible with the observation. The re-

sults suggest that the time unit used is signiﬁcantly

shorter than a day and thus the observation period is

shorter than ten years. This implies that the given RKI

r-value becomes more stable without its fundamen-

tal problem changing. The r-value varies at relative-

ly short intervals between values below and above 1.

The simulations show that it has no signiﬁcance for

the formation of new regional clusters. Regional clu-

sters can trigger regional interventions with a suitable

scaling, which, however, as already mentioned, can

be counterproductive to the development of immune

behavior. To make matters worse, only the I

1

com-

partment is visible, but the dynamics are essentially

determined by the entire I compartment. This natu-

rally also raises the question of whether the r-value

(real and in the simulation) should be calculated with

the entire I compartment or only with the I

1

compart-

ment. The entire I compartment is useful for the si-

mulation, but only the I

1

compartment is practical-

ly measurable. A further complication arises becau-

se there are no representative estimated values for the

population, but Covid-19 test results are only availa-

ble through measures that are essentially politically

selective. Tracking Covid-19 contact chains with ex-

tensive quarantine can limit the positive development

of immunity by the I

2

population. Since this happens

predominantly, only selectively, it will only lead to a

few spatial disparities that are already present through

pattern creation processes.

Infectious diseases often vary from season to sea-

son. This can be implemented in the model in that the

transition probability used are subject to a seasonal

course. However, this only makes sense if the time

unit resulting from the combination of the parameters

used has been adapted sufﬁciently reliably to real ti-

me units. The synchronization of different time sca-

les poses is a challenge for future research. The mo-

deling on the torus could be extended by bifurcation

at certain points or intervals, which leads to a direc-

ted graph in the graph-theoretical interpretation. This

could be used to model the interaction of distant con-

tinents.

We considered a delay model in the simulation un-

der consideration. In a subsequent analysis it is ex-

amined whether the qualitative behavior of the simu-

lation is retained if the length of stay in the compart-

ments is replaced by transition probabilities. In mo-

dels of ordinary and partial differential equations, de-

lays generally lead to qualitatively signiﬁcantly diffe-

rent results.

If Markov models lead to qualitatively similar re-

sults in the context used for the epidemiological situa-

tion under consideration, eigenvalues and eigenfunc-

tions of the corresponding analytical manifolds could

be helpful for analyzing the stability of the parameters

used.

With suitable boundary conditions, eigenvalues

and eigenfunctions of the Laplace operator are well

known for the torus under consideration. The Laplace

operator is decisive for diffusion processes that play

a decisive role and leads to morphological develop-

ments in pattern formation, cf. (Murray, 2002), (Mur-

ray, 2003). Mean value operators in the sense of diffe-

rential geometry have not yet been used in mathemati-

cal biology. In differential geometry and the theory of

relativity, they are used to characterize spatial structu-

res, cf. (G

¨

unther and Pr

¨

ufer, 1999), (Schuster, 1988),

(Schuster, 1997). In the simulation under considerati-

on, we used mean value operators with a uniform dis-

tribution in the maximum metric used. The equal dis-

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480

tribution can be replaced by other distributions which,

in a possibly more realistic way, reduce the risk of in-

fection with increasing distance. Again, this is only

partially realistic because life does not take place on

a torus and there are different near-far relationships

that can be analyzed with graph-theoretic methods, cf.

(Solimano and Bretta, 1982), (R

¨

udiger et al., 2020),

(Welch et al., 2011), (Boguna et al., 2003), (Barrat

and Vespignani, 2008).

It is also of essential importance in which way the

”

small world model“ is used, in which it is described

in which

”

small“ number of steps each individual can

be reached by any other individual, here relevant for

chains of infection. In the present context, however,

a theoretically possible path must be supplemented

with a probability with which this path will be im-

plemented in practice.

In a subsequent paper we will analyze the consi-

dered epidemiological development on real life graph

networks. As a small network we will use anonymi-

zed physicians as nodes of the graph, the edges are gi-

ven by a through a neighborhood relationship in terms

of the number of common patients. As a large network

we will use anonymized patients related by physicians

visited by both patients using a random selection. In

graph theory one also can use the Laplace operator

and mean value operators.

Current virological results on Covid-19, as stated

in (Chinazzi et al., 2020), (Rosenbaum, 2020), (Pan

et al., 2020), (Behrens et al., 2020), (Tang et al.,

2020), (Xue et al., 2020), (Kucharski et al., 2020) may

inﬂuence the spatial-temporal modeling on the level

of individuals in future and will give new insights in

the dynamics of propagation. But the dynamic of the

contagion occurs to a certain extent independently of

the knowledge of the known details on an individual

level. The inﬂuence of measurable variables in indivi-

dual contacts on global expansion and their validation

is a challenge for future research.

It could be that the currently predominantly used

PCR tests identify the individuals in the I

1

compart-

ment in our interpretation or the individuals that have

already passed into the R compartment. The dynamics

observed could indicate that, due to cross immunity

or other immunological mechanisms, that the I

2

com-

partment, which could be very important for the dy-

namics of spread, has so far been little identiﬁed in

practice.

In different countries there were different drama-

tic initial situations in the early phase of Covid-19.

So far, this has been largely discussed as the result of

various politically decided preventive measures. But

it could also be that different initial conditions exi-

sted with regard to the R-compartment as a result of

pre-existing immunity. As already stated, an overar-

ching cross-immunity with regard to Covid-19 could

also be modeled in the S compartment. This would ha-

ve theoretical advantages, but would make the search

for meaningful parameters for modeling based on the

current state of knowledge more difﬁcult.

Differing spreading dynamics, regardless of the

measures taken, may also have played a role.

If measures are aimed at greatly reducing the con-

tagion, this subsequently leads to poor immunity at

least without effective vaccinations. Since the contact

restriction was practically inconsistent, sufﬁcient im-

munity was nevertheless not prevented with the ex-

ception of very restrictive measures in some coun-

tries, which may have since led to a high second wave.

The amount of people in overcrowded local transport

means may be seen in different interpretations. Cur-

rent observations can be interpreted to the effect that

particularly drastic contact restrictions due to immu-

nity not being achieved could lead to a second wave.

The compartment status has a discrete value (S, I

1

,

I

2

, R

1

, R

2

), while eigenfunctions and their discretiza-

tions have a real values. If one intends to use the me-

thod of separating the variables to carry out a series

expansion using the eigenvalues of the Laplace opera-

tor, additional considerations are necessary. A similar

problem arises in the bisection problem of graphs. A

good starting solution is to separate the eigenfunction

for the largest non-trivial eigenvalues into points with

positive and negative values. In this respect, it could

be a sensible approach to use the eigenfunctions for

suitable eigenvalues in a hierarchical procedure (since

there are more than two discrete states) for the identi-

ﬁcation of inherently periodic solutions.

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