Updates daily near 1pm NZST
Progression of the COVID outbreak in New Zealand
See also: Australian vaccination rollout
Reff estimate from all cases
New Zealand is currently experiencing an outbreak of COVID, due to the delta variant. On August 17th, Auckland entered lockdown restrictions under Alert Level 4. Since then restrictions have been loosened to Alert Level 3, and further loosened under "Phase 1" of a reopening plan.
How have restrictions in New Zealand affected the spread of the virus? The below plot shows how the effective reproduction number of the virus, Reff has changed over time in New Zealand, as well as how the daily cases have changed over time. A trendline shows the approximate trajectory daily case numbers would follow, were the reproduction number of the virus to remain at its current level.
The same plot is shown twice, the first with cases on a linear scale, and the second on a log scale—the latter showing how consistently the caseload has followed exponential growth (which forms a straight line on a log scale).
Projected effect of vaccine rollout
The trendlines in the above plots simply project forward case numbers assuming that Reff remains at its current estimated value. What about the effect of vaccines? Below is a plot with the projected trend taking into account an estimated reduction of spread due to vaccination levels increasing in New Zealand through the remainder of the year. This projection is a stochastic SIR model, and thus also takes into account immunity due to infections, which would decrease transmission once a non-neglibible fraction of the population has been infected.
The rate of vaccination assumed in the model 1.5 doses per 100 population per day, up to a maximum of 85% of the population. See below for plots of how these assumed rates compare to the vaccination rollout in New Zealand so far. The vaccines are assumed to reduce Reff by 40% per dose per capita—that is, overall spread is reduced by 40% for a partly-vaccinated individual and 80% for a fully vaccinated individual.
This projection shows what might be possible if all other factors affecting Reff—such as restrictions—are held constant. If New Zealand eases restrictions, however, then Reff will likely increase and the effect of vaccines in reducing spread will be delayed compared to this projection. On the other hand, if contact tracing is successful in decreasing Reff, case numbers may decline faster than this projection.
The same plot is shown twice, the first with cases on a linear scale, and the second on a log scale.
Below are plots of the vaccination rollout in New Zealand to date, in terms of daily and cumulative doses per 100 population, with the assumed future rate used for the above projections also shown.
Animated projections over timeHow have the above projections changed over time? Below are animated versions of the above projection, one with cases on a linear scale, and one with cases on a log scale, run on old data to show how the projections have changed over time. Note that these are not 100% identical to the projections actually made on previous days, as there have been some slight methodology changes - but they should be very close.
Disclaimer on trends
The plotted trendlines are simple extrapolations of what will happen if Reff remains at its current value. This does not take into account that things are in a state of flux. As restrictions take effect, the virus should have fewer opportunities for spread, and Reff will decrease. If restrictions are eased, it may increase. Contact tracing may suppress spread to a greater or lesser degree over time. The above plots specifically showing the effect of vaccines do take into account a reduction in Reff as vaccination coverage increases, but ignore any other possible future changes in Reff.
Furthermore, when case numbers are small, the random chance of how many people each infected person subsequently infects can cause estimates of Reff to vary randomly in time. As such the projection should be taken with a grain of salt—it is merely an indication of the trend as it is right now.
Smoothing, calculating Reff and projections
Daily case numbers have been smoothed with 4-day Gaussian smoothing:
Nsmoothed(t) = N(t) ∗ (2πTs2)-1/2 exp(-t2 / 2Ts2)
where Ts = 4 days and ∗ is the convolution operation.
Before smoothing, the daily case numbers are padded on the right with an extrapolation based on a exponential fit to the most recent 14 days of data.
Reff is then calculated for each day as:
Reff(ti) = (Nsmoothed(ti) / Nsmoothed(ti-1))Tg
where Tg = 5 days is the approximate generation time of the virus.
The uncertainty in Reff has contributions from the uncertainty in the above-mentioned exponential fit, as well as uncertainty in daily case numbers. The latter is considered to be Poisson noise scaled by a constant, chosen so as to make the reduced chi squared between raw and smoothed daily case numbers equal to 1.0.
The extrapolation of daily case numbers is based on exponential growth/decay using
the most recent value of Reff and its uncertainty range:
Nextrap(ti) = Nsmoothed(ttoday) Reff(ttoday) (ti - ttoday)/Tg
Note 2021-07-30: I have changed the model used for the projections of the vaccine rollout to a stochastic SIR model, and the description below is out of date. I haven't yet documented the new model on this page, but the code can be seen here in the meantime. The projected outcomes with the new model are very similar to the old model, which was valid in the limit of infections only reaching a small fraction of the total population.
To model vaccines taking effect after a delay, daily vaccine dose numbers
V(t) are convolved with a Gaussian offset 1.5
weeks in the future, with standard deviation 0.5 weeks:
Veffective(t) = V(t) ∗ (2πσ2)-1/2 exp(-(t - μ)2 / 2σ2)
where μ = 10.5 days, σ = 3.5 days, and ∗ is the convolution operation. This causes the effect of a vaccine dose to over the course of the second week after it is adminsitered, reaching nearly full effectiveness approximately two weeks after administration.
I assume that one dose of any vaccine reduces spread by 40%, and two doses reduce
spread by 80%. The proportion of the population susceptible to the virus is then:
s(ti) = 1 - 0.4 × D(ti)
where D(ti) is the cumulative number of doses per capita on each day.
Reff is then estimated for any future date as:
Reff(ti) = Reff(ttoday) × s(ti) / s(ttoday)
And case numbers extrapolated from one day to the next, beginning with
Nsmoothed(ttoday), according to:
Nextrap(ti) = Nextrap(ti-1) Reff(ti) 1/Tg
Data sources and contact
Source for case numbers: New Zealand Ministry of Health
Plots/analysis by Chris Billington. Contact: chrisjbillington [at] gmail [dot] com
Python script for producing the plots can be found at GitHub.