What’s new?
Today’s announcement was 608 new local cases in NSW is highly consistent with the new models. The new models have been performing well with 7-day and 14-day projections within 1-2% error, with the Gompertz model slightly outperforming the Richards’ growth curve model.
The new Richards’ growth curve model estimates that we will see around 3200-3700 cases over the next 7 days. The new Gompertz model estimates 3400-3900. The models are highly compatible with each other.
Context and timeline of the Sydney 2021 outbreak
I started these analysis in early July with the new COVID-19 (delta variant) outbreak in Sydney NSW. The purpose of these charts is that they provide some projections into the near future. Hopefully this allows for some data driven expectations. The Sydney 2021 series starts on 12 July 2021 and are available here: https://vitualis.com/?page_id=4071
The current Sydney outbreak started on 17 June 2021 – the first day with reports of community cases in Bondi. The NSW Government formally commenced stay-at-home orders (“lockdown”) for a number of inner-Sydney regions on 25 June 2021 at 2359, having announced this earlier in the day. This was then broadened 18 hours later to cover the whole of Greater Sydney. The initial plan was a two-week lockdown, with it potentially being lifted on 9 July 2021.
By 9 July 2021, there were early signs that COVID-19 transmission was worsening. On 15 July 2021, the lockdown was again extended until the end of July, and then extended again, this time for four weeks on 28 July 2021. Conditions at the end of the month for people living with South Western, and Western Sydney included a limit to travelling only 5 km from home, masks at all times outside, with only essential workers allowed to leave the local government area, with the requirement of a COVID-19 swab every 72 hours.
On 14 August 2021, in the setting of rapidly rising case counts, further restrictions were announced. The 5 km radius limit (from 10 km) was extended to all residents in Greater Sydney, starting 16 August 2021. Permits were also now required to leave the Greater Sydney area. Furthermore, stay-at-home orders were extended to across the entire state of NSW.
On 20 August 2021, the lockdown in Greater Sydney was extended to the end of September, and a range of new restrictions stated on 23 August 2021 in the “hot” LGAs including an overnight curfew, restricting outdoor exercise to an hour a day, closure of most businesses except for click-and-collect, and increased policing powers.
Projection of new daily cases, and cumulative counts of COVID-19 with data up to 5 October 2021
What is this?
The green charts are the Richards’ growth curve model, and the blue charts are the Gompertz model. The top image of each colour is a chart of the cumulative (total) COVID-19 cases in NSW, starting from 17 June 2021, and the lower image is a chart of the daily new cases. Only local cases are included (i.e., excluding cases identified in quarantine). Projections are given for the next 7 days. It should be noted that estimates have high levels of uncertainty beyond a few days and must be interpreted cautiously.
The projections are made using a model by fitting the cumulative case data since 23 August 2021 to the respective equations using non-linear regression. The regression uses the black data points, and excludes the grey data points. The dark central dashed lines are the model estimates, with 95% confidence intervals of the estimate. On the lower chart, the colour gradations can be understood as the degree of uncertainty in the model projections.
Gompertz and Richards’ growth curve
The Gompertz function is a type of sigmoid, or “S”-shaped curve. It’s been around since the early 19th century and was initially used to describe and model demographic mortality curves, and hence, well known to actuaries. The Gompertz function can also be used to accurately model biological growth (e.g., epidemics, tumour size, enzymatic reactions). I have chosen to use this model to help with creating insights as earlier in the pandemic, it was found to be useful in modelling cumulative cases of COVID-19 from the Chinese outbreaks (Jia et al. arXiv:2003.05447v2 [q-bio.PE]).
The Richards’ growth curve (or the generalised logistic function), which is a broad family of sigmoid (S-shaped) curves that can describe well many types of growth, including epidemics. It has also been demonstrated to have utility in modelling COVID-19 outbreaks in 2020 (Lee et al. PLoS One 2020 doi: 10.1371/journal.pone.0236860).
Gompertz equation:
Richard’s growth curve / generalised logistic function:
- B, C, D, E, F are the parameters of the model
- e is Euler’s number
- x in the model is the date (day number)
- These are the parameterisation of these functions used in the drc package: https://cran.r-project.org/web/packages/drc/drc.pdf
Why the changes?
I’ve undertaken some assessment of the degree of predictive error in both the Gompertz and Richards’ growth curve models. These charts compare the 7- and 14-day total case projections of the models, to what actually occurred in reality 7 and 14 days later. For interpretation, above the 0% error line means that the model provided an over-estimate compared to reality, and below the 0% error line an underestimate.
Both new models, restricted to regression to cumulative case data from 23 August 2021 onward, have demonstrated excellent projection accuracy. The Gompertz model has been slightly outperforming the Richards’ curve model.
Daily case trends
Comparison between the Gompertz and Richards’ growth curve model projections, along with smoothed data trends (7-day simple moving average, and GAM) with data up to 5 October 2021
The simple moving average and generalised additive model gives a descriptive “reality check” to the models. The GAM can be considered as an advanced smoothed trend of the daily counts. The new Gompertz and Richards’ growth curve models seem to be performing very well.
Model summaries
Gompertz model
summary(model.g)
Model fitted: Gompertz (4 parms)
Parameter estimates:
Estimate Std. Error t-value p-value
b:(Intercept) -5.7964e-02 8.2730e-04 -70.064 < 2.2e-16 ***
c:(Intercept) 7.2365e+03 3.7605e+02 19.243 < 2.2e-16 ***
d:(Intercept) 7.2688e+04 3.9601e+02 183.551 < 2.2e-16 ***
e:(Intercept) 8.3259e+01 1.2453e-01 668.576 < 2.2e-16 ***
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Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
145.4707 (40 degrees of freedom)
Richards’ growth curve model
summary(model.r)
Model fitted: Generalised logistic (ED50 as parameter) (5 parms)
Parameter estimates:
Estimate Std. Error t-value p-value
b:(Intercept) -6.2700e-02 1.0054e-03 -62.3628 < 2.2e-16 ***
c:(Intercept) 4.7785e+03 8.9855e+02 5.3180 4.571e-06 ***
d:(Intercept) 7.1557e+04 3.9712e+02 180.1901 < 2.2e-16 ***
e:(Intercept) 5.9418e+01 3.5388e+00 16.7907 < 2.2e-16 ***
f:(Intercept) 4.5649e+00 9.6150e-01 4.7477 2.767e-05 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
166.9253 (39 degrees of freedom)
Want to know more?
Primary data source is from NSW Health for daily new cases. The analysis is performed using RStudio Cloud using R version 4.1.0.