What’s new?
This will be the final set of projections for the Sydney/NSW COVID-19 outbreak that started in mid-June 2021, initially with the first cases identified in Bondi. Today is day 5 into the reduction of public health restrictions to contain the outbreak, and it is likely that moving forward, the model will no longer be a good description of the new case series. With the change in context, including further data into the model doesn’t make theoretical sense.
Today’s announcement of 399 new local cases in NSW remains highly consistent with the models. The Gompertz model has been performing especially well with less than 1% error for both 7-day and 14-day projections of the cumulative cases.
The very low rates of error in the projections for a number of weeks means that the estimates of the final model can be compared with future case numbers. Divergence from the model estimates give an indication of the effect of some of the changes in context. It is expected that case numbers are likely to rise over the next few weeks, though the impact of this will be uncertain.
I’ve included charts today that have projections up to the end of the year (31 December 2021) so that future comparisons can be made.
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.
On the 6 October 2021, the proportion of people aged 16 years and older in NSW who were fully vaccinated reached 70%.
On 11 October 2021, some public health orders were relaxed for people who were vaccinated, including allowing up to 10 visitors at home, group bookings of up to 20 people at hospitality venues, and masks no longer mandated in outdoor areas.
Projection of new daily cases, and cumulative counts of COVID-19 with data up to 15 October 2021
What is this?
The blue charts are the Gompertz model. The top image is a chart of the cumulative (total) COVID-19 cases in NSW, starting from 17 June 2021, and the middle image is a chart of the daily new cases. Only local cases are included (i.e., excluding cases identified in quarantine). With these final charts, projections are given until the end of the year. The bottom image zooms in on the new daily case projections from 1 October 2021 to 31 December 2021.
It should be noted that projections into the future at this point are indicative of what we could expect if the reduction in restrictions did not occur on 11 October 2021. We should, therefore, expect to see some variance, most likely a rise in the number of cases, starting in the next week and moving forward.
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.
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 15 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
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) -6.0195e-02 5.7442e-04 -104.7919 < 2.2e-16 ***
c:(Intercept) 4.6213e+03 7.1812e+02 6.4352 4.976e-08 ***
d:(Intercept) 7.2561e+04 1.9090e+02 380.0978 < 2.2e-16 ***
e:(Intercept) 5.4714e+01 3.1496e+00 17.3718 < 2.2e-16 ***
f:(Intercept) 5.6652e+00 1.0444e+00 5.4245 1.778e-06 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
154.4248 (49 degrees of freedom)
Gompertz model
summary(model.g)
Model fitted: Gompertz (4 parms)
Parameter estimates:
Estimate Std. Error t-value p-value
b:(Intercept) -5.7190e-02 4.8774e-04 -117.254 < 2.2e-16 ***
c:(Intercept) 6.9700e+03 2.8954e+02 24.073 < 2.2e-16 ***
d:(Intercept) 7.3124e+04 1.8526e+02 394.719 < 2.2e-16 ***
e:(Intercept) 8.3253e+01 1.1722e-01 710.239 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
134.9367 (50 degrees of freedom)
Model projections of daily cases
model.g bootstrap simulation derivatives
[1] NA NA NA NA NA NA NA
[8] NA NA NA NA NA NA NA
[15] NA NA NA NA NA NA NA
[22] NA NA NA NA NA NA NA
[29] NA NA NA NA NA NA NA
[36] NA NA NA NA NA NA NA
[43] NA NA NA NA NA NA NA
[50] NA NA NA NA NA NA NA
[57] NA NA NA NA NA NA NA
[64] NA NA NA NA NA 859.958584 924.081589
[71] 985.943734 1044.931873 1100.323661 1151.769766 1198.739454 1240.889222 1277.950406
[78] 1309.773479 1336.321009 1357.545950 1373.589818 1384.500612 1390.417495 1391.571582
[85] 1388.241857 1380.683823 1369.275237 1354.266365 1336.004611 1314.744937 1290.921341
[92] 1264.798767 1236.655300 1206.801297 1175.538077 1143.169547 1109.863776 1075.851476
[99] 1041.423107 1006.634362 971.778311 936.995959 902.438237 868.145558 834.345486
[106] 801.139743 768.493618 736.531199 705.317802 674.912989 645.350069 616.635269
[113] 588.828352 561.926236 535.942519 510.884381 486.741089 463.502337 441.206726
[120] 419.792012 399.230910 379.547337 360.696493 342.646894 325.407397 308.941704
[127] 293.215743 278.220025 263.910287 250.280527 237.295835 224.942900 213.196087
[134] 202.020649 191.392219 181.286455 171.693017 162.587369 153.934533 145.718525
[141] 137.918980 130.526446 123.516587 116.876383 110.572335 104.599956 98.943589
[148] 93.582717 88.505764 83.701127 79.149754 74.835971 70.756721 66.894345
[155] 63.237328 59.779149 56.507085 53.412236 50.483533 47.712281 45.093988
[162] 42.615319 40.272107 38.055569 35.960176 33.979757 32.106584 30.336436
[169] 28.663596 27.080168 25.584436 24.171148 22.836974 21.574958 20.383140
[176] 19.256059 18.191322 17.184625 16.233776 15.335050 14.485991 13.683953
[183] 12.925775 12.209248 11.533069 10.893476 10.289838 9.718820 9.179590
[190] 8.670300 8.189373 7.734982 7.305662 6.900173 6.516900 6.155019
[197] 5.813454 5.490806
Note: Day 121 = 15/10/2021; Day 127 = 21/10/2021; Day 134 = 28/10/2021; Day 141 = 4/11/2021; Day 148 = 11/11/2021; Day 155 = 18/11/2021; Day 162 = 25/11/2021; Day 169 = 2/12/2021; Day 176 = 9/12/2021; Day 183 = 16/12/2021; Day 190 = 23/12/2021; Day 197 = 30/12/2021
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, models g EOY