Progression of the COVID outbreak in New Zealand

Last updated: 2022-09-13 13:18 NZ time
Updates daily after 1pm NZ time

R_eff estimate from all cases

New Zealand is currently experiencing an outbreak of COVID, due to the omicron variant. On August 17th, Auckland entered lockdown restrictions under Alert Level 4 during an outbreak of the delta variant. Since then restrictions have been loosened a number of times, and then tightened in late January 2022 as the omicron variant began to spread in New Zealand.

How have restrictions in New Zealand affected the spread of the virus? The below plot shows how the effective reproduction number of the virus, R_eff 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).

SIR model projection

The trendlines in the above plots simply project forward case numbers assuming that R_eff remains at its current estimated value. What about the effect of built-up immunity? Below is a plot with the projected trend taking into account an estimated reduction of spread due to increasing immunity from infections, which slows transmission once a non-negligible fraction of the population has been infected. The projection assumes that only 35% of infections are captured by testing, as such, for every projected case there are an additional three to four infections implied that are not detected through testing.

This projection shows what might be possible if all other factors affecting R_eff—such as restrictions—are held constant.

The same plot is shown twice, the first with cases on a linear scale, and the second on a log scale.

Animated projections over time

How have the above projections changed over time? Below are animated versions of previous projections, 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.

Note that previously, the main factor affecting these projections was an estimated reduction in the rate of spread due to vaccines. The model is still the same, but there is no longer any expected future effect from increasing vaccines, as the 1st and 2nd dose rollout is mostly complete. 3rd are not taken into account in the model, and during the omicron wave, the primary reason R_eff will decline over time is due to increasing immunity due to infections.

R_eff in Auckland vs rest of NZ

Below are two plots showing case numbers and an estimated R_eff value in Auckland (the Auckland, Counties Manukau, and Waitematā DHBs), and the rest of New Zealand excluding Auckland.

Note: case numbers on these plots are shown on a log scale.

Disclaimer on trends

The plotted exponential trendlines in the above plots are simple extrapolations of what will happen if R_eff remains at its current value. the SIR model projection, although it accounts for the effect of increasing immunity, still assumes that R_eff otherwise remains constant.

This does not take into account that things are in a state of flux. If restrictions are tightened, the virus should have fewer opportunities for spread, and R_eff 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 population immunity do take into account a reduction in R_eff as immunity due to infections increases, but ignores any other possible future changes in R_eff.

Furthermore, when case numbers are small, the random chance of how many people each infected person subsequently infects can cause estimates of R_eff to vary randomly in time. As such the projections should be taken with a grain of salt—it is merely an indication of the trend as it is right now.

Methodology

Padding, smoothing, and calculating R_eff

Before calculating R_eff, the daily case numbers are padded on the right and smoothed. A symmetric smoothing method is used, which is why padding on the right is necessary before smoothing.

The padding is an extrapolation based on a fit to the most recent 14 days of daily case numbers n(t), with the following fit function:

n(t) = A exp[ k( t-t_today) + 1/2 k̇ (t-t_today )² ]

where A and k are the fit parameters for the current daily caseload and exponential growth rate, and k̇ is the current expected rate of change of the exponential growth rate due to vaccines:

k̇ = 1 / 𝜏 (s_vax(t_today) / s_vax(t_yesterday) - 1)

where 𝜏 = 5 days is the approximate mean generation time of the virus, and s_vax(t_today) and s_vax(t_yesterday) are the fraction of the population not protected by vaccination from infection today and yesterday. See vaccination assumptions below for how s_vax(t) is calculated from vaccination numbers.

After padding, the daily case numbers—after adding an offset and taking their log— are smoothed with 4-day Gaussian smoothing. The result is then exponentiated and the offset subtracted off again to obtain smoothed daily case numbers. This is a compromise between ordinary Gaussian smoothing for small case numbers, and Gaussian smoothing in log space for larger case numbers.

log(n_smoothed(t)+10) = log(n(t)+10) ∗ (2𝜋T_s²)^-1/2 exp(-t² / 2T_s²)

where T_s = 4 days and ∗ is the convolution operation.

R_eff is then calculated for each day as:

R_eff(t_i) = (n_smoothed(t_i) / n_smoothed(t_i-1))^𝜏.

The uncertainty in R_eff has contributions from the uncertainty in the above-mentioned 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.

Trend based on R_eff only

The plot(s) with a simple exponential extrapolation of daily case numbers, without taking into account the expected effects of vaccines or population immunity, is exponential growth/decay using the most recent value of R_eff and its uncertainty range:

n_extrap (t_i) = n_smoothed(t_today) R_eff(t_today) ^{(t_i - t_today) / 𝜏}

SIR model with projected effect of vaccines

To project the effect of vaccines and community immunity on the progression of daily cases in the future, I use a stochastic SIR model. The model is run 1000 times to project daily case numbers, and the median result and 68% range of outcomes shown in the projection.

In each run of the model, a random value for each of R_eff(t_today) and n_smoothed(t_today) is drawn from a multivariate normal distribution, using the mean values and covariance of the estimates of the two parameters.

The initial infectious population is taken to be:

i(t_today) = 𝜏 n_smoothed(t_today)

The initial recovered population r(t_today) is taken to be the cumulative number of cases so far in the outbreak, minus the currently infectious population.

Each day, the fraction of the population susceptible to infection s(t) is taken to be

s(t) = s_vax(t) [1 - (r(t) + i(t)) / N]

where s_vax(t) is the fraction of the population not protected from infection by vaccines (see See below for how this is calculated), and N is the total population multiplied by the fraction of infections assumed to come forward for testing (35%).

Assuming that R_eff is proportional to the susceptible fraction of the population, we can estimate what the reproduction number of the virus would be in the absence of any vaccines or immunity from infection:

R_naive = R_eff(t_today) / s(t_today)

And compute an R_eff for each future day based on the susceptible fraction of the population that day:

R_eff(t) = s(t) R_naive

Each day of the model, the infectious population is increased by the number of new infections, and decreased by the number of new recoveries. The recovered population is increased by the number of new recoveries.

The number of new infections each day is drawn from a Poission distribution with mean equal to i(t) R_eff(t) / 𝜏.

The number of new recoveries each days is drawn from a binomial distribution with number of trials equal to the infectious population i(t), and with probability 1 / 𝜏.

Vaccination assumptions

To model vaccines taking effect after a delay, daily vaccine dose numbers v(t) are convolved with a Gaussian function offset 1.5 weeks in the future, with standard deviation 0.5 weeks:

v_effective(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 not protected by the vaccine against infection is then:

s_vax(t_i) = 1 - 0.4 × d(t_i)

where d(t_i) is the cumulative number of doses per capita on each day.

As of Dec 2021, these vaccine assumptions are not particularly relevant to the projections, as the 1st and 2nd dose rollouts are mostly complete, and omicron likely escapes most vaccine-induced immunity against infection in any case. I am not yet including a modelled effect of 3rd doses, but may do so if the omicron waves last long enough for it to plausibly make a difference to spread, or if 3rd doses are brought forward in time.

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.