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Inference under Superspreading: Determinants of SARS-CoV-2 Transmission in Germany
At the point of writing this article, the world records a million deaths associated with Covid-19 and over 30 million people have tested positive for the SARS-CoV-2 virus. Societies around the world have responded with unprecedented policy interventions and changes in behavior. The reduction of transmission has arisen as a dominant strategy to prevent direct harm from the newly emerged virus. Yet, quantitative evidence on the determinants of transmission remains scarce.
Individual variation in transmission (dispersion or potential for so called superspreading lloyd2005superspreading) is one of the reasons why the collection of adequate evidence is challenging. Overdispersion (high likelihood of superspreading) is well documented for SARS-CoV-1 riley2003transmission,lipsitch2003transmission and is shown to be a dominant feature of SARS-CoV-2 (see also adam2020clustering). Under superspreading single cases convey little information as transmission kinetics are driven by few outliers. Thus, conclusions drawn from anecdotal evidence with few primary cases (e.g., hendrix2020absence and khan2020transmission) are subject to substantial uncertainty.
In the absence of sufficiently large and detailed contact tracing data, some research has relied on surveillance data with aggregated cases sehra2020maximum,salje2020estimating,CHERNOZHUKOV2020 or deaths flaxman2020estimating,CHERNOZHUKOV2020. However, such data provide additional challenges to inference, including a time lag in reporting, under-reporting of cases, and a lack of established methods to account for superspreading.
I address those issues with the following three points. First, by aggregating cases based on symptom onset instead of reporting date, the timing of transmission is estimated more sharply. Second, I account for underreporting by modelling infections as compartmentalized unobservable latent process. Age compartments allow to identify age specific growth rates if reporting differs between age groups. Finally, the paper proposes a probabilistic infection process, that extends well-established models for superspreading lloyd2005superspreading to aggregated case counts.
I apply the model to German surveillance data and find that transmission was reduced predominantly by public awareness rising and voluntary behavioral adaptations to publicly reported local incidence. Furthermore, extrapolation suggests strong seasonal effects and potential for testing and tracing.
In the following, I introduce the model. Afterwards, the data is described, and estimation results are illustrated and discussed. A supplementary document contains additional detail in Sections (ref) through (ref).
The model distinguishes between cases and infections. Infections at time $t$ are denoted by $i_t$. Infections are unobserved. Instead, we observe the number of reported cases $c_t$ with symptom onset at time $t$. The full model (Supplementary Section (ref)) uses daily data and accounts for timing as standard in the literature cori2013new,flaxman2020estimating. In this section, I present a simplified version to illustrate the implications of superspreading for statistical inference. In particular, the generation time (from primary infection to secondary infection) and incubation period (from infection to symptom onset) are fixed to one time step. Further, indices for age and location are dropped.
This simplified model is illustrated in Figure (ref). A primary infection causes new infections in the following time interval. The average number of secondary infections caused by one primary infection is denoted by the reproductive number $R_t$. The probability of an infection being reported as a case is denoted by the reporting rate $r_{t}$.
In a popular transmission model lloyd2005superspreading secondary infections follow a negative binomial distribution with mean $R_t$ and dispersion $\Psi$, i.e. $NB(R_t,\Psi)$. Dispersion $\Psi$ describes the individual variation in transmission, where small values are associated with high variation. The reproductive number $R_t$ is determined by the conditions that transmissions at time $t$ are subject to. Given those conditions, I assume that the number of secondary infections caused by one primary infection is independent across individuals. This implies for infections $i_t$ that $$i_t \sim NB(i_{t-1} R_t,i_{t-1} \Psi).$$ Thus, infections are more dispersed under a small infection count $i_{t-1}$. For large infection counts $i_{t-1}$ the model converges to a Poisson model with no overdispersion.
The growth rate $\frac{i_t}{i_{t-1}}$ constitutes a random variable with mean $R_t$ and variance $\frac{R_t (\Psi+R_t)}{i_{t-1}\Psi}$. Variation in the growth rate is caused by variation in the mean $R_t$ and noise influenced by individual dispersion $\Psi$. We can separate the two factors, as the former influences growth rates irrespective of the current infection count, while the latter has less impact for many infections $i_{t-1}$. Empirical weekly growth rates in Germany fit this model and can be used to estimate the dispersion parameter (see Section (ref)).
The model presented in this study deviates from the literature, where overdispersion is often ignored CHERNOZHUKOV2020,hsiang2020effect,davies2020effects or assumed constant flaxman2020estimating. A constant dispersion of aggregated cases arises under the assumption that the number of secondary infections across primary cases are identical. In this case it follows that $i_t \sim NB(i_{t-1} R_t, \Psi)$. This alternative assumption of a constant dispersion is inherent, but often unappreciated, in inference based on negative binomial regression, the endemic/epidemic model introduced in held2005statistical, and epidemiological models with random effects (e.g., the model in fisher2020ecological).
The model in this study does not feature infections between compartments. Supporting this simplification, a large scale contact tracing study in India found that most transmissions occur within the same age group Laxminarayaneabd7672. I discus extensions with importation in Section (ref). The remaining parts of the model are standard. Cases $c_t$ constitute a sum of Bernoulli trials that is approximated by a Poisson distribution for computational convenience.
The reproductive number is modelled as a function of covariates, where the effect is assumed to be multiplicative. Each location and age group features a basic reproductive number $R_0$ in the absence of all interventions and under average weather conditions. Effect estimates can be interpreted as the ratio of prevented/added infections at a particular time.
The main obstacles for inference include high correlation between covariates, unobservables, and an unknown reporting rate (see Section (ref) for details).
I apply the model to German surveillance data, which features date of symptom onset, age groups (15-34, 35-59, 60-79, and above 80), and geographical units (county or city). Daily location-specific covariates include 21 policy interventions, average temperature, relative humidity, the ratio of traced infectious, local incidence as known at that point in time, local cumulative incidence, weekday fixed effects, and daily, age and location specific error terms. I classified interventions for 10 German states until May 2020. Estimation is based on cases in 112 counties until mid May 2020, which encompasses more than 60,000 symptomatic cases. See the Supplementary Section (ref) for additional details.
Reproductive numbers are identified even if only a fraction of infections is ascertained as cases wallinga2004different. Importantly, said fraction has to be constant over time. As the likelihood of developing symptoms changes with age (Figure (ref)), the reporting rate $r_t$ cannot be assumed to be constant across age groups.
I chose a reporting rate of $25\%$ for the model. Case fatality rates can provide some support for this model assumption. If the infection fatality rate is constant over time, the case fatality rate identifies changes in the reporting rate. The observed case fatality rate in Figure (ref) does not suggest major changes over time. A comparison to the age-specific infection fatality rates (estimated based on multiple international serological studies levin2020assessing) indicates a reporting rate close to $25\%$. Similar reporting rates arise based on first evidence of unpublished serological studies in Germany rkisero. Identification of the reproductive number relies on the assumption that symptomatic infections were equally likely to be reported over time within a specific age group and location.
I estimate the model in a Bayesian framework and chose weakly informative priors on transmission characteristics (see Section (ref)). Estimation results are informative about transmission in the context of this study.
The average incubation period was $5.30$ days ($95\%$-CI: $[5.16, 5.45]$). The average generation time was slightly longer with $5.84$ days ($95\%$-CI: $[5.68, 6.01]$). Assuming independence this corresponds to $42\%$ of secondary infections occurring before symptom onset.
The basic reproductive number denotes the average number of secondary infections in the absence of interventions and under average weather conditions. There is no evidence for an age dependent basic reproductive number (Table (ref)). Population density was positively correlated with the basic reproductive number (Table (ref), supplementary material).
For the age groups 15-34 and 35-59 we observe high dispersion, with more than half of all cases infecting nobody and $20\%$ of primary cases initiating $70$ to $80\%$ of secondary cases. Older age groups show less tendency for superspreading. Previous work studied dispersion abstracting from age differences. For comparison, I simulated the marginal distribution of secondary infections, which is a mixture negative binomial distribution furman2007convolution. Assuming equally distributed infections across age groups, its mean variance ratio is $0.378$ ($95\%$-CI: $[0.33,0.44]$), equivalent to a dispersion parameter of $0.61$ ($95\%$-CI: $[0.49,0.77]$). Contact tracing studies obtained similar results. A study from Hongkong estimated a dispersion parameter of $0.33$ ($95\%$-CI: $[0.14, 0.98]$) or $0.19$ ($95\%$-CI: $[0.13,0.26]$) when assuming a single primary case for an unresolved large cluster. A study from Shenzhen (China) reported $0.58$ ($95\%$-CI: $[0.35, 1.18]$) and a study in two Indian states $0.51$ ($95\%$-CI: $[0.49, 0.52]$). Global data-sets of outbreaks suggested a higher potential for superspreading with a dispersion parameter of $0.10$ ($95\%$-CI: $[0.05, 0.20]$) endo2020estimating. Noteworthy, the aforementioned studies do not account for changing reproductive numbers and are prone to overestimate the prominence of superspreading. Further, clusters might be more likely to be traced, while diffuse community spread is harder to identify.
Other limitations apply to the approach presented here. The model does not allow for overdispersion in reporting and is therefore prone to overestimate the dispersion in transmission. Moreover, changes in the reproductive number might be inadequately modelled. If the reproductive number is over-fitted, individual variation is under-estimated (and vice versa).
Figure (ref) shows average effects of the most important determinants of transmission. Age-specific effects for all covariates are available in Supplementary Figure (ref). Interventions change the reproductive number by a fixed ratio. Interaction effects are ignored and estimates evaluate the effect of an intervention under the circumstances it was implemented.
Many covariates are strongly correlated (Figure (ref)). However, the correlation matrix of effect estimates suggests that effects are identified (Figure (ref)).
The strongest effect was associated with public awareness rising. Transmission reduced by $58\%$ ($95\%$-CI: $[53\%,62\%]$) when government officials gave their first speeches asking for decided behavioral changes from the public. Simultaneously implemented changes, like the staggered declaration of international risk areas, might bias this result.
The closure of restaurants was associated with additional reductions in transmission of $15\%$ ($95\%$-CI: $[5\%,23\%]$). The effect was strongest for the age group 15-59.
School and daycare closures were associated with a $12\%$ ($95\%$-CI: $[-5\%,28\%]$) reduction in transmission. The intervention affected the age group 15-79 equally, which suggests that the reduction stems from behavioral changes, instead of the absence of transmission in the educational setting, which would affect other age groups one generation time later.
Limiting sports was associated with additional reductions in transmission. The closing of non-essential shops, mandatory distancing and limitation of gatherings in public spaces, however, showed no evidence for reducing transmission. In the age group 15-34, transmission actually increased by $25\%$ ($95\%$-CI: $[-1\%,54\%]$) when public distancing became mandatory. This might explain why no significant effect of stay-at-home orders nor of business closure was found in the United States CHERNOZHUKOV2020. The regulation could induce more private contacts with higher transmission risk. In some states a mild stay-at-home order was imposed, allowing individual sport and work, which was associated with a $9\%$ ($95\%$-CI: $[4\%,13\%]$) reduction in transmission across all age groups.
Initially, testing was covered for patients with exposure or within a risk group. Later, any symptomatic case, regardless of exposure, was eligible. The narrow testing regime was associated with a $16\%$ ($95\%$-CI: $[7\%,26\%]$) increase in transmission. This effect is driven by younger cohorts, whereas transmission in the age group 80+ was reduced.
The reopening of daycare and churches with precautionary measures had no significant impact. Some evidence for increased transmissions was associated with school reopenings.
Masks (mouth and nose cover including cloth masks) became mandatory in supermarkets and public transport. Compliance was found to be high betsch2020social and timing of implementation varied across states and counties. A small effect of $-6\%$ ($95\%$-CI: $[-17\%,7\%]$) was associated with this policy. If only a small ratio of transmissions occurs in these settings, this is consistent with a household study in China, where mask wearing before symptom onset reduced transmission by $80\%$. As a comparison, mandatory masks for employees were associated with a reduction in transmission by $10\%$ in the United States CHERNOZHUKOV2020.
Testing and tracing has been argued to be crucial ferretti2020quantifying. As symptom onset date and reporting date (when health departments were informed about positive test) are available, we know for each case the days of potential infectiousness and when testing and tracing was initiated (for details see the supplementary material). This allows to construct a daily location specific proxy for the ratio of traced infectious and to provide empirical evidence on the effectiveness of testing and tracing that corroborates results of modelling studies kretzschmar2020impact.
The ratio of traced infectious was associated with a reduction of transmission by $33\%$ ($95\%$-CI: $[22\%, 43\%]$). As the ratio does not reflect unreported infections, the impact of testing and tracing on an infectious individual arises after adjusting for the reporting rate. Extrapolating to unobserved infections, testing and tracing reduced secondary infections by $84\%$ ($95\%$-CI: $[35\%, 100\%]$) for the age group 15-59. The effect was strongest for the high risk group over 60 years, where the ratio of traced infectious reduced transmission by $44\%$ ($95\%$-CI: $[27\%, 59\%]$). Adjusting for unreported infections, this effect corresponds to an eradication of transmission to older age groups for tested and traced infectious individuals. The hypothetical extrapolation to all infections should be interpreted with caution.
Recent modelling studies for Germany concluded that 30-50$\%$ of transmission can be reduced by testing and tracing contreras2020challenges. I find that the total effect of testing and tracing increased from March to May (Panel A of Figure (ref)). In May 2020, testing and tracing was attributed an average transmission reduction of $15\%$ ($95\%$-CI: $[9\%,20\%]$). For the age group over 80 years, the impact is larger with a reduction of $28\%$ ($95\%$-CI: $[17\%,37\%]$).
In general testing and tracing capabilities are limited. If infections surpass capacities, the ratio of traced infectious decreases. The results from above imply that unmitigated spread increases the speed of transmission, especially in older age groups.
Another potential factor for transmission is voluntary behavioral response to risk of infection CHERNOZHUKOV2020. Individual risk of infection depends on local incidence. In line with this argument, I find that publicly reported incidence reduced transmission. This information effect was highest in April (Panel B in Figure (ref)) when estimates indicate a reduction of $44\%$ ($95\%$-CI: [$40\%$, $48\%$]).
An interesting counterfactual is the level of incidence that is sufficient to stop growth in the absence of policy interventions. The results suggest that a reported incidence of 300 to 1000 weekly cases in 100,000 induces sufficient voluntary behavioral change. This incidence corresponds to 3,600 to 12,000 weekly deaths in Germany (based on a reporting rate of $50\%$ and an infection fatality rate of $0.75\%$ guided by levin2020assessing and the German age distribution from 1 to 80 years). The result is speculative, as it extrapolates beyond the support of the data (the $99\%$ quantile of incidence is 160 cases in 100,000).
Cumulative incidence (in percentage points) was negatively associated with transmission. Random mixing, full immunity after infection, and the absence of underreporting would be consistent with an effect of $-1\%$. The data does not allow a sharp identification of this effect as the highest measured cumulative incidence was $1.2\%$, which corresponds to a reduction of $20\%$ ($95\%$-CI: $[4\%,35\%]$).
Seasonal effects have been discussed to play a role in SARS-CoV-2 spread baker2020susceptible,carlson2020misconceptions. I find that low temperature was associated with higher transmission. Relative humidity was associated with a small increase in transmission. As the latter effect is mostly driven by the age group 60-79, there is no strong evidence for relative humidity being an important determinant of transmission in Germany. The extrapolated change in transmission due to seasonality is $53\%$ ($95\%$-CI: $[43\%, 64\%]$) (based on average temperature and relative humidity in January and July). The expected seasonal effects over the year are denoted in Panel C of Figure (ref). These findings are consistent with other studies. In an international cross-city comparison, low temperature and humidity was associated with community spread sajadi2020temperature. A cross state comparison in the United States found that high temperature and UV-light reduced incidence sehra2020maximum.
The effect estimate for seasonality conflates the interaction of the virus with environmental circumstances and behavioral changes due to weather carlson2020misconceptions. Moreover, extrapolating effects identified by daily weather to entire seasons can induce substantial biases. In line with the findings presented here, a study based on monthly case data in countries on both hemispheres found that the season of other human corona viruses was associated with low temperature and high relative humidity Li2020seasonality.
The effect estimates presented so far are subject to substantial uncertainty and suffer from high correlation (compare Figure (ref)). Model misspecification may play a role as other factors can influence behavior contemporaneously.
The main model is estimated on data until May 2020 as detailed intervention data is not available for the remaining time period. However, the covariates for weather, information, and testing and tracing are available. As robustness check, I consider data from May to August 2020 including the age group 5-14 and with the same model specifications, but substituting policy interventions with week fixed effects. This study covers another 30,000 cases in 141 counties. See the supplementary material for details.
I find that results are consistent for information on incidence, temperature, and testing and tracing. Seasonality effects were less strong which might indicate non-linear effects. School children below 15 years exhibited higher transmission during school holidays. This suggests that reopened schools constituted a relatively low risk of transmission.
Empirical evidence is paramount to inform modelling studies and policy decisions. I rely on features of German surveillance data to improve upon early empirical studies analyzing SARS-CoV-2 transmission flaxman2020estimating,hsiang2020effect,salje2020estimating,dehning2020inferring. In particular, symptom onset allows sharper estimation of infection timing. Moreover, age groups improve identification of growth rates and enable the estimation of age-dependent effects. Inference in such highly compartmentalized data should account for superspreading. As with any structural estimation, results should be interpreted with caution and a keen eye on the model's limitations. The Bayesian framework allows to integrate alternative assumptions (e.g., priors on the reporting rate) to asses their impact on the results presented here.
A large set of covariates was analyzed. Previous studies found stronger effects of policy interventions without controlling for information, tracing, or seasonality. For example, the shut-down in France was attributed a $77\%$ reduction in transmission based on hospitalization and death data salje2020estimating. An international comparison of death data estimated an effect of $81\%$ flaxman2020estimating. A reduced form approach on case growth found heterogeneous effects of policy interventions in China, South Korea, Italy, Iran, France and the United States hsiang2020effect. As noted in other studies, growth rates in Germany substantially reduced before major policy interventions were put in place dehning2020inferring. If not controlled for, adaptation to local risk of infection can be wrongly attributed to policy interventions CHERNOZHUKOV2020.
I find that higher reported incidence reduced transmission. This explains the common observation that incidence rarely exhibits prolonged exponential growth. Spread is unmitigated as long as it is undetected. In Germany, the reduction in transmission was driven mainly by behavioural adaptations to reported incidence instead of immunity by infection. If transmission keeps decreasing with higher incidence, there exists an equilibrium value for reported incidence where the reproductive number is $1$. If information or behavioral adaptation is delayed, incidence moves in waves around this equilibrium value.
Individual risk of contracting (or spreading) Covid-19 is a function of behavior and local incidence. If behavior is constant, incidence grows until immunity slows further expansion. Behavioural changes to prevent transmission are costly to individuals and their peers, but decrease incidence and are therefore potentially beneficial to society. Thus, policy interventions can be warranted to share and direct the costs of transmission prevention. As behavior was found to change with local incidence, indirect social and economic consequences of the pandemic can be expected even in the absence of policy interventions. Ultimately the impact of policy interventions depends on their effect on transmission and their ability to allocate costs of prevention effectively.
Estimating the costs of interventions is beyond the scope of this study, but some remarks regarding effectiveness seem warranted: Public awareness rising was crucial, while little effect of closing shops and public distancing rules was found. Daycare reopening showed no, school reopening relatively little impact. Testing and tracing had a large effect. As many infections occur before or around symptom onset, timely testing and fast turnaround is important. These results support the call for investments in alternative testing technologies mina2020rethinking.
Seasonality of SARS-CoV-2 transmission has been a highly politicized topic carlson2020misconceptions. Given the high rate of susceptibility, seasonal effects on SARS-CoV-2 kinetics are limited baker2020susceptible. While weather was only one of the key determinants of transmission in the study period, its extrapolated effect could be decisive. Importantly, seasonality increases the equilibrium incidence where information stalls growth. As testing and tracing capacities are limited and crucial to prevent transmission (especially to older age groups), higher incidence could lead to an additional increase in growth rates and an unproportional increase of exposure in high risk groups.
For helpful discussions and comments I thank Johannes Bracher, Holger Brandt, Timo Dimitriadis, Simon Heß, and Zachary Roman. Yalini Ahrumukam provided excellent research assistance.\\ Funding: The work of Patrick Schmidt has been funded by the SNF.\\ Competing interests: The author declares that he has no competing interests. \\ Data and materials availability: Data was obtained from the Robert Koch Institute and the German Weather Service. All data is available in the manuscript or the supplementary materials. The data and the replication code are available at \url{https://github.com/Schmidtpk}.
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