Latest update: September 21st, 2020.

Khailaie^1,* Sahamoddin, Mitra^1,* Tanmay, Bandyopadhyay^1,§ Arnab, Schips^1,§ Marta, Mascheroni¹ Pietro, Vanella² Patrizio, Lange² Berit, Binder¹ Sebastian, Meyer-Hermann^1,3,4,# Michael

¹Department of Systems Immunology and Braunschweig Integrated Centre of Systems Biology (BRICS), Helmholtz Centre for Infection Research, Rebenring 56, D-38106 Braunschweig, Germany
²Department of Epidemiology, Helmholtz Centre for Infection Research, Inhoffenstraße 7, D38124 Braunschweig, Germany
³ Institute for Biochemistry, Biotechnology and Bioinformatics, Technische Universität Braunschweig, Braunschweig, Germany
⁴Cluster of Excellence RESIST (EXC 2155), Hannover Medical School, Carl-Neuberg-Strasse 1, 30625 Hannover, Germany
^*shared first author, alphabetic order
^§equal contribution, alphabetic order
^# corresponding author

This report is a frequently updated analysis of the model described in our article.

• Time-varying reproduction number for Germany
  • Prospective daily new infected cases
  • Current reproduction number (R_t) for federal states
• Evolution of reproduction number for federal states of Germany
  • R_t over time for individual federal states
• Our model
  • Model structure
  • Model equations
  • Parameter values
  • Reproduction number

The basic reproduction number (R_0) is a good measure for the long-term evolution of an epidemic that can be derived from models. However, it assumes constant conditions over the whole period analysed. We opted for a shifting time window in each of which the reproduction number (R_t) is determined (time-window of 1 week and shifting time of 1 day). We developed an automatized algorithm for the fast analysis of the current R_t. Importantly, each time window is not analysed independently but includes the history of the epidemic by starting from the saved state of the simulation at the beginning of each time window. This analysis was developed for the sake of providing a daily updated evaluation of the reproduction number suitable to support political decisions on non-pharmaceutical interventions in the course of the CoV-outbreak and applied to German data.

The estimate of the reproduction number is based on confirmed cases exclusively and ignores any estimate of unreported cases.

Because of dynamic changes in the reported case numbers of the final days as the result of delayed reporting, calculating Rt on those might be misleading. Therefore, we do not show the result for the two last days.

We are not considering any delay in reporting for the model. This issue is resolved over time by frequent updates in the data and simulation.

Interpretation of the reproduction number: It is important to note that, while the rate of new infections is decreasing for values of R_t<1, achieving a value of slightly below 1 by itself is not sufficient but means that no uncontrolled outbreak is imminent. In particular, it is not an indication that contact restrictions should be released, as this will most likely increase its value to greater than 1 again. In fact, the value should be substantially lower than 1, as the time to eradication of the virus from the population can still be very long for values of slightly below 1.

Time-varying reproduction number for Germany

Data for Germany were fitted to the cumulative number of reported cases in a sliding time window with a size of one week. The transmission rate R_1 was varied to fit to the data. Parameter sets were randomly sampled within the specified ranges (see parameter values, parameter set from literature) and, upon refitting, this induced a variability of reported R_t values. The box plot shows the 25 and 75 percentiles as well as the min and the max values. Outliers were removed.

Our method to calculate R_t​ was kept the same during the whole epidemic period.

Prospective daily new infected cases

The simulations from the previous section are continued from the final date assuming a stable R_t value. The prospective number of daily new cases (symptomatic and reported) is shown for the next three weeks. The box plot shows the 25 and 75 percentiles as well as the min and the max values. Outliers were removed.

Current reproduction number (R_t) for federal states

Evolution of reproduction number for federal states of Germany

The same analysis as in the previous section was performed for all federal state in Germany separately. Only median values are shown.

R_t over time for individual federal states

Baden-Württemberg

Bayern

Berlin

Brandenburg

Bremen

Hamburg

Hessen

Mecklenburg-Vorpommern

Niedersachsen

Nordrhein-Westfalen

Rheinland-Pfalz

Saarland

Sachsen

Sachsen-Anhalt

Schleswig-Holstein

Thüringen

Our model

Based on a classical model of infection epidemics, we developed a mathematical model particularly adapted to the requirements and specificities of the CoV-outbreak (SECIR model). An in-depth description can be found in our article.

Model structure

The scheme of the SECIR model. The model distinguishes healthy individuals without immune memory of CoV (S), infected individuals without symptoms but not yet infectious (E), infected individuals without symptoms who are infectious (C_{I,R}), and detected (I_{H,R}) and undetected (I_X) symptomatic patients. Further, compartments for hospitalization (H_{U,R}) and intensive care units (U_{D,R}) were introduced to monitor the load on the healthcare system. Detected patients recover from different states of the disease (R_Z) or die (D). Undetected individuals who went through the infection are monitored (R_X).

Model equations

April 10th, 2020 - The model equations read:

Parameter values

Reproduction number

The basic reproduction number (R_0) represents how many new infections are caused at the start of an epidemic by one individual capable of transmitting the disease to other susceptible persons. The chance of transmission is directly proportional to the contact frequency (\lambda), i.e., how many close contacts a person makes on an average per day. If we assume that the virus has an intrinsic transmission probability of v during each of these close contacts, the parameter R_1 depicts the overall chance of transmission per day due to close contacts and is given by R_1= \lambda v. Obviously, to calculate R_0, we also need to consider the duration for which an individual remains infectious.

At the start of a new epidemic, it is fair to assume that nobody in the population of size N_0 is immune to the disease and hence, the number of susceptible population S_0 at the beginning is same as N_0. Hence, the fraction of individuals who can catch infection initially is one w.r.t. the overall population. As the disease progresses, some people get immune to the disease following their recovery and some of them die due to the infection. This impacts the ratio of the susceptible population and the overall population as both face a reduction due to recovery and death, respectively. Moreover, due to various political measures and people’s responsiveness to them, the contact frequency (\lambda) also decreases thereby impacting R_1. Hence, effectively the reproduction number becomes a time-varying number mainly dependent on (S(t))/(N(t)) and R_1(t). If the infectivity period of an exposed person is I_p, the person would cause new R_t= (S(t))/(N(t)) R_1 (t) I_p infections.

Let’s now investigate what the infectivity period is for the group of individuals of different compartments of our model. The asymptomatic carriers (C_R) who consist of \alpha fraction of the exposed population, are capable of infecting others before they recover, i.e., for 1/R_9 days on an average. The pre-symptomatic carriers (C_I), representing (1-\alpha) fraction of the exposed population can infect others for an average duration of 1/R_3 days. All the undetected symptomatic cases (I_U), i.e., the fraction of (1-\alpha)(1-\mu) among the exposed remain infectious for a period of 1/R_4 days following the onset of their symptoms. Similarly, the symptomatic individuals who eventually get detected but don’t require hospitalization (I_R), i.e., the fraction of (1-\alpha)(1-\rho)\mu among the exposed persons remain infectious for 1/R_4 days. On the other hand, the detected symptomatic persons eventually requiring hospitalization (I_H) who represents the fraction (1-\alpha)\rho\mu among the exposed people can spread the infection for 1/R_6 days on average following the onset of their symptoms and before getting admitted to hospital. We assume that once somebody is admitted to a hospital cannot infect others because of suitable measures, proper isolation, and PPEs given to healthcare workers. Once somebody is detected to have an infection but still at home quarantine can still pose some risk to the susceptible population depending on how strict regulation the person follows while in-home quarantine. This risk factor is represented by \beta.

Hence, on average, the infectivity period of the disease is given by the summation of fractions of exposed present in a particular compartment multiplied by its average infectivity period. For the detected symptomatic cases, we also need to take into account the risk factor to spread the infection while in-home quarantine. This would give an average infectivity period, I_p = \left[ \frac{\alpha}{R_9} + \frac{1-\alpha}{R_3} + \frac{(1-\alpha)(1-\mu)}{R_4} + \frac{\beta (1-\alpha) (1-\rho) \mu}{R_4} + \frac{\beta (1-\alpha) \rho \mu}{R_6} \right] days for an exposed person in the population. This would mean that an exposed person would cause new R_t infections in the population,

R_t = R_1(t) \frac{S(t)}{N(t)} \left( \frac{\alpha}{R_9} + \frac{1-\alpha}{R_3}  +  \frac{(1-\alpha)(1-\mu)}{R_4} + \frac{\beta (1-\alpha) (1-\rho) \mu}{R_4}   + \frac{\beta (1-\alpha) \rho \mu}{R_6} \right)

and can be mathematically derived using the next generation matrix method from our model equations.

A detailed description of deriving R_t formula can be found in this document.