INTRODUCTION

Background

Currently, our daily life is profoundly impacted by the outbreak of the Coronavirus disease 2019 (COVID-19), and the number of confirmed cases has quickly increased in the past few weeks. Starting mid-March, states, companies, businesses, and schools are beginning to take action to respond to the outbreak. States are promoting residents to stay at home, companies are promoting remote working, businesses are starting to suspend their operations, and schools are beginning to transition to e-learning. All of these are acting as a part of social distancing. Research shows that social distancing can highly reduce the infection rate of infectious diseases. A model shows that if 25% of the residents reduce daily social interactions by 50%, the number of primary infections can be reduced by 81% (Maharaj).

Orienting Material

a. Using a Quadratic Least Squares polynomial to discover the formula modeling the new cases of COVID-19 over a period of time.
b. Using Richardson’s Extrapolation to the 3-point midpoint formula to calculate the increased rate of new cases of COVID-19.
c. Generating a report for the time period every 7 days starting January 21st.

Thesis

Even though some people believe the Coronavirus is less serious or that quarantine at home has no effect, the purpose of this report is to raise more people’s attention to the new Coronavirus and encourage people to isolate at home. Compared with before, people began to pay attention to the severity of the Coronavirus, and many people began to isolate themselves, resulting in a significant drop in the rate of new patients.

ANALYSIS

Data Collection

a. Our data are collected from the 2019 Novel Coronavirus COVID-19 (2019-nCoV) Data Repository by Johns Hopkins CSSE, which aggregated data from the WHO, CDC, ECDC, NHC, DXY, 1point3acres, Worldometers.info, BNO, the COVID Tracking Project, state and national government health departments, and local media reports (Dong). They also promise that the data will be updated frequently. The data provided contains the Federal Information Processing Standards (FIPS) code (Dong); County name; Province, state, or dependency name; Update time; Geocode; Confirmed case number; Deaths case number; Recovered number; and Active number. In summary, the source is credible and usable.

b. Data Table

Week Number Start Date End Date New Cases Total Cases
Week1 01-21-20 01-27-20 5 5
Week2 01-28-20 02-03-20 6 11
Week3 02-04-20 02-10-20 1 12
Week4 02-11-20 02-17-20 0 12
Week5 02-18-20 02-24-20 2 14
Week6 02-25-20 03-02-20 43 57
Week7 03-03-20 03-09-20 588 645
Week8 03-10-20 03-16-20 3728 4373
Week9 03-17-20 03-23-20 38378 42751
Week10 03-24-20 03-30-20 117935 160686
Week11 03-31-20 04-06-20 202357 362952
Week12 04-07-20 04-13-20 215226 578178
Week13 04-14-20 04-20-20 197672 775850
Week14 04-21-20 04-27-20 206800 982650

^Download daily/monthly table

c. Data Graph

weekly table

Quadratic Least Squares Polynomial

Quadratic_least_squares_polynomial.m
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%% Quadratic least squares polynomial
close all
clear
clc

x = [50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 ]';
y = [291 269 393 565 662 676 872 1291 2410 3948 5417 6271 8631 10410 9939 12226 17050 19046 20093 19118 20463 25396 26756 28879 32525 33725 25717 29359 30512 32282 34243 33578 31869 28057 24685 25987 29465 32076 30915 28084 26013 25132 25178 29246 32331 36137 34956 26929 22023]';

A = [x.^2 x ones(size(x))];
d = ((A'*A)^(-1)) * A' * y;
a = d(1); b = d(2); c = d(3);
fprintf('Model: f(x) = (%.5f)*x^2 + (%.5f)*x + (%.5f)\n', a, b, c)

hold on
plot(x,y,'r*')
pp=@(t) a.*t.^2+b.*t+c;
plot(x,pp(x))
xlabel('Days')
ylabel('New Confirmed Cases')
title('Days Vs. New Confirmed Cases')
legend('Actual data','Quadratic least-squares fit','location','best')
box on; grid on;

^Download MATLAB Code for Quadratic least squares polynomial
^Download MATLAB Code for Cubic least squares polynomial

Model: $$f(x)=(-25.91048)(x)^2+(4572.54925)x-171258.00999$$

model

Although the data we have is somewhat noisy, we can still generate a reasonable model formula by using the Quadratic Least Squares polynomial. We can use the generated polynomial and apply it to Richardson’s Extrapolation using the 3-point midpoint formula.

Richardson’s Extrapolation to the 3-point Midpoint

Once we get a formula using the Quadratic Least Squares polynomial, we can use Richardson’s Extrapolation to the 3-point midpoint formula to find its first derivative. In numerical analysis, the Richardson extrapolation method can improve the convergence efficiency of a series or sequence. Furthermore, Richardson’s extrapolation method can generate high-precision results even when using low-order formulas (Burden, 185).

The formula we have is $f(x)=(-25.91048)(x)^2+(4572.54925)x-171258.00999$.

Richardson_3_point_midpoint.java
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import java.util.Scanner;

class ThreePointMidpoint {
	static double f(double x) {
    	return (-25.91048)*Math.pow(x,2)+(4572.54925)*x-171258.00999;
	}

	public static void main(String[] args){
    	double h;
    	double firstDerivative;
    	double inputDay;
    	Scanner scnr = new Scanner(System.in);
    	System.out.println("Please input your initial h:");
    	h = scnr.nextDouble();
    	System.out.println("Please input your Day (x):");
    	inputDay = scnr.nextDouble();
    	for(int i = 0; i < 3; i++){
        	double coefficient = 0.5 / h;
        	firstDerivative = coefficient * (f((inputDay + h)) - f((inputDay - h)));
        	System.out.println("N1(" + h + ") result: " + firstDerivative);
        	h = h / 2;
    	}
   }
}

^Download Python Code for Richardson 3-point midpoint formula

Input day = 60, with h = 4:

result: 1463.2916425999992
result: 1463.2916499999992
result: 1463.2916500000138

$N_1 (4)=1463.2916425999992$
$N_1 (2)=1463.2916499999992$
$N_1 (1)=1463.2916500000138$
$N_2 (2)=N_1 (1)+\frac{1}{3} [N_1 (1)-N_1 (2)]=1463.29165+\frac{1}{3} [1463.29165-1463.29165]=1463.29165$
$N_2 (4)=N_1 (2)+\frac{1}{3} [N_1 (2)-N_1 (4)]=1463.29165+\frac{1}{3} [1463.29165-1463.29164]=1463.29165$
$N_3 (4)=N_2 (2)+\frac{1}{5} [N_2 (2)-N_2 (4)]=1463.29165+\frac{1}{5} [1463.29165-1463.29165]=1463.29165$

Input day = 70, with h = 4:
$N_1 (4)=945.0820499999973$
$N_1 (2)=945.0820499999973$
$N_1 (1)=945.0820499999973$
$N_2 (2)=N_1 (1)+\frac{1}{3} [N_1 (1)-N_1 (2)]=945.08205+\frac{1}{3} [945.08205-945.08205]=945.08205$
$N_2 (4)=N_1 (2)+\frac{1}{3} [N_1 (2)-N_1 (4)]=945.08205+\frac{1}{3} [945.08205-945.08205]=945.08205$
$N_3 (4)=N_2 (2)+\frac{1}{5} [N_2 (2)-N_2 (4)]=945.08205+\frac{1}{5} [945.08205-945.08205]=945.08205$

Input day = 80, with h = 4:
$N_1 (4)=426.8724499999953$
$N_1 (2)=426.87245000000985$
$N_1 (1)=426.87244999998074$
$N_2 (2)=N_1 (1)+\frac{1}{3} [N_1 (1)-N_1 (2)]=426.87245+\frac{1}{3} [426.87245-426.87245]=426.87245$
$N_2 (4)=N_1 (2)+\frac{1}{3} [N_1 (2)-N_1 (4)]=426.87245+\frac{1}{3} [426.87245-426.87245]=426.87245$
$N_3 (4)=N_2 (2)+\frac{1}{5} [N_2 (2)-N_2 (4)]=426.87245+\frac{1}{5} [426.87245-426.87245]=426.87245$

Input day = 90, with h = 4:
$N_1 (4)=-91.33715000000302$
$N_1 (2)=-91.33715000000666$
$N_1 (1)=-91.33715000000666$
$N_2 (2)=N_1 (1)+\frac{1}{3} [N_1 (1)-N_1 (2)]=-91.33715+\frac{1}{3} [-91.33715+91.33715]=-91.33715$
$N_2 (4)=N_1 (2)+\frac{1}{3} [N_1 (2)-N_1 (4)]=-91.33715+\frac{1}{3} [-91.33715+91.33715]=-91.33715$
$N_3 (4)=N_2 (2)+\frac{1}{5} [N_2 (2)-N_2 (4)]=-91.33715+\frac{1}{5} [-91.33715+91.33715]=-91.33715$

CONCLUSION

Summary

Two methods are applied in this project: the Quadratic Least Squares polynomial and Richardson’s Extrapolation to the 3-point midpoint formula. The Quadratic Least Squares polynomial is used to discover the underlying formula, and Richardson’s Extrapolation to the 3-point midpoint formula is used to discover the case increase rate. A graph is generated when the Quadratic Least Squares polynomial is calculated in MATLAB, which clearly shows the polynomial approximation of the COVID-19 outbreak over time.

Accuracy

By analyzing the data, we can easily find that the data is extremely noisy, which limits the usage of certain methods in Numerical Analysis. However, polynomial least squares regression can solve this problem; it can generate a low-degree polynomial that can be used to estimate an underlying polynomial trend (Kalman).
When we use Richardson’s extrapolation method for our data analysis, we can see that the results are increasingly converging to the real derivative solution of the generated formula.

Conclusion

From our analysis results above, the Quadratic Least Squares polynomial can provide us with the best-fit line even though we have noisy data. Since many organizations and schools encourage people to isolate themselves, the results from the 60th, 70th, 80th, and 90th days—which give us the derivative of the formula (representing the rate of increase)—show that the rate of new cases is continuously decreasing. We can easily conclude that with the effort of social distancing, including the hard work of government officials, medical workers, and all residents in the US, the infection rate will drop into a stable stage and will gradually disappear.

Reference

  1. Burden, Richard L., and J. Douglas. Faires. Numerical Analysis. Brooks/Cole, Cengage Learning, 2011.
  2. Dong, Ensheng, et al. “An Interactive Web-Based Dashboard to Track COVID-19 in Real Time.” The Lancet Infectious Diseases, 19 Feb. 2020, www.thelancet.com/journals/laninf/article/PIIS1473-3099(20)30120-1.
  3. Kalman, R. E. “A New Approach to Linear Filtering and Prediction Problems.” Journal of Basic Engineering, American Society of Mechanical Engineers Digital Collection, 1 Mar. 1960, https://asmedigitalcollection.asme.org/fluidsengineering/article-abstract/82/1/35/397706/A-New-Approach-to-Linear-Filtering-and-Prediction.
  4. Maharaj, Savi, and Adam Kleczkowski. “Controlling Epidemic Spread by Social Distancing: Do It Well or Not at All.” BMC Public Health, 20 Aug. 2012, https://bmcpublichealth.biomedcentral.com/articles/10.1186/1471-2458-12-679.