What’s new?
Today’s announcement was 1083 new local cases in NSW is consistent with the observation that we have crossed the peak number of new cases, and consistent with the expectation that the model has been overly optimistic with the “descending limb” in the fall in cases.
The flexible nature of the Richards’ growth curve model at this point means that there is some significant uncertainty regarding the trajectory of the growth in the near future. Any estimates, even in the immediate days needs to be taken cautiously. The rapid fall in the number of cases as suggested by the model estimates is not plausible and are likely underestimates. Effectively, the falling “limb” of the curve will likely be pushed to the right, that is, towards the future. We can see this effective already when comparing the model today, to that from 7 days ago (see below). This decline will most likely be slower, possibly much slower.
The model suggests that we are likely to see around 4000-5000 new cases over the next 7 days. Accounting for the underestimate, the model would suggest around 7000 cases over the next week.
It should be noted that the fall in the number of cases is on the assumption that the factors which limit transmission of infection remain in place. The model cannot take into account the effect of “taking the foot off the brake” prematurely.
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 19 September 2021
What is this?
The top image 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 17 June 2021 to a Richards’ growth curve using non-linear regression. 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 models were in early July providing substantial underestimates, which is what I suspected when I started this series. The Richards’ growth curve model has struggled with the fit to the data, with poor predictive accuracy in July. Both models have been gradually underestimating further since the beginning of August. By mid-late August, the Richards’ growth curve model was clearly outperforming the Gompertz model. The current likely underestimation of cases (i.e., the Richards’ growth curve model being overly optimistic) is demonstrated by the ~5% underestimate of cumulative cases from the projection 7 days ago.
The Richards’ growth curve model – today vs projections from 7-, 14-days, and 21-days ago
I don’t publish the future projections beyond 7 days as these are uncertain and I’m concerned about the risk of misleading members of the public. The further one goes into the future, the more likely that the assumptions of the model are a poor fit and description for reality. On this occasion, I’m going to retrospectively compare the projections from 7-, 14-, and 21-days ago to the projection from today, with axes at the same scale. The data points including today are included in all the charts, but the projections from the past only use the data up to that date. The Richards’ growth curve model has actually been relatively stable over the last 3 weeks, including the identification of a peak arriving in September. What has been somewhat unstable is the timing and size of the peak – the model tends to be sensitive to changes in the data series.
The overall the trend we can see is as the time progresses with more data, the confidence intervals narrow. The projection from a week ago was overly optimistic, and the current model is likely to be biased in that direction as well. The projection from 2-3 weeks ago were overly pessimistic.
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 19 September 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. At present, there is notable incompatibility between the Gompertz and Richards’ growth curve models. As both the simple moving average and GAM demonstrate the appearance of the peak, the Gompertz model is likely incorrect.
Model summaries
Gompertz model
summary(model.g)
Model fitted: Gompertz (4 parms)
Parameter estimates:
Estimate Std. Error t-value p-value
b:(Intercept) -2.1327e-02 3.1790e-04 -67.0860 < 2.2e-16 ***
c:(Intercept) 4.3526e+02 8.6562e+01 5.0283 2.468e-06 ***
d:(Intercept) 2.4646e+05 8.7322e+03 28.2247 < 2.2e-16 ***
e:(Intercept) 1.1810e+02 1.2802e+00 92.2509 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
530.1828 (91 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) -1.5325e-01 5.7403e-03 -26.6976 < 2.2e-16 ***
c:(Intercept) -2.2987e+02 3.7172e+01 -6.1838 1.802e-08 ***
d:(Intercept) 5.4568e+04 6.6851e+02 81.6263 < 2.2e-16 ***
e:(Intercept) 9.0162e+01 1.5126e-01 596.0771 < 2.2e-16 ***
f:(Intercept) 4.0942e-01 1.8534e-02 22.0899 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
147.89 (90 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.
Today’s charts
Data: au_covid
R code: models, model-r0-7-14-21