# Using Mathematical Model to Predict the Impact of COVID-19 Pandemic
###### tags:`Project`
A. Wu *, Mt. San Antonio College, ENGR 7, Prof. Eugene Mahmoud, 01/31/2021*
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***Abstract*—The current research is dedicated of elaborating the pattern of COVID-19 between different regions across the United States including New York County, NY. King County, WA. And New Orleans, LA. By using the SIRD model as the reasonable foundation. By using the SIRD model, we are allowed to find the correlation within these major cities and to develop a tractable prediction of the on-going pandemic by the pattern found with mathematic tools.**
***Index Terms*—COVID-19, SIRD Model, Prediction, New York County, Kings County, New Orleans, Transmission Rate, Statistics.**
## I. INTRODUCTION
Since the end of 2019, the world was suffering from the pandemic caused by the Corona Virus 2019, or COVID-19. It causes severe damage to the global society thousands of patients could not survived from this lethal event. In this study, we used the SIRD model, which is one of a standard model in epidemiology, to fit and develop a tractable model of deaths, positive COVID-19 tests, total tests, and the hospitalization numbers from early 2020 to Jan 24, 2022. Analyzing the data for conducting further predictions to prevent the sorrows happen again in this pandemic. [1] We collected the real data in New York County, NY. King County, WA. And New Orleans, LA. for creating mathematical models with the parameters of the infection rate, incubation period, and death rate in local outbreaks to fit the SIDR model.
Description automatically generated](Aspose.Words.74bf02c1-bf65-4813-baa3-39e151833d5b.001.png)Figure 1: SIRD Model of New York County
## II. Mathematical operations
The SIRD model is consisted with 4 differential equations S(t), I(t), R(t), D(t). Each of them represents the capital letter of the name of this model, Susceptible, Infections, Recovered, and Deaths.
Description automatically generated](Aspose.Words.74bf02c1-bf65-4813-baa3-39e151833d5b.002.png)Figure 2: SIRD Model
The correlations of them could be represented in terms of the three parameters β, μ, and Pd as shown:
Note that we are only considering the that the patient gets either recovered or death in the end of the infection stages. The cross infections and repeatedly infections are not considered in this model. [2]
### A. *The Coefficients*
𝜷 represents the contacts per day between the infected and the susceptible that are sufficient to spread the disease.
𝝁 is the fixed fraction of the infected group that will recover during any given day.
pd is the possibility of death. In this model, one can only be either recovered or dead from the infections, so we can use this parameter represents the death rate and recovery rate.
**Ro** is the ratio of 𝜷 to 𝝁, is known as the contact number where we can calculate it as:
The contact number measures the relative contagiousness of the disease, because it tells us indirectly how many the contacts are close enough to actually spread the disease.
### B. *Finding S(t), I(t), R(t), and D(t)*
***S(t)*** represents the susceptible population at time t, it is given that the total population – the initial infected population (I0) of the region equals S(0) since we assume that everyone lives in the selected region is susceptible.
***I(t)*** represent the infected population at time t. It is a value being hypothetically made for the mathematical models. We set the value of initial infected population I(0) differently instead, assuming that the I(0) values for all three counties we are discussing as the table shown:
|New York County|King County|New Orleans|
| :- | :- | :- |
|150|8|1|
***R(t)*** and ***D(t)*** separately represents the recovery population and deaths population on time t. Both of them are set equal to 0 since it is the beginning of the pandemic.
### C. *Scaling*
In order to obtain a valid and reliable results in terms of data plotting, using the scale to reflect and represent the real situation is more reasonable than directly using the original numerical results. We can obtain the scaled data by the following set of equations, where **N** represents the total population of the selected regions [3]:
## III. REAL DATA
To start calculating the model and to form the plots, we must first figure out the existing data sets we could possibly obtain.
### A. *Total Populations*
|New York County|King County|New Orleans|
| :- | :- | :- |
|7900000|2253000|390144|
### B. *Total Deaths*
|New York County|King County|New Orleans|
| :- | :- | :- |
|5207|2244|1032|
Figure 3: Total Deaths
### C. *Total Positive Tests*
|New York County|King County|New Orleans|
| :- | :- | :- |
|385249|300503|76754|
Figure 4: Total Positive Tests
### D. *Total Tests*
|New York County|King County|New Orleans|
| :- | :- | :- |
|33837495|4229452|1770108|
Figure 5: Total Tests Conducted
### E. *Total Hospitalization*
|New York County|King County|New Orleans|
| :- | :- | :- |
|141320|10488|155948|
Figure 6: Total Hospitalizations
## IV. Data Models & Analysis
The data in the current research are collected from the official database of the government in each different region. [4] [5] [6] Once we obtained all the necessary information and real data that is needed, we can use the model and stretch it out to 1400 days after the day when the first death was recorded. The stretch will allow the prediction data to be examined and estimate a date when the death ratio increase reaches a plateau. The plateau stage of the plot will suggest the coming of the end of the pandemic.
As we can see, the slope in Figure 7 is decreasing per wave. Each wave the slope is massively decreased after the first wave. Which indicates that the COVID-19 in this model is gradually hitting its limit as the time keeps move on. Note that although the slope itself is decreasing, the change in y values remains almost the same in each wave.
Figure 7: New Orleans Death Prediction
The parameters μ and β could be calculated by equation 5 into Ro as shown:
Table F: The Ro of New Orleans
|Days|Ro|
| - | - |
|1 ~ 300|1.94|
|301 ~ 500|15.33|
|501~733|9.33|
` `In the model of New York County, we can obviously find the same pattern of the flattening curve even easier. The change in y values is massive in this case, which indicates that not only does the virus kill people slower, but also it is more difficult to kill people.
Figure 8: New York County Death Prediction
Table G: The Ro of New York County
|Days|Ro|
| - | - |
|1 ~ 250|2.19|
|251 ~ 550|3.00|
|551~733|1.64|
As for the King County model, it is the least obvious one in terms of massively decreasing slope in these three models. The second wave of King County death model actually has a slightly bigger slope value compares to the slope 1. But it is indeed decreasing the slope after wave 3.
Figure 9: King County Death Prediction
Table H: The Ro of King County
|Days|Ro|
| - | - |
|1 ~ 225|7.8|
|226 ~ 525|8.9|
|526~733|9.7|
The instructor Eugene Mahmoud elaborated the importance of the consideration of population density that the population density in fact has a higher impact potential on transmission rate than total population[7].
Table I: The Population Density
|County|Population Density (ppl/sq mi)|
| - | - |
|New York County|71510 |
|King County|1073|
|New Orleans|2292|
From the table we realize that the density of New York County is 70 times than King County, 35 times than New Orleans County.
Figure 10: Total Deaths D(t) Comparison
Figure 11: Total Death Ratio d(t) Comparison
From the figure 10 and 11, we can see that the New Orleans County is the most serious county in this case. The total death ratio up to January 24, 2022:
Table J: The Death Ratio
|County|Overall Death Ratio **d(t)**|
| :- | :- |
|New York County|0.063629%|
|King County|0.096005%|
|New Orleans |0.2645%|
Figure 12: Monte Carlo Simulation for New Orleans
Figure 13: Monte Carlo Simulation for NYC
Figure 14: Monte Carlo Simulation for King County
Figure 15: New Orleans SIRD Model
Figure 16: New York County SIRD Model
Figure 17: King County SIRD Model [8]
## V. Conclusion
### A. *The end of the pandemic*
From figure 7, 8, and 9, we can clarify that it takes approximately 220 days to complete one cycle, the plateau stage of the curve in the death plots tells us great about the end of the pandemic. All of the evidence are showing that we are at the beginning of the “end cycle” of COVID-19, and all of the predictions indicate that there might not be another wave started before day 1400 after the first day of virus outbreaking.
### B. *The Transmission and Death Ratio*
The New York County has the highest population density compares to other two county. Yet it has the least death ratio and the lowest transmission rate.
### C. *The R0 value*
From Table F, G, and H. We can see the pattern of the transmission rate R0. In table F, the transmission rate goes up to 15.33 and backdown to 9.33, indicating that the peak of virus transmission peak is over. In table G, the transmission rate follows the same patter as table F, the virus transmission peak is over as well. As for table H, the R0 value is constantly increasing, which implies that the transmission rate is steadily increasing.
### D. *The peak of the pandemic*
From figure 7, 8, and 9, we can observe that the first wave is usually the biggest wave of causing deaths. The reason why we use the death data is because that the data for death will be the most reliable indicator in all the SIRD variables since the uncertainty of death is the lowest. Total cases (infected population) will have a high uncertainty since not all COVID-19 positive populations are tested and displayed in the data collection.
By observing the figure 8 to 10, we can conclude and generate a table indicating the peak in each wave:
||Wave 1 peak|Wave 2 peak|Wave 3 peak|
| :- | :- | :- | :- |
|NYC|6/3/20|4/28/21|1/24/22|
|King County|5/21/20|3/23/21|1/7/22|
|New Orleans|5/21/20|3/17/21|21/10/21|
## VI. Discussion
### A. *Recovery*
From figure 14 and 15, we can observe that there’s a massive recovery around day 170 ~ 220 from February 1, 2020. Since the vaccine doesn’t cure the decease, it only prevents from getting a decease. We do not currently have any cure for COVID-19 virus, the only reasonable hypothesis is that around that time, there exist a massive number of people who previously got infected at the same time. We can rational that by clarifying the covid’s virus incubation period is steady. The SIRD models work here as delayed indicators.
### B. *Policies*
I surprisingly find out that the New York County has the highest population density, which is one of the most important factors in the pandemic cases like this, but at the same time it has the lowest transmission rate, lowest deaths, and relatively low hospitalization rate. We observed the correlation between total tests rate and the total deaths rate for New York County and New Orleans, we found that the reason why New Orleans has a death rate about 0.2645%, which is 4 times than the death rate in New York County; and New Orleans also has a 3.5 times R0 value than New York County, is because that the tests rate in New Orleans is really low. We can now specify that the death rate and transmission rate have a positive correlation. The tests rate and the deaths rate have a negative correlation.
### C. *Errors*
There are several expected errors in this study, including the accuracy of modeling, the reliability of the sample data, and the statistics errors.
1. For the accuracy of modeling, we are assuming that the people who got infected would either die or recover, and never get infected again. Which means that the virus’ elimination is only a time manner after all of the susceptible are infected and have gone through that stage. Moreover, we assumed that the susceptible is the total population, which is also incorrect. Some of the nerdy students who don’t even have a friend and stay in their room all the time would almost be impossible to be infected. Plus, the SIRD model does not consider the manner of exposure or vaccination.
1. For the reliability of sample data, there are some of the people who doesn’t want to get tested, this could not be diagnosed. This significantly change the data collection because that means the real data will never be possibly known. That is also why we often choose the death data as the predicting model. However, the death data aren’t always correct. Because when one is observed or declared as dead would not be instantly clarified if him or her death is caused by COVID-19. This means it usually take more time to determine and reflect to the data. Which might possibly cause the inaccuracy and errors.
1. As for the statics errors, like I previously mentioned about the death reflection on the data, there always have some delay between the reality tests and the data. For example, the data provided by New Orleans government isn’t correct on every weekend, since there’s no one working there. The data were intentionally remained as the same value they have on Friday all over the weekend. The best way to solve this problem is by looking at 7-days average data instead of daily data.
### D. *Vaccination*
` `Consider about the vaccination, the impact of the future vaccine on population susceptibility would be decreasing the S(0) value. Like I previously mentioned, we are not considering any vaccination in the current model, which means most of the populations are considered susceptible. The impact that caused by vaccination here is to drastically reduce the susceptible potential as much as possible. If we consider the R0 value as 5, we can put it as an equation that:
And we also have to assume that the vaccination’s protectability is about 100% as well, we will have 80% of the population who need to be vaccinated in order to fit the size of inoculation.
## VII. References
[1] Author InformationARTICLE SECTIONSJump ToCorresponding AuthorDebasis Sen - SolidState Physics Division, Author, C., Debasis Sen - SolidState Physics Division, [email protected], E., Author, Devosmita Sen - Departmentof Chemical Engineering, Notes, & The authors declare no competing financial interest. (n.d.). *Use of a modified SIRD model to analyze COVID-19 data*. ACS Publications. Retrieved January 31, 2022, from https://pubs.acs.org/doi/10.1021/acs.iecr.0c04754
[2] *Power bi report*. Covid-19 New Orleans Dashboard. (n.d.). Retrieved January 31, 2022, from https://app.powerbigov.us/view?r=eyJrIjoiMWUxZjFjM2ItOTI0ZS00MTcxLWJjYjgtODQwNzg2MDRhMmU3IiwidCI6IjA4Y2JmNDg1LTFjYjctNGEwMi05YTIxLTBkZDliNDViOWZmNyJ9&pageName=ReportSection
[3] Author InformationARTICLE SECTIONSJump ToCorresponding AuthorDebasis Sen - SolidState Physics Division, Author, C., Debasis Sen - SolidState Physics Division, [email protected], E., Author, Devosmita Sen - Departmentof Chemical Engineering, Notes, & The authors declare no competing financial interest. (n.d.). *Use of a modified SIRD model to analyze COVID-19 data*. ACS Publications. Retrieved January 31, 2022, from https://pubs.acs.org/doi/10.1021/acs.iecr.0c04754
[4] *Power bi report*. Covid-19 New Orleans Dashboard. (n.d.). Retrieved January 31, 2022, from https://app.powerbigov.us/view?r=eyJrIjoiMWUxZjFjM2ItOTI0ZS00MTcxLWJjYjgtODQwNzg2MDRhMmU3IiwidCI6IjA4Y2JmNDg1LTFjYjctNGEwMi05YTIxLTBkZDliNDViOWZmNyJ9&pageName=ReportSection
[5] Nychealth. (n.d.). *Nychealth/coronavirus-data: This repository contains data on coronavirus disease 2019 (COVID-19) in New York City (NYC), from the NYC Department of Health and Mental Hygiene.* GitHub. Retrieved January 31, 2022, from https://github.com/nychealth/coronavirus-data
[6] *US covid-19 cases and deaths by State*. USAFacts.org. (2022, January 29). Retrieved January 31, 2022, from https://usafacts.org/visualizations/coronavirus-covid-19-spread-map/
[7] E. Mahmoud, general assistance, January 2022
[8] D. Lu, general assistance, January 2022
## VIII. Biography
**First Author:** Aaron Wu is from Kaohsiung, Taiwan. Graduated from Kaohsiung Senior High School, professional music program, majoring in Trumpet.
After graduating from high school, he worked as a research assistant at the Space and Rocket Propulsion Laboratory at National Cheng-Kung University (NCKU) with Professor Yei-Chin Chao, focusing on an H2O2 P-Class hybrid rocket project.
After having gained experience in rocket engineering, he founded and led the Institute of Space Propulsion at NCKU in 2019.
He is a sophomore majoring in mechanical engineering at Mount. San Antonio College (Mt.SAC) from 2020.
He is also participating in the Mt.SAC Hybrid Rocket team as the current leader with Professor Martin S. Mason in 2022 Spring.
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