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Publicly Available Published by De Gruyter September 15, 2020

Mathematical modeling the epicenters of coronavirus disease-2019 (COVID-19) pandemic

  • Babak Jamshidi , Mansour Rezaei EMAIL logo , Shahriar Jamshidi Zargaran and Farid Najafi
From the journal Epidemiologic Methods

Abstract

In epidemiology, the modeling of epicenters is important both conceptually and mathematically. This paper is an attempt to model epicenters mathematically. We present an algorithm to find new epicenters. Applying our model for the data related to COVID-19 pandemic, we obtain epicenters in China, South Korea, Iran, Italy, France, Germany, Spain, the USA, and Switzerland, on the days 1, 35, 42, 42, 49, 50, 50, 50, and 56, respectively. Although the number of these epicenters is less than 5% of all contaminated countries across the globe, as of March 22, 2020, they make up 74% of new cases and over 80% of total confirmed cases. Finally, we conclude that we expect to face three new epicenters between March 22 and April 1, 2020.

Introduction

Dr. Li Wenliang, a 34-year-old ophthalmologist, warned his colleagues and set the alarm to the society in December 2019 in Wuhan, China. The ongoing pandemic is an infectious disease caused by the coronaviruses, a large family that cause illness ranging from the common cold to severe diseases such as Middle East Respiratory Syndrome (MERS) and Severe Acute Respiratory Syndrome (SARS). On January 30, 2020, the World Health Organization (WHO) declared it a public health emergency of international concern.

Generally, nowadays, infectious diseases are spreading around the world faster than ever. This unprecedented rate is the result of some factors such as the increasing ease of international travel, population growth, resistance to antibiotic and antiviral drugs, and degradation of the environment. This high rate of new emerging infectious diseases stresses the importance of research on the spreading of communicable diseases.[1] Particularly, about the ongoing communicable disease, the indicators of the speed of transmission such as the relative increment, the rate of growth, and the reproductive number are higher than for other diseases caused by the coronaviruses.

This high rate of spreading is so worrying and even frightening that these days coronavirus disease-2019 (COVID-19) is the main issue in almost all countries, both developed and developing countries. Some countries have even warned that over 50% of their population could contract the disease in the near future. On March 11, 2020, the WHO declared the disease to become a pandemic based on its geographic spread, severity of illnesses it causes, and also its impact on society.

As of 22 March 2020, the disease has spread to over 185 countries and infected around 3,00,000 cases. Up to the mentioned date, the death toll had risen about 13,000 worldwide in 81 countries.[2] , [3]

Undoubtedly, due to the lack of treatment or vaccine, severity of the disease, and spreading to the wide geographical areas, this communicable disease is a major challenge for the human being in the 21st century. One of the aspects of this challenge is modeling the global propagation of the disease mathematically. Regarding a typical mathematical global model, emerging epicenters are of the utmost importance.

These days, the media are using the word “epicenter” repeatedly all over the world to describe the center of spreading, the territory with a relatively higher rate of spreading, the geographical area where an outbreak began, or wherever the communicable disease propagates remarkably. As you see, the concept of epicenter is somehow ambiguous and not well-defined.

Therefore, we need to provide a specified definition to deal with this concept in scientific texts. It is worth saying that the word ‘epicenter’ is borrowed from the field of seismology. The epicenter is the point on the Earth’s surface directly above a hypocenter or focus, the point where an earthquake or an underground explosion originates.[4] This definition dates back to 1880.[5] This paper is the first attempt to offer a scientific frame for this concept in epidemiology. Our goal is to mathematize an epidemic concept and derive a solid definition, which, based on it, we can improve the mathematical models and epidemic concepts, consecutively.

In epidemiology, the modeling of epicenters is important both conceptually and mathematically. It is conceptually important because communicable diseases are no longer behind the borders, so we should have a global viewpoint. Modeling epicenters is essential to provide global models and is the missing link between local and global modeling. Accordingly, the concept of epicenter and its modeling are matters of concern of the organizations and research projects studying the spreading of a disease worldwide such as WHO. Generally, about all pandemics, and particularly, about COVID-19, we face emerging epicenters. We need a common language; therefore, providing a clear definition is helpful.

Mathematically, a suitable model for a communicable disease should have a decelerating growth (Britton et al. 2014). Emerging epicenters significantly accelerate the spread of the disease. This acceleration contrasts with the downward trend of growth.

When a new epicenter emerges, the relative increment of it is usually extreme while the acceleration of growth worldwide is not that high and sometimes is remarkably lower. Overall, the presence of the epicenters can severely disrupt the decreasing pattern of the relative propagation of the disease. To the extent of our knowledge, the model presented by Jamshidi, Rezaei, and Rezaei (2020) is the first study that considers the role of the epicenters in a global model. In stochastic processes and mathematical modelling words, this concept is essential to deal with the self-similarity of the global and local patterns. By using this notion, we can approach better to the problem of extension of a local model to a global one.

Definition

We define the epicenters according to the following algorithm:

Step 1.

first country whose the new confirmed cases exceeds 100 is taken as the first epicenter ( i = 1 ,     T i = 1 ).

The index i represents the number of the new emerging epicenter (the first, second, third, fourth, or i-th known epicenter), and T denotes the time (the first, second, third, or i-th day of spreading). Accordingly, the notation T 3 = 12 denotes that the third epicenter emerged on the twelfth day.

Step 2.

i = i   +   1 .

Step 3.

We calculate the threshold which is equal to the first 3-quantile or the lower first 3-quantile of the increments of the existing epicenters based on the following formulation for t > T i     1 ;

Thresh i , t = { Q 1 3 ( Δ x 1 ,   Δ x 2 , ,   Δ x i 1 ) f o r t > T i 1 } ,

wherex j denotes the increment of the number of confirmed cases regarding the j-th epicenter. In order to calculate Q 1 3 ( Δ x 1 ,   Δ x 2 , ,   Δ x i 1 ) , we first sort them ascendingly, and label them asx (1), ∆x (2), …, ∆x ( 1) where ∆x (1)≤∆x (2)≤…≤∆x ( 1). Now, Δ x ( i 3 ) = ( 1 ( ( i 3 ) [ i 3 ] ) ) Δ x [ i 3 ] + ( ( i 3 ) [ i 3 ] ) Δ x [ i 3 ] + 1 is considered as the T h r e s h i , t .

Step 4.

The territories whose number of new cases exceed the threshold become the novel epicenters, and the time this happens is taken as T i .

Step 5.

Go to Step 2.

We use a recursive method based on the behavior of existing epicenters to determine new epicenters. We define the new epicenters as the regions that exceed the least one-third of the existing epicenters in new cases. Applying the algorithm, the following countries are obtained as the epicenters:

China, South Korea, Iran, Italy, France, Spain, Germany, the USA, and Switzerland with emerging time on the days 1, 35, 42, 42, 49, 50, 50, 50, and 56, respectively. Figure 1 illustrates how the new epicenters exceed the threshold and join to the existing epicenters. The first individual infected by COVID-19 was identified in Wuhan, China, and it was the first country that exceeds 100 daily cases. Therefore, China is recognized as the first epicenter. The propagation of the disease in China is illustrated by Figure 1A. Based on the first epicenter and the time series of new confirmed cases, the threshold of the second epicenter is obtained. Figure 1B shows that the number of new cases in South Korea exceeded the threshold on the 35th day (Feb 24, 2020). Based on our model, if we have two epicenters, the minimum of the number of new cases of these two epicenters is the threshold. Figure 1C shows that Iran and Italy exceed the threshold at the same time. They both exceeded China in the number of new cases on the 42nd day, March 1, 2020. France with 180 new confirmed cases on March 8 became the fourth epicenter (Figure 1D). One day later, Germany, Spain, and the USA joined the mentioned epicenters simultaneously (Figure 1E). Finally, on March 8th, as you see in Figure 1F, because of 842 new cases, Switzerland got the eighth epicenter of COVID-19.

Figure 1: 
The thresholds and the time series of the emerging epicenters for the first eight epicenters.
(A) China as the first epicenter. (B) South Korea exceeded two-third of the number of new cases of China on the day 35. (C) After the day 42, the increments of Iran and Italy are both greater than the minimum of the increments of China and South Korea. (D) France passes the weighted mean of the new cases of South Korea and China as the threshold on the day 49. (E) Germany, Spain, and the USA pass the weighted mean of the new cases of South Korea and China as the threshold simultaneously on the day 50. (F) The number of new cases in Switzerland exceeds the threshold on the day 56.
Figure 1:

The thresholds and the time series of the emerging epicenters for the first eight epicenters.

(A) China as the first epicenter. (B) South Korea exceeded two-third of the number of new cases of China on the day 35. (C) After the day 42, the increments of Iran and Italy are both greater than the minimum of the increments of China and South Korea. (D) France passes the weighted mean of the new cases of South Korea and China as the threshold on the day 49. (E) Germany, Spain, and the USA pass the weighted mean of the new cases of South Korea and China as the threshold simultaneously on the day 50. (F) The number of new cases in Switzerland exceeds the threshold on the day 56.

It is noticeable that, until the end of the studied period, as of 22 March, the epicenters we obtain are exactly the same as the countries with the most confirmed cases (Table 1).

Table 1:

The information over the spreading COVID-19 by country sorted based on the total confirmed cases as of March 22nd, 2020.3

Country Total cases New cases Total deaths New deaths Total recovered Active cases Serious critical
China 81,008 41 3,255 7 71,740 6,013 1,927
Italy 53,578 6,557 4,825 793 6,072 42,681 2,857
Spain 25,496 3,925 1,381 288 2,125 21,990 1,612
USA 24,207 4,824 302 46 176 23,729 637
Germany 22,364 2,516 84 16 209 22,071 2
Iran 20,610 966 1,556 123 7,635 11,419
France 14,459 1,847 562 112 1,587 12,310 1,525
S. Korea 8,799 147 102 8 2,612 6,085 59
Switzerland 6,863 1,248 80 24 131 6,652 141
UK 5,018 1,035 233 56 93 4,692 20
Netherlands 3,631 637 136 30 2 3,493 354
Austria 2,992 343 8 2 9 2,975 15
Belgium 2,815 558 67 30 263 2,485 288

It is worth saying that as time goes by, the threshold endures some fluctuations.

Accordingly, the threshold is a nonstationary stochastic process determined by varying laws. Each law is valid until the appearance of the next epicenter. According to our model, we get the realization of this stochastic process regarding COVID-19 illustrated by Figure 2.

Figure 2: 
The graph of threshold of the epicenters regarding the spreading of COVID-19 worldwide.
Figure 2:

The graph of threshold of the epicenters regarding the spreading of COVID-19 worldwide.

In order to highlight the epidemiological importance of the epicenters, Notice that the epicenters obtained by our model over the period January 21st–March 21st are between 5 and 10% of all the countries involved with the infection (Figure 3A).

Figure 3: 
The proportion of epicenters of COVID-19 between January 21st and March 21st.
(A) The number of countries involved with COVID-19 and the number of epicenters. (B) The cumulative number of confirmed cases belonging to epicenters from the beginning and from becoming epicenter, and worldwide. (C) The number of new confirmed cases worldwide, belonging to epicenters from the beginning and from becoming epicenter.
Figure 3:

The proportion of epicenters of COVID-19 between January 21st and March 21st.

(A) The number of countries involved with COVID-19 and the number of epicenters. (B) The cumulative number of confirmed cases belonging to epicenters from the beginning and from becoming epicenter, and worldwide. (C) The number of new confirmed cases worldwide, belonging to epicenters from the beginning and from becoming epicenter.

For example, on March 21st, 2020, there are more than 180 countries and only nine epicenters (less than 5% of the countries are recognized as epicenters). Although the proportion of the number of the epicenters is less than 10%, they possess a great proportion of the statistics regarding the pandemic COVID-19. Figure 3B represents that the epicenters make up a large proportion of the cumulative number of confirmed cases. The proportion of the epicenters is varying between 80 and 100% from all confirmed cases. On March 21, there were 29,434 new cases. Over 74% of these new cases belonged to the nine epicenters. According to the statistics,[6] , [7] the nine epicenters make up more than 80% of the death toll from the pandemic COVID-19.

To justify the model, we present some evidence that the obtained dates are special for the countries correspond to:

  1. On 12 March (three days after joining the epicenters), by arriving COVID-19 to Melilla and Cueta, all of the regions of Spain were contaminated by the pandemic.[8]

  2. On 10 March, Saxony-Anhalt as the last state confirmed the seven cases infected by COVID-19, then, one day after getting epicentre, all of the states of Germany had gotten confirmed cases.[9]

  3. By the first confirmed case in West Virginia, the USA on 17 March reported arrival of the disease to all of the states.[10]

  4. Up to 14 March (one day before joining the epicenters of our model), all of the 26 cantons of Switzerland had declared the confirmed cases infected by COVID-19.[11]

  5. In France, since 10 March (two days after becoming epicenter), all of the 13 regions have faced new daily confirmed cases in all of the days.

  6. On 21, 22, and 27 Feb 2020, South Korea experienced more than 100, 200, and 500 new daily confirmed cases for the first time, respectively. On 27 Feb, the number of new cases in South Korea exceeded this number about China and remains over it up to 20 March. Accordingly, three days after qualifying as an epicenter according to our model, South Korea became the country with the most new cases in the world.

  7. By the beginning of March, the virus had spread to all regions of Italy (Coronavirus 2020).

  8. As of 3 March, two days after getting epicenter according to our model, all of the provinces of Iran had been polluted.

Regarding the level of spreading disease in the epicenters, in all of them, the time of becoming epicenter is extremely close to contamination of all the regions of the country. In addition, the ranking of the countries by the cumulative number of confirmed cases is exactly in line with our model. It means that the epicenters of our model have the highest cumulative number of confirmed cases. Finally, they comprise a remarkable proportion of daily cases, daily deaths, and total deaths as well.

Generalization

The following alternatives can be suggested to extend or modify this model:

  1. We can take a conservative approach and consider two or three consecutive days of exceeding the threshold as the criterion rather than the first time that the number of new cases goes beyond the threshold.

  2. We can replace the first 3-quantile with the first quartile, the first quintile, the second quintile, the 31st percentile, or other quantiles. We can even use a scale varying over time.

  3. We can base our model on the daily relative increment or the cumulative number of confirmed cases as well.

  4. We can apply this model for countries as the global area and the provinces or cities as local areas.

Discussion and conclusion

The present paper models epicenters mathematically for the first time. We offered an algorithm to identify new epicenters. Applying our model for the data related to the COVID-19 pandemic, we identified China, South Korea, Iran, Italy, France, Germany, Spain, the USA, and Switzerland as the epicenters up to 21 March 2020. We must add that these countries became the epicenter of COVID-19 since the days 1, 35, 42, 42, 49, 50, 50, 50, and 56 from the first situation report of the WHO, respectively. In fact, the investigation about the new emerging epicenters can help the health policy-makers have their plans to lockdown such areas in advance. Such interventions are crucial for controlling the epidemic. In addition, informing the policy-makers about the next epicenters is one of the key steps to provide response plan against an epidemic. Although the number of these epicenters is less than 5% of all countries across the globe that the disease spread, as of March 22, 2020, they make up 74% of new cases and over 80% of total confirmed cases. In addition, it seems that coming epicenters are probably of European countries like the UK. Finally, the track of the emerging epicenters indicates that the present epicenters are situated on the same latitude, and, the next epicenters may be of the lower latitudes and in long term, they are going to the southern hemisphere.

Also, we suggest the following ways to modify existing models or extend the introduced model:

  1. considering two or three consecutive days of exceeding the threshold as the criterion,

  2. using quantiles other than the least 3-quantile,

  3. basing a model on daily relative increment, and

  4. basing a model on the cumulative number of confirmed cases.

In addition to the alternatives to extend the model, there are some suggestions to research on including the similar pattern of epicenters in connection with a special disease like COVID-19 and modeling the threshold process in terms of time series or stochastic processes.


Corresponding author: Mansour Rezaei, Professor of Biostatistics, Social Development and Health Promotion Research Center, Kermanshah University of Medical Sciences, Kermanshah, Iran, E-mail:

  1. Research funding: None declared.

  2. Author contributions: BJ: Idea, models, simulation and first manuscript; Final manuscript; SJZ: Simulation and calculation; FN: Literature.

  3. Competing interests: Authors state no conflict of interest.

References

Britton, T., T. House, A. L. Lloyd, D. Mollison, S. Riley, and P. Trapman. 2014. “Five Challenges for Stochastic Epidemic Models Involving Global Transmission.” Epidemics 10: 54–7. https://doi.org/10.1016/j.epidem.2014.05.002.Search in Google Scholar PubMed PubMed Central

Jamshidi, B., M. Rezaei, and K. Rezaei. 2020. “A New Model for Epidemic Spreading with Focus on COVID-19.” Health Scope 9 (3). https://doi.org/10.5812/jhealthscope.102837.Search in Google Scholar

Coronavirus. Colpite tutte le regioni. La Protezione civile: ecco i numeri aggiornati. Avvenire (in Italian). 5 March 2020. Retrieved 19 March 2020. Also available at: https://www.avvenire.it/attualita/pagine/coronavirus-aggiornamento-5-marzo-2020.Search in Google Scholar

Received: 2020-05-07
Accepted: 2020-08-26
Published Online: 2020-09-15

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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