Back to the roots: A discrete Kermack-McKendrick model adapted to Covid-19

Nearly a decade ago Breda/Diekmann et al. made the sharp remark that even experienced experts seem often to believe that Kermack and McKendrick’s famous paper of 1927 [14] is just about the standard SIR or SEIR model described by the well known ODE system [4, p. 105]. Taking this criticism seriously, the present paper proposes a Kermack-McKendrick type model adapted to Covid-19, which differs from the widely used S(E)IR models and which in our eyes has a greater plausibility and applicability than the latter.

For doing so we work with a discrete time parameter. After all, people are not participating in the spread of the disease during every part of the day, e.g. when they are asleep. And also the data are communicated on a daily basis. In addition, discrete models are easy to program. So we adapt the Kermack-McKendrick approach and replace functions depending on a continuous time $t$ by a functions depending on integers $k$.

We will describe the model in the next section and then explain the role of its parameters (sections 1 and 2). Then we describe how one applies the model starting from the empirical data and demonstrate the result for Germany (sections 3 and 4). We give a striking application showing what happens if people are sent earlier to quarantine or hospital (section 5), before we discuss how vaccination, the rise of new mutants, and mass testing can be implemented in the model (section 6). Finally, we add some remarks on the relationship between the reproduction numbers as they appear in our model to the ones given by the Robert-Koch-Institut for Germany (section 9). At the end we discuss some aspects of our model.

If one wants to model an epidemic one has to find the appropriate parameters. If, for the sake of argument, we remove the dark sector for a moment, there are two types of parameters in the model: parameters that are of a medical nature such as the duration of the exposed people period $e$ and the infectivity function $\gamma(k)$, and those that can be influenced by politicians, such as the duration $p_{c}$ between the onset of infectivity and the beginning of quarantine or isolation, or the daily contact rate $\kappa(k)$. Coming back to the dark sector, one has to know in addition the ratio $\alpha$ between the later recorded people and the rest, and the duration $p_{d}$ of the infectious phase of the dark people. Amongst these parameters we assume that $e$ and the strength function $\gamma$ are essentially unchanged during the course of the epidemic, as well as $p_{c}$, $p_{d}$ and even $\alpha$ if we abstract from changes in the ratio due to rising unspecific testing. The parameter which definitely changes over time is $\kappa(k)$.

Thus the choice of $\gamma(k)$ is essential for obtaining a realistic model for Covid-19. This is a delicate point since data from which one can derive $\gamma$ are not easy to access. From our point of view $\gamma$ reflects medical data, more precisely the viral load and the probability of successful cell culture isolation (in the following “culture probability”). From the point of view of the paper by Sofonea et al. [23], which we shortly discussed above, $\gamma$ is replaced in their model by the serial time distribution $\zeta$. So it is useful to compare them. In addition we will discuss a third way to obtain information about $\gamma$, which is just based on observations when the force of infection starts to leave the level $0$, when it obtains its maximum and when it decreases to $0$ again.

We found two studies [24], [13] by virologists which give very similar values for $\gamma$. Simplifying a bit, this leads to the following picture. The length of $e$ is approximately $2$ days, so $\gamma(j)=0$ for $1\leq j\leq 2$. Within another 2 days it reaches a maximum close to the moment where symptoms show up, before it starts to decrease, slowly at the beginning and then faster until, after about further $10$ days, it reaches a value where people are practically no longer infectious. This indicates that $p_{d}$ is $11$. Curves which show this increase and decrease for a small number of different infected people can be found in [24]. They allow one to infer (to “guess”) a typical course of the strength of infectivity $\gamma(k)$ for Covid-19 from the data on culture probabilities depending on the time after infection [24, fig. 1.f]. A recent evaluation of the data of $25,381$ positive subjects comes to a similar course of positive culture probability [13, fig. 4.c]. Using the information of these studies we establish the following table:

We use a third way to obtain information about $\gamma$ which just uses the time span until the maximum is reached and the time needed to fall down again to zero after a short phase of staying close to the maximum. If one requires that such a transition is modelled by a function which is arbitrary often differentiable, and at the interval ends all derivatives are zero, one can use a natural standard candidate frequently used in mathematics as transition function. Normalized to the interval $[0,1]$ it is:

where

Using this function and the corresponding function for the decrease and adapting both to the data curve for Covid-19 one obtains another candidate for $\gamma$ expressed by

which is rather close to that obtained by the virological studies mentioned above, as figure 1 (top right) shows. Summarizing we come to the conclusion that, although it is impossible to derive $\gamma$ (or $\zeta$) precisely, there are strong reasons to believe that our choice for $\gamma$ is not far off. We will later compare our model with the standard SIR model. Here we only compare our input function $\gamma$ fig. 1 (top) with the input function for the latter (bottom).

We remark that this is the point where one can see why in [4] the authors speak about an“incessant misconception” if people apply the SIR-model indiscriminately (and the situation is not so different for the SEIR model). The function $\gamma(t)=\lambda e^{-\mu t}$ – at least for Covid 19 – has nothing to do with the medical data observed by the virologists. This may have considerable consequences for conditional predictions; see section 6. Other possibilities for adapting the model to Covid-19 specific data can be found in the literature. For example, Ferretti et al. try to reconstruct the distribution of “age” (time after infection) dependent generation time from the analysis of data of 40 source-recipient transmission pairs documented in the literature by May 2020 [8]. They conclude that among several distributions used for comparison the Weibull distribution with shape parameter $k=2.826$ and scale $\lambda=5.665$ fits best. The choice of the parameters suppresses the tail of the distribution for $t>15$ strongly enough to distinguish it qualitatively from standard SIR distributions. It is difficult to make sure that the contact data used for the evaluation stem from comparable conditions of the social contact regime. It is not clear to us whether this condition is satisfied. In our approach, on the other hand, we rely on the medical data for the $\gamma$-function and include the effects of social (and seasonal) transmission conditions by a proportionality factor $\kappa(t)$ determined from the data of recorded newly infected people (see next section).

A model using the biological input function $\gamma$ above will be called a K-McK-model adapted to Covid-19, or for short an adapted K-McK-model. To specify such a model one has to estimate three further parameters $p_{c}$, $\alpha(k)$ and $\xi$.

The next parameter one has to chose is $p_{c}$. It is difficult to estimate and depends on the country and the action mode of the health institutions. For Germany it is estimated to lie between 5 and 8; where for the year 2020 officials of the health system estimate it in the upper range. So we work with $p_{c}=7$. There are data that indicate that the ratio $\alpha$ of recorded new infections among all new infections was about $0.25$ for Germany at the beginning of the epidemic, but changed to approximately $0.5$ during the course of the year, probably due to the increasing test rates [12]. Since the influence of the dark sector was negligible in the first months we have chosen $\alpha=0.5$ for the whole period. Finally, it is difficult to estimate the relative strength of infectivity $\xi$ of the unrecorded infected people from empirical data, so we take $\xi=1$, i.e., we assume that people in the dark sector (including the asymptomatic ones) are basically as infectious as those in $P_{c}$.

In this paper we start from the data given by the Humdata project of the Johns Hopkins University.¹¹1https://data.humdata.org/dataset/novel-coronavirus-2019-ncov-cases More precisely, we estimate the values for $S(k)$ from the listed confirmed cases and the ratio $\alpha$. In this step the estimated time translation $e+p_{c}$ between the times of infection and recording has to be taken into account. Moreover, there are weekly fluctuations of the values of newly recorded cases (the first differences of the data lists of confirmed cases) due to weekends where data taking and transmission is usually slowed down considerably. Since we are interested in the central development of the epidemic we replace the daily values of the newly recorded by (centered) 7-days averages and also the values of the susceptible people $S_{7}(k)$.

In the next step we compute the daily contact rates $\kappa(k)$ according to the observation above. This allows one to give the tautological model picture of the values of $S_{7}(k)$ according to the observation above; but of course this is not what we want. We therefore investigate the values of $\kappa(k)$ and look for periods where they are more or less stationary and can be approximated by a constant. Such time intervals may be interpreted as periods where the behaviour of the population is roughly the same and no crucial change of the virus occurs. The transition times between such stationary phases seem to characterize periods in which the contact behaviour of the population changes. This is often the result of political measures (non-pharmaceutical interventions) reducing the number of contacts or allowing more contacts, which in many cases may lead to new levels where the contact rates can again be approximated by a constant. Besides this, eyeballing will indicate further levels of near-constancy that then need to be justified. For example, the explanation might be that the changed contact rate is the result of a climate change from summer to winter where people meet more in closed rooms, or the other way round. It could also be that the infectivity of the virus changes (e.g. with the rise of a more aggressive variant). In our view the most important criterion for the applicability of a Kermack/McKendrick type model to a real-life epidemic boils down to the question of whether or not replacing the contact rates during periods where they stay nearly constant by their averages, or a value close to it inside the respective $\sigma$-interval, leads to a good approximation of the data curves.

In the case of Germany, one observes seven rather obvious periods between March 2020 and mid-January 2021 in which $\kappa$ can be approximated by a constant, ignoring fluctuations (see fig. 2). Some of these constancy intervals can be interpreted as results of interventions, others not: There was a series of three interventions in March 2020 resulting in the first constancy interval from ($t_{1}=$) March 24, 2020, to ($t_{2}=$) April 26, a rather short interval. Around April 26 the interventions were reduced and the result is a long period until ($t_{3}=$) July 3. Already in this phase people were presumably getting more careless, the vacation period started and contacts increased, which lead to the third period, lasting until ($t_{4}=$) September 27. In this phase the mean reproduction rate rose noticeably above 1. With the beginning of cooler temperatures and life moving more to closed rooms, the reproduction number rose to about 1.5 in the fourth interval until ($t_{5}=$) October 31. This provoked new containment measures, at first at Oct. 16 and Nov. 2 which are reflected by the fifth interval lasting until Nov. 26, in which the daily new infections went down. They started to rise again, probably because of early Christmas shopping on ($t_{6}=$) Nov. 26. A partial lockdown (schools, restaurants, cultural activities) on December 16 (our $t_{7}$) brought the reproduction rate below 1 and resulted in, on average, falling numbers of new infections. This period lasts until the end of the data considered here in mid-January (Jan. 15).

The model values of the contact rates $\kappa_{j}$ in the intervals $[t_{j},\,t_{j+1}-\Delta_{j+1}]$ ($1\leq j\leq 7$), with varying durations $\Delta_{j}$ of the transition periods, are chosen close to the mean values, with small deviations inside the $\sigma$-interval allowed if this improves the fit considerably. The model $\kappa_{j}$ and the (effective) model reproduction numbers $\rho_{j}$ at the left edge of the intervals are given in the following table:

Using this input for our recursion formula determines the model numerically. The values for the recorded daily new infections $Q_{\text{new}}$ are shown in fig. 3, the numbers of recorded recovered people $R_{c}$ and of the recorded actual infected people $Q$ in figur 4. All three of them show a convincing agreement between the model and the empirical data. The overshooting of the data curve of $Q(t)$ over the model in November and December 2020 seems to be due to a higher mean time of recorded illness due to an increased number of hospitalized and severe cases.

In this section we compare the frequently used standard SIR model with our $\gamma$-K-McK model. Since in the standard SIR model only the three compartments S, I and R occur we reduce the number of compartments in the $\gamma$-K-McK model by setting $e=0$, since there is no compartment of exposed in the standard SIR model and $q=0$, since no quarantine is built in. Then we have to prolong $p_{c}$ to the length of the interval given by the support of $\gamma$. In addition we remove compartment $P_{d}$ since there is no dark sector built into the standard SIR model. To distinguish the functions $S$, $I$ and $R$ of the two models we add the suffix $\mathit{st}$ (like standard) to the functions in the standard SIR model: $S_{st}$, $I_{st}$ and $R_{st}$.

If one looks at models based on different input functions $\gamma$ it was observed by Feßler [9] and Diekmann et. al. [7, sec. 7], that there are differences in the long term behaviour if one assumes constant contact rates, even if one is only interested in the number of actually infected people. For short periods the difference is negligible since in all models this is an exponential growth which can be adjusted as long as contact rates don’t change. But if non-pharmaceutical interventions are imposed contact rates jump. Thus it is interesting to compare the two models in this situation.

To do this we consider the initial course where $s$ is approximately equal to $1$ and try to adjust the model parameters to achieve a good fit of the model curves. So given a function $\gamma$ with corresponding $\gamma$-K-McK model we can ask whether there are choices of the parameters $\alpha$ and $\beta$ for the standard SIR model so that the model curves essentially agree for some while ($S_{st}^{\prime}=-\alpha I_{st}S_{st}$ and $R^{\prime}_{st}=\beta I_{st}$). In both models the contact rate enters, in the standard SIR model it is proportional to $\alpha$. The role of $\beta$ is often that $\frac{1}{\beta}$ is interpreted as the duration of the infectious period. In our notation this corresponds to $p_{c}$. But this parameter is not so sharp, for example one can prolong $\gamma$ by a very small constant function so that the support grows but the dynamics determining $S$ is not so much changed. Thus equating $\frac{1}{\beta}$ to $p_{c}$ is somewhat problematic.

For adjusting the models one might therefore prefer to use the mean generation time $\tau=\frac{\sum_{i}\gamma_{i}i}{\sum_{i}\gamma_{i}}$ instead of the duration of the infectious period.²²2We thank an anonymous referee for this proposal and Stefan Müller for enlightening discussions on this point. If in addition one insists that $N-S$, which approximately is an exponential function at the beginning, agrees for both models then for each $\gamma$-K-McK model one can derive $\alpha$ and $\beta$ such that the mean generation time and the growth rate of $N-S$ or, equivalently, the growth rate of $-S_{st}^{\prime}$ and $I_{new}$ of the K-McK model agree at the beginning.

With this choice of the parameters of the SIR model we observe in numerical tests that a change of the contact rate by factor $0.6$ after 50 days shows a very similar course of $-S^{\prime}_{st}$ and $I_{new}$ of the K-McK mdoel for some time (see fig. 6 left).³³3Parameters: $\gamma$-function see sec. 3, $p_{d}=11,\,\kappa_{1}=0.228$, $\kappa_{2}=c\,\kappa_{1}$ with $c=0.6$, $\tau=4.47$; initial exponential growth rate $\eta_{1}=\alpha_{1}-\beta=0.0688$, $\beta=\tau^{-1}=0.224$, $\alpha_{2}=c\,\alpha_{1}$; $I_{0}=1000,\,N=80\cdot 10^{6}$. Here we chose the parameters $\gamma$ as in the table in section 3 and $p_{c}=11$, the length of the support of $\gamma$. This might be a general fact for reasonable kernels and moderate growth rates as Stefan Müller has pointed out to us.

On the other hand, if one wants to control an epidemic one needs to know more than the number of newly infected, which corresponds to $-S^{\prime}$, one also wants to know the number $I(k)$ of actually infected (a certain portion of which needs intensive care). So one may ask whether with the above choice for $\alpha$ and $\beta$ besides the curves for $-S^{\prime}$ also the curves for $I$ are close to each other for both models. Numerical tests show that this is not the case, the curves differ drastically already in the initial course (fig. 6 right).

If one wants to control both $-S^{\prime}=-(I^{\prime}+R^{\prime})$ and $I$, or equivalently $I$ and $R$, at least in the initial course and then check what happens after changing the contact rates, one asks whether there are values for $\alpha$ and $\beta$ such that during that time both curves agree approximately. This is of course possible, since both curves are approximately exponential functions with same growth rate. Then the question is, what happens under a change of the contact rates. Again we have tested this numerically by changing the contact rate at day 50 by a factor of $0.6$ and found that then both, $I$ and $R$, diverge drastically (fig. 7).⁴⁴4Parameters of K-McK model and initial exponential growth rate see fn. 3; $\beta=0.223$, $\alpha_{1}=\beta+\eta_{1}$, $\alpha_{2}=c\,\alpha_{1}$, $I_{0}=1000,\,N=80\cdot 10^{6}$.

The upshot is: If one is only interested in the course of $S$ or $-S^{\prime}$ numerical calculations indicate that one can choose $\alpha$ and $\beta$ such that as long as $s$ is nearly equal to $1$ both curves are similar even after changes of the contact rates. But the prize one has to pay for this is that then the curves for $I$ diverge even in the initial phase. Alternatively one can chose $\alpha$ and $\beta$ such that in the initial course $-S^{\prime}$ and $I$ are nearly the same, but then both curves diverge after changes of the contact rate.

So our conclusion is that – as expected – the choice of $\gamma$ plays a strong role. Needless to say that we are convinced that a more realistic guess like we made in this paper leads to a better model. This is also supported by our observation indicated at the end of section 4 that the adapted $\gamma$-K-McK model behaves well with regard to all three functions over the range of the epidemic in Germany with several change points. We expect that the same holds if one replaces the medical function $\gamma$ by the generation time distribution $\zeta$ as in the paper by Sofonea et al. [22]. Also a K-McK model with a Weibull distribution with a shape parameter which suppresses the tail of the distribution sufficiently strong after roughly 12-15 days would not behave too differently.

We strongly doubt that the SIR model can achieve an adequate modelling of all three curves over the whole time of the epidemic, even with complicated manipulations. The more convincing one can relate $\gamma$ (or $\zeta$) to the reality of Covid 19 the more reliable the $\gamma$-K-McK model is.

In this section we extend our model step by step to include the effects of vaccination, new mutants and mass testing. Please note that all our functions $S(k)$, $E(k)$ etc. have new definitions in this section and should not be mixed up with the previous formulas.

Once the past and the expected future rates of daily vaccination are known or estimated, it is simple to extend the model to take vaccination into account. If $V_{\text{new}}(k)$ is the number of vaccinated persons on day $k$ and $V(k)$ the total number of immunized by vaccination on day $k$,

equ. (1) turns into

The recursion equation (9) has to be amended accordingly by the additional summand $V_{\text{new}}(k)$ on the right hand side (see the Theorem below in the special case $\zeta=1$ and $\beta=0$), and the terms $\mathbb{P}_{c}$ and $\mathbb{P}_{d}$ have to be adapted:

Making provision for the appearance of a new mutant of the virus with stronger or weaker infectivity is slightly more involved. If we assume that the function of infectivity changes only up to a time dependent scalar factor $\zeta(k)$, with $k$ the epidemic time scale, $\gamma(j)$ (as above $j$ denotes here the day after the onset of infectivity) has to be replaced by

In this way the coefficient of the right hand side of eq. 9 is enriched by the factor $\zeta(k)$ (see eq. 27).

Now we consider (in addition to vaccination and new mutants) reliable unspecific daily mass testing of a specified subset $M_{\text{t}}$ (with cardinality $N_{\text{t}}$) of the total population $M$ (with cardinality $N$). Such testing may be realistic if based on next generation sequencing as proposed in [16, 21]. We denote the complement of $M_{\text{t}}$ by $M_{\text{n-t}}$ and assume that the decomposition $M=M_{\text{t}}\cup M_{\text{n-t}}$ is kept fixed over the whole period of testing. To simplify the considerations we assume that testing takes place every day and is error free (sensitivity 100% and specificity 100%). If in applications the testing starts later, or takes place on selected days a week only, one has to modify the algorithm appropriately. We call $\beta:=N_{\text{t}}/N$ the test ratio. Assuming that the tests are reliable implies that there is no dark sector within $M_{\text{t}}$. Moreover, we assume that the test is sensitive from day $l$ of the period of infection on. A positively tested person is assumed to be sent to quarantine on the following day.

We further assume perfect contact mixing between $M_{\text{\,t}}$ and $M_{\text{n-t}}$. If we denote the number of newly infected at day $k$ amongst people in the tested group $M_{\text{t}}$ by $E_{\text{t,new}}(k)$ and in the non-tested group by $E_{\text{n-t,new}}(k)$, we have:

where $E_{\text{new}}(k)$ is given by formula (19).

Now we count the number $P_{\text{t}}(k)$ of persons from $M_{\text{t}}(k)$ who are effectively propagating the virus under the test regime on day $k$. Since it takes at most $l$ days until a newly infectious person is either positively or negatively tested we obtain:

This formula has to be amended for weekly tests cycles in which tests are not taken daily but only on specified week days.

We define $Q_{\text{t}}(k)$ as the number of people from $M_{\text{t}}(k)$ who are in quarantine or isolation on day $k$. Observing that the quarantine starts the day after positive testing we find for the number of people newly sent to quarantine on day $k$:

The corresponding number for the people who are not tested is

where again $E_{\text{new}}(k)$ is given by formula (19).

Next we define

where $E_{\text{new}}(k)$ is given by formula (19).

This number counts only the propagating people who are tested (those in $M_{\text{t}}$); the corresponding number for those who are are not tested (those from $M_{\text{n-t}}$) has to be added. It is essentially given by 20 and 21, but the expressions have to be multiplied by the factor $(1-\beta)$, i.e. the percentage of non-tested.

This allows to formulate our final result including the effects of vaccination, new mutants and mass testing as described above

In this case the method for determining the empirical values for the $\kappa(k)$ described in the Observation above finds the values of the product $\kappa(k)\zeta(k)$ rather than $\kappa(k)$ itself. The latter can be isolated by correcting for the factor $\zeta(k)$, which is to be determined from the data. In the following we apply this theorem and study the effect of mutants, of vaccination, and a scenario of mass testing for Germany.

We start with considering the rise of the variant B.1.1.7 in Germany. It was first identified in Germany in late December 2020 and the sequencing data of [20] show that the transition ends in early April 2021. In the German case, we know that the share of infections carrying the mutant B.1.1.7, determined by genetic sequencing, rose approximately in a linear progression from roughly 6% in calendar week (CW) 4, the last week of January 2021, to approximately 70% in CW 10 and 88% in CW 12 [20]. We may safely assume that about 2 or 3 weeks after CW 10, i.e. in early April, B.1.1.7 will be dominating the infections observed in Germany. Given this one has to find the factor $c$ by which the infectivity increases. This could either be determined by medical studies which we didn’t find or by epidemiological studies. We found two such studies (we would like to thank the referee for these references) in England and France the increase of infectivity of B.1.1.7 has been estimated to be characterized by a factor $c$ with $1.43\leq c\leq 1.9$ [5] or $1.52\leq c\leq 1.69$ respectively [11]. In agreement with these data we may safely assume that also in Germany the final scaling factor is constrained by $1.4\leq c\leq 1.9$.

Using the transition function (17) introduce above we model the rise of the factor of infectivity $\zeta$ in dependence of $c$ by the function

with $d=t_{a}-t_{b}$, where $t_{a}$ parametrizes the beginning of the spread of B.1.1.7 in Germany (December 25, 2020), $t_{b}$ the date of final dominance (April 20, 2021), and $c\in[1.4,\,1.9]$. The case $c=1.5$ is shown in fig. 8.

To constrain $c$ more narrowly we investigate the $\kappa(k)$ in early March, determined from empirical data as described in the Observation of sec. 3 in dependence of the choice of $c$. The case $c=1$ is shown for comparison (fig. 9, top left); it would correspond to the case of no new mutant. The resulting rise of $\kappa(k)$ would be completely implausible as no relaxation was announced by the German authorities and the majority of the population accepted the restriction measures issued in December 2020. In early March a partial opening of shops was permitted and a slightly disorganized discourse started on possible relaxation of restrictions. This is sufficient to explain a moderate rise of $\kappa$ at the beginning of March by 5 - 10 %, as we find it for $c\approx 1.4$ (fig. 9, top right) and $c\approx 1.6$ (bottom left) . The hypothesis $c\approx 1.9$ appears again implausible, as it would imply that the early March partial opening had no effects on the mean contact rate or would even be correlated with a small decrease of $\kappa$ (fig. 9, bottom right).

Summing up, we obtain a consistent and plausible choice for the beginning of the transition at December 25, 2020, a duration $d\approx 100$ and an up-scaling factor $1.4\leq c\leq 1.6$. In the following we work with $c=1.5$.

If we now factor in the available data on vaccinations in Germany,⁵⁵5https://impfdashboard.de/static/data/germany_vaccinations_timeseries_v2.tsv we can analyse the impact of vaccination on the course of the epidemic in Germany. Figure 10 shows the numbers of daily new infections (7-day averages); red the JHU data, black dashed the model reconstruction taking the vaccinations into account, pink dotted the model values under the counterfactual assumption that no vaccination had taken place in early 2021.

In fig. 11 one can inspect the change of the effective reproduction numbers for Germany in early 2021 (data available until end of March) and a conditional prediction of the future months April and May under the assumption of no essential changes in the contact rates (orange – empirical values, black dashed and dotted model with vaccination, pink dotted model without vaccination). Under this assumption the vaccinations will push the effective reproduction rate below the critical value 1 at the turn from April to May 2021. If no vaccination were to take place this would happen only in late May – after a catastrophic increase in the number of daily new infected people was followed by herd immunity.

We would like to demonstrate the strong effect of vaccination by showing the catastrophic rise of daily new infections without vaccination, assuming no essential change of contact behaviour of the German population (in comparison with March 2021). During April the number of daily new infections would rise above 100.000, see fig. 12, dotted pink curve. But because of the ongoing vaccinations our model lets us expect the local maximum of the third wave for daily new infections at the turn from April to May 2021 with values (7-day averages) about 25 000 (black dashed; dotted are the boundaries of the 1 sigma interval for the empirical $\kappa(k)$ in the last constancy interval). Note that the data available on April 09, 2021, are subject to an artificial drop of $Q_{\text{new}}(k)$ induced by Easter (April 2–5); this leads to additional uncertainties in the data evaluation and conditional predictions. In any case the figures 11 and 12 demonstrate the important role of vaccination in turning the tide of the epidemic in Germany during the months April and May 2021.

Finally we apply our theorem to study the effectiveness of mass testing techniques, e.g., by new generation sequencing proposed in [16, 21]. The question is how fast testing on a daily basis of a certain subset $M_{\text{t}}$ can be expected to suppress an epidemic of the Covid-19 type. We model here an ideal mass test (sensitivity and specificity both 100%) of a fixed subset $M_{\text{t}}$ containing 60% of the total population on a daily basis, starting on October 15, 2020. Moreover we even assume (counter-factually) that the contact regime of early October is not being changed by further restrictive measures later in the year, while the effects of vaccinations and the rise of the new mutant B.1.1.7 since late December 2020 have been included.

In other words, we use the right hand side (rhs) of eq. 9 for $t<$ Oct 15 with the contact coefficients of the German model and the rhs of eq. (27) with constant contact coefficient $\kappa_{4}$ (see sec. 4). Using the characteristic function $f_{\text{test}}(t)=\chi[t,J]$ of the interval $J=[t_{a},\infty)$ the recursion is then

Figure 13 shows the suppression of $Q_{\text{new}}(k)$ which is to be expected in terms of our model calculation.

It demonstrates a high in-principle effectiveness of such an approach, although in an idealised scenario. Under a test regime similar to the one above the rise of the new mutant would only lead to a slight increase of registered new infections starting in early March 2021. But it would be held below roughly 1000 by the increase of vaccinations in April and May (with actual data on vaccination for Germany). We have checked that for a moderately reduced reliability of the test with a ratio of 0.9 recognized infectious persons (sensitivity 90%) and daily testing the picture would not change significantly (the number of false positives for genetic screening tests is expected to be extremely low; specificity 1 with extremely high precision).

We would like to remark that the effect of mass testing has a similar effect as the shortening of the time between the offset of symptoms and sending people to quarantine or isolation as discussed in our striking application. Mass testing is a particular effective way to shorten this period for a certain part of the population.

The Robert Koch-Institut (RKI), Berlin, publishes the daily data for an epidemic in Germany and determines the daily reproduction numbers from these data. The RKI calculation uses a method described in [2] for a stochastic estimation of the numbers of newly infected, called $E(t)$, from the raw data of newly reported cases,. The calculation of the reproduction numbers works with these $E(t)$ and assumes constant generation time and serial intervals of equal lengths $\tau_{g}=\tau_{s}=4$ [19]. Two versions of reproduction numbers are being used, a day-sharp and therefore “sensitive” one $\rho_{rki,\,1}(t)=\frac{E(t)}{E(t-4)}$, and a weekly averaged one,

which we refer to in the following simply as $\rho_{\text{RKI}}(t)$.

The paper remarks that the RKI reproduction numbers (“$R$-values”) $\rho_{\text{RKI}}(u)$ indexed by the date $u$ of calculation refer to a period of infection which, after taking the incubation period $\iota$ between 4 and 6 days into account, lies between $u-16,\ldots,u-8$ (with central day $u-12$ in the interval). The reproduction numbers calculated in the adapted K-McK approach are close to the RKI reproduction numbers. The main differences lie in the usage of different raw data bases (RKI versus JHU) and the adjustment of the raw data (stochastic redistribution $E(t)$ versus sliding 7-day averages $\hat{Q}_{new,7}$). After a reasonable time shift the agreement between the reproduction numbers of the adapted $\gamma$-K-McK approach and the $\rho_{\text{RKI}}(k)$ are close. The stochastic smoothing of the RKI data seems to lead to smaller amplitudes of the fluctuations which otherwise are in striking coincidence (fig. 14).

We formulate four points with regard to the adapted K-McK-model, which at present seem the most important ones to us.

(1) A model has to be useful for analysing an epidemic on the basis of the available data. The most important datum for this purpose is the number of newly infected people each day; but one is interested in other numbers also, such as the number of actually counted as infected or those in hospital, or how many people are actually infectious or how many people are already immune. In particular one is interested in the daily reproduction number (see formula (2). All these numbers are given by our model. Knowing these numbers is important information for politicians as well as for citizens. For politicians such numbers provide a basis for thinking about regulations and for citizens this information is a helpful guide for how to behave.

• The present model is a useful tool for determining key figures of an epidemic such as the reproduction number and the number of people in quarantine or hospital.

(2) A reliable model can do more. The optimum would be that it allows one to predict the future. But in the strict sense this is impossible, since important parameters such as the contact rate changes depending on the behaviour of the population. If one knows the development of the epidemic in the past (in terms of a model) and assumes that the contact rate is not changing drastically, it is possible to extrapolate the development for some time. We indicate how this may be done by looking at the situation in Germany at the beginning of April 2020. In a series of preliminary steps a lockdown was imposed on March 22. Only a few days later it pushed the reproduction number below 1, and about a week later down to about $\rho=0.7$ (see fig. 2). It fluctuated around that number for roughly a month. This was observable in the data about a fortnight later, so that around April 10 the number was stable already for about a week. If at that moment one were to assume that the behaviour of the population is not changing very much, one could use the model to predict the future for the next weeks quite successfully until the contact rate was changing substantially. Figure 3 shows how well this worked until the end of April when the restrictions were partially relaxed and the reproduction number rose. In other words:

• The adapted K-McK-model is a reliable tool for extrapolating the development in times where the reproduction number is stable and there are reasons to assume that this stays so for a while.

(3) At the beginning of the epidemic there was no evidence of what the effect of interventions would be. But over the time there was a chance of using the model to observe the effects and, if possible, to draw realistic conclusions. A quantitative estimate of the effectiveness of non-pharmaceutical interventions in terms of the change of contact rates (or reproduction numbers) determined in the model framework is a delicate point. But keeping this in mind, a realistic model such as the adapted K-McK-model may be used heuristically for deriving useful information about how certain interventions will influence the development by comparison with other ones in the past which took place under similar conditions. Like any realistic model, the present one may be used as a tool for learning how certain interventions under specified conditions (such as climate) influence the development. This was also tried in [6] for the three interventions in Germany in March 2020, claiming a direct relation between each of these and a new level of the reproduction number. In the light of the data evaluation represented in figure (2) we are not able to confirm this result. In the decline of the contact rates, respectively reproduction numbers, in Germany during March 2020 no clear intermediate steps can be discerned which could be read as the signature of the first two interventions. To assume the existence of constancy intervals between two non-pharmaceutical interventions per se seems doubtful to us. In this phase we see only a cumulative effect of all three interventions.

On the other hand, if the data evaluation shows a relative stability of the contact rate $\kappa(k)$, a conditional prediction for the coming development is possible, sometimes even for several weeks. “Conditional” means here under the assumption of no essential change of the behaviour of people, relative stability of climatic conditions and no mutation of the virus. In section 4 time intervals in which such conditional predictions could be made have been discussed. For example, one could recognize a stable trend of increasing numbers of daily newly infected people for Germany already in late July and early August 2020 and could have foreseen the approach of the second wave long before autumn (see fig. 3, bottom, and table in Section 4).

In this context we would like to point to two papers which are of interest in this context. The authors of [10] study the effects of non-pharmaceutical interventions on COVID-19 in 11 European countries. It would be very interesting to apply our model to the data and compare the results with theirs. The other paper studies how epidemic control can be obtained by contact tracing [8]. If such data are available one could improve the application of our model by improved estimates of the contact rates.

As conditional predictions with the present model turn out to be convincing and reliable, we also consider future scenarios for the overall development of the epidemic for up to a few months as informative. For an example of a scenario that can be checked against the real course of the epidemic see the end of section 7. Such scenarios may be helpful for exploring alternative actions (e.g. non-pharmaceutical interventions) and for identifying necessary steps to avoid the breakdown of the health system.

• The study of future scenarios with the $\gamma$-K-McK model may be useful for persons or institutions looking for orientation with regard to alternative actions and/or external changes of condition (rise of new mutants, climate change).

(4) As we have seen, there is a parameter which can, in principle, be influenced by socio-political decisions comparatively easily, the time between infection and the day people go to quarantine. We therefore emphasize again:

• The model shows that reducing the time until entering quarantine by one day leads to a drastic improvement.

Whether one can reduce it depends on various factors, in particular the infrastructure and effectiveness of the health system. Discussions with experts about the German health system have convinced us that there are good chances for lowering the time until quarantine by at least one day. Of course, this needs a great effort: One has to enable the health system to carry it out and one has to convince the population to follow the corresponding rules. But the latter should have a good chance of success since this is a restriction which hits only a small number of the population: those who show first symptoms or are identified as Covid-positive in an unspecified test.

(5) A comparison between different models is difficult. If models are based on clear principles the first thing one could do is to compare the principles and discuss their strengths and weaknesses. If one wants to compare the standard SIR model with our adapted model this is simplified by the fact that they are both based on the same principle, the Kermack-McKendrick idea. So the first think is to compare the input functions. But it still could be that although they are very different both models are approximately equivalent. In section 6 we have shown that this is not the case. If one takes both the compartments $I$ and $R$ into account ($S$ is determined by $I$ and $R$), the difference is drastic. We conclude that a model based on a more realistic $\gamma$ function, like the one we have chosen, seems to be more useful.

Saying this, one should not forget that no model is a mirror of reality. It can even happen that a model based on less realistic principles may work better, but so far we don’t see a sign of this in the comparison of the adapted model with the standard SIR model. So we conclude:

• Our comparison of the adapted model with the standard SIR model indicates that the former one is based on more realistic assumptions. Since the differences in the long run or when the contact rates jump are large, the adapted model should be the first choice.

We thank Odo Diekmann for discussing our thoughts. His help was invaluable to us, non-experts in the field, for coming to a proper understanding of compartment models. Similarly we thank Stefan Müller for extensive discussions in particular concerning the comparison of models. Section 6 is greatly influenced by thoughts he and one of the referees shared with us. Moreover, we thank Martin Bootsma, Robert Feßler, Jan Mohring, Robert Schaback, Hans Lehrach for valuable discussions, the referees for very useful hints, Jeremy Gray for tidying up our English of the final version, and Don Zagier for his intensive support in the preparation of the manuscript.
Calculations and graphics have been made with Mathematica 12.