Did Lockdowns Truly Help During COVID-19? A Mathematical Dive into the Pandemic Response
This write-up borrows significantly from the insights shared in this post: https://www.freecodecamp.org/news/how-to-model-an-epidemic-with-r/
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The COVID-19 pandemic disrupted the world in ways few could have predicted. Among the arsenal of responses, lockdowns stood out as a controversial yet widely adopted strategy to contain the spread. Governments imposed restrictions, hoping to "flatten the curve" and buy time for healthcare systems. But did lockdowns truly deliver on their promise? Using the SEIR model—a mathematical framework for infectious disease dynamics—we can explore the impact of lockdowns and draw meaningful insights about their effectiveness and consequences.
At its core, the SEIR model divides a population into compartments: Susceptible (S), Exposed (E), Infectious (I), and Recovered (R). Each individual transitions between these compartments based on specific rates that capture the biology of the disease and the social dynamics of transmission. For COVID-19, we extended this model to include a mortality compartment (M) to account for fatalities.
The transitions between these compartments are governed by differential equations. For example, the rate at which susceptible (S) individuals become exposed (E) is determined by the equation:
Here, β represents the transmission rate, and N is the total population.
Note: The transmission coefficient β is calculated using the formula:
β = (average number of new infections / number of susceptible individuals) * (1 / duration of the time interval)
This formula gives you the average rate at which susceptible individuals become infected per unit of time. Let's illustrate this with an example:
Suppose you have data for a population of 10,000 individuals over a week. During that week, 300 new infections occurred. At the beginning of the week, there were 9,500 susceptible individuals.
Thus, average number of new infections = 300/7 = 42.86; & β = (42.86/9500)* (1/7) = 0.000476
So, in this example, the estimated transmission rate (β) is 0.000476 per susceptible individual per day.
Similarly, the flow of individuals from the exposed (E) to the infectious (I) compartment is described by:
where σ represents the rate at which exposed individuals become infectious. For example, if an individual becomes infectious after 4 days of exposure, σ will be 1/4 (or 0.25).
Once infectious, individuals either recover or succumb to the disease, captured by the equations:
Here, γ is the recovery rate, and μ is the mortality rate.
Note: If we divide the transmission rate (β) by the recovery rate (γ), it gives us R0 or the "basic reproduction number" - this number estimates the number of people who will be infected by the average infectious individual. In viral epidemiology;
if R₀ > 1, then an outbreak of the virus is likely to become an epidemic
If R₀ < 1, then an outbreak is likely to be contained
By solving these equations, we can simulate how a disease spreads through a population under different conditions. Let’s consider two conditions involving lockdown.
We first simulated the model under a "no lockdown" scenario (see code below), where the transmission rate (β) remained constant. The results were alarming but not unexpected. In a population of 1 million, infections peaked on day 112, with 126,516 individuals simultaneously infectious—a surge that would overwhelm any healthcare system. By the end of the simulation, 887,300 individuals recovered, 108,264 avoided infection, and 4,436 succumbed to the disease. This rapid rise and fall of infections demonstrated the virus’s natural progression when left unchecked.
library(deSolve)
library(ggplot2)
library(dplyr)
library(tidyr)
library(knitr)
# SEIR model function
SEIR <- function(time, current_state, params){
with(as.list(c(current_state, params)), {
N <- S + E + I + R
dS <- -(beta * S * I) / N
dE <- (beta * S * I) / N - sigma * E
dI <- sigma * E - gamma * I - mu * I
dR <- gamma * I
dM <- mu * I
return(list(c(dS, dE, dI, dR, dM)))
})
}
# Parameters
params <- c(beta = 0.5, sigma = 0.25, gamma = 0.2, mu = 0.001)
# Initial state
initial_state <- c(S = 999999, E = 1, I = 0, R = 0, M = 0)
# Time points
times <- seq(0, 365, by = 1)
# Solve the model
model <- ode(initial_state, times, SEIR, params)
# Convert to data frame
model_df <- as.data.frame(model) %>%
rename(Time = time) %>%
pivot_longer(cols = -Time, names_to = "Compartment", values_to = "Population")
# Visualization
ggplot(model_df, aes(x = Time, y = Population, color = Compartment)) +
geom_line(size = 1) +
labs(title = "SEIR Model: Dynamics Over Time",
x = "Time (Days)",
y = "Population",
color = "Compartment") +
theme_minimal()
# Peak Infection
peak_infection <- model_df %>%
filter(Compartment == "I") %>%
slice(which.max(Population))
print(peak_infection)
Next, we introduced a lockdown into the model (see code) by reducing β during a 60-day period (day 90 to day 150). This adjustment reflected reduced human interactions during the lockdown. The impact was significant: infections peaked much later, on day 223, with a lower peak of 72,444 simultaneous infections. Additionally, the total number of deaths dropped to 4,195, and 156,886 individuals avoided infection entirely. The curve flattened, showcasing the immediate benefits of lockdowns in reducing strain on healthcare resources.
# Lockdown scenario
SEIR_lockdown <- function(time, current_state, params) {
with(as.list(c(current_state, params)), {
beta <- ifelse(time >= start_lockdown & time <= end_lockdown, 0.1, 0.5)
N <- S + E + I + R
dS <- -(beta * S * I) / N
dE <- (beta * S * I) / N - sigma * E
dI <- sigma * E - gamma * I - mu * I
dR <- gamma * I
dM <- mu * I
return(list(c(dS, dE, dI, dR, dM)))
})
}
# Lockdown Parameters
lockdown_params <- c(sigma = 0.25, gamma = 0.2, mu = 0.001, start_lockdown = 90, end_lockdown = 150)
lockdown_model <- ode(initial_state, times, SEIR_lockdown, lockdown_params)
# Convert to data frame
lockdown_df <- as.data.frame(lockdown_model) %>%
rename(Time = time) %>%
pivot_longer(cols = -Time, names_to = "Compartment", values_to = "Population")
# Comparison Visualization
combined_df <- bind_rows(
model_df %>% mutate(Scenario = "No Lockdown"),
lockdown_df %>% mutate(Scenario = "Lockdown")
)
ggplot(combined_df, aes(x = Time, y = Population, color = Compartment, linetype = Scenario)) +
geom_line(size = 1) +
labs(title = "SEIR Model: Impact of Lockdown",
x = "Time (Days)",
y = "Population",
color = "Compartment",
linetype = "Scenario") +
theme_minimal()
# Summary Statistics
summary_stats <- combined_df %>%
group_by(Scenario, Compartment) %>%
summarize(
Peak_Population = max(Population),
Peak_Time = Time[which.max(Population)],
.groups = "drop"
)
kable(summary_stats, caption = "Summary Statistics of SEIR Model with and without Lockdown")
An intriguing observation from this analysis was the emergence of a "second wave." When the lockdown ended, interactions resumed, leading to a resurgence of cases. While the peak of this second wave was lower than the first, it highlighted a key limitation of lockdowns: they delay the inevitable unless accompanied by long-term strategies such as vaccination, mass testing, and effective contact tracing. This phenomenon mirrored real-world patterns, where countries experienced multiple waves of infection due to a combination of virus evolution, inconsistent policies, and human behavior.
The mathematical derivation of the SEIR model offers a powerful lens to understand disease dynamics. Lockdowns, modeled as a reduction in β, act as a brake on the epidemic’s acceleration. However, the energy of the system—analogous to the remaining susceptible population—remains, ready to reignite as restrictions are lifted. This underscores a critical inference: lockdowns buy time, but they must be leveraged to implement sustainable solutions.
Take home message:
Our SEIR model shows that while lockdowns effectively slow the spread of infectious diseases, they are not a silver bullet. Just so you know, the real-world dynamics involve many more factors besides those described in this model, such as vaccination, variants, and human behavior. The insights, however, remain valuable in understanding the principles of disease spread.
Ultimately, it’s not just about flattening the curve but about strengthening the system.
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