key: cord-0638215-nstnhmla
authors: Hong, Hyokyoung G.; Li, Yi
title: Estimation of time-varying transmission and removal rates underlying epidemiological processes: a new statistical tool for the COVID-19 pandemic
date: 2020-04-13
journal: nan
DOI: nan
sha: dce4aec3e326de362e6505ad430738984835c31e
doc_id: 638215
cord_uid: nstnhmla

The coronavirus pandemic has rapidly evolved into an unprecedented crisis. The susceptible-infectious-removed (SIR) model and its variants have been used for modeling the pandemic. However, time-independent parameters in the classical models may not capture the dynamic transmission and removal processes, governed by virus containment strategies taken at various phases of the epidemic. Moreover, very few models account for possible inaccuracies of the reported cases. We propose a Poisson model with time-dependent transmission and removal rates to account for possible random errors in reporting and estimate a time-dependent disease reproduction number, which may be used to assess the effectiveness of virus control strategies. We apply our method to study the pandemic in several severely impacted countries, and analyze and forecast the evolving spread of the coronavirus. We have developed an interactive web application to facilitate readers' use of our method.

Coronaviruses are enveloped single-stranded positive-sense RNA viruses belonging to a broad family of coronaviridae and are widely harbored in animals [1] [2] [3] . Most of the coronaviruses only cause mild respiratory infections, but SARS-CoV-2, a newly identified member of the coronavirus family, initiated the very contagious and lethal coronavirus disease 2019 in December 2019 [4, 5] . Since the detection of the first case in Wuhan, the COVID-19 pandemic has evolved into a global crisis within only four months. As of 4/9/2020, the virus has infected about 1.6 million individuals, caused more than 100,000 deaths [6] , and altered the life of billions of people.

The pandemic has been closely monitored by the international society. Since 1/22/2020, daily numbers of infectious and recovered cases, and deaths have been reported for nearly every country. Much effort has been devoted by the affected countries to battling the disease. However, the crisis has yet been mitigated, with new infections detected every day. To forecast when the pandemic gets controlled, it is imperative to develop appropriate models to describe and understand the change trend of the pandemic [7] [8] [9] [10] .

The susceptible-infectious-removed (SIR) model was utilized to explain the rapid rise and fall of the infected individuals from the epidemics of severe acute respiratory syndrome (SARS), influenza A virus subtype (H1N1) and middle east respiratory syndrome (MERS) [11] [12] [13] [14] [15] . The key idea is to divide a total population into three compartments: the susceptible, S, who are healthy individuals capable of contracting the disease; the infectious, I, who have the disease and are infectious; and the removed, R, who have recovered from the disease and gained immunity or who have died from the disease [16] . The model assumes a one-way flow from susceptible to infectious to removed, and is reasonable for infectious diseases, which are transmitted from human to human, and where recovery confers lasting resistance [17] .

The following ordinary differential equations (ODE) describe the change rates of s(t), i(t) and r(t):

with an initial condition: i(0) = i 0 and r(0) = r 0 , where i 0 > 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), we derive an important quantity, which is the time-dependent reproduction number

Indeed, dividing the second equation by the third equation in (1) leads to

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 (2) , 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 t 0 such that R 0 (t 0 ) = 1, when the outbreak is controlled with (di/dr)(t 0 ) < 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) as

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 (3) with finite differences, we can consider a discrete version of (3):

where β(t) and γ(t) are positive functions of t. We set i(1) = i 0 > 0 and r(1) = r 0 , with t = 1 being the starting date. We assume that s(t) . = 1, or i(t) + r(t) . = 0, for t = 1, . . . , T , that is, the portion of the infectious and removed is minor compared to the susceptible in a general population. This seems true before the pandemic infects a sizable portion of the population. For example, even in the countries severely attacked, such as Italy, the US and Spain, s(t) > 99.5% as of 4/9/2020 [6] . We then propose a modified version of (4):

As the first equation of (5) implies i(t + 1) = {1 + β(t) − γ(t)}i(t), (5) admits a "closed-form" solution:

for t = 2, . . . , T . Equation (6) implies that when R 0 (t) = β(t)γ −1 (t) < 1, 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.

Nonparametric techniques, such as splines [38] , local polynomial regression [39] and reproducible kernel Hilbert space method [40] , can be used to model β(t) and γ(t). Based on our numerical experiences, however, simple polynomial approximations can work well. In particular, we specify

where C 0 is a large constant relative to T (e.g. 2T ) to enhance numerical stability. As such, the polynomials can be regarded as truncated Taylor expansions of log β(t) and log γ(t) around 0. As p and q are unknown, we can choose them by using the Akaike information criterion (AIC) or the Bayesian information criterion (BIC). When p = q = 0, the model reduces to a constant SIR model. Denote by β = (β 0 , . . . , β p ) and γ = (γ 0 , . . . , γ q ) 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)) and Z I (t) ∼ Pois(I(t)).

We also assume that, given I(t) and R(t), the observed (Z I (t), Z R (t)) are independent across t = 1, . . . , T , meaning the random reporting errors are "white" noise. With (5), (6) and (7), 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 likelihood

or, equivalently, maximizing the log likelihood function

where C is a constant free of β and γ. To solve this optimization problem, we differentiate (β, γ) with respect to (β, γ). Then (β,γ) solves the following estimating equations:

where the summation starts from t = 2 as the term corresponding to t = 1 is 0 by using the initial condition and, for t ≥ 2,

Here, ∂ ∂β β(j) = β(j) × (1, j/C 0 , . . . , (j/C 0 ) p ) T and ∂ ∂γ γ(j) = γ(j) × (1, j/C 0 , . . . , (j/C 0 ) q ) T . We then estimate the variance-covariance matrix of (β,γ) by inverting the second derivative of − (β, γ) evaluated at (β,γ). Finally, for t = 1, . . . , T , we estimate I(t) and R(t) byÎ(t) = Nî(t) and R(t) = Nr(t), whereî(t) andr(t) are obtained from (6) with all unknown quantities replaced by their estimates; estimate β(t) and γ(t) byβ(t) andγ(t), obtained by using (7) with (β, γ) replaced by (β,γ); and estimate R 0 (t) byR 0 (t) =β(t)/γ(t).

Summary of estimation and inference for β(t), γ(t), R 0 (t), I(t), R(t) Estimation: Let N be the size of population of a given country. The date when the first case was reported is set to be the starting date with t = 1, i 0 = Z I (1)/N and r 0 = Z R (1)/N . The observed data are {Z I (t), Z R (t), t = 1, . . . , T }. (7), we set p = q in practice for computational convenience and maximize (9) to obtainβ = (β 0 ,β 1 , . . . ,β p ) andγ = (γ 0 ,γ 1 , . . . ,γ p ). The optimal p, denoted by p * , minimizes the AIC or BIC criterion:

When there is no confusion, in the following we denote byβ = (β 0 ,β 1 , . . . ,β p * ) andγ = (γ 0 ,γ 1 , . . . ,γ p * ). We also calculateβ(t),γ(t),R 0 (t),R(t),Î(t) with p = p * .

Inference: The estimated variance-covariance matrix of (β,γ), denoted byV (β,γ), is obtained by inverting the second derivative of − (β, γ) evaluated at (β,γ). For each t, asβ(t),γ(t),R 0 (t),R(t) andÎ(t) are smooth functions ofβ andγ, we apply the delta method [41] to estimate their variances and obtain the confidence intervals. As an illustration, we compute var(R(t)) =Ṙ(t) TV (β,γ)Ṙ(t) and var(Î(t)) =İ(t) TV (β,γ)İ(t), whereṘ(t) andİ(t) are the partial derivative vectors ofR(t) and I(t) with respect to (β,γ).

3.1 Effects of mis-specifications of i 0 and r 0 on estimation An important question to address is what roles the initial values r 0 and i 0 play in estimation. Accurate assessment of i 0 can be problematic as in the beginning of an epidemic, cases are likely to be underreported or unreported because of lack of awareness or lack of testing. For a deterministic SIR model (5) , r(t) and i(t) may heavily depend on their initial values. As shown in Figure 1 , when the initial value i 0 is mis-specified to be 5 times of the truth, the curves of i(t) and r(t) are biased to the left. On the other hand, our proposed statistical model (8) , by accounting for the randomness of the observed data, is more robust toward the mis-specification of i 0 and r 0 , and estimates r(t) and i(t) with negligible biases even with mis-specified initial values. We also mis-specify i 0 to be only twice of the truth, and obtain the same results, which are omitted. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q True q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q (5) with true initials ("True"), and with the mis-specified initials ("Mis-specified"), and the estimated i(t) and r(t) using the proposed model (8) with the mis-specified initials ("Proposed"). The i(t) and r(t) are generated using (β, γ) = (e 0.35 , e −1.95 ) and (i 0 , r 0 ) = (10 −6 , 10 −6 ) in (5) . The mis-specified initials are (5 × 10 −6 , 5 × 10 −6 ).

We explore the robustness of the estimates of r(t) and i(t) when β(t) and γ(t) are mis-specified. We generate Z R (t) and Z I (t) with Model (8), where R(t) and I(t) are specified by the SIR model (5) with r 0 = i 0 = 10 −6 , β(t) = 1 − 0.005t and γ(t) = 0.5 + 0.002t, for t = 1, . . . , 70. A total of 1,000 datasets are generated. We apply the proposed method to analyze each dataset and estimate i(t) and r(t) for t = 1, . . . , 70. Figure 2 examines the relative biases of the estimates of i(t), r(t) and R 0 (t) at each t. In the first few days of the time series, i(t) and r(t) are overestimated, but as t gets large the biases become negligible. On the other hand, the estimates of R 0 (t) incur few biases. All suggest the robustness of our method. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q (8), where R(t) and I(t) are specified by (5) . Both AIC and BIC, for the majority of times, choose 0 (884 out of 1,000 for AIC, 986 out of 1,000 for BIC) and only occasionally choose 1.

We evaluate the performance of the proposed model using the bias and the agreement between the empirical and model based standard errors, based on 1,000 independently simulated datasets. The estimates are nearly unbiased and the model based standard errors match with the empirical standard errors; see Table 1 .

Example 2. We modify Example 1 by specifying the true β(t) and γ(t) to be time-dependent with β(t) = exp(β 0 + β 1 t/C 0 ) and

We observe that both AIC and BIC choose p * = 1 with an accuracy of 100%, the estimates are with negligible biases, and the model based standard errors agree to the empirical standard errors; see Table 1 .

4 Analysis of the COVID-19 pandemic among severely impacted countries

The Johns Hopkins University's Coronavirus Resource Center [6] hosts the country level data. We derive the number of active COVID-19 cases, I(t), and the cumulative number of combined recovered We first fit the model with constant transmission and removal rates. Though the model may not fit the data well, the estimated transmission and removal rates can roughly be interpreted as the averages of time-varying transmission and removal rates over the period of the observation time, and may give a simple exposition of how the countries fared during this crisis so far. As of 4/9/2020, the basic reproduction number R 0 has been computed for 17 severely impacted countries ( Figure 3 ). Among them, the estimated R 0 ranges from 1.16 to 17.40, with an average of 5.7. Countries, such as China and Korea, which developed the virus outbreak early, have a relatively lowR 0 of 1.16 and 2.44, meaning the outbreak has been reasonably controlled. Other countries, such as India, the US and Brazil, which were hit hard by the pandemic recently, have seen a ramp-up in virus testing with more cases detected, and the estimated R 0 is alarmingly high with 8.99, 14.1 and 17.4, respectively. However, we caution that R 0 is sensitive to the observation period and should be interpreted within a timing context [42] .

Since the first case of COVID-19 was detected in Wuhan, China, it quickly spread to nearly every part of the world [6] . COVID-19, conjectured to be more contagious than the previous SARS and H1N1 [43] , has put great strain on healthcare systems worldwide, especially among the severely impacted countries [44] . We apply our method to assess the epidemiological processes of COVID-19 in some of these countries; see Figure 4 for the estimated country-specific transmission, removal rates, and R 0 (t).

In January 2020, the transmission rate in China was high with a low removal rate and rapidly rising cases, resulting in a high R 0 . Because of extremely stringent mitigation policies such as city lockdown and mandatory mask-wearing implemented in the end of January, China brought its epidemic under control with a quickly dropping R 0 in February. On Feb 15, R 0 touched 1, or equivalently, log R 0 crossed 0, indicating that China has contained the epidemic and more people removed from infectious status than those who became infectious.

Korea followed a similar pattern. The sudden outbreak with a massive cluster of more than 5,000 cases was linked to a minor Christian sect [45] , which explains an extremely high R 0 in the early phase of the epidemic. Since then, Korea appeared to have greatly slowed its epidemic, likely due to expansive testing programs and extensive efforts to trace and isolate patients and their contacts [46] . Around 3/17/2020, R 0 dropped below 1.

Since the early March, the US has seen soaring infectious cases, and R 0 reached the peak around 3/10/2020. Around that time, the federal government and several states have issued mandatory or Estimates of β, γ, and R 0 for model (5) with time-independent parameters, based on the data up to 4/9/2020. advisory stay-home orders, which seemed to have lessened the virus spread. The transmission rate started decreasing around the mid-March and R 0 dropped from 38.04 on 3/10/2020 to 3.89 on 4/9/2020. More broadly, Figure 4 categorizes countries into two groups based on the shape of R 0 . One group features a monotone decreasing R 0 . The countries, such as China and Korea, took aggressive actions after the outbreak and present sharper slopes of the curves. Some European countries, such as France, Iran, Italy and Spain, which were hit later than the Asian countries, share a similar pattern, though with much flatter slopes. On the other hand, bell-shaped curves are observed for the US and UK, likely because these governments were initially slow to react to the virus attack, but did take strong measures later to fight the disease. Nevertheless, almost all of the countries are featuring decreasing R 0 , which soon will likely drop below 1, and may be able to declare the containment of the epidemic in the near future.

Our model enables estimation of I(t) and R(t). As an illustration, Figures 5 and 6 depict the estimated I(t) and R(t) curves for the selected countries. The red and green curves represent the observed data and the model-based predictions, respectively, with 95% confidence intervals. With a large N , the model-based standard errors are substantially small compared toÎ(t) andR(t), making the confidence interval bounds hardly distinguished fromÎ(t) andR(t). The estimates of I(t) and R(t) seem to be fairly close to the observed numbers of infectious and removed cases, especially when the observed curves are smooth. As expected, larger prediction errors happen at where sudden jumps or drops occur. With extrapolations, our model can make reasonably accurate short-term predictions; however, for long-term predictions, we are cautious as model-based predictions cannot reflect the consequences of the future policy changes.

We created an interactive web application (https://younghhk.shinyapps.io/tvSIRforCOVID19/) to facilitate users' application of the proposed method to compute the time-varying reproduction number, and to predict the daily numbers of active cases and removed cases for selected countries; see Figure 7 for an illustration. 

The rampaging pandemic of COVID-19 has called for developing proper computational and statistical tools to understand the trend of the spread of the disease and evaluate the efficacy of mitigation measures [47] [48] [49] [50] . We propose a Poisson model with time-dependent transmission and removal rates. Our model accommodates possible random errors in the number reporting, and estimates a timedependent disease reproduction number, R 0 (t), which can serve as a metric for timely evaluating the effects of health policies. Applications of our method to study the epidemics in several selected countries illustrate the results of the virus containment policies implemented in these countries, and may serve as the epidemiological benchmarks for the future preventive measures. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqqqqqqqqqqq qqq qq qq qq q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q qq q qq qq qqq qq qqq qqqqq qqqqq qqqqqqqqqqqqq qqq qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq qq qq qqq qqqq qqqqqqqq qqqqqqqqqqqqq qqq qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq qq qq qqq qqqq qqqqqqqq qqqqqqqqqqqqq qqq qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq qq qqq qqqq qqqqqqq qq China R q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q France R q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Italy R q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Spain R q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Korea R q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Several methodological questions need to be addressed. First, we analyzed each country separately, without considering the traffic flows among these countries. We will develop a joint model for the global epidemic, which accounts for the geographic locations of and the connectivity among the countries. Second, we have ignored the birth and natural death processes (e.g. deaths due to non-COVID-19 causes), and also combined the recovered and deaths in our model. We will extend our model by considering them as separate compartments. Finally, to reduce the computational complexity, we have assumed that log β(t) and log γ(t) are polynomials and with the same order. Though the performance of our parametric approach is adequate, the model can be refined with more flexible functions or by using nonparametric approaches. In particular, model (5) implies R 0 (t) = 1 + i(t + 1) − i(t) r(t + 1) − r(t) , naturally leading to a nonparametric estimator R 0 (t) = 1 + Z I (t + 1) − Z I (t) Z R (t + 1) − Z R (t) .

We will pursue this.

Containment of COVID-19 requires the concerted effort of health care workers, health policy makers as well as citizens. Measures, e.g. self-quarantine, social distancing, and shelter in place, have been executed at various phases by each country to prevent the community transmission. Timely and effective assessment of these actions constitutes a critical component of the effort. SIR models have been widely used to model this pandemic. However, constant transmission and removal rates may not capture the timely influences of these policies. We propose a time-varying SIR Poisson model to assess the dynamic transmission patterns of COVID-19. With the virus containment measures taken at various time points, R 0 may vary substantially over time. Our model provides a systematic and daily updatable tool to evaluate the immediate outcomes of these actions. As some countries in Asia are now shifting gear to battle the second waves of virus attack induced by the imported cases [51, 52] , our tool may also shed light on and aid the implementation of future containment strategies.

Coronaviruses: an overview of their replication and pathogenesis

Bats are natural reservoirs of SARS-like coronaviruses

Discovery of seven novel Mammalian and avian coronaviruses in the genus deltacoronavirus supports bat coronaviruses as the gene source of alphacoronavirus and betacoronavirus and avian coronaviruses as the gene source of gammacoronavirus and deltacoronavirus

Human coronavirus and severe acute respiratory infection in Southern Brazil. Pathogens and Global Health

Evolving epidemiology and impact of non-pharmaceutical interventions on the outbreak of coronavirus disease 2019 in Wuhan

Johns Hopkins Cornonavirus Resource Center

Real-time epidemic forecasting for pandemic influenza

Mathematical models of infectious disease transmission

Modelling transmission and control of the COVID-19 pandemic in Australia

Challenges in control of Covid-19: short doubling time and long delay to effect of interventions

Individual vaccination as Nash equilibrium in a SIR model with application to the 2009-2010 influenza A (H1N1) epidemic in France

Estimating epidemic parameters: Application to H1N1 pandemic data

Bayesian estimation of the dynamics of pandemic (H1N1) 2009 influenza transmission in Queensland: A space-time SIR-based model

Modeling super-spreading events for infectious diseases: case study SARS

Deterministic SIR (Susceptible-Infected-Removed) models applied to varicella outbreaks

An introduction to compartmental modeling for the budding infectious disease modeler

A contribution to the mathematical theory of epidemics

Statistics based predictions of coronavirus 2019-nCoV spreading in mainland China

A time delay dynamical model for outbreak of 2019-nCoV and the parameter identification

Epidemic analysis of COVID-19 in China by dynamical modeling

Preliminary prediction of the basic reproduction number of the Wuhan novel coronavirus 2019-nCoV

Effective containment explains sub-exponential growth in confirmed cases of recent COVID-19 outbreak in mainland China

Lessons from the history of quarantine, from plague to influenza A. Emerging Infectious Diseases

A Time-dependent SIR model for COVID-19 with undetectable infected persons

SIR model with time dependent infectivity parameter : approximating the epidemic attractor and the importance of the initial phase

An epidemiological forecast model and software assessing interventions on COVID-19 epidemic in China

Modeling Count Data

Methods for estimating disease transmission rates: Evaluating the precision of Poisson regression and two novel methods

Fitting outbreak models to data from many small norovirus outbreaks

Multi-species SIR models from a dynamical Bayesian perspective

The estimation of the basic reproduction number for infectious diseases

Mathematical Epidemiology of Infectious Diseases: Model Building

Transmission potential of smallpox: estimates based on detailed data from an outbreak

Measurability of the epidemic reproduction number in data-driven contact networks

A Time-dependent SIR model for COVID-19 with undetectable infected persons

Notes on R0

A Practical Guide to Splines

Empirical-bias bandwidths for local polynomial nonparametric regression and density estimation

New reproducing kernel functions. Mathematical Problems in Engineering

A note on the jackknife, the bootstrap and the delta method estimators of bias and variance

Complexity of the basic reproduction number (R0)

COVID-19, chronicle of an expected pandemic

Covid-19: how doctors and healthcare systems are tackling coronavirus worldwide

Normile D. Coronavirus cases have dropped sharply in South Korea. What's the secret to its success?

Current status of global research on novel coronavirus disease (COVID-19): A bibliometric analysis and knowledge mapping

Investigating the cases of novel coronavirus disease (COVID-19) in China using dynamic statistical techniques

The impact of social distancing and epicenter lockdown on the COVID-19 epidemic in mainland China: A data-driven SEIQR model study

COVID-19 Italian and Europe epidemic evolution: A SEIR model with lockdown-dependent transmission rate based on Chinese data

As China's virus cases reach zero, experts warn of second wave

Asian nations face second wave of imported cases

This work was supported in part by the National Science Foundation and the National Institutes of Health.