The following ordinary differential equations (ODE) describe the change rates of s(t), i(t) and r(t): d s ( t ) d t = - β ( t ) s ( t ) i ( t ) , (1) d i ( t ) d t = β ( t ) s ( t ) i ( t ) - γ ( t ) i ( t ) , (2) d r ( t ) d t = γ ( t ) i ( t ) , (3) with an initial condition: i(0) = i₀ and r(0) = r₀, where i₀ > 0 in order to let the epidemic develop [36]. Here, β(t) > 0 is the time-varying transmission rate of an infection at time t, which is the number of infectious contacts that result in infections per unit time, and γ(t) > 0 is the time-varying removal rate at t, at which infectious subjects are removed from being infectious due to death or recovery [33]. Moreover, γ⁻¹(t) can be interpreted as the infectious duration of an infection caught at time t[37].

From (1)–(3), we derive an important quantity, which is the time-dependent reproduction number R 0 ( t ) = β ( t ) γ ( t ) .

To see this, dividing (2) by (3) leads to R 0 ( t ) = 1 s ( t ) { d i d r ( t ) + 1 } , (4) where (di/dr)(t) is the ratio of the change rate of i(t) to that of r(t). Therefore, compared to its time-independent counterpart, R 0 ( t ) is an instantaneous reproduction number and provides a real-time picture of an outbreak. For example, at the onset of the outbreak and in the absence of any containment actions, we may see a rapid ramp-up of cases compared to those removed, leading to a large (di/dr)(t) in (4), and hence a large R 0 ( t ). With the implemented policies for disease mitigation, we will see a drastically decreasing (di/dr)(t) and, therefore, declining of R 0 ( t ) over time. The turning point is t₀ such that R 0 ( t 0 ) = 1 , when the outbreak is controlled with (di/dr)(t₀) < 0.

Under the fixed population size assumption, i.e., s(t) + i(t)+ r(t) = 1, we only need to study i(t) and r(t), and re-express (1)–(3) as d i ( t ) d t = β ( t ) i ( t ) { 1 - i ( t ) - r ( t ) } - γ ( t ) i ( t ) , d r ( t ) d t = γ ( t ) i ( t ) , (5) with the same initial condition.

As the numbers of cases and removed are reported on a daily basis, t is measured in days, e.g. t = 1, …, T. Replacing derivatives in (5) with finite differences, we can consider a discrete version of (5): i ( t + 1 ) - i ( t ) = β ( t ) i ( t ) { 1 - i ( t ) - r ( t ) } - γ ( t ) i ( t ) , r ( t + 1 ) - r ( t ) = γ ( t ) i ( t ) , (6) where β(t) and γ(t) are positive functions of t. We set i(0) = i₀ and r(0) = r₀ with t = 0 being the starting date.

Model (6) admits a recursive way to compute i(t) and r(t): i ( t + 1 ) = { 1 + β ( t ) - γ ( t ) } i ( t ) - β ( t ) i ( t ) { i ( t ) + r ( t ) } , r ( t + 1 ) = r ( t ) + γ ( t ) i ( t ) (7) for t = 0, …, T − 1. The first equation of (7) implies that β(t) < γ(t) or R 0 ( t ) = β ( t ) γ - 1 ( t ) < 1 leads to that i(t + 1) < i(t) or the number of infectious cases drops, meaning the spread of virus is controlled; otherwise, the number of infectious cases will keep increasing.

To fit the model and estimate the time-dependent parameters, we can use nonparametric techniques, such as splines [38–43], local polynomial regression [44] and reproducible kernel Hilbert space method [45]. In particular, we consider a cubic B-spline approximation [46].

Denote by B(t) = {B₁(t),…,B_q(t)}_T the q cubic B-spline basis functions over [0, T] associated with the knots 0 = w₀ < w₁ < … < w_q−2 < w_q−1 = T. For added flexibility, we allow the number of knots to differ between β(t) and γ(t) and specify log β ( t ) = ∑ j = 1 q 1 β j B j ( t ) , log γ ( t ) = ∑ j = 1 q 2 γ j B j ( t ) . (8) When β 1 = ⋯ = β q 1 and γ 1 = ⋯ = γ q 2, the model reduces to a constant SIR model [46]. We use cross-validation to choose q₁ and q₂ in our numerical experiments.

Denote by β = ( β 1 , … , β q 1 ) and γ = ( γ 1 , … , γ q 2 ) the unknown parameters, by Z_I(t) and Z_R(t) the reported numbers of infectious and removed, respectively, and by z_I(t) = Z_I(t)/N and z_R(t) = Z_R(t)/N, the reported proportions. Also, denote by I(t) and R(t) the true numbers of infectious and removed, respectively at time t. We propose a Poisson model to link Z_I(t) and Z_R(t) to I(t) and R(t) as follows: Z R ( t ) ∼ Pois { R ( t ) } , Z I ( t ) ∼ Pois { I ( t ) } . (9)

We also assume that, given I(t) and R(t), the observed daily number {Z_I(t), Z_R(t)} are independent across t = 1, …, T, meaning the random reporting errors are “white” noise. We note that (9) is directly based on “true” numbers of infectious cases and removed cases derived from the discrete SIR model (6). This differs from the Markov process approach, which is based on the past observations.

With (6), (7) and (8), R(t) and I(t) are the functions of β and γ, since R(t) = N × r(t) and I(t) = N × i(t). Given the data (Z_I(t), Z_R(t)), t = 1, …, T, we obtain ( β ^ , γ ^ ), the estimates of (β, γ), by maximizing the following likelihood L ( β , γ ) = ∏ t = 1 T e - R ( t ) R ( t ) Z R ( t ) Z R ( t ) ! × ∏ t = 1 T e - I ( t ) I ( t ) Z I ( t ) Z I ( t ) ! , or, equivalently, maximizing the log likelihood function ℓ ( β , γ ) = N ∑ t = 1 T { - r ( t ) + z R ( t ) log r ( t ) - i ( t ) + z I ( t ) log i ( t ) } + C , (10) where C is a constant free of β and γ. See the S1 Appendix for additional details of optimization.

We then estimate the variance-covariance matrix of ( β ^ , γ ^ ) by inverting the second derivative of −ℓ(β, γ) evaluated at ( β ^ , γ ^ ). Finally, for t = 1, …, T, we estimate I(t) and R(t) by I ^ ( t ) = N i ^ ( t ) and R ^ ( t ) = N r ^ ( t ), where i ^ ( t ) and r ^ ( t ) are obtained from (7) with all unknown quantities replaced by their estimates; estimate β(t) and γ(t) by β ^ ( t ) and γ ^ ( t ), obtained by using (8) with (β, γ) replaced by ( β ^ , γ ^ ); and estimate R 0 ( t ) by R ^ 0 ( t ) = β ^ ( t ) / γ ^ ( t ).

Summary of estimation and inference for β(t), γ(t), R 0 ( t ), I(t), R(t)

The country-specific time-series data of confirmed, recovered, and death cases were obtained from a GitHub data repository website (https://github.com/ulklc/covid19-timeseries). This site collects information from various sources listed below on a daily basis at GMT 0:00, converts the data to the CSV format, and conducts data normalization and harmonization if inconsistencies are found. The data sources include

In particular, the current population size of each country, N, came from the website of WorldoMeter. Our analyses covered the periods between the date of the first reported coronavirus case in each nation and June 30, 2020. In the beginning of the outbreak, assessment of i₀ and r₀ was problematic as infectious but asymptomatic cases tended to be undetected due to lack of awareness and testing. To investigate how our method depends on the correct specification of the initial values r₀ and i₀, we conducted Monte Carlo simulations. As a comparison, we also studied the performance of the deterministic SIR model in the same settings. Fig 1 shows that, when the initial value i₀ was mis-specified to be 5 times of the truth, the curves of i(t) and r(t) obtained by the deterministic SIR model (6) were considerably biased. On the other hand, our proposed model (9), by accounting for the randomness of the observed data, was robust toward the mis-specification of i₀ and r₀: the estimates of r(t) and i(t) had negligible biases even with mis-specified initial values. In an omitted analysis, we mis-specified i₀ and r₀ to be only twice of the truth, and obtain the similar results.

Our numerical experiments also suggested that using the time series, starting from the date when both cases and removed were reported, may generate more reasonable estimates.

Using the cubic B-splines (8), we estimated the time-dependent transmission rate β(t) and removal rate γ(t), based on which we further estimated R 0 ( t ), I(t) and R(t). To choose the optimal number of knots for each country when implementing the spline approach, we used 5-fold cross-validation by minimizing the combined mean squared error for the estimated infectious and removed cases.

Fig 2 shows sharp variations in transmission rates and removal rates across different time periods, indicating the time-varying nature of these rates. The estimated I(t) and R(t) overlapped well with the observed number of infectious and removed cases, indicating the reasonableness of the method. The pointwise 95% confidence intervals (in yellow) represent the uncertainty of the estimates, which may be due to error in reporting. Fig 3 presents the estimated time-varying reproduction number, β ^ ( t ) γ ^ ( t ) - 1, for several countries. The curves capture the evolving trends of the epidemic for each country.

In the US, though the first confirmed case was reported on January 20, 2020, lack of immediate actions in the early stage let the epidemic spread widely. As a result, the US had seen soaring infectious cases, and R 0 ( t ) reached its peak around mid-March. From mid-March to early April, the US tightened the virus control policy by suspending foreign travels and closing borders, and the federal government and most states issued mandatory or advisory stay-home orders, which seemed to have substantially contained the virus.

The high reproduction numbers with China, Italy, and Sweden at the onset of the pandemic imply that the spread of the infectious disease was not well controlled in its early phases. With the extremely stringent mitigation policies such as city lockdown and mandatory mask-wearing implemented in the end of January, China was reported to bring its epidemic under control with a quickly dropping R 0 ( t ) in February. This indicates that China might have contained the epidemic, with more people removed from infectious status than those who became infectious.

Sweden is among the few countries that imposed more relaxed measures to control coronavirus and advocated herd immunity. The Swedish approach has initiated much debate. While some criticized that this may endanger the general population in a reckless way, some felt this might terminate the pandemic more effectively in the absence of vaccines [50]. Fig 3 demonstrates that Sweden has a large reproduction number, which however keeps decreasing. The “big V” shape of the reproduction number around May 1 might be due to the reporting errors or lags. Our investigation found that the reported number of infectious cases in that period suddenly dropped and then quickly rose back, which was unusual.

Around February 18, a surge in South Korea was linked to a massive cluster of more than 5,000 cases [51]. The outbreak was clearly depicted in the time-varying R 0 ( t ) curve. Since then, South Korea appeared to have slowed its epidemic, likely due to expansive testing programs and extensive efforts to trace and isolate patients and their contacts [52].

More broadly, Fig 3 categorizes countries into two groups. One group features the countries which have contained coronavirus. Countries, such as China and South Korea, took aggressive actions after the outbreak and presented sharper downward slopes. Some European countries such as Italy and Spain and Mideastern countries such as Iran, which were hit later than the East Asian countries, share a similar pattern, though with much flatter slopes. On the other hand, the US, Brazil, and Sweden are still struggling to contain the virus, with the R 0 ( t ) curves hovering over 1. We also caution that, among the countries whose R 0 ( t ) dropped below 1, the curves of the reproduction numbers are beginning to uptick, possibly due to the resumed economy activities.

We have developed a web application (https://younghhk.shinyapps.io/tvSIRforCOVID19/) to facilitate users’ application of the proposed method to compute the time-varying reproduction number, and estimated and predict the daily numbers of active cases and removed cases for the presented countries and other countries; see Fig 4 for an illustration.

Our code was written in R [53], using the bs function in the splines package for cubic B-spline approximation, the nlm function in the stats package for nonlinear minimization, and the jacobian function in the numDeriv package for computation of gradients and hessian matrices. Graphs were made by using the ggplot2 package. Our code can be found on the aforementioned shiny website.