Estimating the Growth Rate and Doubling Time During the COVID-19 Pandemic with a SAS Macro

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Estimating the Growth Rate and Doubling Time to Assist Short-Term Prediction During the COVID-19 Pandemic with a SAS Macro

Stanley Xu^1,2, Christina Clarke¹, Susan Shetterly¹, Komal Narwaney¹

¹ The Institute for Health Research, Kaiser Permanente Colorado, 2550 S. Parker Road, Aurora, CO 80014

² School of Public Health, University of Colorado, 13001 E 17th Pl, Aurora, CO 80045

1. BACKGROUND

In December 2019, an outbreak of coronavirus disease (COVID-19) caused by the novel coronavirus (SARS-CoV-2) began in Wuhan, China and has now spread across the world [1,2]. In the United States, the cumulative number of identified COVID-19 cases was 186,101 as of March 31st, 2020; among the identified cases, 3603 died [3]. To slow the spread of COVID-19, federal and local governments issued mitigation measures such as case isolation, quarantine, school closures and closing non-essential businesses. The COVID-19 pandemic imposes tremendous challenges to the US health care system, particularly given concerns that the need for hospital beds and ICU beds could exceed capacity [4-6]. Predicting the future numbers of COVID-19 cases and healthcare utilization is critical for governments and health care systems preparation plans [4,6,7]. Two useful and critical quantities for prediction are the growth rate [8] and the doubling time of number of events [9]. The growth rate is the percent change of daily events (e.g, COVID-19 cases, number of patients hospitalized or number of deaths).  The doubling time is the length of time required to double the number of daily events.

Our goal was to develop an approach and create a SAS macro using observed data to estimate the growth rate and doubling time in days for short-term prediction.

2. METHODS

 2.1 A rolling growth curve approach (RGCA)

In the United States, there were several barriers for testing people for COVID-19 such as shortages of swabs and testing kits and restrictions on who should get tested. Therefore, the number of COVID-19 cases was often under-identified and under-reported. However, the number of hospitalized COVID-19 patients and number of deaths due to COVID-19 were more reliable than the reported number of COVID-19 cases [10]. In this paper, we used the number of daily deaths to calculate the growth rate and doubling time in days.

We assumed a growth curve of daily deaths over a period of  days from day   (start day) to day (). Let

  denote the daily deaths at day , . Based on the growth model, we have

                                                            (1)         

where   is the number of deaths at the start day ;   is the growth rate. When the growth rate , the number of daily deaths increases. For example, if  =0.4, the growth rate of deaths is 40% more for each day. When growth rate , the number of daily deaths has no change. When growth rate , the number of daily deaths declines.  When the number of deaths doubles at , that is , we have, 

                   .

Further, it can be shown that

                                                                          (2)

We fit two models: a) using equation (1) which estimates the growth rate ; b) using equation (1) with  substituted with   from equation (2). The second model estimates the doubling time in days , meaning that it takes D days from the start day   for the number of daily deaths to double.  We used SAS PROC NLIN [11] to fit these two nonlinear models. Note that equation (2) is valid for . When , one can use ; the estimated  represents the days required to reduce the number of deaths by half.

Because the growth rate and doubling time may change over time, we used a rolling growth curve approach (RGCA). For example, we set the length of the period to be 7 days ( days). We estimated the growth rate and the doubling time in days for the following periods for death data from New York State from March 14th – March 31st [12]: March 14^th-20^th,15^th-21^st, 16^th-22^nd. . ., 25^th-31^st.

2.2 Short-term prediction

The estimated growth rate from the last period of the RGCA approach (e.g., March 25^th-31^st) can be used for future short-term prediction of deaths. Let  denote the last day of the last period,  is the number of deaths on this day. For the New York death data in this analysis,  is March 31^st, . Let  denote the date after date , then the predicted  is

                         

where  is the estimated growth rate from the last period. As the growth rate changes over time, the prediction is only appropriate for short-term prediction (e.g., within 7 days) and updated growth rates should be used.

3. RESULTS

While the growth rate had a spike at March 17th-19th, subsequent days are lower with a decreasing trend for March 20th-25th (Figure 1).  Consistently, the doubling time in days had an increasing trend most evident in the later time as well (Figure 2).

While the graphs provide a useful visual tool, the macro supports calculating rates within defined time intervals of interest. At the beginning of this observation period (March 14^th-March 20^th), the growth rate was 0.48 (95% CI, 0.39-0.57) and the doubling time was 2.77 days (95% CI, 2.49-3.04). At the end of this observation period (March 25^th-March 31^st), the growth rate decreased to 0.25 (95% CI, 0.22-0.28) and the doubling time increased to 4.09 days (95% CI, 3.73-4.44).

Using the estimated growth rate from the last period March 25^th-March 31^st, , the predicted numbers of daily deaths for April 1^st and 2^nd were 468 and 586, respectively. The observed number of deaths in New York State was 498 on April 1^st.

4. DISCUSSION

These models can be similarly applied to hospitalization data if those data are available. When COVID-19 testing is widely available to the public and the number of COVID-19 testing is less selective, these models can also be used to directly estimate the growth rate and the doubling time for COVID-19 cases. Due to a lag in reporting death, it is recommended to exclude the recent 1-2 days’ death data in fitting the growth curves. This paper illustrates that death data can be used to estimate the growth rate and doubling time to aid predicting future deaths, hospitalizations and COVID-19 cases. Because a series of growth curves were fit, the RGCA approach can also be used for real-time monitoring of the epidemic trend as shown in Figure 1.

Acknowledgements

This research was supported by the Institute for Health Research, Kaiser Permanente Colorado. Xu was also supported by NIH/NCRR Colorado CTSI Grant Number UL1 RR025780.

REFERENCES

1. CDC. 2019 Novel Coronavirus, Wuhan, China. Available at https://www.cdc.gov/coronavirus/2019-ncov/
2. WHO. WHO Director-General’s opening remarks at the media briefing on COVID-19 – 11 March 2020
3. CDC. https://www.cdc.gov/coronavirus/2019-ncov/cases-updates/cases-in-us.html
4. IHME COVID-19 health service utilization forecasting team. Forecasting COVID-19 impact on hospital bed-days, ICU-days, ventilator days and deaths by US state in the next 4 months. MedRxiv. 26 March 2020. doi:10.1101/2020.03.27.20043752.
5. Ferguson NM, Laydon D, Nedjati-Gilani G, et al. Impact of non-pharmaceutical

• interventions (NPIs) to reduce COVID-19 mortality and healthcare demand. Imp Coll COVID-19 Response Team. March 2020:20. doi:https://doi.org/10.25561/77482

6. Tsai TC, Jacobson B, Jha AK. American hospital capacity and projected need for COVID-19 patient care. Health Aff (Millwood). March 2020. doi:10.1377/hblog20200317.457910
7. Petropoulos F, Makridakis S (2020) Forecasting the novel coronavirus COVID-19. PLoS ONE 15(3): e0231236. https://doi.org/10.1371/journal.pone.0231236
8. Du Z, Xu X, Wu Y, Wang L, Cowling BJ, Ancel Meyers L. Serial interval of COVID-19 among publicly reported confirmed cases. Emerg Infect Dis. 2020 Jun [date cited]. https://doi.org/10.3201/eid2606.200357
9. Nunes-Vaz, R., 2020. Visualising the doubling time of COVID-19 allows comparison of the success of containment measures. Global Biosecurity, 1(3), p.None. DOI: http://doi.org/10.31646/gbio.61
10. https://covid19.healthdata.org/projections
11. SAS Institute, version 9.4, Cary, NC
12. https://coronavirus.1point3acres.com/en

Figure 1. Estimated growth rate with 95% CIs over time using death data from New York State.

Figure 2. Estimated doubling time in days with 95% CIs over time using death data from New York State.

APPENDIX

CreateSampleDatasets_RunMacro.sas

Macro_COVID_GrowthRate.sas

[Rick_SAS]

See also the article, "Estimates of doubling time for exponential growth."