Key Epidemic Parameters of the SIRV-Model Determined from Past COVID-19 Mutant Waves (preprint)

Monitored infection and vaccination rates during past past Corona waves are used to infer a posteriori two key parameter of the SIRV-epidemic model, namely the real time variation of the (i) ratio of recovery to infection rate and (ii) ratio of vaccination to infection rate. We demonstrate that using the classical SIR model the ratio between recovery and infection rates tends to overestimate the true ratio, that is of relevance in predicting the dynamics of an epidemics in the presence of vaccinations.

Determination of a Key Pandemic Parameter of the SIR-Epidemic Model from Past Covid-19 Mutant Waves and Its Variation for the Validity of the Gaussian Evolution (preprint)

Monitored differential infection rates of past Corona waves are used to infer, a posteriori, the real time variation of the ratio of recovery to infection rate as key parameter of the SIR-epidemic model. From monitored Corona waves in five different countries it is found that this ratio exhibits a linear increase at early times below the first maximum of the differential infection rate before the ratios approach a nearly constant value close to unity at the time of the first maximum with small amplitude oscillations at later times. The observed time dependencies at early times and at times near the first maximum agree favorably well with the behavior of the calculated ratio for the Gaussian temporal evolution of the rate of new infections, although the predicted linear increase of the Gauss ratio at late times is not observed.

Forecast of omicron wave time evolution (preprint)

Adopting an early doubling time of three days for the rate of new infections with the omicron mutant the temporal evolution of the omicron wave in different countries is predicted. The predictions are based on the susceptible-infectious-recovered/removed (SIR) epidemic compartment model with a constant stationary ratio k=mu(t)/a(t) between the infection (a(t)) and recovery (mu(t)) rate. The fixed early doubling time then uniquely relates the initial infection rate a0 to the ratio k, which therefore determines the full temporal evolution of the omicron waves. For each country three scenarios (optimistic, pessimistic, intermediate) are considered and the resulting pandemic parameters are calculated. These include the total number of infected persons, the maximum rate of new infections, the peak time and the maximum 7-day incidence per 100000 persons. Among the considered European countries Denmark has the smallest omicron peak time and the recently observed saturation of the 7-day incidence value at 2478 is in excellent agreement with the prediction in the optimistic scenario. For Germany we predict peak times of the omicron wave ranging from 32 to 38 and 45 days after the start of the omicron wave in the optimistic, intermediate and pessimistic scenario, respectively, with corresponding maximum SDI values of 7090, 13263 and 28911, respectively. Adopting Jan 1st, 2022 as the starting date our predictions implies that the maximum of the omicron wave is reached between Feb 1 and Feb 15, 2022. Rather similar values are predicted for Switzerland. Due to an order of magnitude smaller omicron hospitalization rate, due to the high percentage of vaccinated and boostered population, the German health system can cope with maximum omicron SDI value of 2800 which is about a factor 2.5 smaller than the maximum omicron SDI value 7090 in the optimistic case. By either reducing the duration of intensive care during this period of maximum, and/or by making use of the nonuniform spread of the omicron wave across Germany, it seems that the German health system can barely cope with the omicron wave avoiding triage decisions. The reduced omicron hospitalization rate also causes significantly smaller mortality rates compared to the earlier mutants in Germany. In the optimistic scenario one obtains for the total number of fatalities 7445 and for the maximum death rate 418 per day which are about one order of magnitude smaller than the beta fatality rate and total number.

Reasonable Limiting of 7-Day Incidence per Hundred Thousand and Herd Immunization in Germany and Other Countries (preprint)

Based on hospital capacities, facts from past experience with the coronavirus disease 2019 (COVID-19) virus and the number of dark infections during the second wave (DII = 2D2), a reasonable limiting value of 140/D2 for the 7-day incidence per 100,000 persons (MSDIHT) and a second wave herd immunization threshold fraction value of 0.26 in Germany were calculated. If the MSDIHT is held below this limiting value, the German hospital system can cope with the number of new seriously infected persons without any triage decisions. On the basis of the SIRV epidemics model, the classical threshold values for herd immunization were calculated for 18 countries. For these countries, the dates regarding when herd immunization against the second COVID-19 wave will be reached were estimated

Analytical Modeling of the Temporal Evolution of Epidemics Outbreaks Accounting for Vaccinations (preprint)

With the now available vaccination against Covid-19 it is quantitatively explored how vaccination campaigns influence the mathematical modeling of epidemics. The standard susceptible-infectious-recovered/removed (SIR) epidemic model is extended to the fourth compartment V of vaccinated persons and the vaccination rate v(t) that regulates the relation between susceptible and vaccinated persons. The vaccination rate v(t) competes with the infection (a(t)) and recovery (\mu(t)) rates in determining the time evolution of epidemics. In order for a pandemic outburst with rising rates of new infections it is required that k+b<1-2\eta, where k=\mu_0/a_0 and b=v_0/a_0 denote the initial ratios of the three rates, respectively, and \eta << 1 is the initial fraction of infected persons. Exact analytical inverse solutions t(Q) for all relevant quantities Q=[S,I,R,V] of the resulting SIRV-model in terms of Lambert functions are derived for the semi-time case with time-independent ratios k and b between the recovery and vaccination rates to the infection rate, respectively. These inverse solutions can be approximated with high accuracy yielding the explicit time-dependences Q(t) by inverting the Lambert functions. The values of the three parameters k, b and \eta completely determine the reduced time evolution the SIRV-quantities Q(\tau). The influence of vaccinations on the total cumulative number and the maximum rate of new infections in different countries is calculated by comparing with monitored real time Covid-19 data. The reduction in the final cumulative fraction of infected persons and in the maximum daily rate of new infections is quantitatively determined by using the actual pandemic parameters in different countries. Moreover, a new criterion is developed that decides on the occurrence of future Covid-19 waves in these countries. Apart from Israel this can happen in all countries considered.

Forecast for the second Covid-19 wave based on the improved SIR model with a constant ratio of recovery to infection rate (preprint)

We start out by deriving simple analytic expressions for all measurable amounts of cases and fatalities during a pandemic evolution exhibiting multiple waves, described by the semi-time SIR model. The approximant shares all relevant features with the exact solution, including time and position of the peak of daily new infections, as well as the asymptotic behaviors at small and large times. We derive exact analytic expressions for the early doubling time, late half decay time, and a half-early peak law, characterizing the dynamical evolution. We show, in particular, how the asymmetry of the first epidemic wave and its exponential tails are affected by the initial conditions; a feature that has no analogue in the all-time SIR model. We apply the approach to available data from different continents. Our analysis reveals that the immunity is very strongly increasing during the 2nd wave, while it was still at a very moderate level of a few percent in several countries at the end of the first wave. The wave-specific SIR parameters describing the infection and recovery rates we find to behave in a similar fashion, while their ratio k was decreasing only by a about 5% for most countries. Still, an apparently moderate change of k can have significant consequences for the relevant numbers like the final amount of infected or deceased population. As we show, the probability for an additional wave is however low in several countries due to the fraction of immune inhabitants at the end of the 2nd wave, irrespective the currently ongoing vaccination efforts. We compare with alternate approaches.

Reasonable Limiting 7-day Incidence per Hundred Thousand Value in Germany (preprint)

Based on the hospital capacities, facts from the past experience with the Covid-19 virus and the dark number of infections D=10D_{10} a reasonable limiting value of 170/D_{10} for the monitored 7-day incidence per 100000 persons value (MSDIHT) in Germany is calculated. If the MSDIHT is held below this limiting value the German hospital system can cope with the number of new seriously infected persons without any triage decisions. A significant improvement to an almost complete testing of the population would lead to dramatic reduction of the current dark numer value to D_{10}=0.1 so that ten times higher MSDIHT values of 1700 are acceptable. Such a high limiting value would spare Germany from its currently imposed strict lockdown. The costs for such extensive and complete testing campaigns are highly justified as they are orders of magnitudes below the estimated economical costs of more than 3.6 billion Euros for each lockdown day.

Analytical Solution of the SIR-Model for the Temporal Evolution of Epidemics. Part B. Semi-Time Case (preprint)

The earlier analytical analysis (part A) of the Susceptible-Infectious-Recovered (SIR) epidemics model for a constant ratio k of infection to recovery rates is extended here to the semi-time case which is particularly appropriate for modeling the temporal evolution of later (than the first) pandemic waves when a greater population fraction from the first wave has been infected. In the semi-time case the SIR model does not describe the quantities in the past; instead they only hold for times later than the initial time t=0 of the newly occurring wave. Simple exact and approximative expressions are derived for the final and maximum values of the infected, susceptible and revovered/removed population fractions as well the daily rate and cumulative number of new infections. It is demonstrated that two types of temporal evolution of the daily rate of new infections j(tau) occur depending on the values of k and the initial value of the infected fraction I(0)=eta: in the decay case for k > 1-2 eta the daily rate monotonically decreases at all positive times from its initial maximum value j(0)=eta (1-eta). Alternatively, in the peak case for k<1-2 eta the daily rate attains a maximum at a finite positive time. By comparing the approximated analytical solutions for j(tau) and J(tau) with the exact ones obtained by numerical integration, it is shown that the analytical approximations are accurate within at most only 2.5 percent. It is found that the initial fraction of infected persons sensitively influences the late time dependence of the epidemics, the maximum daily rate and its peak time. Such dependencies do not exist in the earlier investigated all-time case.

Epidemics forecast from SIR-modeling, verification and calculated effects of lockdown and lifting of interventions (preprint)

Due to the current COVID-19 epidemic plague hitting the worldwide population it is of utmost medical, economical and societal interest to gain reliable predictions on the temporal evolution of the spreading of the infectious diseases in human populations. Of particular interest are the daily rates and cumulative number of new infections, as they are monitored in infected societies, and the influence of non-pharmaceutical interventions due to different lockdown measures as well as their subsequent lifting on these infections. Estimating quantitatively the influence of a later lifting of the interventions on the resulting increase in the case numbers is important to discriminate this increase from the onset of a second wave. The recently discovered new analytical solutions of Susceptible-Infectious-Recovered (SIR) model allow for such forecast and the testing of lockdown and lifting interventions as they hold for arbitrary time dependence of the infection rate. Here we present simple analytical approximations for the rate and cumulative number of new infections.

Gaussian Doubling Times and Reproduction Factors of the COVID-19 Pandemic Disease (preprint)

The Gauss model for the time evolution of the first corona pandemic wave rendered useful in the estimation of peak times, amount of required equipment, and the forecasting of fade out times. At the same time it is probably the simplest analytically tractable model that allows to quantitatively forecast the time evolution of infections and fatalities during a pandemic wave. In light of the various descriptors such as doubling times and reproduction factors currently in use to judge about lock-downs and other measures that aim to prevent spreading of the virus, we hereby provide both exact, and simple approximate relationships between the two relevant parameters of the Gauss model (peak time and width), and the transient behavior of two versions of doubling times, and three variants of reproduction factors including basic reproduction numbers.

Clues from the First Covid-19 Wave and Recommendations for Social Measures in the Future (preprint)

The Gauss model for the time evolution of the first corona pandemic wave allows to draw conclusions on the dark number of infections, the amount of heard immunization, the used maximum capacity of breathing apparati and the effectiveness of various non-pharmaceutical interventions in different countries. In Germany, Switzerland and Sweden the dark numbers are 7.4 +/- 6.1, 11.1 +/- 8.5 and 25 +/- 25, respectively. Our method of estimating dark numbers from modeling both, infection and death rates simultaneously spares these countries the laborious, time-consuming and costly medical testing for antibodies of large portions of the population. In Germany the total number of infected persons, including the dark number of infections by the first wave is estimated to be 1.06 +/- 0.60 million, corresponding to 1.28 +/- 0.72 percent of the German population. We work out direct implications from these predictions for managing the 2nd and further corona waves.

Covid-19 predictions using a Gauss model, based on data from April 2 (preprint)

We propose a Gauss model (GM), a map from time to the bell-shaped Gauss function to model the casualties per day and country, as a quick and simple model to make predictions on the coronavirus epidemic. Justified by the sigmoidal nature of a pandemic, i.e. initial exponential spread to eventual saturation, we apply the GM to the first corona pandemic wave using data from 25 countries, for which a sufficient amount of not yet fully developed data exists, as of April 2, 2020, and study the model's predictions. We find that logarithmic daily fatalities caused by Covid-19 are well described by a quadratic function in time. By fitting the data to second order polynomials from a statistical chi2-fit with 95% confidence, we are able to obtain the characteristic parameters of the GM, i.e. a width, peak height and time of peak, for each country separately. We provide evidence that this supposedly oversimplifying model might still have predictive power and use it to forecast the further course of the fatalities caused by Covid-19 per country, including peak number of deaths per day, date of peak, and duration within most deaths occur. While our main goal is to present the general idea of the simple modeling process using GMs, we also describe possible estimates for the number of required respiratory machines and the duration left until the number of infected will be significantly reduced.

Covid-19 Predictions Using a Gauss Model, Based on Data from April 2 (preprint)

We propose a Gauss model (GM), a map from time to the bell-shaped Gauss function to model the deaths per day and country, as a quick and simple model to make predictions on the coronavirus epidemic. Justified by the sigmoidal nature of a pandemic, i.e. initial exponential spread to eventual saturation, we apply the GM to existing data, as of April 2, 2020, from 25 countries during first corona pandemic wave and study the model's predictions. We find that logarithmic daily fatalities caused by Covid-19 are well described by a quadratic function in time. By fitting the data to second order polynomials from a statistical chi2-fit with 95\% confidence, we are able to obtain the characteristic parameters of the GM, i.e. a width, peak height and time of peak, for each country separately, with which we extrapolate to future times to make predictions. We provide evidence that this supposedly oversimplifying model might still have predictive power and use it to forecast the further course of the fatalities caused by Covid-19 per country, including peak number of deaths per day, date of peak, and duration within most deaths occur. While our main goal is to present the general idea of the simple modeling process using GMs, we also describe possible estimates for the number of required respiratory machines and the duration left until the number of infected will be significantly reduced.

A Gaussian model for the time development of the Sars-Cov-2 corona pandemic disease. Predictions for Germany made on March 30, 2020 (preprint)

For Germany it is predicted that the first wave of the corona pandemic disease reaches its maximum of new infections on April 11th, 2020 +5.4-3.4 days with 90 percent confidence. With a delay of about 7 days the maximum demand on breathing machines in hospitals occurs on April 18th, 2020 +5.4-3.4 days. The first pandemic wave ends in Germany end of May 2020. The predictions are based on the assumption of a Gaussian time evolution well justified by the central limit theorem of statistics. The width and the maximum time and thus the duration of this Gaussian distribution are determined from a statistical {Xi}2-fit to the observed doubling times before March 28, 2020.