ACADEMIA Letters
A network-based computational model showing responses
to changing environmental constraints similar to patterns
observed in the dynamics of the Covid-19 event
PETER KRALL
Abstract
The paper presents models of the dynamics of virus spread in scenarios where social
contacts are severely reduced but contact networks in some fractions of the population
are fairly resilient. It is demonstrated that this can result in the existence of multiple
meta-stable states and changes of environmental constraints trigger transition from one
such state to another but fail to cause reduction of activity beyond the level of the target
state.
Introduction
The Covid-19 event is special with respect to two aspects: First, the contacts between people
are strongly reduced[RW1] . Second, curfews and similar interventions often have a strong
initial effect but then rather suddenly stop to yield further reduction. For example, the 7d
average of new cases in Portugal came down from 12890 January 28th 2021 to 1005 February
28th. But it was 513 March 16th and 512 April 16th, and remained in the band between
420 and 614 over the period before Delta changed the picture, and, notwithstanding effects of
vaccination and change of rules, was 641 June 11th after a minimum of 327 May 11th. Similar
patterns were observed elsewhere, e.g. in Belgium and Ireland[5]. The curves look like what
would be expected from a system that has different steady states, depending on environmental
constraints. In such systems, change of the environment will trigger a possibly strong initial
Academia Letters, July 2021
©2021 by the author — Open Access — Distributed under CC BY 4.0
Corresponding Author: PETER KRALL, peter.krall.it@googlemail.com
Citation: Krall, P. (2021). A network-based computational model showing responses to changing environmental
constraints similar to patterns observed in the dynamics of the Covid-19 event. Academia Letters, Article 1581.
https://doi.org/10.20935/AL1581.
1
response but as the steady state corresponding to the new constraints is approached, the delta
will become small.
Explanations based on the assumption of anticipatory compliance before imposing restrictions and reduced compliance are highly implausible. Why should people immediately
stop doing things when told they can do them for some weeks more but will be punished
thereafter? And how do they manage to adjust their behavior in such a way that activity just
remains constant?
Conversely, it has been demonstrated that network effects can be the reason for patterns
in the dynamics of virus spread that differ significantly from compartment models[1,2, 3,
4]. This inspires the question how mathematical or computational models can reproduce the
dynamics of response to changing environmental parameters appearing in the C-19 event.
Another metaphor
Classical epidemiological models are analogous to reactions in constantly stirred liquid. But
the numbers in the C-19 event look more like what one would expect when position effects
play a strong role, like in the wildfire burning in a strip of woodland where the active zone
moves but does not necessarily change in size. Growth rates of expanding rings or sphere
shells also will tend to approach 1 under constant conditions and may expand or narrow under
changing conditions but will not have the potential for prolonged exponential growth.
However, it is difficult to see how to align a model based on pure propagation between
neighbors in networks with very strong initial effects but quasi-steady states reached after
some time. A possible solution is the embedding of a network of resilient chains of groups
into an environment where effects of interventions are strong. The metaphor then is a strip
of burning woodland that cannot be extinguished but is embedded into an area of scrubland
where follow-up activity may spread or be contained, depending on constraints.
The model
The model illustrates network effects on patterns of spread dynamics in a rather abstract framework. There is no direct correspondence between generations and real-world periods of infection.
The model works with discrete generations and cases are active for one generation. After
having been active, individuals are no longer susceptible. The dynamics is based on superposition of several components.
Academia Letters, July 2021
©2021 by the author — Open Access — Distributed under CC BY 4.0
Corresponding Author: PETER KRALL, peter.krall.it@googlemail.com
Citation: Krall, P. (2021). A network-based computational model showing responses to changing environmental
constraints similar to patterns observed in the dynamics of the Covid-19 event. Academia Letters, Article 1581.
https://doi.org/10.20935/AL1581.
2
1. In every generation, there is a number R(t) * AC * (Sus/Pop) of randomly selected
new cases with R(t) being a time-dependent parameter representing random contagion
probability under current environmental conditions, AC=active cases, Sus=susceptible
population size, Pop=population size.
2. There are a number of groups, each associated with a time-dependent ‘conditional activation probability (CAP), defined by: p(act) = 1-(1-CAP(t))AC/G, where p(act) is the
probability of a susceptible group member to become active by in-group transmission,
if not activated according to random spread (defined by 1), and AC/G is the number of
active cases in the group, with canonical generalization for members of several groups.
The model is not bound to any specific assumptions concerning the mechanisms underlying the CAP. In-group transmission is a possible interpretation but exposition to the
same super-spreader or other possibilities can also be considered.
Groups can overlap. A group-chain is a set of groups, such that for some indexing groups
with consecutive index overlap. Groups in the chain with non-consecutive do not overlap, but
groups in the chain may overlap with other groups.
Some results
Computer simulations show that the presence of group chains results in long periods of little
or erratic change when R-value is low. Changes of R-value often have a strong immediate
effect but then activity levels become more or less stable. Some examples are shown in the
curves below:
Figure 1. Frequencies of active cases with constrains as desceibed in the text. Numbers refer
to the highlighted curve. The dashed curve is the activity in group-chains.
The runs shown share the following constraints:
• Population size is 8m
• Initial R(t) is 0.8.
• R(t) is reduced to 0.1 for t after a threshold of activity is passed.
• There are 2M disjoint 4-member ‘family’ groups. CAP(t) for these groups is constant
at 0.15.
Academia Letters, July 2021
©2021 by the author — Open Access — Distributed under CC BY 4.0
Corresponding Author: PETER KRALL, peter.krall.it@googlemail.com
Citation: Krall, P. (2021). A network-based computational model showing responses to changing environmental
constraints similar to patterns observed in the dynamics of the Covid-19 event. Academia Letters, Article 1581.
https://doi.org/10.20935/AL1581.
3
• There are 28.6k 14-member ‘office’-groups. CAP(t) is initially 0.15 but after the activation threshold is passed, ¾ are ‘sent to home-office’, reducing the CAP to zero.
• There are 14-member groups in group-chains with overlap 3, length 18 and CAP 0.15.
These groups are orthogonal to office-groups.
What differs between the runs is the number of groups, respectively individuals in such
groups. In the red and the magenta scenario, there are 3600 groups in chains; 7200 are in
the blue and 10800 in the black scenario. Moreover, in the magenta scenario, the original
constraints are temporarily restored after activity falls below 100, and set back to the reduced
value after the activation threshold is passed again (but not restored once more after activity
falls below 100 second time).
The red/magenta scenarios show a rapid reduction after change of constraints but the decline slows down and at some point or the other becomes erratic. In the highlighted example,
a activity falls rapidy from 1816 in generation 8 to 555 in generation 11 after reduction of
R and deactivating most groups outside the chains. The decline slows down, reaching 226
in generation 28. A period of erratic fluctuation follows, with the last maximum above 226
reached in generation 60 (334).
The blue and black scenarios show that higher numbers of group-chains increase unpredictability and amplitude of swings. What the model runs with high number of group chains
have in common is an immediate fall after reduction of R, followed by a rebound that will
result in much higher numbers than the local minimum immediately after reduction.
Discussion
One aspect of the dynamics shown in the model runs is trivial: if the dynamics of spread
of activity results from superposition of several components, then some parameter modifications may affect some but not all components directly. Other components may or may not
be affected indirectly. They will be indirectly affected if activation chains necessarily include
directly affected edges,– if household-to-household activations are reduced, then follow-up
in-household infections will also go down, even if transmission probabilities in households
remain unchanged. But there can be also resilient nets where transmission is neither directly
nor indirectly affected.
The less obvious finding is, that networks including only small fractions of the population can modulate the dynamics, resulting in patterns completely different from compartment
models. Particularly, long periods of erratic fluctuations tend to appear under constant environmental conditions. Change of constraints will often trigger a transition to another level
Academia Letters, July 2021
©2021 by the author — Open Access — Distributed under CC BY 4.0
Corresponding Author: PETER KRALL, peter.krall.it@googlemail.com
Citation: Krall, P. (2021). A network-based computational model showing responses to changing environmental
constraints similar to patterns observed in the dynamics of the Covid-19 event. Academia Letters, Article 1581.
https://doi.org/10.20935/AL1581.
4
of activity but the effect seen initially cannot be extrapolated – rather, the system can swing
into erratic fluctuations, may rebound to a level of activity somewhere between the point of
an intervention and the minimum reached shortly thereafter.
It is also apparent that turning points are unpredictable, even if the constraints are known.
While fluctuations may be confined to some band, what happens within the band is coincidental.
References
[1] Dekker, Anthony. (2008). Network Effects in Epidemiology. https://www.researchgate.
net/publication/228988928_Network_Effects_in_Epidemiology
[2] Ganesh, Ayalvadi & Massoulié, Laurent & Towsley, Donald. (2005). The effect of network topology on the spread of epidemics. Proceedings - IEEE INFOCOM. 2. 1455-1466.
10.1109/INFCOM.2005.1498374. https://www.researchgate.net/publication/221241705_
The_effect_of_network_topology_on_the_spread_of_epidemics
[3] Moreno, Y. et al (2002) Epidemic outbreaks in complex heterogeneous networks. The
European Physical Journal B - Condensed Matter and Complex Systems. https://doi.org/
10.1140/epjb/e20020122
[4] Thurner, Stefan et al (2020). A network-based explanation of why most COVID-19 infection curves are linear 10.1073/pnas.2010398117 Proceedings of the National Academy of
Sciences p 22684-22689. https://www.pnas.org/content/117/37/22684
[5] Worldometer (2021) https://www.worldometers.info/coronavirus/country/portugal/
Academia Letters, July 2021
©2021 by the author — Open Access — Distributed under CC BY 4.0
Corresponding Author: PETER KRALL, peter.krall.it@googlemail.com
Citation: Krall, P. (2021). A network-based computational model showing responses to changing environmental
constraints similar to patterns observed in the dynamics of the Covid-19 event. Academia Letters, Article 1581.
https://doi.org/10.20935/AL1581.
5
ACADEMIA Letters
A network-based computational model showing responses
to changing environmental constraints similar to patterns
observed in the dynamics of the Covid-19 event
PETER KRALL
Abstract
The paper presents models of the dynamics of virus spread in scenarios where social
contacts are severely reduced but contact networks in some fractions of the population
are fairly resilient. It is demonstrated that this can result in the existence of multiple
meta-stable states and changes of environmental constraints trigger transition from one
such state to another but fail to cause reduction of activity beyond the level of the target
state.
Introduction
The Covid-19 event is special with respect to two aspects: First, the contacts between people
are strongly reduced[RW1] . Second, curfews and similar interventions often have a strong
initial effect but then rather suddenly stop to yield further reduction. For example, the 7d
average of new cases in Portugal came down from 12890 January 28th 2021 to 1005 February
28th. But it was 513 March 16th and 512 April 16th, and remained in the band between
420 and 614 over the period before Delta changed the picture, and, notwithstanding effects of
vaccination and change of rules, was 641 June 11th after a minimum of 327 May 11th. Similar
patterns were observed elsewhere, e.g. in Belgium and Ireland[5]. The curves look like what
would be expected from a system that has different steady states, depending on environmental
constraints. In such systems, change of the environment will trigger a possibly strong initial
Academia Letters, July 2021
©2021 by the author — Open Access — Distributed under CC BY 4.0
Corresponding Author: PETER KRALL, peter.krall.it@googlemail.com
Citation: Krall, P. (2021). A network-based computational model showing responses to changing environmental
constraints similar to patterns observed in the dynamics of the Covid-19 event. Academia Letters, Article 1581.
https://doi.org/10.20935/AL1581.
1
response but as the steady state corresponding to the new constraints is approached, the delta
will become small.
Explanations based on the assumption of anticipatory compliance before imposing restrictions and reduced compliance are highly implausible. Why should people immediately
stop doing things when told they can do them for some weeks more but will be punished
thereafter? And how do they manage to adjust their behavior in such a way that activity just
remains constant?
Conversely, it has been demonstrated that network effects can be the reason for patterns
in the dynamics of virus spread that differ significantly from compartment models[1,2, 3,
4]. This inspires the question how mathematical or computational models can reproduce the
dynamics of response to changing environmental parameters appearing in the C-19 event.
Another metaphor
Classical epidemiological models are analogous to reactions in constantly stirred liquid. But
the numbers in the C-19 event look more like what one would expect when position effects
play a strong role, like in the wildfire burning in a strip of woodland where the active zone
moves but does not necessarily change in size. Growth rates of expanding rings or sphere
shells also will tend to approach 1 under constant conditions and may expand or narrow under
changing conditions but will not have the potential for prolonged exponential growth.
However, it is difficult to see how to align a model based on pure propagation between
neighbors in networks with very strong initial effects but quasi-steady states reached after
some time. A possible solution is the embedding of a network of resilient chains of groups
into an environment where effects of interventions are strong. The metaphor then is a strip
of burning woodland that cannot be extinguished but is embedded into an area of scrubland
where follow-up activity may spread or be contained, depending on constraints.
The model
The model illustrates network effects on patterns of spread dynamics in a rather abstract framework. There is no direct correspondence between generations and real-world periods of infection.
The model works with discrete generations and cases are active for one generation. After
having been active, individuals are no longer susceptible. The dynamics is based on superposition of several components.
Academia Letters, July 2021
©2021 by the author — Open Access — Distributed under CC BY 4.0
Corresponding Author: PETER KRALL, peter.krall.it@googlemail.com
Citation: Krall, P. (2021). A network-based computational model showing responses to changing environmental
constraints similar to patterns observed in the dynamics of the Covid-19 event. Academia Letters, Article 1581.
https://doi.org/10.20935/AL1581.
2
1. In every generation, there is a number R(t) * AC * (Sus/Pop) of randomly selected
new cases with R(t) being a time-dependent parameter representing random contagion
probability under current environmental conditions, AC=active cases, Sus=susceptible
population size, Pop=population size.
2. There are a number of groups, each associated with a time-dependent ‘conditional activation probability (CAP), defined by: p(act) = 1-(1-CAP(t))AC/G, where p(act) is the
probability of a susceptible group member to become active by in-group transmission,
if not activated according to random spread (defined by 1), and AC/G is the number of
active cases in the group, with canonical generalization for members of several groups.
The model is not bound to any specific assumptions concerning the mechanisms underlying the CAP. In-group transmission is a possible interpretation but exposition to the
same super-spreader or other possibilities can also be considered.
Groups can overlap. A group-chain is a set of groups, such that for some indexing groups
with consecutive index overlap. Groups in the chain with non-consecutive do not overlap, but
groups in the chain may overlap with other groups.
Some results
Computer simulations show that the presence of group chains results in long periods of little
or erratic change when R-value is low. Changes of R-value often have a strong immediate
effect but then activity levels become more or less stable. Some examples are shown in the
curves below:
Figure 1. Frequencies of active cases with constrains as desceibed in the text. Numbers refer
to the highlighted curve. The dashed curve is the activity in group-chains.
The runs shown share the following constraints:
• Population size is 8m
• Initial R(t) is 0.8.
• R(t) is reduced to 0.1 for t after a threshold of activity is passed.
• There are 2M disjoint 4-member ‘family’ groups. CAP(t) for these groups is constant
at 0.15.
Academia Letters, July 2021
©2021 by the author — Open Access — Distributed under CC BY 4.0
Corresponding Author: PETER KRALL, peter.krall.it@googlemail.com
Citation: Krall, P. (2021). A network-based computational model showing responses to changing environmental
constraints similar to patterns observed in the dynamics of the Covid-19 event. Academia Letters, Article 1581.
https://doi.org/10.20935/AL1581.
3
• There are 28.6k 14-member ‘office’-groups. CAP(t) is initially 0.15 but after the activation threshold is passed, ¾ are ‘sent to home-office’, reducing the CAP to zero.
• There are 14-member groups in group-chains with overlap 3, length 18 and CAP 0.15.
These groups are orthogonal to office-groups.
What differs between the runs is the number of groups, respectively individuals in such
groups. In the red and the magenta scenario, there are 3600 groups in chains; 7200 are in
the blue and 10800 in the black scenario. Moreover, in the magenta scenario, the original
constraints are temporarily restored after activity falls below 100, and set back to the reduced
value after the activation threshold is passed again (but not restored once more after activity
falls below 100 second time).
The red/magenta scenarios show a rapid reduction after change of constraints but the decline slows down and at some point or the other becomes erratic. In the highlighted example,
a activity falls rapidy from 1816 in generation 8 to 555 in generation 11 after reduction of
R and deactivating most groups outside the chains. The decline slows down, reaching 226
in generation 28. A period of erratic fluctuation follows, with the last maximum above 226
reached in generation 60 (334).
The blue and black scenarios show that higher numbers of group-chains increase unpredictability and amplitude of swings. What the model runs with high number of group chains
have in common is an immediate fall after reduction of R, followed by a rebound that will
result in much higher numbers than the local minimum immediately after reduction.
Discussion
One aspect of the dynamics shown in the model runs is trivial: if the dynamics of spread
of activity results from superposition of several components, then some parameter modifications may affect some but not all components directly. Other components may or may not
be affected indirectly. They will be indirectly affected if activation chains necessarily include
directly affected edges,– if household-to-household activations are reduced, then follow-up
in-household infections will also go down, even if transmission probabilities in households
remain unchanged. But there can be also resilient nets where transmission is neither directly
nor indirectly affected.
The less obvious finding is, that networks including only small fractions of the population can modulate the dynamics, resulting in patterns completely different from compartment
models. Particularly, long periods of erratic fluctuations tend to appear under constant environmental conditions. Change of constraints will often trigger a transition to another level
Academia Letters, July 2021
©2021 by the author — Open Access — Distributed under CC BY 4.0
Corresponding Author: PETER KRALL, peter.krall.it@googlemail.com
Citation: Krall, P. (2021). A network-based computational model showing responses to changing environmental
constraints similar to patterns observed in the dynamics of the Covid-19 event. Academia Letters, Article 1581.
https://doi.org/10.20935/AL1581.
4
of activity but the effect seen initially cannot be extrapolated – rather, the system can swing
into erratic fluctuations, may rebound to a level of activity somewhere between the point of
an intervention and the minimum reached shortly thereafter.
It is also apparent that turning points are unpredictable, even if the constraints are known.
While fluctuations may be confined to some band, what happens within the band is coincidental.
References
[1] Dekker, Anthony. (2008). Network Effects in Epidemiology. https://www.researchgate.
net/publication/228988928_Network_Effects_in_Epidemiology
[2] Ganesh, Ayalvadi & Massoulié, Laurent & Towsley, Donald. (2005). The effect of network topology on the spread of epidemics. Proceedings - IEEE INFOCOM. 2. 1455-1466.
10.1109/INFCOM.2005.1498374. https://www.researchgate.net/publication/221241705_
The_effect_of_network_topology_on_the_spread_of_epidemics
[3] Moreno, Y. et al (2002) Epidemic outbreaks in complex heterogeneous networks. The
European Physical Journal B - Condensed Matter and Complex Systems. https://doi.org/
10.1140/epjb/e20020122
[4] Thurner, Stefan et al (2020). A network-based explanation of why most COVID-19 infection curves are linear 10.1073/pnas.2010398117 Proceedings of the National Academy of
Sciences p 22684-22689. https://www.pnas.org/content/117/37/22684
[5] Worldometer (2021) https://www.worldometers.info/coronavirus/country/portugal/
Academia Letters, July 2021
©2021 by the author — Open Access — Distributed under CC BY 4.0
Corresponding Author: PETER KRALL, peter.krall.it@googlemail.com
Citation: Krall, P. (2021). A network-based computational model showing responses to changing environmental
constraints similar to patterns observed in the dynamics of the Covid-19 event. Academia Letters, Article 1581.
https://doi.org/10.20935/AL1581.
5