Exponential Growth of Covid-19
Christina Daniels
2023-05-18
Growth of Covid-19 in March 2020
In March 2020, the Covid-19 pandemic began to take hold in the United States. In that first month, the virus spread exponentially as more and more people tested positive for the virus. My project focuses primarily on the state of New York where some of the highest numbers of positive Covid-19 tests were seen in the US.
Cited from https://coronavirus.health.ny.gov/positive-tests-over-time-region-and-county
x <- seq(1,31,1)
x # x = number of days## [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
## [26] 26 27 28 29 30 31y <- c(0,1,0,2,22,11,24,28,63,44,56,102,164,131,294,432,1009,1769,2950,3254,4812,5707,4790,5145,6448,7379,7681,7195,6984,9298,7917)
y # y = number of positive Covid-19 tests in New York State## [1] 0 1 0 2 22 11 24 28 63 44 56 102 164 131 294
## [16] 432 1009 1769 2950 3254 4812 5707 4790 5145 6448 7379 7681 7195 6984 9298
## [31] 7917plot(x, y, main="Growth of Covid-19 March 2020",
xlab= "Time (days)", ylab="Positive Covid-19 Tests",
xlim = c(0,31), ylim=c(0,10000))
Exponential and Logistic Growth Models
In the following program, we will use the calculated half-life of the virus to create an additional exponential growth as well as logistic model for the data set.
t <- 0 # initial time
P <- 10000 # number of people
N = 31 # Number of time increments
Time = 31 # each time is one day
dt <- Time/N
k = 0.3 # rate constant per day
#P = 10000, Pini = 1, t = 31 for equation ln(P/Pini)=k*t
# t[1] and P[1] are initialized
for(i in 2:N)
{ t[i] <- t[i-1] + dt
P[i] <- P[i-1] + k*P[i-1]*dt
print(P)
}## [1] 10000 13000
## [1] 10000 13000 16900
## [1] 10000 13000 16900 21970
## [1] 10000 13000 16900 21970 28561
## [1] 10000.0 13000.0 16900.0 21970.0 28561.0 37129.3
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30 48268.09
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30 48268.09 62748.52
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30 48268.09 62748.52
## [9] 81573.07
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30 48268.09
## [8] 62748.52 81573.07 106044.99
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30 48268.09
## [8] 62748.52 81573.07 106044.99 137858.49
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30 48268.09
## [8] 62748.52 81573.07 106044.99 137858.49 179216.04
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30 48268.09
## [8] 62748.52 81573.07 106044.99 137858.49 179216.04 232980.85
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30 48268.09
## [8] 62748.52 81573.07 106044.99 137858.49 179216.04 232980.85 302875.11
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30 48268.09
## [8] 62748.52 81573.07 106044.99 137858.49 179216.04 232980.85 302875.11
## [15] 393737.64
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30 48268.09
## [8] 62748.52 81573.07 106044.99 137858.49 179216.04 232980.85 302875.11
## [15] 393737.64 511858.93
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30 48268.09
## [8] 62748.52 81573.07 106044.99 137858.49 179216.04 232980.85 302875.11
## [15] 393737.64 511858.93 665416.61
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30 48268.09
## [8] 62748.52 81573.07 106044.99 137858.49 179216.04 232980.85 302875.11
## [15] 393737.64 511858.93 665416.61 865041.59
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30
## [7] 48268.09 62748.52 81573.07 106044.99 137858.49 179216.04
## [13] 232980.85 302875.11 393737.64 511858.93 665416.61 865041.59
## [19] 1124554.07
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30
## [7] 48268.09 62748.52 81573.07 106044.99 137858.49 179216.04
## [13] 232980.85 302875.11 393737.64 511858.93 665416.61 865041.59
## [19] 1124554.07 1461920.29
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30
## [7] 48268.09 62748.52 81573.07 106044.99 137858.49 179216.04
## [13] 232980.85 302875.11 393737.64 511858.93 665416.61 865041.59
## [19] 1124554.07 1461920.29 1900496.38
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30
## [7] 48268.09 62748.52 81573.07 106044.99 137858.49 179216.04
## [13] 232980.85 302875.11 393737.64 511858.93 665416.61 865041.59
## [19] 1124554.07 1461920.29 1900496.38 2470645.29
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30
## [7] 48268.09 62748.52 81573.07 106044.99 137858.49 179216.04
## [13] 232980.85 302875.11 393737.64 511858.93 665416.61 865041.59
## [19] 1124554.07 1461920.29 1900496.38 2470645.29 3211838.88
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30
## [7] 48268.09 62748.52 81573.07 106044.99 137858.49 179216.04
## [13] 232980.85 302875.11 393737.64 511858.93 665416.61 865041.59
## [19] 1124554.07 1461920.29 1900496.38 2470645.29 3211838.88 4175390.54
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30
## [7] 48268.09 62748.52 81573.07 106044.99 137858.49 179216.04
## [13] 232980.85 302875.11 393737.64 511858.93 665416.61 865041.59
## [19] 1124554.07 1461920.29 1900496.38 2470645.29 3211838.88 4175390.54
## [25] 5428007.70
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30
## [7] 48268.09 62748.52 81573.07 106044.99 137858.49 179216.04
## [13] 232980.85 302875.11 393737.64 511858.93 665416.61 865041.59
## [19] 1124554.07 1461920.29 1900496.38 2470645.29 3211838.88 4175390.54
## [25] 5428007.70 7056410.01
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30
## [7] 48268.09 62748.52 81573.07 106044.99 137858.49 179216.04
## [13] 232980.85 302875.11 393737.64 511858.93 665416.61 865041.59
## [19] 1124554.07 1461920.29 1900496.38 2470645.29 3211838.88 4175390.54
## [25] 5428007.70 7056410.01 9173333.02
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30
## [7] 48268.09 62748.52 81573.07 106044.99 137858.49 179216.04
## [13] 232980.85 302875.11 393737.64 511858.93 665416.61 865041.59
## [19] 1124554.07 1461920.29 1900496.38 2470645.29 3211838.88 4175390.54
## [25] 5428007.70 7056410.01 9173333.02 11925332.93
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30
## [7] 48268.09 62748.52 81573.07 106044.99 137858.49 179216.04
## [13] 232980.85 302875.11 393737.64 511858.93 665416.61 865041.59
## [19] 1124554.07 1461920.29 1900496.38 2470645.29 3211838.88 4175390.54
## [25] 5428007.70 7056410.01 9173333.02 11925332.93 15502932.80
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30
## [7] 48268.09 62748.52 81573.07 106044.99 137858.49 179216.04
## [13] 232980.85 302875.11 393737.64 511858.93 665416.61 865041.59
## [19] 1124554.07 1461920.29 1900496.38 2470645.29 3211838.88 4175390.54
## [25] 5428007.70 7056410.01 9173333.02 11925332.93 15502932.80 20153812.64
## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30
## [7] 48268.09 62748.52 81573.07 106044.99 137858.49 179216.04
## [13] 232980.85 302875.11 393737.64 511858.93 665416.61 865041.59
## [19] 1124554.07 1461920.29 1900496.38 2470645.29 3211838.88 4175390.54
## [25] 5428007.70 7056410.01 9173333.02 11925332.93 15502932.80 20153812.64
## [31] 26199956.44t## [1] 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
## [26] 25 26 27 28 29 30P## [1] 10000.00 13000.00 16900.00 21970.00 28561.00 37129.30
## [7] 48268.09 62748.52 81573.07 106044.99 137858.49 179216.04
## [13] 232980.85 302875.11 393737.64 511858.93 665416.61 865041.59
## [19] 1124554.07 1461920.29 1900496.38 2470645.29 3211838.88 4175390.54
## [25] 5428007.70 7056410.01 9173333.02 11925332.93 15502932.80 20153812.64
## [31] 26199956.44plot(t,P,type = "p", main="Exponential Growth of Covid-19 March 2020", xlab= "Time (days)", ylab="Positive Covid-19 Tests",xlim = c(0,Time), ylim = c(10000,15000000), col = "red")
Pc <- P[1] * exp(k*t)
lines(t,Pc)
# Logistic equation
Cap <- 1000000
t <- 0 # initialize time
P <- 10000 # number of people
N = 31 # Number of time increments
Time = 31 # each time is one day
dt <- Time/N
k = 0.3 # rate constant sec-1
# t[1] and P[1] are initialized
for(i in 2:N)
{ t[i] <- t[i-1] + dt
P[i] <- P[i-1] + k*P[i-1]* (1-P[i-1]/Cap) * dt # we did by hand
print(P)
}## [1] 10000 12970
## [1] 10000.00 12970.00 16810.53
## [1] 10000.00 12970.00 16810.53 21768.92
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08 60284.78
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08 60284.78
## [9] 77279.93
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08 60284.78
## [9] 77279.93 98672.26
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97 198206.02
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97 198206.02 245882.14
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97 198206.02 245882.14
## [15] 301509.38
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97 198206.02 245882.14
## [15] 301509.38 364689.82
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97 198206.02 245882.14
## [15] 301509.38 364689.82 434197.17
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97 198206.02 245882.14
## [15] 301509.38 364689.82 434197.17 507898.16
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97 198206.02 245882.14
## [15] 301509.38 364689.82 434197.17 507898.16 582879.45
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97 198206.02 245882.14
## [15] 301509.38 364689.82 434197.17 507898.16 582879.45 655818.75
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97 198206.02 245882.14
## [15] 301509.38 364689.82 434197.17 507898.16 582879.45 655818.75 723534.90
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97 198206.02 245882.14
## [15] 301509.38 364689.82 434197.17 507898.16 582879.45 655818.75 723534.90
## [22] 783544.55
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97 198206.02 245882.14
## [15] 301509.38 364689.82 434197.17 507898.16 582879.45 655818.75 723534.90
## [22] 783544.55 834425.29
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97 198206.02 245882.14
## [15] 301509.38 364689.82 434197.17 507898.16 582879.45 655818.75 723534.90
## [22] 783544.55 834425.29 875873.21
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97 198206.02 245882.14
## [15] 301509.38 364689.82 434197.17 507898.16 582879.45 655818.75 723534.90
## [22] 783544.55 834425.29 875873.21 908489.01
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97 198206.02 245882.14
## [15] 301509.38 364689.82 434197.17 507898.16 582879.45 655818.75 723534.90
## [22] 783544.55 834425.29 875873.21 908489.01 933430.03
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97 198206.02 245882.14
## [15] 301509.38 364689.82 434197.17 507898.16 582879.45 655818.75 723534.90
## [22] 783544.55 834425.29 875873.21 908489.01 933430.03 952071.55
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97 198206.02 245882.14
## [15] 301509.38 364689.82 434197.17 507898.16 582879.45 655818.75 723534.90
## [22] 783544.55 834425.29 875873.21 908489.01 933430.03 952071.55 965760.95
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97 198206.02 245882.14
## [15] 301509.38 364689.82 434197.17 507898.16 582879.45 655818.75 723534.90
## [22] 783544.55 834425.29 875873.21 908489.01 933430.03 952071.55 965760.95
## [29] 975680.97
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97 198206.02 245882.14
## [15] 301509.38 364689.82 434197.17 507898.16 582879.45 655818.75 723534.90
## [22] 783544.55 834425.29 875873.21 908489.01 933430.03 952071.55 965760.95
## [29] 975680.97 982799.25
## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97 198206.02 245882.14
## [15] 301509.38 364689.82 434197.17 507898.16 582879.45 655818.75 723534.90
## [22] 783544.55 834425.29 875873.21 908489.01 933430.03 952071.55 965760.95
## [29] 975680.97 982799.25 987870.72t## [1] 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
## [26] 25 26 27 28 29 30P## [1] 10000.00 12970.00 16810.53 21768.92 28157.42 36366.80 46880.08
## [8] 60284.78 77279.93 98672.26 125353.07 158244.97 198206.02 245882.14
## [15] 301509.38 364689.82 434197.17 507898.16 582879.45 655818.75 723534.90
## [22] 783544.55 834425.29 875873.21 908489.01 933430.03 952071.55 965760.95
## [29] 975680.97 982799.25 987870.72plot(t,P,type = "p",main="Logistic Model of Covid-19 Growth",xlim = c(0,Time), ylim = c(10000,1000000),
col = "red")
Pc <- P[1] * exp(k*t)
lines(t,Pc)
Pex = Cap / (1+(Cap/100+1) * exp(k*t)) Non-Linear Least Square (nls)
x <- seq(15,31,1)
x # x = number of days## [1] 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31y <- c(294,432,1009,1769,2950,3254,4812,5707,4790,5145,6448,7379,7681,7195,6984,9298,7917)
y # y = number of positive Covid-19 tests in New York State## [1] 294 432 1009 1769 2950 3254 4812 5707 4790 5145 6448 7379 7681 7195 6984
## [16] 9298 7917tryfit <- nls(y~a * x ^ 3 + b * x + c,
start = list(a = 1, b = 2, c = 1))
plot(x, y, col.lab ="black",
col.axis ="black")
lines(x, predict(tryfit))
# error value
print(sum(resid(tryfit)^2))## [1] 5756980summary(tryfit)##
## Formula: y ~ a * x^3 + b * x + c
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## a -2.971e-01 1.054e-01 -2.820 0.0136 *
## b 1.024e+03 1.746e+02 5.861 4.14e-05 ***
## c -1.455e+04 2.603e+03 -5.589 6.67e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 641.3 on 14 degrees of freedom
##
## Number of iterations to convergence: 1
## Achieved convergence tolerance: 1.025e-07