key: cord-335886-m0d72ntg
authors: Tomie, Toshihisa
title: Relations of parameters for describing the epidemic of COVID―19 by the Kermack―McKendrick model
date: 2020-03-03
journal: nan
DOI: 10.1101/2020.02.26.20027797
sha: 
doc_id: 335886
cord_uid: m0d72ntg

In order to quantitatively characterize the epidemic of COVID―19, useful relations among parameters describing an epidemic in general are derived based on the Kermack-McKendrick model. The first relation is 1/τgrow=1/τtrans−1/τinf, where τgrow is the time constant of the exponential growth of an epidemic, τtrans is the time for a pathogen to be transmitted from one patient to uninfected person, and the infectious time τinf is the time during which the pathogen keeps its power of transmission. The second relation p(∞) ≈1−exp(−(R0−1)/0.60) is the relation between p(∞), the final size of the disaster defined by the ratio of the total infected people to the population of the society,and the basic reproduction number, R0, which is the number of persons infected by the transmission of the pathogen from one infected person during the infectious time. The third relation 1/τend=1/τinf−(1−p(∞))/τtrans gives the decay time constant τend at the ending stage of the epidemic. Derived relations are applied to influenza in Japan in 2019 for characterizing the epidemic.

We reported the understanding of the present status and forecasting of pneumonia by in China which is supposed to have originated in Wuhan by analyzing the data up to February 11 (ref.1). In ref. 1 , we clarified that the behavior of the epidemic was different in different regions and that the outbreak was well described by a Gaussian as was for influenza in Japan. We reported the following; 1. the epidemic in China passed the peak in the beginning of February, 2. the date of the epidemic peak was different by region and was the latest in Wuhan. For more than 10 days after our forecast, the epidemic closely followed our forecast. The new patients outside Wuhan decreased to less than 4 % of that at the peak around February 5.

Although the epidemic in China is near the end, patients by COVID-19 are found in many other countries, and COVID-19 is still a big fear of the people over the world. In order to forecast the epidemic of COVID-19 in other countries, we need to theoretically characterize the COVID-19 epidemic by fitting a model calculation to the data observed in China. We choose the Kermack-

2) for our analysis. The Kermack-McKendrick model was proposed as early as 1927, but still, it is the basis of many modified models for describing epidemics. We want to analyze the epidemic of COVID-19 in detail by applying a model. As a preparation of the analysis, useful relations of parameters describing an epidemic are derived in the present paper.

Useful relations of parameters of an epidemic are Eqs. (7) to (10) in the following, which are derived from the Kermack-McKendrick model (ref.2). The number of susceptible people in the group is set as S 0 , the number of infected persons is I(t), the number of persons who have been infected and recovered to obtain immunity or have died is R(t), the number of persons who is susceptible but not yet infected is S(t). S 0 = S(t) + I(t) + R(t). When the transmission power of the disease is set as β and the recovery rate from the infection is γ, the epidemic of the disease is given by the following three differential equations. the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint (which was not peer-reviewed) is . https://doi.org/10.1101/2020.02.26.20027797 doi: medRxiv preprint

We define the initial transmission time constant, τ trans , after which a pathogen is transmitted to one susceptible host in the early stage of the epidemic and the infectious time of a pathogen, τ inf , after which the pathogen loses the power of transmission as follows,

The basic reproduction number, R 0 , which is the number of persons infected by the transmission of a pathogen from one infected person during the infectious time, is given by

In the following, we see that an epidemic starts with exponential growth with a time constant τ grow given by

and decays exponentially with a time constant τ end given by

Here, p(∞) is the final size of the disaster which is defined by the ratio of the total number of the infected people, R(∞), to the population of the society, S 0. As shown later, p(∞) is approximated as,

By using R 0 , Eq. (7) is rewritten as

R 0 , τ inf, and p(∞) are parameters describing an epidemic. The above relations, Eqs. (7) to (10), are derived as follows.

In the beginning of the epidemic, I(t) and R(t) are negligible compared to S 0 , and Eq. the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint (which was not peer-reviewed) is which gives exponential decay of the patients.

The time constant of the decay τ end is given by

Thus, Eq. (8) is derived.

The final size equation was numerically solved and the result is shown by the solid curve in Fig.2 .

As R 0 increases from 1, p (∞) increases from zero and saturates at a large R 0. When R 0 is larger than 2.5, more than 90 % of people in the group are 

We apply the above model to the epidemic of influenza in Japan as cited in ref.1 as a reference for the general epidemic. Figure 3 shows the influenza epidemic in Japan over the past decade (ref. the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint (which was not peer-reviewed) is from 0.52 weeks, the time constant of the rising of the epidemic can be increased to 1.5 weeks in the model calculation, but the width of the epidemic in the model calculation was too wider than the real one as shown in Fig.6 .

In all the cases of the epidemic, there should be a difference between the reported number of patients and the real number of infected people. Not all infected people will go to a hospital and not all patients are not inspected by medical institutes. In analyzing statistics, we assume the ratio of the reported number to the real number is constant during the epidemic, but often the assumption can be wrong.

We expect the ratio is smaller in the beginning and at the ending of an epidemic. Then, the time constant of the "apparent rise" and "apparent ending" will be shorter than the real ones. However, this expectation was the opposite in Fig.5 .

The slower increase and the slower decrease at the skirt of the epidemic could suggest the transmission power β of the virus may change in time, which, we think, is not plausible. At present, we do not know the reason why the simple Kermack-McKendrick cannot reproduce the whole epidemic including the skirt parts. Thus, from the analysis of JpnInf2019 by the model described above, we learn that it is important to remember that the time constant of the "apparent" increase of the epidemic in the early stage does not re-produce the whole epidemic.

The above information helps greatly to understand the to-be-planned analysis of COVID-19.

Useful By applying the model, we found that the epidemic of influenza in Japan in 2019 was re-produced by the parameters;τ trans = 0.52 week and τ inf = 1 week and that τ grow observed in the early stage can be different from τ grow for re-producing the overall epidemic.

Understanding the present status and forecasting of COVID-19 in Wuhan

A contribution to the mathematical theory of epidemics

National Health Commission of the People's Republic of China, Prevention of Epidemics

Health Commission of Hubei Province, Anti-Prevention New Coronary Disease Infection, Information

Number of reports at fixed disease points-Comparison with the past 10 years

National Institute of Infectious Diseases and Ministry of Health, Labor and Welfare Tuberculosis Infectious Disease Division