The Basic Model for Spread of COVID-19
A salute to the heroes of coronavirus: Health care workers on the frontline
Dennis Hou, Philip Hou, Raymond Hou and Janpu Hou
Hou Research Institute
Hou Research Institute
April 4, 2020
Introduction
The new name of this disease is COronaVIrus Disease 2019, abbreviated as COVID-19. In COVID-19, ‘CO’ stands for ‘corona,’ ‘VI’ for ‘virus,’ and ‘D’ for disease.
On March 23, more than 35,000 people in the United States have been infected with the new coronavirus and death toll reached 555, and Ray began to work in COVID only ICU. To understand his risk, we has begun our preliminary studies to try to determine how many Americans have already been infected with SARS-CoV-2, the virus that causes Covid-19. By April 3rd, nearly 240,000 people in the country have been infected with the virus and nearly 5,500 have died.
Basic Theory
Epidemiologists routinely turn to models to predict the progression of an infectious disease. Building on the research of Ronald Ross and Hilda Hudson, A. G. McKendrick and W. O. Kermack published their theory in a set of three articles from 1927, 1932, and 1933. For more details on “A contribution to the mathematical theory of epidemics” see https://royalsocietypublishing.org/doi/10.1098/rspa.1927.0118.
The problem may be summarised as follows: One (or more) infected person is introduced into a community of individuals, more or less susceptible to the disease in question. The disease spreads from the affected to the unaffected by contact infection. Each infected person runs through the course of his sickness, and finally is removed from the number of those who are sick, by recovery or by death.
These models are not snapshots of the future. They always describe a range of possibilities-and those possibilities are highly sensitive to our actions.
df <- read.csv(file = 'c:/R353/us-counties.csv')
head(df)## date county state fips cases deaths
## 1 2020-01-21 Snohomish Washington 53061 1 0
## 2 2020-01-22 Snohomish Washington 53061 1 0
## 3 2020-01-23 Snohomish Washington 53061 1 0
## 4 2020-01-24 Cook Illinois 17031 1 0
## 5 2020-01-24 Snohomish Washington 53061 1 0
## 6 2020-01-25 Orange California 6059 1 0tail(df)## date county state fips cases deaths
## 30838 2020-04-03 Sheridan Wyoming 56033 10 0
## 30839 2020-04-03 Sublette Wyoming 56035 1 0
## 30840 2020-04-03 Sweetwater Wyoming 56037 3 0
## 30841 2020-04-03 Teton Wyoming 56039 32 0
## 30842 2020-04-03 Uinta Wyoming 56041 1 0
## 30843 2020-04-03 Washakie Wyoming 56043 2 0nycdf <- df[grep("New York City", df$county),]
case_df <- data.frame(nycdf$date,nycdf$cases)
death_df<- data.frame(nycdf$date,nycdf$deaths)
tail(death_df)## nycdf.date nycdf.deaths
## 29 2020-03-29 776
## 30 2020-03-30 914
## 31 2020-03-31 1096
## 32 2020-04-01 1374
## 33 2020-04-02 1562
## 34 2020-04-03 1867tail(case_df)## nycdf.date nycdf.cases
## 29 2020-03-29 33768
## 30 2020-03-30 38087
## 31 2020-03-31 43139
## 32 2020-04-01 47440
## 33 2020-04-02 51810
## 34 2020-04-03 57160day = seq(from = as.Date("2020-03-02"), to = as.Date("2020-04-03"), by = 'day')
x <- apply( death_df[-1] , 2 , diff )
daily_df <- data.frame(day,x)Daily death count
plot(daily_df,type="h",lwd= 10,
col="red", ylab="daily death count",xlab="date",main="Daily death count in New York City")
day1 = seq(from = as.Date("2020-03-01"), to = as.Date("2020-04-03"), by = 'day')
death_df <- data.frame(day1,nycdf$deaths)Accumulative death count
plot(death_df,type="h",lwd= 10,
col="blue", ylab="Accumulative death count",xlab="date",main="Accumulative death count in New York City")
case_df <- data.frame(day1,nycdf$cases)Accumulative confirmed cases
plot(case_df,type="h",lwd= 10,
col="yellow", ylab="Confirmed case",xlab="date",main="Accumulative confirmed cases in New York City")
Key: Rate of Change
case_df <- data.frame(day1,log(nycdf$cases))
plot(case_df,type="h",lwd= 10,
col="yellow", ylab="Log(Confirmed case)",xlab="date",main="Log(confirmed cases) in New York City")
Methodology
Ordinary differential equations for this SIR model are given by:
S(t) = at day t, number of subsceptible people can catch the COVID-19
I(t) = at day t, number of infectious people have caught the COVID-19 and can spread it to subsceptible
R(t) = at day t, number of recovered people have recorved from the COVID-19 and are immune
Total population N = S(t) + I(t) + R(t), since the population is a constant, therefore, dN/dt = 0 imply dS/dt + dI/dt + dR/dt = 0.
Beta is the transmission rate, gama is the recovery rate.
The Mathematical Model
The Mathematical Model
With these equations, we can build our model for the coronavirus itself to try to better understand how it might spread.
Some recent studies of COVID-19 estimates some of these valuse for us, so we can have some initial estimate such as ?? = 0.5.?? = 1.75. as starter. For example: “Early dynamics of transmission and control of COVID-19: a mathematical modelling study” by Kucharski et.al Published on THE LENCET infectious disease March 11, 2020 https://www.thelancet.com/journals/laninf/article/PIIS1473-3099(20)30144-4/fulltext; “Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts” by Hellewell et al. The Lencet VOLUME 8, ISSUE 4, PE488-E496, APRIL 01, 2020: https://www.thelancet.com/journals/langlo/article/PIIS2214-109X(20)30074-7/fulltext
The COVID-19 Dataset
The dataset provided by New York Times : https://github.com/nytimes/covid-19-data
On MARCH 27, 2020, The New York Times has made one of the most comprehensive datasets of coronavirus cases in the United States publicly available in response to requests from researchers, scientists, government officials and businesses who would like access to the data to better understand the virus and model what may come next.
The Times initially began tracking cases in late January after it became clear that no federal government agency was providing the public with an accurate, up-to-date record of cases, tracked to the county level, of people in the U.S. who had tested positive for the virus.
The Times led effort has grown from a handful of correspondents to a team of several dozen journalists, including data scientists and student journalists from Northwestern University, the University of Missouri and the University of Nebraska-Lincoln, working around the clock to record details about every case. The Times is committed to collecting as much data as possible in connection with the outbreak and is collaborating with the University of California, Berkeley, on an effort in California.
# install.packages("deSolve")
library(deSolve)
sir_equations <- function(time, variables, parameters) {
with(as.list(c(variables, parameters)), {
dS <- -beta * I * S
dI <- beta * I * S - gamma * I
dR <- gamma * I
return(list(c(dS, dI, dR)))
})
}
parameters_values <- c(
beta = 1.047, # infectious contact rate (/person/day)
gamma = 0.5 # recovery rate (/day)
)
initial_values <- c(
S = 1-1e-06, # number of susceptibles at time = 0
I = 1e-06, # number of infectious at time = 0
R = 0 # number of recovered (and immune) at time = 0
)
time_values <- seq(1, 61,1)
daym = seq(from = as.Date("2020-03-01"), to = as.Date("2020-04-30"), by = 'day')
df2 <- ode(
y = initial_values,
times = time_values,
func = sir_equations,
parms = parameters_values
)
dft<-as.data.frame(df2)
m_dft <- data.frame(daym,dft$I)
parameters_values <- c(
beta = 1.047, # infectious contact rate (/person/day)
gamma = 0.75 # recovery rate (/day)
)
df2 <- ode(
y = initial_values,
times = time_values,
func = sir_equations,
parms = parameters_values
)
dft<-as.data.frame(df2)
m2_dft <- data.frame(daym,dft$I)Results and Discussion
Flatten the Curve
This so-called exponential curve has experts worried. If the number of cases were to continue to double every three days, there would be about a hundred million cases in the United States by May.
That is math, not prophecy. The spread can be slowed, if the rate of inflection is reduced.
Apex of the Outbreak
The worst of the coronavirus outbreak is yet to come and extra hospital space is being built and converted around New York in preparation.
The apex of the outbreak is when the highest number of COVID-19 patients will need care and that’s still an estimated two weeks away as our model suggested.
We send our best wishes to exhausted medical workers. By April 4, 2020, at least three nurses, a well-known pediatric neurosurgeon, and an ER doctor have died in New York. We hope our effort on search of the apex and to show that the other side of the mountain is near and we can starting to see light at the end of the tunnel may be confort to some.
It is an emotional and powerful moment to see and feel we come together as one to reassure each other that we will get through this. It is a moment where we can show our gratitude to all the health-care workers!!!
