Results from an explorative study revealing spatio-temporal patterns of the SARS-CoV-2/COVID-19 epidemic in Germany are presented. We dispense with contestable model assumptions and show the intrinsic spatio-temporal patterns of the epidemic dynamics. The analysis is based on COVID-19 incidence data, which are age-stratified and spatially resolved at the county level, provided by the Federal Government’s Public Health Institute of Germany (RKI) for public use. Although the 400 county-related incidence time series show enormous heterogeneity both with respect to temporal features as well as spatial distributions, the counties’ incidence curves organise into well distinguished clusters that coincide with East and West Germany. The analysis is based on dimensionality reduction, multidimensional scaling, network analysis, and diversity measures. Dynamical changes are captured by means of difference-in-difference methods which are related to fold changes of the effective reproduction numbers. The age-related dynamical patterns suggest a considerably stronger impact of children, adolescents and seniors on the epidemic activity than previously expected. Besides these concrete interpretations, the work mainly aims at providing an atlas for spatio-temporal patterns of the epidemic which serves as a basis to be further explored with the expertise from different disciplines, particularly sociology and policy makers. The study should also be understood as a methodological contribution to getting a handle on the unusual complexity of the COVID-19 pandemic.

Based on comprehensible non-parametric methods, estimates of crucial parameters that characterise the COVID-19 pandemic with a focus on the German epidemic are presented. Where appropriate, the estimates for Germany are compared with the results for seven other countries (FR, IT, US, UK, ES, CH, BR) to get an idea of the breadth of applicability and a relational understanding. Thereby, only prevalence data of daily reported new counts of diagnosed cases and fatalities provided by the Johns Hopkins University are used. Drawing on uncertain a priori knowledge is avoided. Specifically, we present estimates resulting from delay-time correlations for the duration from diagnosis to death being 13 days for Germany and Switzerland. The delay-time correlation applied to time series from other countries exhibit less pronounced peaks suggesting high variabilities for the corresponding time-to-death durations. With respect to the German data, the two time series of new cases and fatalities exhibit a strong coherence within the frequency range of interest, which backs our findings. Furthermore, based on the knowledge of this time lag between diagnoses and deaths, properly delayed asymptotic as well as instantaneous fatality-case ratios are calculated having superiority compared to the commonly published case-fatality rate. The temporal median of the instantaneous fatality-case ratio with proper delay of 13-days between cases and deaths for Germany turns out to be 0.02. Time courses of asymptotic fatality-case ratios are presented for other countries which substantially differ during the first half of the pandemic, however, converge to a narrow range with standard deviation 0.57% and mean 2.4%. Additionally, the time courses of instantaneous fatality-case ratios with optimal delay for the 8 exemplarily chosen countries are calculated and compared by means of the temporal medians. Similarly to the asymptotic fatality-case ratios, the differences are much smaller than expected from mass media reports. The basic reproduction number, R₀, for Germany is estimated to be between 2.4 and 3.4. The uncertainty stems from uncertain knowledge of the generation time. A delay autocorrelation shows resonances at about 4 days and 7 days, where the latter resonance is at least partially attributable to the sampling process with weekly periodicity. The calculation of the basic reproduction number is based on an evaluation of cumulative numbers of cases yielding time-dependent doubling times as an intermediate step. This allows to infer to the reproduction number during the early phase of onset of the epidemic. In a second approach, the instantaneous reproduction number is derived from the incident (counts of new) cases and allows, in contrast to the first version, to infer to the temporal behaviour of the reproduction number during the later epidemic course. The time course of the reproduction number is compared to an alternative control measure given by the per capita growth, which largely confirms the conclusions drawn from the reproduction number. To conclude, by avoiding complicated parametric models we provide insights into basic features of the COVID-19 epidemic in an utmost transparent and comprehensible way. The perhaps most striking insight is that the fatality-case ratios do not differ between countries as much as previously suspected.

Socioeconomic conditions and social attitudes are known to represent epidemiological determinants. Credible knowledge on socioeconomic driving factors of the COVID-19 epidemic is still incomplete. Based on a linear random effects regression, a predictive model is derived to estimate COVID-19 incidence in German rural districts from local socioeconomic factors and popularity of political parties in terms of their share of vote. Thereby, time series provided by Germany's public health institute (Robert Koch Institute) of weekly notified 7-day incidences per 100,000 inhabitants per district from the outset of the epidemic in 2020 up to December 1, 2021, have been used to construct the dependent variable. Local socioeconomic conditions including share of votes, retrieved from the Federal Statistical Office of Germany, have been used as potential risk factors. Popularity of the right-wing party Alternative for Germany (AfD) bears a considerable risk of increasing COVID-19 incidence both in terms of predicting the maximum incidences during three epidemic periods (alternatively, cumulative incidences over the periods are used to quantify the dependent variable) and in a time-continuous sense. Thus, districts with high AfD popularity rank on top in the time-average regarding COVID-19 incidence. The impact of the popularity of the Free Democrats (FDP) is markedly intermittent in the course of time showing two pronounced peaks in incidence but also occasional drops. A moderate risk emanates from popularities of the Green Party (GRÜNE) and the Christian Democratic Union (CDU/CSU) compared to the other parties with lowest risk level. Socioeconomic parameters like \emph{per capita} income, proportions of protection seekers and social benefit claimants, and educational level have negligible impact. To the contrary, incidence significantly increases with population density. In order to effectively combat the COVID-19 epidemic, public health policymakers are well advised to account for social attitudes and behavioural patterns reflected in local popularities of political parties, which are conceived as proper surrogates for these attitudes. Whilst causal relations between social attitudes and the presence of parties remain obscure, the political landscape in terms of share of votes constitutes at least viable predictive "markers" relevant for public health policy making.

Diversity as well as temporal and spatial changes of the proportional abundances of different antibiotics (cycling, mixing or combinations thereof) have been hypothesised to be an effective administrative control strategy in hospitals to reduce the prevalence of antibiotic-resistant pathogens in nosocomial or community-acquired infections. However, a rigorous assessment of the efficacy of these control strategies is lacking. The main purpose here is to present a mathematical framework for the assessment of control stategies from a processual stance. To this end, we adopt diverse measures of heterogeneity and diversity of proportional abundances based on the concept of entropy from other fields and adapt them to the needs in assessing the impact of variations in antibiotic consumption on antibiotic resistance. Thereby, we derive a family of diversity measures whose members exhibit different degrees of complexity. Most important, we extent these measures such that they account for the assessment of temporal changes in heterogeneity including otherwise undetected diversity-invariant permutations of antibiotics consumption and prevalence of resistant pathogens. We apply a correlation analysis for the assessment of associations between changes of heterogeneities on the antibiotics and on the pathogen side. As a showcase, which serves as a proof-of-principle, we apply the derived methods to records of antibiotic consumption and prevalence of antibiotic-resistant germs from University Hospital Dresden. Besides the quantification of heterogeneities of antibiotics consumption and antibiotic resistance, we show that a reduction of prevalence of antibiotic-resistant germs correlates with a temporal change of similarity with respect to the first observation of antibiotics consumption, although heterogeneity remains approximately constant. Although an interventional study is pending, our mathematical framework turns out to be a viable concept for the assessment and optimisation of control strategies intended to reduce antibiotic resistance.