We estimated the infection rate from the reported cases over a 7-day window to obtain a continuous estimation of R t. However, the 7-day forecast mean absolute prediction error approached The maximal R t displayed a wide range of 2. We found that the aSIR model can rapidly adapt to an increase in the number of tests and an associated increase in the reported cases of infection.
Our results also suggest that intensive testing may be an effective method of reducing R t. The aSIR model provides a simple and accurate computational tool for continuous R t estimation and evaluation of the efficacy of mitigation measures. Modeling studies are key to understanding the factors that drive the spread of the disease and for developing mitigation strategies. Early modeling efforts forecasted very large numbers of infected individuals, which would overwhelm health care systems in many countries [ 1 - 3 ].
These forecasts served as a call to action for policymakers to introduce policy measures including social distancing, travel restrictions, and eventually lockdowns to avoid the predicted catastrophe [ 4 - 6 ]. In , Hamer [ 8 ] speculated that the course of an epidemic is determined by the rate of contact between susceptible and infectious individuals. The R t depends on three factors: 1 the likelihood of infection per contact, 2 the period during which infectious individuals freely interact with susceptible individuals and spread the disease, and 3 the rate of contact.
The likelihood of infection per contact factor 1 is determined on the basis of pathogen virulence and protective measures such as social distancing or wearing masks. Free interactions between infectious and susceptible individuals factor 2 occur until the infectious individual is self-quarantined or hospitalized, either when the individual tests positive or experiences severe symptoms. Finally, the rate of contact factor 3 is strongly affected by public health measures to mitigate risk [ 10 ], such as lockdowns during the COVID pandemic.
Thus, R t is determined on the basis of the biological properties of the pathogen and multiple aspects of social behavior. Real-time R t estimation is critical for determining the effect of implemented mitigation measures and future planning. We propose a method for continuous estimation of the infection rate and R t to investigate the effect of mitigation measures and immunity acquired by those who recover from the disease.
The SIR model is one of the simplest epidemiological models that still captures the main properties of an epidemic [ 11 , 12 ], and it has been widely used in epidemic modeling studies. In most SIR modeling studies, the model parameters were constant. An SIR model with constant parameters, however, cannot be applied for the COVID pandemic because various mitigating measures were introduced during pandemic progression.
In these modeling studies, the rate of infection spread was assumed to be piece-wise linear among the 3 dates of the implementation of policy changes. In another approach, continuous estimation of R t and an assessment of the effect of mitigation measures were carried out on the basis of estimates of the distribution of the serial intervals between symptom onset in the primary and secondary cases [ 15 - 17 ].
Bayesian inference and methods based on estimations of the serial interval include multiple parameters whose values are not estimated from the data. In contrast, we propose an adaptive SIR aSIR model in which only one parameter—the removal rate—is determined from the literature, while the second parameter—the infection rate—is continuously estimated from the data through a sliding window approach.
A continuous R t estimate is then obtained using the infection rate estimate. The SIR model is described as a system of differential equations, and the key idea in our proposed method is that the initial conditions for each window are considered as values estimated for the previous window.
The only additional hyperparameter is the length of the sliding window. The proposed method retains the conceptual and computational simplicity of SIR-type models and can be easily extended through the introduction of additional compartments supported by data.
Data on daily and cumulative confirmed cases between February 29 and September 2, , were obtained from John Hopkins University JHU , and the dates of interventions by state eg, state of emergency and stay-at-home orders were obtained from Wikipedia. The JHU data were available at 2 levels of aggregation: county and state. Here, S is the number of susceptible individuals, I is the number of infectious individuals who freely interact with others and can transmit the infection, R is the number of individuals excluded from the other 2 compartments because they are quarantined or hospitalized, have recovered and acquired immunity, or have died.
Several sources of government data on COVID provide the daily number of newly confirmed cases and a cumulative number of confirmed cases. Careful consideration is required to determine whether these numbers should be attributed to the I or R compartment. Therefore, we assigned the data on confirmed cases to the R compartment, and we fit the model to the cumulative number of confirmed cases.
Therefore, we assumed the duration of the infectious period as the average time taken for the infected individual to be isolated, not the overall time for recovery. The average time to symptom onset is days [ 21 - 23 ]. For the first window, we determined the date when the number of confirmed cases began to increase exponentially. This is important because for many states or counties, very few confirmed cases were initially reported for a number of days or even weeks, which suggests that either the epidemic had not started or the true number of infected individuals was unknown.
It is not reasonable to apply an SIR model for this initial period. We considered the onset of the pandemic as the first of the 4 consecutive days in which the number of reported confirmed cases increased in at least 3 days. The initial conditions for system 1 for window 0 were as follows:. The R t. To obtain a smooth estimate of R t , we used a rolling average of 5 points.
We fit the model for each state and county in the United States. Model performance was evaluated by calculating the quality of fit as the root mean squared error between the actual and fitted R data for all windows concatenated wRMSE. Furthermore, we calculated 1-day, 3-day, and 7-day forecasts of R after each window Figure 1 A.
The mean absolute prediction error for the forecasts is provided in Table 1. The 1-day forecast error did not exceed 2. In particular, the 7-day forecast strongly overestimated the number of cases when R t was rapidly decreasing Figure 1. A Estimated Infectious and forecast Removed.
B Estimated reproduction number R t. The shaded region indicates the dates of the lockdown. The estimated daily number of infectious individuals rapidly increased and then gradually declined after the lockdown was implemented on March 22, Figure 1 A.
The estimated R t also declined upon implementation of the lockdown Figure 1 B. The time course of R t exhibits weekly seasonality, which likely reflects the effect of social interactions and possibly the effect of fluctuations in case reporting on weekdays vs weekends.
For New York and Nassau county, R t initially increased, which may reflect the fact that the pandemic in New York was continuously seeded by travelers arriving at John F Kennedy International Airport until a ban on international travel was implemented on March 12, This may also reflect the fact that not all severe cases were initially recognized and reported as COVID cases.
In June , Florida authorities introduced more stringent measures to control the pandemic, which is reflected in the reduction in R t in the second half of July The opening of multiple states since June has been accompanied by an increase in R t beyond 1 data not shown , and close monitoring of R t is needed to contain another wave of the pandemic.
In EpiEstim, we assumed an equal probability of infection within the infectious period of 6 days, the R t estimate was smoothed with a 7-point rolling average window, same as that in aSIR. While all 3 models show similar estimates when R t approaches 1, their estimates differ considerably in the beginning of the pandemic. In particular, the rt. The EpiEstim and aSIR models estimated similar peak values of R t , and both models estimated that R t decreased and approached 1 in the first week of April A hard-hitting detective who fearlessly dives into his work, Seo Do Won is relentless when it comes to bringing criminals to justice.
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