Berlin - When will this finally be over? Everyone will have asked themselves this question more than once in the past 18 months. In Germany, things were looking good lately. Summer had come, and so had the vaccine. But Britain, the country with the fastest vaccine programme in Europe, is reporting a spectacular rise in new infections due to the Delta variant. One is never quite sure with this virus.
Matthias Kreck’s Covid story begins with the question of the pandemic’s ending. On 19 March 2020, like so many others, he was watching the Maybrit Illner talk show. The virologist Christian Drosten was there. Asked when restrictions and the pandemic in general might recede, Drosten said, "You'll have to ask the mathematicians." In January 2020, Drosten’s team at Charité hospital in Berlin was the first to publish instructions on PCR-detection of the novel coronavirus. Few people know more than Drosten about how Sars-Cov-2 transmits, how it attacks the body and how it mutates. But when it comes to lockdown measures, virologists are not the specialists. If you want to estimate which restrictions should be tightened and how they can be loosened again, you’ll have to ask epidemiologists. They’re the ones who need sound mathematical models.
Kreck listened carefully to Drosten. He is a mathematician, and a famous one at that: he won the Cantor Medal, Germany's most important maths prize. He was the founding director of the Hausdorff Institute at the University of Bonn and presided over the Mathematical Research Institute in Oberwolfach for eight years. Few places in the world attract as much mathematical talent and genius as these.
Kreck wrote to Drosten, offering his help and his network. His first e-mail is a little long-winded. He starts to explain the basics of epidemiological modelling, then apologises in a subsequent e-mail, expresses his respect for the virologist, says that he doesn’t want to seem pushy. Drosten writes back, asking Kreck who the leading experts of mathematical epidemiology in Germany are. He says he’s surprised how little input has come from that community. Kreck has to think. He is a pure mathematician. Most of his research does not even require a computer. "For German researchers, mathematical epidemiology is not an attractive field," he says. Those who want to push their career work on other topics. Kreck eventually recommended some colleagues. He never heard from Drosten again.
A good year later, Matthias Kreck says that the models of the "state modellers" – as he sometimes grumbles – are based on mathematical "nonsense".
To understand what Kreck means by these accusations, and to grasp the vehemence with which he criticises some of Germany’s most prominent Covid modellers, one has to look at his entire pandemic year. Kreck’s corona story is one of dry mathematics, forgotten research papers from the 1920s, fumbling e-mails, jovial replies, overflowing inboxes and wounded pride. It is also about a working group of Germany’s NoCovid initiative from which Kreck was expelled, and about a diagramme that sought but failed to find its way into the Chancellor’s Office. It simulated a measure that would put the peak of the second wave before Christmas at under 6,000 new daily infections in Germany and not, as in reality, at around 30,000.
Kreck’s criticism is directed at three research groups that, in his view, have dominated the epidemiological discourse in Germany: one led by Dirk Brockmann, professor at Humboldt University and head of epidemiological modelling at the Robert Koch Institute (RKI), the federal agency for disease control and prevention; another led by Michael-Meyer Hermann, director of Systems Immunology at the Helmholtz Centre in Braunschweig and professor at the Technical University there; and a third one led by Viola Priesemann at the Max-Planck-Institute for Dynamics and Self-Organisation in Göttingen.
In autumn 2020, Germany was no longer the "pandemic world champion" it had thought it was in the spring. It was just another western country that had Covid-19 as little under control as its neighbours. Almost overnight, the three modellers became the faces of German epidemiology: on Twitter, on the evening news, within the NoCovid initiative. All three are trained physicists. All of them published scientific papers on the spread of Covid-19 last year; Brockmann and Priesemann each published a paper in Science.
There was a reason why they only really came to the public's attention in the autumn: they had to fill the gap that Drosten and Kreck had identified in the spring. In contrast to Britain, the Netherlands or Switzerland, Germany has neither a lively tradition in public health research nor in mathematical epidemiology. Dirk Brockmann from the RKI regularly complains about this in interviews. The Braunschweig research group usually specialises in cell biology, the Göttingen group in brain research. In March 2020, they rapidly converted their mathematical models to Covid.
Why does Kreck call them, sometimes jokingly, sometimes polemically, the "state modellers"? Why is he launching a frontal attack against scientists who fought the good fight and have already received plenty of hate and malice in the public debate?
Kreck has the impression, he says, that the government and the media listened to these groups in particular, even if they themselves complained that they were not listened to enough. He rejects the logic of hyper-polarised discourse where criticism of certain scientific positions is mechanically associated with the camp of trolls and deniers. He was never against the lockdown. Unlike the Leipzig mathematician Stephan Luckhaus, who resigned from Leopoldina, Germany’s national academy of science, over scientific disagreements, Kreck never said that the virus should run free among youths to build up herd immunity. Kreck’s point is purely mathematical. He says that with more accurate models, better lockdowns and more efficient containment measures would have been possible.
Together with his friend and colleague Erhard Scholz, a mathematics historian and professor emeritus from Wuppertal, Kreck has been developing his own corona model since spring 2020. The two have published three preprints, one of which is under peer-review at the Bulletin of Mathematical Biology. They also sent a version of their model to Science. After two days, they received a message stating that "the scope and focus" of their paper made it "more appropriate for a more specialised journal." Kreck cannot take a review in two days seriously. The 48-page research paper on Surgery and Duality which made him one of the world’s leading specialists in algebraic topology was in peer review for four years.
"Matthias, as a mathematician, sees modelling more dogmatically than I am used to from the history of science," says Scholz. After tough discussions and reading the research literature, Kreck convinced him that a deeper theoretical justification of the models was necessary.
Kreck’s criticism is directed against the SIR model, which is not the only model in epidemiology, but certainly one of the dominant ones. Hundreds of research papers that have been published in prestigious journals in the past year work with SIR. To be able to describe Kreck’s critique, one has to do something that almost never happens in the media. Data journalism is catchy, it can be presented in ways that readers intuitively understand. What is almost never addressed, though, are the formulas and calculations behind the charts. For science communication and for a policy that wants to rely entirely on scientific evidence, this hermeticism creates problems.
The French philosopher Bruno Latour has argued for years that in response to post-truth populism, scientists should be more forthcoming about their methods and disclose the devices and techniques, collegial networks and corrective loops through which scientific evidence is produced, validated and renewed. In all these processes, mistakes happen and disputes arise. But the disclosure of the process can itself be a persuasive argument in public debate. In most scientific disciplines, the procedures that lead to validated knowledge remain unknown to the outside world. Mathematical knowledge is perhaps the most extreme example of this: it eludes discourse because it is shared by so few.
I know Matthias Kreck through his son Benjamin with whom I briefly studied mathematics when I was 20. I went on to study other subjects and later returned to maths for another three semesters. For this article, I spoke with mathematicians in Utrecht, Leipzig, Stockholm, Kaiserslautern, Bonn and London. At first, most of them hesitate to engage too deeply with Kreck and Scholz’s "discrete Kermack-McKendrick model adapted to Covid-19". Most are even more cautious when it comes to being quoted. When it comes to the topic of corona, there is immense fear of appearing in a context that one does not fully control.
But if you dig deeper with theorists of epidemiological modelling, it becomes clear that Kreck and Scholz’ central points cannot be refuted easily. The SIR model contains assumptions about the transmission behaviour of the virus that are very unrealistic. Good modellers know this and consider this weakness when they evaluate their results. In certain situations, working with unrealistic assumptions can be perfectly legitimate, because one cannot know the complete data of reality anyway, for example. Or because a model may be particularly well adaptable to the specific segment of reality that one wants to study. A lack of precision is accepted in exchange for a gain in practicability.
The standard SIR model was developed in the 1920s by Anderson Gray McKendrick, a British military doctor, and William Ogilvy Kermack, a biochemist at the Royal Society of Edinburgh. SIR is a special case of their more general theory on the spread of infectious disease. To predict the number of infections, Kermack and McKendrick divided the population into three groups: S for people who are susceptible to the virus, I for the infected and R for those who are removed from the infection process by death or immunisation. In the beginning, almost the entire population is susceptible. Then the number of new infections increases and begins to decrease again when the pathogen runs out of targets due to a growing number of removed people. This results in a characteristic wave of infection (shown here in red):
Such population curves were shown at the beginning of the pandemic. Of course, no one thought that a wave would run through the population so unchecked. It was expected that people would change their behaviour on the basis of such shocking diagrammes and anticipate in private what would become public policy shortly thereafter.
But the unrealistic aspects of the SIR model are not only due to the effect of feedback loops of public messages on human behaviour. They also lie in a mathematical function that is not so readily apparent. If one wants to calculate the number of new infections at a point in time t, the only decisive factor is that one knows the contribution of an infected person to the further spread of the virus in the days after their own infection.
The probability that an infected person will infect others is not constant throughout their infectious phase. It depends on changes in the viral load in their throat, on contact rates and on the amount of time that passes before they go into isolation. The equations of the standard SIR model can only be derived if one assumes that this background function g(t), i.e. the contribution of an infected person, is exponentially distributed. That means that g(t) has to start with a maximum and decrease quite rapidly towards zero without ever reaching it. Therefore, the background function, which is built into the SIR model as a premise, assumes that an infected person is most dangerous to their fellow human beings at the very beginning of their own infection.
Virologists know how the viral load of Sars-Cov-2 actually behaves in the throat of most infected people. For about two days after infection, it remains below the infectivity threshold. Then it rises, reaches its maximum at the onset of symptoms about five days after infection, and falls below the critical threshold again after another five to seven days. If the virus mutates, this typical course can also shift. The course of infectivity is something that makes Covid a very special disease. The more deadly Sars-1 and Ebola viruses did not trigger a pandemic because the viral load only really rose when the patients were already seriously ill.
When Kreck showed me the discrepancy between g(t), the background function of SIR, and the biologically detectable infectivity, I could hardly believe him at first. g(t) massively overestimates the infection rate in the initial phase, when an infected person does not yet show any symptoms. Conversely, the dangerousness of infected persons around the onset of symptoms is greatly underestimated.
One can try to counteract this distortion by including additional population groups into the model, for example, a group E for "exposed" persons who are already infected but not yet infectious themselves. One can integrate a delay that makes the exponentially decreasing background function kick in only on the second day after infection. There are countless ways to adapt the SIR model. One simple way to bring it closer to reality is to use what is called "fitting". In the background function g(t), two parameters can be chosen freely (they determine the height at which the curve starts and the speed at which it falls). By choosing these parameters, one optimises the model in such a way that the resulting population curves fit the real data of the past, on the assumption that one can also use them to predict the future. Nevertheless, the assumed course of infectivity remains unrealistic.
The respective scientific papers by Brockmann and Priesemann use adapted SIR models. Two publications by Meyer-Hermann do as well, as do works by such renowned epidemiologists as Sebastian Funk, Matt Keeling and Alex Vespignani. Do these models therefore all have a flaw?
Meyer-Hermann, who has been advising German government officials since May 2020, clarifies that no SIR modelling was used in his policy advice, but another, agent-based model, to which Kreck’s criticism would not apply. Dirk Brockmann from RKI says that he and the epidemiological community are fully aware of the limitations of the SIR model. But he underlines that plenty of research has been performed on the conditions under which SIR can still be used. SIR models, says Brockmann, do not take into account factors such as age structures, contact networks, mobility and changes in behaviour. But this was no reason to disqualify the model design as such, he says. For many questions, such factors would simply not play a role in the model results. Kreck, says Brockman, had "focused on a factor that is really very insignificant in most scenarios." Moreover, he adds, it was obvious that Kreck and Scholz ignored vast parts of the scientific literature.
Kreck and Scholz’s own corona model does not have the problem of the unrealistic distribution of infectivity because it uses the Kermack and McKendrick theory in a different way. Kreck has taken a deep dive into this problem over the past year. He is of course aware of the accusation that he has not read the literature. The question is which literature one means: the theoretical foundations, or the latest applications of the them. In March 2020, Kreck began reading through textbooks, recent papers, but especially Kermack and McKendrick's forgotten theory itself. Kreck’s son Benjamin, who lives in Hamburg and holds a PhD in bioinformatics himself, says his father camped out on his couch for weeks at a time, homeschooling the grandchildren during the day and perfecting his Covid model at night.
Kreck and Scholz developed a "discrete" mathematical model that directly incorporates the biologically detectable viral load and other real-world data that can easily be collected such as the daily number of new infections, the time between the onset of symptoms and quarantine, and the variable rates of contact. "We don’t have to fit our model using arbitrary parameters," says Kreck. "We model closer to reality from the start." No SIR model could take into account the real course of the viral load, he says, despite all corrective factors.
Jan Mohring, who developed his own Covid model at the Fraunhofer Institute for Industrial Mathematics in Kaiserslautern and corresponded with both Kreck and a host of other German modellers, confirms this general tendency of SIR. "The question is how to rebalance the model-induced overemphasis on infectivity at the beginning of the infectious phase," he says. Thanks to special funding, Mohring and his colleague Robert Feßler were able to develop a model that has a box-shaped background function and that reflects the effects of large scale testing, vaccination and of different age groups. In his experience, there has hardly been a profound exchange about the problems and strengths of different modelling approaches within the German scientific community.
Kreck wrote to Brockmann and Meyer-Hermann as early as April 2020. He says he has still not received an answer to his technical questions. An exchange of e-mails with the Göttingen group in autumn was fruitless. Looking through the messages, one gets the impression that highly specialised people were talking past each other, at first in a friendly manner and then increasingly annoyed. Kreck and Scholz were made to feel what they actually are on paper: two emeritus professors who do not belong to the established modelling community and have not yet published anything in epidemiological journals. At first look, they are hardly distinguishable from dozens of other older men who rushed into the public eye during the pandemic with a lot of self-confidence and questionable expertise. The duo both turn 74 this summer. This experience is also part of their pandemic year: they suddenly found themselves in the role of scientific outsiders who, in the eyes of many, seem per se suspicious because of their age, gender and a certain stubbornness.
Dirk Brockmann says that he and his colleagues received dozens of e-mails like Kreck’s last spring. "It is extremely rare that scientists from outside the field really contribute anything new," he says. At best, such contributions would represent long-known insights, says the RKI expert. Engaging with them would take time and was "simply not feasible in an urgent phase of the pandemic."
Odo Diekmann disagrees with the view that Kreck and Scholz’s work is banal or irrelevant. When it comes to the theory behind epidemiological modelling, Diekmann is one of the most recognised experts worldwide. He has written major theoretical papers and textbooks on the subject, and his institute at Utrecht University trained some of the researchers who are shaping the epidemiological discussion in Switzerland and the UK. Diekmann has read the work of Kreck and Scholz. He thanks them and quotes them in his own latest paper. One must take their model seriously, Diekmann says, because a discrete Kermack McKendrick approach like the one Kreck and Scholz developed had not been formulated in the epidemiological literature before. From the formal correctness of their model, however, Diekmann does not want to conclude that their criticism of the prognostic abilities of other modellers is also correct. Diekmann does not want to say anything more on this: it is not his field of research.
At some point, one has to go from the theory of modelling to the models’ application: mathematical skill flows into policy recommendation. Then it depends on what questions can be asked of a model and what answers it can give.
Two questions that can be answered well with Kreck and Scholz's model are: What happens if you maintain a certain contact regime over a long period of time and if you take action to reduce the overall contact rate by a certain percentage? Here, their model comparison shows that SIR models significantly overpredict infection rates. Logically, SIR modellers concluded that contact rates must come down even further. Moreover, Kreck and Scholz’s model calculates what is plausible even without a model but not easy to quantify: the effect of shortening the time to quarantine.
Kreck and Scholz’s model presents the time between infection and isolation separately. Various sources cite seven days in real terms for this period in Germany. According to Kreck/Scholz, if it had been possible to shorten this period to an average of six days, all other factors remaining equal, then the peak of the second wave of infection before Christmas would have been just under 6,000 new daily infections in Germany, and not around 30,000, as in reality. What that would have meant for hospital occupancy and death figures is not hard to understand.
In January, Kreck sent this "striking application" of their model to Karl Lauterbach. Lauterbach, from the SPD, a party in the governing coalition, is a simple member of parliament. But his training as an epidemiologist and relentless media presence have turned him into a shadow health minister. Lauterbach shared the following graph presenting the findings of Kreck and Scholz’s model on Twitter:
Because of this tweet, Kreck was invited to a NoCovid working group at the end of January when the initiative was just taking shape. To this day it remains an informal association of health practitioners and scientists – including Brockmann, Meyer-Hermann and the virologist Melanie Brinkmann – who are calling for a low-incidence strategy. Michael Meyer-Hermann says he had already discussed the acceleration of quarantine through a strengthening of the local health authorities with politicians in May 2020. The striking application of Kreck and Scholz's model was nothing new in January 2021, he claims. Everything Kreck wanted to contribute to the community had long been known to all involved, he says. "We didn't think about whether the shortest possible time to quarantine could help with pandemic control. We were assuming that. We were thinking about how to implement it concretely." Was he also able to quantify the effect, such as Kreck’s model was? Meyer-Hermann would not comment on this.
An e-mail invitation to Kreck in January said that NoCovid was developing a strategy "for the Chancellery2 (that there was an official mandate is disputed by those involved today). Kreck presented his model in several Zoom meetings. "We finally have the key," an enthusiastic e-mail to him read. But shortly thereafter, Kreck was excluded from the group – as he says, with an absurd justification. His diagramme could not be taken into account, because if it were included, it would soon end up on the Chancellery's table, an e-mail says. But his model had not yet been peer-reviewed. And he had to understand that only people who harmonised well could work together in such a group.
To this day, Kreck has not gotten over this setback. The fact that scientists whose models had also not yet been peer-reviewed offered exclusive advice to policy-makers is something that Kreck finds problematic, just like the fact that the federal government never convened an official scientific counsel that would have included mathematicians (Meyer-Hermann’s agent-based model, which integrates SIR-type models on local scales with mobility and age group analyses, was published in Mathematical Biosciences only on June 30, 2021.) Kreck sees the RKI forecasts from the end of March – predicting 100,000 new cases per day and an incidence number of 2,000 – as the result of bad modeling. These forecasts helped push through the "federal emergency brake", a policy that concentrated public health legislation – usually the domain of the 16 German states – on the federal level and enacted much stronger contact restrictions than would otherwise have been in place.
Kreck maintains that with the insights of other models, the German corona strategy could have been more nuanced and more effective. The idea seems plausible: Only what can be clearly quantified can be elevated to the top strategic priority. And with his model, shortening the time to quarantine would have clearly emerged as the strongest lever against the second and third waves.
Of course there are a lot of ifs, woulds and coulds in this reasoning. What is the point of debating last winter now? Isn’t Kreck overestimating the influence of modellers who were allowed to speak at ministerial conferences but whose recommendations were repeatedly ignored? Weren’t the health authorities completely overwhelmed by contact tracing anyway, meaning a shortening of the period before quarantine wasn’t even realistic?
There is a view in political communication that one should limit oneself to a few messages that are as concise as possible. In the German corona winter, this was: Everything that reduces contacts will save us. Turned into a moral principle, this meant that anyone who still saw friends was a nefarious egoist and a driver of the pandemic. Only in second place came the message: Quarantine as soon as possible. Anyone who feels the slightest symptom must isolate themselves immediately and radically.
There are a few videos of Kreck on YouTube. Most show him playing the cello, his passion for 63 years. A million people have watched a video in which he mathematically proves why a wobbly four-legged table can be stabilised with a slight rotation.
When Kreck was stuck in his house in Mainz in the spring of 2020, he started a video series called "Mathematics for Everyone". Maths is a special science in that it presupposes almost nothing except a few axioms. After the 1 comes 2, and the distance to the next number is always the same. Everything more abstract is built from such basic elements. "A theorem, once correctly proven, is valid from eternity to eternity," Kreck says in his video and then pauses, making his statement sound even more biblical. Kreck's father was a Protestant theologian, and Kreck himself studied theology for four years after receiving his doctorate in mathematics in 1972. For him, it’s a matter of principle: either the pandemic models are clearly formulated and conclusively proven, or they are not.
I talked to Kreck for almost two months for this article and exchanged dozens of e-mails with mathematicians. There is hardly a doubt that he is right in his theoretical points. Diekmann et al. point out that Kreck and Scholz’s discrete modelling approach is mathematically innovative. It uses data inputs that are easy to collect and it is structured in such a way that it can be implemented with simple computer tools. Kreck would never call himself a modelling expert and considers his own contribution to be mathematically rather simple. The problem with SIR, he says, is too big to let go. "I don't deny that you could get acceptable approximations with SIR in the beginning. But as soon as the initial data changes, for example because a lot of people are recovering or get vaccinated or because contact behaviour changes a lot, SIR gets totally bogged down."
How Kreck and Scholz's model, had it been heeded, would have actually fared in the public arena remains speculation. What Kreck does not see, and perhaps cannot see, is the polarising dynamic which affects every contribution to the pandemic debate. In terms of public discourse, the coronavirus produced the exact opposite of a new expertocracy: people’s screen time exploded, social media became even more central to the formation of opinions than before. Covid strategies were discussed on Twitter, in a medium made for abbreviated, impulsive communication.
Kreck is not on Twitter. He would never comment there, he says, nor does he know any serious mathematician who ever would. For a full year, he tried to talk to crucial experts who themselves had to improvise and read up on epidemiology and was simply rejected. How this could happen to a specialist of his stature is beyond me. But pandemic modelling is a field of applied mathematics. There is a reason that Odo Diekmann would never say anything about data and forecasts. For that you need good empirical data, a sense of the instability of human behaviour, a perhaps more pragmatic approach to a science that by definition wants to eliminate the grey area between "right" and "wrong". "Nothing is harder than mathematics," Kreck said in one of our recent conversations.
Christian Althaus, an epidemiologist at the University of Bern with whom I spoke at length for this article, doesn’t think that Kreck's submitted paper contains any new insights on controlling the pandemic. He also points out that it ignores most of the existing literature on the effectiveness of isolation and quarantine strategies. Althaus finds it worrying that during the past year, more and more scientific papers have become the subject of public debate whose peer review had not yet been completed. "Whether a paper contributes to a better understanding of the pandemic is not decided by the media, but by the normal procedures of science. If someone has a valid point, they will make it through peer review, publications in prestigious journals and citations. This process is the same for all researchers."
Does peer review allow the most important viewpoints to rise to the top even under the extreme conditions of mass production during the pandemic? Is a journalistic article like this one allowed to report the rifts of an ongoing scientific dispute? Would it have been better to not write about two outsiders like Matthias Kreck and Erhard Scholz at all, despite their arguments and mathematical merits?
I believe that such texts need to be written and that the inner workings of science be made visible. The pandemic has given science a new political role. Measures were adopted because scientists presented evidence of their efficacy. After the first wave of infections in spring 2020, many Germans believed that their country had entered an expertocracy of a whole new quality. One year and a series of failed state premier conferences later, the public knows that this is not the case. The failure of a corona policy based on scientific expertise alone is symptomatic of a dilemma that every politics of truth must face. Political speech always needs both truth and persuasion. If you are right but cannot make a convincing case for yourself in the public arena, all you are left with is the argument of pure technocratic authority. In many cases, such authority can be secured by collegial consensus and institutional credentials. But it is not persuasive in the political arena.
Every argument that claims scientific evidence for itself ultimately runs into this problem: At the end of the day, it simply has to be believed. This does not make it easier to distinguish good from bad, or well-founded from abstruse criticism of accepted scientific opinion. Corona experts, who in talk shows and social media could not help but engage with dissenters, those outside their field or even deniers, were therefore in a doubly difficult position. Unlike in a professional dispute, they could not simply end nonsensical discussions by referring to accepted basic knowledge in their field. But when dissenting opinions did make sense – and this is certainly the case with Kreck and Scholz – many apparently considered it too risky to bring the full complexity of the controversy out into the open. To do so weakens the evidence claim of science. And it disproves the myth that the scientific community only speaks with one voice on the issue of corona.
Tobias Haberkorn is a literary scholar and translator. His book Das Problem des Zuviel in der Literatur (The Problem of Too Much in Literature) is forthcoming at lmverlag Berlin.
Matthias Kreck was born in 1947. He earned his PhD at Bonn University in 1972 and studied theology for another four years. He was professor of mathematics at the Universities of Mainz, Heidelberg, Bonn and Frankfurt am Main. From 1994 to 2002, he was the director of the MFO Oberwolfach Research Institute for Mathematics. His ongoing research focuses on algebraic topology.
Erhard Scholz was born in 1947. He studied mathematics and physics in Bonn and Warwick and earned his PhD in 1979 with a study on the history of mathematics since 1800. He is one of the editors of the works of Felix Hausdorff and has been a professor at the University of Wuppertal, where he conducts research at the The Interdisciplinary Center for Science and Technology Studies, since 1979.
A selection of sources for this article:
Kreck/Scholz: Research notebooks.
Kreck/Scholz: Back to the roots: A discrete Kermack-McKendrick model adapted to Covid-19, arXiv:2104.00786.
Diekmann/Othmer/Planqué/Bootsma: On discrete time epidemic models in Kermack-McKendrick form, medRxiv 2021.03.26.21254385.
Meyer-Hermann et al.: Assessment of effective mitigation and prediction of the spread of SARS-CoV-2 in Germany using demographic information and spatial resolution, 10.1016/j.mbs.2021.108648.
Maier/Brockmann: Effective containment explains subexponential growth in recent confirmed COVID-19 cases in China, 10.1126/science.abb4557.
Dehning et al.: Inferring change points in the spread of COVID-19 reveals the effectiveness of interventions, 10.1126/science.abb9789.