Germany's fourth COVID-19 wave was mainly driven by the unvaccinated.

Background: While the majority of the German population was fully vaccinated at the time (about 65%), COVID-19 incidence started growing exponentially in October 2021 with about 41% of recorded new cases aged twelve or above being symptomatic breakthrough infections, presumably also contributing to the dynamics. So far, it remained elusive how significant this contribution was and whether targeted non-pharmaceutical interventions (NPIs) may have stopped the amplification of the crisis.
Methods: We develop and introduce a contribution matrix approach based on the next-generation matrix of a population-structured compartmental infectious disease model to derive contributions of respective inter- and intragroup infection pathways of unvaccinated and vaccinated subpopulations to the effective reproduction number and new infections, considering empirical data of vaccine efficacies against infection and transmission.
Results: Here we show that about 61%-76% of all new infections were caused by unvaccinated individuals and only 24%-39% were caused by the vaccinated. Furthermore, 32%-51% of new infections were likely caused by unvaccinated infecting other unvaccinated. Decreasing the transmissibility of the unvaccinated by, e. g. targeted NPIs, causes a steeper decrease in the effective reproduction number R than decreasing the transmissibility of vaccinated individuals, potentially leading to temporary epidemic control. Reducing contacts between vaccinated and unvaccinated individuals serves to decrease R in a similar manner as increasing vaccine uptake. Conclusions: A minority of the German population-the unvaccinated-is assumed to have caused the majority of new infections in the fall of 2021 in Germany. Our results highlight the importance of combined measures, such as vaccination campaigns and targeted contact reductions to achieve temporary epidemic control.

Introduction

Vaccines are the most powerful pharmaceutical tool to prevent infections with SARS-CoV-2 and combat the COVID-19 pandemic. Fast vaccine uptake by as many individuals as possible saves lives, people’s health, and livelihoods. Despite large-scale vaccine roll-out campaigns, many countries, most prominently in Europe, have experienced a rise in case numbers in the late summer and early fall of 2021 and reported effective reproduction numbers above one for an extended period of time[1]. This means that on average, every infected person infected more than one other person, thus causing exponentially rising incidences[2]. Since the beginning of this pandemic, such resurgences have, in part, been mitigated by harsh non-pharmaceutical interventions (NPIs) such as lockdowns or curfews that limit the population’s contacts, thereby decreasing the effective reproduction number and relieving overburdened public health systems[3,4]. Measures that affect large parts of the general population over a long period of time can have devastating effects, such as increasing social inequality and domestic violence, detrimental impacts on mental health, or economic disruptions[5-9]. Such harsh restrictions should therefore be considered a last resort of pandemic control.

During the onset of the fourth COVID-19 wave in Germany, many hospitals and intensive care units (ICUs) were operating at maximum capacity or were projected to do so at a later point[10]. In the four weeks between Oct 11, 2021, and Nov 7, 2021, Germany’s central public health institute, the Robert Koch Institute (RKI) reported 250,552 new symptomatic infections in individuals with known vaccination status, 90,471 of which were attributed to vaccinated individuals, i.e. 36% were symptomatic breakthrough cases (41% in age groups eligible for vaccination)[11]. During this time, the average vaccination rate in different age groups [0,12), [12,18), [18,60), and 60+ were 0%, 40.1%, 72.4%, and 85.1%, respectively, leading to 0%, 4.8%, 41.6%, and 61.9% of new cases being classified as symptomatic breakthrough cases within the respective age groups[11], Table 1. Simultaneously, the effective reproduction number remained at a relatively stable value of (under the assumption of a generation time of four days)[12].

Table 1

Share of breakthrough infections in the age groups eligible for vaccination according to official estimates by the Robert Koch Institute (RKI)[11] and the model for “low efficacy”, “medium efficacy”, and “high efficacy” scenarios.

Age group	RKI report (symptomatic cases)	Model (“high eff.”)	Model (“medium eff.”)	Model (“low eff.”)
adolescents	4.8%	5.1%	21.1%	25%
adults	41.6%	42.3%	51.2%	57%
elderly	61.9%	61.5%	74.1%	77.4%

Given that breakthrough cases are a challenge both for communication and vaccine acceptance[13] and that harsh NPIs may be illegitimate for vaccinated individuals, the above situation raises two important questions: How much does the unvaccinated population contribute to the infection dynamics despite being in the minority? And could targeted NPIs aiming at reducing the contacts of unvaccinated individuals temporarily and sufficiently suppress the infection dynamics such that harsh, large-scale NPIs could be avoided?

To address these questions, we establish the contribution matrix approach, a theoretical concept derived from the next-generation matrix framework[14]. The contribution matrix quantifies the contributions to caused by the infection pathways from un-/vaccinated individuals to other un-/vaccinated individuals, considering the age and contact structure of the population, vaccination rates, as well as expected vaccine efficacies regarding susceptibility and transmission reductions, respectively. In its general form, it quantifies the contributions made by any combination of two subpopulations.

Based on this approach, we estimate that in October 2021, around 32%–51% (depending on vaccine efficacy) of the effective reproduction number was caused by unvaccinated individuals infecting other unvaccinated individuals (see Fig. 1). Since unvaccinated individuals have a higher probability of suffering from severe disease[15-17], this contribution is the major factor that drove the public health system into a crisis characterized by hospitals and ICUs reaching maximum capacity. In contrast, we estimate that only 15%–18% of the reproduction number were attributable to vaccinated individuals infecting unvaccinated individuals. In October 2021, about 65% of the German population was fully vaccinated, implying that the majority of the overall population contributed little to the amplification of the crisis. In total, we estimate that the vaccinated population contributed 24%–39% to while the unvaccinated population contributed the remaining 61%–76%, despite the fact that unvaccinated individuals have been in the minority in Germany. 9%–21% of new infections would be caused by vaccinated individuals infecting other vaccinated people. In total, we estimate that unvaccinated individuals were involved in 8–9 out of 10 new infections, either as infecting, acquiring infection, or both.

Fig. 1

Estimated contributions of infection pathways towards new cases within vaccinated and unvaccinated subpopulations.

Estimated contributions of infection pathways to in the (a) “high efficacy”, (b) “medium efficacy”, and (c) “low efficacy” scenarios as a graphical representation of Tabs. 2–4. The charts can be read as follows: Consider an infected population that caused a new generation of 100 new infecteds. Then for (a), 51 of those newly infected individuals will be unvaccinated people that have been infected by other unvaccinated people. Likewise, 25 newly infected individuals will be vaccinated people that have been infected by unvaccinated individuals. Hence, 76 new infections will have been caused by the unvaccinated. Along the same line, 15 newly infecteds will be unvaccinated people that have been infected by vaccinated individuals and 9 newly infecteds will be vaccinated people that have been infected by other vaccinated individuals, totaling 24 new infections that have been caused by vaccinated individuals.

Table 2

Contribution to from infections between vaccinated and unvaccinated populations for the upper parameter bounds.

	← (u)nvaccinated	← (v)accinated
u ←	51.4%	15.0%
v ←	24.5%	9.1%
total	75.9%	24.1%

Table 4

Relative contributions to from infections between vaccinated and unvaccinated groups for the “low efficacy” scenario.

	← (u)nvaccinated	← (v)accinated
u ←	31.6%	18.2%
v ←	29.5%	20.7%
total	61.1%	38.9%

We further argue that regarding the situation in the fall of 2021, the unvaccinated would have had to reduce their transmissibility two to three times as strongly as the vaccinated in order for the system to reach (and hence containment of the infection wave), if the burden of contact reductions were to be distributed between the two subpopulations according to their respective contributions. Moreover, decreasing mixing between individuals of distinct vaccination status can decrease . Ultimately, a higher vaccine uptake would have led to less unvaccinated being involved in infections, which can not only decrease , but is critical for relieving an overburdened public health system, as they are more likely to suffer from severe disease. Combinations of these interventions that address mainly the unvaccinated might have rendered the dynamics subcritical.

Methods

Mathematical framework

We use a population-structured compartmental infectious disease model that captures a variety of aspects regarding vaccination against COVID-19 (see Supplementary Methods, Sec. 1.1.1). The model’s dynamics are fully described by the next-generation matrix K of small domain (see Supplementary Methods, Sec. 1.1.2), which quantifies the average number of offspring in group j caused by a single infectious individual in group i[14]. Here, the index i (or j, respectively) refers to the subpopulation that is determined by a respective age group and the vaccination status within that group, thus yielding two subpopulations per age group. In the regime of small outbreaks (relative to the total population size), the ordinary differential equations governing the epidemic growth can be linearized, with the dynamics being determined by K, such that the generational growth of the number of infected individuals in group i followsThe incidence approaches the eigenstate y of K that corresponds its spectral radius, which in turn is equal to the effective reproduction number[14]. Hence, the entries of the normalized eigenvector contain the relative frequency of newly infected individuals in age/vaccination group i.

Consequently, the number of j-offspring caused by i-individuals in a dynamical system defined by K is given by the contribution matrixSumming over all matrix elements of C yields the effective reproduction number (see Supplementary Methods, Sec. 1.1.1)). A single matrix element C can thus be considered the contribution of the i → j infection pathway to the reproduction number (a derivation of the concept and an operational definition of C can be found in the Supplementary Methods, Sec. 1.1.1–1.1.2 and Sec. 1.2.4), respectively). The normalized contribution matrix C/ gives the relative contributions of i → j infections towards (and consequently, towards the total number of new infections).

We derive explicit equations for the contributions of un-/vaccinated individuals in the homogeneous case, i.e. ignoring age structure (see Supplementary Methods, Sec. 1.2.3). These contributions arewhere v is the vaccine uptake, s is the susceptibility reduction after vaccination, is the adjusted transmissibility reduction (i.e. it contains the relative increase of the recovery rate after a breakthrough infection b and viral shedding reduction r), is the base transmissibility of unvaccinated infecteds, and is the base transmissibility of vaccinated infecteds (both of which quantify differences in behavior in the respective groups). The total effective reproduction number is given by

Model structure, parameters, and scenarios

In the full model, we construct the next-generation matrix of small domain (see Supplementary Methods, Eq. (S3)) based on the following observations, assumptions, and estimates: We structure the population into four age groups [0,12) (children), [12,18) (adolescents), [18,60) (adults), and 60+ (elderly). Contact numbers between those age groups and subpopulation sizes were constructed based on the POLYMOD (2005) data set[18,19] using the ‘socialmixr’ software package[20] (see Supplementary Methods, Sec. 1.2.1). Since vaccine efficacy was, at the time of writing, estimated only for the status “fully vaccinated” in Germany without distinguishing between different vaccines, we solely distinguish between “unvaccinated” and “vaccinated” individuals in the model, regardless of the make of the received doses (note that by the fall of 2021, a total number of four vaccine types was available in Germany, i.e., Spikevax (Moderna), Ad26.COV2.S (Janssen), Vaxzevria (AstraZeneca), and Comirnaty (BioNTech/Pfizer) with the latter being by far the most used[21]). Following the example of Scholz et al.[22], we further assume that children and adolescents have reduced susceptibility to the virus and a reduced base transmissibility if infected, as was observed in Germany, Israel, and Greece[23-25]. In the discussed time frame, 14.7%, 9.4%, 60.2%, and 15.7% of new cases can be attributed to the respective age groups [0,12), [12,18), [18,60), and 60+[26]. In order to match this distribution approximately, we calibrate the base susceptibility (i. e. susceptibility without vaccination) and infectiousness of our model by assuming that children are 72% as susceptible and 63% as infectious as adults (72% and 81% for adolescents), which is larger than what was observed for the wild type[24,25], see Supplementary Methods, Sec. 1.2. However, since the B.1.617.2 variant (Delta) that was predominant in Germany in October/November 2021 was generally observed to be more infectious than the wild type[27], such an increase is plausible. Note that in principle, heterogeneous ascertainment may lead to a distribution of detected cases that is skewed towards the adult population, as children and adolescents may have higher probability of suffering from an asymptomatic infection[28] and thus are less likely to be detected via symptom-based testing strategies. Yet, by the fall of 2021, Germany made regular screening via rapid antigen tests mandatory in schools across the country, potentially lowering the level of under-ascertainment in these age groups[29]. Nevertheless, we test how our results change by assuming children and adolescents are as susceptible as adults in a sensitivity analysis (see Supplementary Methods, Sec. 1.3.3). Additionally, note that we ignore the number of recovered individuals. Until Oct 10, 2021, about 4.3 million infections were reported in Germany[12], 74% of which likely received a vaccination[30-32] and are therefore considered as vaccinated in our analysis. With an under-ascertainment ratio of about 1.8[33], we estimate that the total number of non-vaccinated recovered individuals was on the order of 2.4% of the population in Germany at the time, and therefore negligible in our analysis.

In Germany, an estimated average vaccine efficacy of 72% against symptomatic COVID-19 in adults and the elderly was found for cases reported between Oct 11, 2021 and Nov 7, 2021[11]. Vaccine efficacy for adolescents was not reported due to the respective data being potentially unreliable (low number of cases). Because these efficacies were computed for symptomatic cases, we use their values as a “high efficacy” scenario regarding vaccine efficacy in our analysis, because unreported and/or asymptomatic breakthrough infections might lower the estimated efficacies (see Supplementary Methods, Sec. 1.2.5). However, note that these observed 72% vaccine efficacy are in line with an estimated population-wide vaccine efficacy against infection based on vaccination time series and waning immunity data that was published in a meta-review by the WHO[34,35]. In order to obtain breakthrough infection rates in adolescents on the order of observed symptomatic breakthrough cases we assume a vaccine efficacy of s = 92% for adolescents. Despite being comparably large, this value seems justified considering that adolescents have been made eligible to receive a vaccine in Germany only shortly prior to the study period, and a high vaccine efficacy against infection with the Delta variant has been reported for this age group[36]. Regarding the infectiousness of individuals suffering from breakthrough infections, viral load of vaccinated patients suffering from symptomatic COVID-19 was reported to be at the same level as of those unvaccinated[37,38]. Another study from the UK found decreased infectiousness in breakthrough infections[39]. Considering both these results, we assume a conservative transmission reduction of r = 10% for breakthrough infections. In agreement with the literature[37,40] we further consider that the average infectious period of breakthrough infections is shorter than for unvaccinated individuals and assume a 50% increase in recovery rate for the vaccinated, amounting to an average infectious period that is 2/3 as long as that of unvaccinated infecteds (b = 3/2) (see Supplementary Methods, Sec. 1.2.2). Such an increased recovery rate can also be caused by deliberate behavior. As individuals that are not opposed to vaccination typically adhere to protection measures more consistently[41], behavioral changes following a breakthrough infection might further decrease the effective infectious period. Note that together with a decreased duration of infection b = 3/2, the adjusted transmission reduction reads , which is lower than a 63% reduction that was observed for household transmissions of the Delta variant between infected vaccinated and susceptible unvaccinated individuals in the Netherlands in August and September 2021, close to our observational period[42]. As this reduction was observed to wane over time[43], is a reasonable assumption.

In a second, “medium efficacy” scenario, we consider that vaccine efficacies against infection are in the range of 50%–60%, i.e. lower than the observed value against symptomatic COVID-19, and lower than vaccine efficacies reported in the UK for the Comirnaty (BioNTech/Pfizer) vaccine[44], considering that partial immunity might have waned over time[45]. Since vaccine efficacy is expected to decrease with age[45,46], we assume an efficacy against infection of s = 60% for adolescents and adults as well as s = 50% for the elderly (see Supplementary Methods, Sec. 1.2.4).

Finally, we also discuss a “low efficacy” scenario where the susceptibility reduction is assumed to be much lower than the observed efficacy against symptomatic COVID-19, namely 50% for adolescents and adults, and 40% for the elderly (see Supplementary Methods, Sec. 1.2.5).

To summarize the main scenarios, for the “high efficacy” the vaccination efficacy s for adolescents, adults, and elderly is assumed to be 92%, 72%, 72%, in the “medium efficacy” scenario 60%, 60%, 50%, and in the “low efficacy” scenario 50%, 50%, 40%, respectively.

Based on these considerations we compute the respective full-model next generation matrices K and numerically find the normalized population eigenvectors corresponding to the respective and the contribution matrices C, which we further reduce to the two-dimensional vaccination status space by summing over the respective contributions of age groups (see Supplementary Methods, Eq. (S2)).

Table 3

Relative contributions to from infections between vaccinated and unvaccinated groups for the “medium efficacy” scenario.

	← (u)nvaccinated	← (v)accinated
u ←	38.1%	17.4%
v ←	28.5%	16.0%
total	66.6%	33.4%