key: cord-259745-69wk591l
authors: Baerwolff, Guenter K.F.
title: A Contribution to the Mathematical Modeling of the Corona/COVID-19 Pandemic
date: 2020-04-06
journal: nan
DOI: 10.1101/2020.04.01.20050229
sha: 
doc_id: 259745
cord_uid: 69wk591l

The responsible estimation of parameters is a main issue of mathematical pandemic models. Especially a good choice of β as the number of others that one infected person encounters per unit time (per day) influences the adequateness of the results of the model. For the example of the actual COVID-19 pandemic some aspects of the parameter choice will be discussed. Because of the incompatibility of the data of the Johns-Hopkins-University to the data of the German Robert-Koch-Institut we use the COVID-19 data of the European Centre for Disease Prevention and Control (ECDC) as a base for the parameter estimation. Two different mathematical methods for the data analysis will be discussed in this paper and possible sources of trouble will be shown. Parameters for several countries like UK, USA, Italy, Spain, Germany and China will be estimated and used in W.O. Kermack and A.G. McKendrick's SIR model. Strategies for the commencing and ending of social and economic shutdown measures are discussed. The numerical solution of the ordinary differential equation system of the modified SIR model is being done with a Runge-Kutta integration method of fourth order. At the end the applicability of the SIR model could be shown. Suggestions about appropriate points in time at which to commence with lockdown measures based on the acceleration rate of infections conclude the paper.

At first I will describe the model. I denotes the infected people, S stands for the susceptible and R denotes the recovered people. The dynamics of infections and recoveries can be approximated by the ODE system

We understand β as the number of others that one infected person encounters per unit time (per day). γ is the reciprocal value of the typical time from infection to recovery. N is the total number of people involved in the epidemic disease and there is N = S + I + R . The empirical data currently available suggests that the corona infection typically lasts for some 14 days. This means γ = 1/14 ≈ 0,07.

The choice of β is more complicated. Therefore we consider the development of the infected persons in Germany. Figs. 1 (and 2) show the history of the last 60 days.

At the beginning of the pandemic the quotient S/N is nearly equal to 1. Also, at the early stage no-one has yet recovered. Thus we can describe the early regime by the equation dI dt = βI with the solution I(t) = I(0) exp(βt) .

To guess values for I(0) and β we fit the real behavior with the function α exp(βt). With a damped Gauss-Newton method [3] we get the value β = 0,218 for the nonlinear approximation and β = 0,175 with a logarithmic-linear regression for Germany. The values of β for Italy and Spain are greater than those for Germany (in Italy:

The resulting exponential curves are sketched in figs. 1 and 2. It is important to note that actual data for Germany can be only coarsely approximated by exponential curves. This reduces the quality of the SIR model, and limits its predictive power.

With the optimmistic choice of β-value 0,175 which was evaluated on the basis of the real data (from the Johns Hopkins University database) one gets the course of the pandemic dynamics pictured in fig. 3 . 1 . R0 is the basis reproduction number of persons, infected by the transmission of a pathogen from one infected person during the infectious time (R0 = β/γ) in the following figures. Neither data from the German Robert-Koch-Institut nor the data from the Johns Hopkins University are correct, for we have to reasonably assume that there are a number of unknown cases. It is guessed that the data covers only 15% of the real cases. Considering this we get a slightly changed result pictured in fig. 4 . The maximum number of infected people including the estimated number of unknown cases is a bit higher than the result showed in fig. 3 . This can be explained by the small reduction of the S stock. 1 I0 denotes the initial value of the I species, that is March 27th 2020. Imax stands for the maximum of I. The total number N for Germany is guessed to be 75 millions.

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With the data β = 0,25 and γ = 0,05 (corresponds to 20 days to heal up or to join the species R), we get the epidemic dynamics showed in fig. 5 . For N we take a value of 70 millons.

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The effects of social distancing to decrease the infection rate can be modeled by a modification of the SIR model. The original ODE system (1)-(3) was modified to

κ is a function with values in [0,1]. For example

is the duration of the temporary lockdown in days). A good choice of t 0 and t k is going to be complicated. Some numerical tests showed that a very early start of the lockdown resulting in a reduction of the infection rate β results in the typical Gaussian curve to be delayed by I; however, the amplitude (maximum value of I) doesn't really change.

The result of an imposed lockdown of 30 days with t 0 = 0 and t 1 = 30 and reduction value κ equal to 0,5 (it means a reduction of contacts to 50 %) is pictured in fig. 6 . There is not a genuine profit for the fight against the disease.

One knows that development of the infected people looks like a Gaussian curve. The interesting points in time are those where the acceleration of the numbers of infected people increases or decreases, respectively.

These are the points in time where the curve of I was changing from a convex to a concave behavior or vice versa. The convexity or concavity can be controlled by the second derivative of I(t). Let us consider equation (2). By differentiation of (2) and the use of (1) we get

With that the I-curve will change from convex to concave if the relation

is valid. For the switching time follows

A lockdown starting at t 0 (assigning β * = κβ, κ ∈ [0,1[) up to a point in time t 1 = t 0 + ∆ t , with ∆ t as the duration of the lockdown in days, will be denoted as a dynamic lockdown (for t > t 1 β * was reset to the original value β). t 0 means the point in time up to which the growth rate increases and from which on it decreases. Fig. 7 shows the result of such a computation of a dynamic lockdown. The result is significant. In fig. 9 a typical behavior of d 2 I dt 2 is plotted. The result of a dynamic lockdown for Italy is shown in fig. 8 Data from China and South Korea suggests that the group of infected people with an age of 70 or more is of magnitude 10%. This group has a significant higher mortality rate than the rest of the infected people. Thus we can presume that α=10% of I must be especially sheltered and possibly medicated very intensively as a highrisk group. Fig. 10 shows the time history of the above defined high-risk group with a dynamic lockdown with κ = 0,5 compared to regime without social distancing. The maximum number of infected people decreases from approximately 1,7 millions of fig. 11 the infection rate κβ which we got with the switching times t 0 and t 1 is pictured.

This result proves the usefulness of a lockdown or a strict social distancing during an epidemic disease. We observe a flattening of the infection curve as requested by politicians and health professionals. With a strict social distancing for a limited time one can save time to find vaccines and time to improve the possibilities to help high-risk people in hospitals.

To see the influence of a social distancing we look at the Italian situation without a lockdown and a dynamic lockdown of 30 days with fig. 12 (κ = 0,5) for the 10% high-risk people.

If we write (2) or (5) resp. in the form

we realize that the number of infected people decreases if

is complied. The relation (8) shows that there are two possibilities for the rise of infected people to be inverted and the medical burden to be reduced.

a) The reduction of the stock of the species S. This can be obtained by immunization or vaccination. Another possibility is the isolation of high-risk people (70 years and older). Positive tests for antibodies reduce the stock of susceptible persons.

b) A second possibility is the reduction of the infection rate κβ. This can be achieved by strict lockdowns, social distancing, or rigid sanitarian moves.

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is the (which was not peer-reviewed) The copyright holder for this preprint . https://doi.org/10.1101/2020.04.01.20050229 doi: medRxiv preprint The results are pessimistic in total with respect to a successful fight against the COVID-19-virus. Hopefully the reality is a bit more merciful than the mathematical model. But we rather err on the pessimistic side and be surprised by more benign developments.

Note again that the parameters β and κ are guessed very roughly. Also, the percentage α of the group of high-risk people is possibly overestimated. Depending on the capabilities and performance of the health system of the respective countries, those parameters may look different. The interpretation of κ as a random variable is thinkable, too. [3] G. Bärwolff, Numerics for engineers, physicists and computer scientists (3rd ed., in German). Springer-Spektrum 2020.

[4] Toshihisa Tomie, Understandig the present status and forcasting of COVID-19 in Wuhan. medRxiv.preprint 2020.

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is the (which was not peer-reviewed) The copyright holder for this preprint . https://doi.org/10.1101/2020.04.01.20050229 doi: medRxiv preprint Figure 12 : Italian history of the infected people of high-risk groups depending on a dynamic lockdown (it)

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is the (which was not peer-reviewed) The copyright holder for this preprint . https://doi.org/10.1101/2020.04.01.20050229 doi: medRxiv preprint

A contribution to the mathematical theory of epidemics