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In response to the COVID-19 global pandemic, New Zealand initially followed an elimination strategy that coupled tough lockdowns with strict border controls. The first COVID-19 case was reported on 28 February 2020 and on 19 March 2020 international borders were closed to all but New Zealand citizens and permanent residents.[[1]] On 25 March 2020, New Zealand moved to Alert Level 4 (Level 4 brings the toughest restrictions in the four-level alert system) with strict lockdown measures including the closure of educational and public facilities along with all non-essential businesses, stay-at-home orders, limits on travel and no gatherings allowed. From 10 April 2020, anyone entering the country had to undergo two weeks of managed isolation and quarantine (MIQ). These measures kept case numbers low with a total of 1504 cases before COVID-19 was declared eliminated in New Zealand on 8 June 2020.[[1]]

Mandatory quarantine of overseas arrivals has been broadly effective at keeping COVID-19 out of New Zealand. In the period up to 15 June 2021 there have been 10 border-related reincursions,[[2]] although these outbreaks were able to be quickly detected and successfully eliminated. On 17 August 2021 a COVID-19 case of the highly transmissible Delta variant, with no clear link to the border, was detected in Auckland. As a result, the entire country moved from Alert Level 1 (minimal restrictions) to Alert Level 4 (toughest restrictions). The lockdown measures and the implementation of an internal boundary around Auckland were largely successful in containing the outbreak to Auckland. However, a small number of cases leaked through the border, leading to community transmission in Northland and Waikato with a limited number of further cases detected around the country, including in the South Island. After 35 days at Alert Level 4 in Auckland, the Government began to ease restrictions and transition from an elimination to suppression strategy.

On 22 October 2021, the Government outlined the COVID-19 Protection Framework, which uses vaccination certificates along with public health measures to manage COVID-19 in the community. This system replaced the Alert Level framework when district health boards reached a vaccination target of 90% of the eligible population. As vaccination rates increased and restrictions eased, as expected, COVID-19 cases spread out of Auckland to other parts of the country (as was previously observed with prior outbreaks in Waikato and Northland). It is important to understand the impact of different vaccination rates on the growth of future outbreaks outside of Auckland along with the number of hospitalisations and stress on the healthcare system.

In this work, a stochastic branching process model[[3–5]] is used to simulate the initial stages of a COVID-19 outbreak within a community. Stochastic models are useful because they incorporate the randomness associated with the initial stages of an outbreak. For some simulations, COVID-19 will spread widely and form an extensive outbreak (eg Auckland in August 2021 where a returnee from Australia sparked an outbreak with >6000 cases at the time of writing), whereas for other simulations, despite the virus having a reproduction number larger than one, random chance will mean that COVID-19 does not spread far beyond the initial seed infection (eg Wellington in June 2021 when a COVID-19 infected traveller visited from Australia but did not infect anyone else). The stochastic model tracks each individual case and becomes computationally expensive for large case numbers. Therefore, while stochastic models are useful for simulating the initial stages of an outbreak, deterministic SEIR (Susceptible, Exposed, Infected, Recovered) models are frequently used for larger and longer-term population level studies of epidemics.[[5,6]]

Here, I use a stochastic model to study the how the number of infections and hospitalisations depend on the vaccination rate and population level controls. I calculate the likelihood that a new infected case or hospitalised individual is vaccinated and determine the relative risk of getting infected or hospitalised with COVID-19 between vaccinated and unvaccinated individuals.

The stochastic model presented here tracks the number of infections in the community and categorises individuals as symptomatic (clinical infections) or asymptomatic (subclinical infections). Each infected individual infects a random number of other individuals, *N*, drawn from a Poisson distribution (Figure 1).[[3]] For a symptomatic individual, the Poisson distribution is defined by *= RC* where *R* is the reproduction number and *C* is the effectiveness of population level controls (eg Level 1, 2, 3 or 4 in the Alert Level Framework or Green, Orange, or Red in the COVID-19 Protection Framework). For an asymptomatic individual, the Poisson distribution is defined by *= RC/2*, which assumes that asymptomatic individuals infect, on average, half as many people as symptomatic individuals.[[7]] This model accounts for “super-spreading” events through the tail of the Poisson distribution.[[8]] A symptomatic individual in an Alert Level 1 environment, for example, has an 8% chance of infecting more than 10 people. It is possible, however, that the spread of COVID-19 is more heterogeneous, potentially with up to 80% of COVID-19 infections caused by only 10% of cases.[[9]] Therefore, while multiple realisations of this model can be used to obtain an average perspective of the initial stages of an outbreak (which is the focus of this work), alternative models (and distributions) should be used to investigate super-spreading events and their impact on the evolution of an outbreak.

Population level controls include public health measures such as physical distancing, wearing of masks, closure of schools and non-essential businesses, and restrictions on gatherings and social activities. The effectiveness of population level controls are taken from Plank et al[[3]] as *C*=1 for Alert Level 1, *C*=0.72 for Alert Level 2, *C*=0.52 for Alert Level 3, and *C*=0.32 for Alert Level 4. These values were estimated for the initial variant of COVID-19, and population level controls may be less effective against the more transmissible Delta variant.[[10]] In addition, the model does not account for illegal gatherings or other non-compliance with restrictions.

The generation times between an individual becoming infected and infecting *N* other individuals are independently sampled from a Weibull distribution with *a*=5.57 and *b*=4.08 where* a* is the scale parameter and *b* is the shape parameter (mean=5.05 days and variance=1.94 days)[[8]] (Figure 1). The model assumes that 33% of new infections are asymptomatic (subclinical) with the remainder symptomatic (clinical).[[11–13]]

I consider a range of vaccination rates, *V*, from 0% to 90% of the total population (rather than the *eligible population*, which at the time of writing is the over 12 years old population). Unlike previous work by Steyn et al,[[5,14]] age is not accounted for in the model, either in the vaccination rollout where older individuals were initially prioritised (at the start of February 2022, two-dose vaccination rates are now relatively similar for all age groups over 12 years old, although older age groups have higher booster rates),[[15]] or in contact rates where younger people are likely to have more contacts, or in the susceptibility where older individuals are more likely to experience severe disease or death (individuals who are 65–69 years old are 19x more likely to be hospitalised than those who are 25–29).[[16]] Therefore, care should be taken when applying the model results across different age bands as there is considerable heterogeneity of risk with age.

The vaccination rate *V* is assumed to be constant throughout the simulated outbreak. Following Steyn et al,[[5,14]] the Pfizer-BioNTech vaccine, which is the only COVID-19 vaccine currently being widely administered in New Zealand, is assumed to be 70% effective against infection and 50% effective against transmission for breakthrough infections (infections in fully vaccinated individuals).[[17]] Throughout the simulations, the model tracks the total number of vaccinated and unvaccinated infections along with the number of symptomatic and asymptomatic cases. Infected individuals are assumed to be equally likely to interact with vaccinated and unvaccinated individuals, with probabilities based solely on the vaccination rate. This may lead to an underestimation of the spread of COVID-19 in unvaccinated communities, as unvaccinated individuals are more likely to have unvaccinated contacts. Other limitations of this model include not accounting for ethnicity, either in vaccination rates or differential risk factors for different ethnic groups[[18]] or socio-economic status. COVID-19 spreads rapidly through overcrowded households as well as posing a greater risk to those who do not have the economic resources to safely isolate or the ability to work-from-home.[[18]]

**Figure 1**:** **Probability distributions used in the stochastic model. (a) Number of infections caused by a symptomatic case and (b) by an asymptomatic case. The number of infections caused by a symptomatic or asymptomatic case is governed by a Poisson distribution and can only take integer values. (c) The generation time, which is the time between an individual getting infected and infecting others, is governed by a Weibull distribution.

The likelihood of hospitalisations is also modelled. Clinical infections are assumed to have a 7.8% probability of being hospitalised.[[3,19]] I note that this hospitalisation rate was estimated from data prior to the emergence of the more severe Delta variant[[24]] and may underestimate the severity of Delta. Nonetheless, the modelled hospitalisation rate agrees with the total number of cases hospitalised during the August 2021 Auckland outbreak (7.6% hospitalisation rate for all cases as of 30 October 2021).[[20,21]] Based on Dagan et al[[22]] who examined the effectiveness of the Pfizer-BioNTech vaccine against hospitalisation in Israel, it is assumed that the vaccine is 87% effective at preventing hospitalisations after two doses. The model only allows for cases to be fully vaccinated (defined as more than two weeks after the second dose of the two-dose Pfizer-BioNTech vaccine) or unvaccinated. It is assumed that vaccine effectiveness does not wane with time. The model does not include the additional complexity of individuals who are partially vaccinated, either by only receiving one dose of the vaccine or by being within two weeks after receiving the second dose (note that this simulation work was perform prior to the booster rollout and hence boosters are not considered). In addition, there is no lag time between becoming infected and becoming hospitalised. The model is only run for a short duration (28 days), and hence I do not simulate the likelihood of hospitalised individuals dying.

Note that the model presented here does not include any testing, contact tracing, or isolation of cases. Instead, I focus on the impact of vaccination rates, particularly on the early stages of an outbreak when cases may be circulating undetected. The reader is referred to Steyn et al[[5]] for a model that includes testing and estimates the number of infections at the time of detection of the outbreak for various vaccination rates and testing scenarios.

The simulations are seeded with one unvaccinated symptomatic individual at* t*=0 where *t* is the time in days. Simulations are run for 28 days with time steps of one day. The model tracks each infected individual, distinguishes between symptomatic cases (clinical infections) and asymptomatic cases (subclinical infections), tracks hospitalisation rates, and distinguishes between vaccinated and unvaccinated individuals. I consider vaccination rates between 0% and 90% of the total population in 5% increments (the vaccination rate referred to here is the total population rather than the eligible population, which is over 12 years old at the time of writing). The Government’s vaccination target of 90% of the eligible population corresponds to 78.7% of the total population.[[20,21]] For each vaccination rate, we consider four different population level controls based on New Zealand’s Alert Level system using the effectiveness values from Plank et al.[[3]] To get a representative sample of the possible outcomes for each scenario, we run the model 100,000 times for each combination of vaccination rate and population level controls.

I consider the impact of the vaccination rate and population level controls on the total number of infections and hospitalisations. Figure 2 shows likelihood of the number of infections 28 days into an outbreak for V=0%, 30%, 60%, and 90% and no population level controls (*C*=1). The histograms indicate the likelihood of each number of infections and illustrate the randomness associated with the initial stages of an outbreak. For some simulations, random chance causes the outbreak to infect a small number of people, whereas for other simulations the outbreak can rapidly grow due to super-spreader events. For V=30%, there is a 10% chance that an outbreak will cause less than 98 infections after 28 days. However, there is also a 10% chance that an outbreak will cause more than 1340 infections. The potential spread of an outbreak is strongly dependent on the vaccination rate. For V=0%, there is a 50% chance than an outbreak will cause less than 2691 infections after 28 days compared to less than 75 infections for V=60%.

The number of infections is strongly dependent on the vaccination rate, with higher vaccination rates decreasing the number of infections by several orders of magnitude. For V=0%, the maximum number of infections after 28 days is 16,198 compared to 819 for V=60% and 101 for V=90%. The strong dependence of the number of infections on the vaccination rate is further visualised in Figure 3, which shows the median number of infections along with the 25% and 75% quartiles as a function of vaccination rate and population level controls. Increasing the vaccination rate can drastically decrease the number of infections and hence minimise the size of an outbreak.

**Figure 2**: Histograms showing the likelihood for a given number of infections 28 days into an outbreak for vaccination rates. (a) 0%, (b) 30%, (c) 60%, and (d) 90%. Red lines indicate the cumulative probability.

Figure 3 also shows the importance of population level controls, especially at low levels of vaccination. For V=20% (the approximate vaccination rate in New Zealand on 17 August 2021 when the Delta outbreak was first detected), the median number of infections after 28 days is 1,011 (542 and 1602 for 25% quantile and 75% quantile, respectively) at Level 1 but only 7 (3 and 17 and for 25% quantile and 75% quantile, respectively) at Level 4. This illustrates how, in the absence of high vaccination rates, population level controls are extremely important in limiting the growth of an outbreak. It is noted that New Zealand only shifted from Level 1 to Level 4 after a case was detected. At this point, the virus had been circulating undetected for approximately a week prior to this and had potentially already seeded 800 to 1000 cases in the community.[[23]] Furthermore, the effectiveness values used here for the population level controls were estimated for the original strain of COVID-19.[[3]] It is unclear if these values are appropriate for the more transmissible Delta variant[[10,24,25]] or if population level controls are less effective against Delta.

**Figure 3**: Number of infections as a function of vaccination rate and population level controls. (a,c) 14 days and (b,d) 28 days after an unvaccinated symptomatic individual is seeded into the community. Solid lines show the median of the 100,000 realisations while the shaded area shows the range between the 25% and 75% quantiles. Top plots (a,b) show the results on a linear scale while the bottom plots (c,d) show the results on a logarithmic scale. The vertical black dashed lines indicate vaccination rate for the total population that corresponds to the 90% target of the over 12 years old population (78.7%).

The results shown in Figures 2 and 3 illustrate how a vaccine can be extremely effective in preventing or limiting an outbreak even if the vaccine does not provide individuals with 100% protection from infection. Here, it is assumed that the Pfizer-BioNTech vaccine is 70% effective in preventing infection and, for breakthrough infections, 50% effective at preventing onward transmission.[[17]] Despite this imperfect protection, at a population level the vaccine drastically reduces the spread of the virus. For V=60%, the mean number of infections 28 days into the outbreak is only 3% of the mean number of infections for V=0%. For V=90%, the mean number of infections further decreases to only 0.3% compared to the unvaccinated scenario.

**Figure 4**: Histograms showing the likelihood for a given number of hospitalizations 28 days into an outbreak for C=1 and vaccination rates. (a) 0%, (b) 30%, (c) 60%, and (d) 90%. Red lines indicate the cumulative probability. (e) Number of hospitalisations 28 days into an outbreak as a function of vaccination rate showing the (solid line) median simulation result and (shaded area) 25th and 75th quantiles. Vertical black line indicates the vaccination rate of the total population that corresponds to 90% of the eligible (over 12 years old) population (78.7%).

The number of hospitalisations is also strongly dependent on the vaccination rate (Figure 4). For V=0%, there is a 50% chance of over 282 hospitalisations 28 days into the outbreak compared to 50% chance of less than six hospitalisations for V=60% and one hospitalisation for V=90%. The hospitalisation results presented in Figure 4 are calculated for *C*=1, which gives a worst-case scenario. Implementation of population level controls (ie Alert Level restrictions or vaccination certificates) will reduce the total number of hospitalisations at all vaccination rates but the general trend is unchanged; high vaccination rates drastically reduce the number of hospitalisations.

I assume that the vaccine is 87% effective against hospitalisation,[[22]] which results in a hospitalisation rate of 1.0% for fully vaccinated individuals. During the Delta outbreak, the hospitalisation rate for fully vaccinated individuals was 2.3% (as of 22 January 2022, prior to the emergence of Omicron in the community).[[21,26]] There are several possible reasons why the model presented here underpredicts the hospitalisation rate for fully vaccinated individuals. Firstly, the hospitalisation rate used in the model was estimated from data prior to the emergence of the more severe Delta variant.[[19]] Hospitalisation rates for vaccinated individuals can be twice as high for the Delta variant as for Alpha.[[27]] Secondly, I assume that the effectiveness of the vaccine is constant in throughout time, whereas recent studies have shown that vaccine effectiveness decreases with time (hence the current “booster” campaign).[[28,29,30]] Thirdly, the Ministry of Health reports the number of individuals in hospital who have COVID-19, rather than the number that are in hospital due to COVID-19,[[26]] and hence may over estimate the number of hospitalisations relative to what is modelled.

As the vaccination rate increases, there are more vaccinated individuals in the population, fewer unvaccinated individuals, and hence more breakthrough infections will occur . This is expected; the model assumes a 30% chance of breakthrough infections in vaccinated individuals.[[17]] As the number of vaccinated individuals increases, the number of cases in vaccinated individuals will increase, as shown in Figure 5a. For V=40%, there is an 83% chance that a new case will be unvaccinated and a 17% chance that they will be vaccinated. For V=80%, this switches to 46% unvaccinated and 54% vaccinated. It is important to note that, even though the number of infections in vaccinated individuals increases, it stays below the population proportion, which is shown by the dashed lines in Figure 5a. This indicates that, per population, infections are still more likely to occur in unvaccinated individuals. It should also be noted that infections in unvaccinated individuals are more likely to lead to onward transmission and hospitalisation than breakthrough infections.[[17,22]]

Figure 5b shows the probability that an infection occurs in a vaccinated or unvaccinated individual normalised by population. At all vaccination rates, an unvaccinated individual is more likely to be infected than vaccinated individual. Dividing the unvaccinated probability normalised by population by the vaccinated probability normalised by population gives the risk factor for unvaccinated individuals, which quantifies the likelihood of an unvaccinated individual getting infected compared to a vaccinated individual. This calculation shows that, even though the number of breakthrough infections increases with increasing vaccination rate, an unvaccinated individual is 3.3x more likely to be infected than a vaccinated individual for all vaccination rates.

**Figure 5**: Probability that an infected or hospitalized individual is (blue)vaccinated or (red) unvaccinated as a function of vaccination rate, as calculated from mean simulation result. (a) and (c) Probabilitythat an (a) infected or (c) hospitalised individual is vaccinated or unvaccinated. Dashed lines indicate proportion of population that are vaccinated or unvaccinated as a function of vaccination rate. (b) and (d) Probability normalised by population that an (b) infected or (d) hospitalised individual is vaccinated or unvaccinated.

I then perform the same analysis for hospitalisations. Figure 5c shows that the probability that a hospitalised case is vaccinated increases slower than the probability that an infected case is vaccinated. This shows that while the vaccine provides substantial protection against infection, it provides even greater protection against hospitalisation. At all vaccination rates, hospitalisations are significantly more likely to be unvaccinated than vaccinated (at V=80%, there is an 86% chance that a hospitalised case will be unvaccinated compared to 14% vaccinated). Figure 5d shows the probability of hospitalisation normalised by population. Unvaccinated individuals are 25x more at risk of hospitalisation than vaccinated individuals. This reinforces the need to vaccinate a large percent of the population to minimise hospitalisations and prevent strain on the healthcare system.

The results shown here are in broad agreement with a CDC study of 43,127 COVID-19 cases in Los Angeles County that showed that infection and hospitalisation rates among unvaccinated individuals were 4.9x and 29.2x, respectively, higher than for fully vaccinated individuals.[[31]] The risk factors calculated here may be lower due to underestimating the protection that the vaccine provides against infection or because I focus on the Delta variant, which transitioned to becoming the dominant variant during the study period of Griffin et al.[[31]]

The COVID-19 outbreak in New Zealand which began in August 2021, had resulted in the highest case counts experienced in New Zealand during the pandemic, prior to the arrival of Omicron in January 2022. Cases were centred in Auckland but spread to Northland and Waikato, with isolated cases spread around the rest of the country, including the South Island. The Government’s switch from an elimination to suppression strategy has emphasised the importance of vaccination in preventing COVID-19 from overwhelming the healthcare system. As restrictions eased, there was a need to understand how different vaccination rates will impact the initial stages of COVID-19 outbreak as cases become seeded in communities around New Zealand.

Here, I use a stochastic branching process model to examine the impact of vaccination rates on the initial spread of an outbreak. I show that increasing vaccination rates greatly decrease the number of infections (1.4% median number of infections 28 days into the outbreak for V=80% compared to V=20%), even if the Pfizer-BioNTech vaccine only provides individuals with imperfect protection (assumed to be 70% effective against breakthrough infection). This illustrates the effectiveness of the vaccine on a population level.

As the vaccination rate increases, the number of breakthrough infections and hospitalisations among vaccinated individuals will increase. This is expected and reflects the increased proportion of vaccinated individuals in the population. Unvaccinated individuals are 3.3x more likely to be infected and 25x more likely to be hospitalised than vaccinated individuals. The model results presented here agree with real-world data[[31]] and highlight how the Pfizer-BioNTech vaccine provides good protection against infection and extremely good protection against hospitalisation. This work illustrates the need for high vaccination rates to reduce infections and prevent the healthcare system from being overrun with COVID-19 patients.

The August 2021 COVID-19 outbreak in Auckland caused the New Zealand Government to transition from an elimination strategy to suppression, which relies heavily on high vaccination rates in the population. As restrictions ease and as COVID-19 spreads throughout New Zealand, there is a need to understand how different levels of vaccination will impact the initial stages of COVID-19 outbreaks that are seeded around the country.

A stochastic branching process model is used to simulate the initial spread of a COVID-19 outbreak for different vaccination rates.

High vaccination rates are effective at minimizing the number of infections and hospitalizations. Increasing vaccination rates from 20% (approximate value at the start of the August 2021 outbreak) to 80% (approximate proposed target) of the total population can reduce the median number of infections that occur within the first four weeks of an outbreak from 1011 to 14 (25th and 75th quantiles of 545–1602 and 2–32 for V=20% and V=80%, respectively). As the vaccination rate increases, the number of breakthrough infections (infections in fully vaccinated individuals) and hospitalisations of vaccinated individuals increases. Unvaccinated individuals, however, are 3.3x more likely to be infected with COVID-19 and 25x more likely to be hospitalised.

This work demonstrates the importance of vaccination in protecting individuals from COVID-19, preventing high caseloads, and minimising the number of hospitalisations and hence limiting the pressure on the healthcare system.

1) History of the COVID-19 Alert System. Unite Against COVID-19. [Accessed 2021 Nov 4]. Avaliable from https://covid19.govt.nz/alert-levels-and-updates/history-of-the-covid-19-alert-system/.

2) Grout L, Katar A, Ait Ouakrim D, et al. Failures of quarantine systems for preventing COVID-19 outbreaks in Australia and New Zealand. Med J Aust. 2021 Oct 4;215(7):320-324. doi: 10.5694/mja2.51240.

3) Plank MJ, Binny RN, Hendy SC, et al. A stochastic model for COVID-19 spread and the effects of alert level 4 in Aotearoa New Zealand. medRxiv. 2020. doi: 10.1101/2020.04.08.20058743.

4) Plank MJ, Hendy SC, Binny RN, Maclaren O. Modeling the August 2021 COVID-19 outbreak in New Zealand. Technical Report, Te Pūnaha Matatini. 2021.

5) Steyn N, Plank MJ, Binny RN, et al. A COVID-19 vaccination model for Aotearoa New Zealand, Technical Report, Te Pūnaha Matatini. 2021.

6) James A, Hendy SC, Plank MJ, Steyn N. Suppression and mitigation strategies for control of COVID-19 in New Zealand. medRxiv. 2020. doi: 10.1101/2020.03.26.20044677.

7) Davies NG, Klepac P, Liu Y, et al.; CMMID COVID-19 working group, Eggo RM. Age-dependent effects in the transmission and control of COVID-19 epidemics. Nat Med. 2020 Aug;26(8):1205-1211. doi: 10.1038/s41591-020-0962-9.

8) Hendy SC, Steyn N, James A, et al. Mathematical modelling to inform New Zealand’s COVID-19 response. J. R. Soc. N. Z. 2021 51(S1), S86-S106, doi: 10.1080/03036758.2021.1876111.

9) Endo A, Centre for the Mathematical Modelling of Infectious Diseases COVID-19 Working Group, Abbott S, Kucharski AJ, Funk S. Estimating the overdispersion in COVID-19 transmission using outbreak sizes outside China. *Wellcome open research* 5 2020.

10) Allen H, Vusirikala A, Flannagan J, et al.; COVID-19 Genomics UK (COG-UK Consortium). Household transmission of COVID-19 cases associated with SARS-CoV-2 delta variant (B.1.617.2): national case-control study. Lancet Reg Health Eur. 2021 Oct 28:100252. doi: 10.1016/j.lanepe.2021.100252.

11) Byambasuren O, Cardona M, Bell K, et al. Estimating the extent of asymptomatic COVID-19 and its potential for community transmission: systematic review and meta-analysis. JAMMI. 2020 Dec;5(4):223-34. doi: 10.3138/jammi-2020-0030

12) Lavezzo E, Franchin E, Ciavarella C, et al.; Imperial College COVID-19 Response Team, Brazzale AR, Toppo S, Trevisan M, Baldo V, Donnelly CA, Ferguson NM, Dorigatti I, Crisanti A; Imperial College COVID-19 Response Team. Suppression of a SARS-CoV-2 outbreak in the Italian municipality of Vo'. Nature. 2020 Aug;584(7821):425-429. doi: 10.1038/s41586-020-2488-1.

13) Pollán M, Pérez-Gómez B, Pastor-Barriuso R, et al; ENE-COVID Study Group. Prevalence of SARS-CoV-2 in Spain (ENE-COVID): a nationwide, population-based seroepidemiological study. Lancet. 2020 Aug 22;396(10250):535-544. doi: 10.1016/S0140-6736(20)31483-5.

14) Steyn N, Plank MJ, Hendy SC. Modelling to support a future COVID-19 strategy for Aotearoa New Zealand, Technical Report, Te Pūnaha Matatini. 2021.

15) Ministry of Health. https://www.health.govt.nz/our-work/diseases-and-conditions/covid-19-novel-coronavirus/covid-19-data-and-statistics/covid-19-vaccine-data. Accessed Feb 8, 2022.

16) Vattiato G, Maclaren O, Lustig A, et al. A preliminary assessment of the potential impact of the Omicon variant of SARS-CoV-2 in Aotearoa New Zealand. Covid-19 Modelling Aotearoa. 2022. https://www.covid19modelling.ac.nz/a-preliminary-assessment-of-the-potential-impact-of-the-omicron-variant/.

17) Scientific Pandemic Influenza Group of Modeling, SPI-M-O. Summary of further modelling of easing restrictions - Roadmap Step 4. Technical Report, Scientific Advisory Group for Emergencies. 2021. Available from https://www.gov.uk/government/publications/spi-m-o-summary-of-further-modelling-of-easing-restrictions-roadmap-step-4-9-june-2021.

18) Steyn, N, Binny RN, Hannah K, et al. Maori and Pacific people in New Zealand have a higher risk of hospitalization for COVID-19. New Zealand Medical Journal, 134(1538),28-43.

19) Verity R, Okell LC, Dorigatti I, et al. Estimates of the severity of coronavirus disease 2019: a model-based analysis. Lancet Infect Dis. 2020 Jun;20(6):669-677. doi: 10.1016/S1473-3099(20)30243-7.

20) COVID-19 Vaccine Data. Ministry of Health. [cited 2021 Nov 1]. Avaliable from https://www.health.govt.nz/our-work/diseases-and-conditions/covid-19-novel-coronavirus/covid-19-data-and-statistics/covid-19-vaccine-data.

21) Singh H. The Spinoff Covid-tracker: the live graphs that tell the story of delta in Aotearoa. 2021. [cited 2021 Nov 1]. Avaliable from https://thespinoff.co.nz/society/30-10-2021/the-spinoff-covid-tracker/.

22) Dagan N, Barda N, Kepten E, et al. BNT162b2 mRNA Covid-19 Vaccine in a Nationwide Mass Vaccination Setting. N Engl J Med. 2021 Apr 15;384(15):1412-1423. doi: 10.1056/NEJMoa2101765.

23) Derek Cheng. Big Read: Tracking the Covid-19 delta outbreak – how it arrived, simmered through lockdown, then spread beyond reach. The New Zealand Herald. 2021. [cited 2021 Nov 19]. Avaliable from https://www.nzherald.co.nz/nz/politics/big-read-tracking-the-covid-19-delta-outbreak-how-it-arrived-simmered-through-lockdown-then-spread-beyond-reach/UM5CJTAGMBFGA5X2ZGJ3XWBTY4/

24) Liu Y, Rocklöv J. The reproductive number of the Delta variant of SARS-CoV-2 is far higher compared to the ancestral SARS-CoV-2 virus. J Travel Med. 2021 Oct 11;28(7):taab124. doi: 10.1093/jtm/taab124.

25) Sonabend R, Whittles LK, Imai N, et al. Non-pharmaceutical interventions, vaccination and the Delta variant: epidemiological insights from modelling England's COVID-19 roadmap out of lockdown. medRxiv. 2021. doi: 10.1101/2021.08.17.21262164

26) Ministry of Health. https://www.health.govt.nz/our-work/diseases-and-conditions/covid-19-novel-coronavirus/covid-19-news-and-media-updates. Accessed Feb 8 2022.

27) Twohig KA, Nyberg T, Zaidi A, Thelwall S, et al. Hospital admission and emergency care attendance risk for SARS-CoV-2 delta (B.1.617.2) compared with alpha (B.1.1.7) variants of concern: a cohort study. The Lancet Infectious Diseases. 2021 22(1):P35-P42. doi: 10.1016/S1473-3099(21)00475-8.

28) Tartof SY, Slezak JE, Fischer H, et al. Effectiveness of mRNA BNT162b2 COVID-19 vaccine up to 6 months in a large integrated health system in the USA: a retrospective cohort study. Lancet. 2021 Oct 16;398(10309):1407-1416. doi: 10.1016/S0140-6736(21)02183-8.

29) Skowronski DM, Setayeshgar S, Febriana Y, et al. Two-dose SARS-CoV-2 vaccine effectiveness with mixed schedules and extended dosing intervals: test-negative design studies from British Columbia and Quebec, Canada. medRxiv, 2021, doi: 10.1101/2021.10.26.21265397

30) Andrews A, Tessier E, Stowe J, et al. Vaccine effectiveness and duration of protection of Comirnaty, Vaxzeria and Spikevax against mild and severe COVID-19 in the UK. medRxiv, 2021. doi: 10.1101/2021.09.15.21263583

31) Griffin JB, Haddix M, Danza P, et al. SARS-CoV-2 Infections and Hospitalizations Among Persons Aged ≥16 Years, by Vaccination Status - Los Angeles County, California, May 1-July 25, 2021. MMWR Morb Mortal Wkly Rep. 2021 Aug 27;70(34):1170-1176. doi: 10.15585/mmwr.mm7034e5.

contact nzmj@nzma.org.nz

In response to the COVID-19 global pandemic, New Zealand initially followed an elimination strategy that coupled tough lockdowns with strict border controls. The first COVID-19 case was reported on 28 February 2020 and on 19 March 2020 international borders were closed to all but New Zealand citizens and permanent residents.[[1]] On 25 March 2020, New Zealand moved to Alert Level 4 (Level 4 brings the toughest restrictions in the four-level alert system) with strict lockdown measures including the closure of educational and public facilities along with all non-essential businesses, stay-at-home orders, limits on travel and no gatherings allowed. From 10 April 2020, anyone entering the country had to undergo two weeks of managed isolation and quarantine (MIQ). These measures kept case numbers low with a total of 1504 cases before COVID-19 was declared eliminated in New Zealand on 8 June 2020.[[1]]

Mandatory quarantine of overseas arrivals has been broadly effective at keeping COVID-19 out of New Zealand. In the period up to 15 June 2021 there have been 10 border-related reincursions,[[2]] although these outbreaks were able to be quickly detected and successfully eliminated. On 17 August 2021 a COVID-19 case of the highly transmissible Delta variant, with no clear link to the border, was detected in Auckland. As a result, the entire country moved from Alert Level 1 (minimal restrictions) to Alert Level 4 (toughest restrictions). The lockdown measures and the implementation of an internal boundary around Auckland were largely successful in containing the outbreak to Auckland. However, a small number of cases leaked through the border, leading to community transmission in Northland and Waikato with a limited number of further cases detected around the country, including in the South Island. After 35 days at Alert Level 4 in Auckland, the Government began to ease restrictions and transition from an elimination to suppression strategy.

On 22 October 2021, the Government outlined the COVID-19 Protection Framework, which uses vaccination certificates along with public health measures to manage COVID-19 in the community. This system replaced the Alert Level framework when district health boards reached a vaccination target of 90% of the eligible population. As vaccination rates increased and restrictions eased, as expected, COVID-19 cases spread out of Auckland to other parts of the country (as was previously observed with prior outbreaks in Waikato and Northland). It is important to understand the impact of different vaccination rates on the growth of future outbreaks outside of Auckland along with the number of hospitalisations and stress on the healthcare system.

In this work, a stochastic branching process model[[3–5]] is used to simulate the initial stages of a COVID-19 outbreak within a community. Stochastic models are useful because they incorporate the randomness associated with the initial stages of an outbreak. For some simulations, COVID-19 will spread widely and form an extensive outbreak (eg Auckland in August 2021 where a returnee from Australia sparked an outbreak with >6000 cases at the time of writing), whereas for other simulations, despite the virus having a reproduction number larger than one, random chance will mean that COVID-19 does not spread far beyond the initial seed infection (eg Wellington in June 2021 when a COVID-19 infected traveller visited from Australia but did not infect anyone else). The stochastic model tracks each individual case and becomes computationally expensive for large case numbers. Therefore, while stochastic models are useful for simulating the initial stages of an outbreak, deterministic SEIR (Susceptible, Exposed, Infected, Recovered) models are frequently used for larger and longer-term population level studies of epidemics.[[5,6]]

Here, I use a stochastic model to study the how the number of infections and hospitalisations depend on the vaccination rate and population level controls. I calculate the likelihood that a new infected case or hospitalised individual is vaccinated and determine the relative risk of getting infected or hospitalised with COVID-19 between vaccinated and unvaccinated individuals.

The stochastic model presented here tracks the number of infections in the community and categorises individuals as symptomatic (clinical infections) or asymptomatic (subclinical infections). Each infected individual infects a random number of other individuals, *N*, drawn from a Poisson distribution (Figure 1).[[3]] For a symptomatic individual, the Poisson distribution is defined by *= RC* where *R* is the reproduction number and *C* is the effectiveness of population level controls (eg Level 1, 2, 3 or 4 in the Alert Level Framework or Green, Orange, or Red in the COVID-19 Protection Framework). For an asymptomatic individual, the Poisson distribution is defined by *= RC/2*, which assumes that asymptomatic individuals infect, on average, half as many people as symptomatic individuals.[[7]] This model accounts for “super-spreading” events through the tail of the Poisson distribution.[[8]] A symptomatic individual in an Alert Level 1 environment, for example, has an 8% chance of infecting more than 10 people. It is possible, however, that the spread of COVID-19 is more heterogeneous, potentially with up to 80% of COVID-19 infections caused by only 10% of cases.[[9]] Therefore, while multiple realisations of this model can be used to obtain an average perspective of the initial stages of an outbreak (which is the focus of this work), alternative models (and distributions) should be used to investigate super-spreading events and their impact on the evolution of an outbreak.

Population level controls include public health measures such as physical distancing, wearing of masks, closure of schools and non-essential businesses, and restrictions on gatherings and social activities. The effectiveness of population level controls are taken from Plank et al[[3]] as *C*=1 for Alert Level 1, *C*=0.72 for Alert Level 2, *C*=0.52 for Alert Level 3, and *C*=0.32 for Alert Level 4. These values were estimated for the initial variant of COVID-19, and population level controls may be less effective against the more transmissible Delta variant.[[10]] In addition, the model does not account for illegal gatherings or other non-compliance with restrictions.

The generation times between an individual becoming infected and infecting *N* other individuals are independently sampled from a Weibull distribution with *a*=5.57 and *b*=4.08 where* a* is the scale parameter and *b* is the shape parameter (mean=5.05 days and variance=1.94 days)[[8]] (Figure 1). The model assumes that 33% of new infections are asymptomatic (subclinical) with the remainder symptomatic (clinical).[[11–13]]

I consider a range of vaccination rates, *V*, from 0% to 90% of the total population (rather than the *eligible population*, which at the time of writing is the over 12 years old population). Unlike previous work by Steyn et al,[[5,14]] age is not accounted for in the model, either in the vaccination rollout where older individuals were initially prioritised (at the start of February 2022, two-dose vaccination rates are now relatively similar for all age groups over 12 years old, although older age groups have higher booster rates),[[15]] or in contact rates where younger people are likely to have more contacts, or in the susceptibility where older individuals are more likely to experience severe disease or death (individuals who are 65–69 years old are 19x more likely to be hospitalised than those who are 25–29).[[16]] Therefore, care should be taken when applying the model results across different age bands as there is considerable heterogeneity of risk with age.

The vaccination rate *V* is assumed to be constant throughout the simulated outbreak. Following Steyn et al,[[5,14]] the Pfizer-BioNTech vaccine, which is the only COVID-19 vaccine currently being widely administered in New Zealand, is assumed to be 70% effective against infection and 50% effective against transmission for breakthrough infections (infections in fully vaccinated individuals).[[17]] Throughout the simulations, the model tracks the total number of vaccinated and unvaccinated infections along with the number of symptomatic and asymptomatic cases. Infected individuals are assumed to be equally likely to interact with vaccinated and unvaccinated individuals, with probabilities based solely on the vaccination rate. This may lead to an underestimation of the spread of COVID-19 in unvaccinated communities, as unvaccinated individuals are more likely to have unvaccinated contacts. Other limitations of this model include not accounting for ethnicity, either in vaccination rates or differential risk factors for different ethnic groups[[18]] or socio-economic status. COVID-19 spreads rapidly through overcrowded households as well as posing a greater risk to those who do not have the economic resources to safely isolate or the ability to work-from-home.[[18]]

**Figure 1**:** **Probability distributions used in the stochastic model. (a) Number of infections caused by a symptomatic case and (b) by an asymptomatic case. The number of infections caused by a symptomatic or asymptomatic case is governed by a Poisson distribution and can only take integer values. (c) The generation time, which is the time between an individual getting infected and infecting others, is governed by a Weibull distribution.

The likelihood of hospitalisations is also modelled. Clinical infections are assumed to have a 7.8% probability of being hospitalised.[[3,19]] I note that this hospitalisation rate was estimated from data prior to the emergence of the more severe Delta variant[[24]] and may underestimate the severity of Delta. Nonetheless, the modelled hospitalisation rate agrees with the total number of cases hospitalised during the August 2021 Auckland outbreak (7.6% hospitalisation rate for all cases as of 30 October 2021).[[20,21]] Based on Dagan et al[[22]] who examined the effectiveness of the Pfizer-BioNTech vaccine against hospitalisation in Israel, it is assumed that the vaccine is 87% effective at preventing hospitalisations after two doses. The model only allows for cases to be fully vaccinated (defined as more than two weeks after the second dose of the two-dose Pfizer-BioNTech vaccine) or unvaccinated. It is assumed that vaccine effectiveness does not wane with time. The model does not include the additional complexity of individuals who are partially vaccinated, either by only receiving one dose of the vaccine or by being within two weeks after receiving the second dose (note that this simulation work was perform prior to the booster rollout and hence boosters are not considered). In addition, there is no lag time between becoming infected and becoming hospitalised. The model is only run for a short duration (28 days), and hence I do not simulate the likelihood of hospitalised individuals dying.

Note that the model presented here does not include any testing, contact tracing, or isolation of cases. Instead, I focus on the impact of vaccination rates, particularly on the early stages of an outbreak when cases may be circulating undetected. The reader is referred to Steyn et al[[5]] for a model that includes testing and estimates the number of infections at the time of detection of the outbreak for various vaccination rates and testing scenarios.

The simulations are seeded with one unvaccinated symptomatic individual at* t*=0 where *t* is the time in days. Simulations are run for 28 days with time steps of one day. The model tracks each infected individual, distinguishes between symptomatic cases (clinical infections) and asymptomatic cases (subclinical infections), tracks hospitalisation rates, and distinguishes between vaccinated and unvaccinated individuals. I consider vaccination rates between 0% and 90% of the total population in 5% increments (the vaccination rate referred to here is the total population rather than the eligible population, which is over 12 years old at the time of writing). The Government’s vaccination target of 90% of the eligible population corresponds to 78.7% of the total population.[[20,21]] For each vaccination rate, we consider four different population level controls based on New Zealand’s Alert Level system using the effectiveness values from Plank et al.[[3]] To get a representative sample of the possible outcomes for each scenario, we run the model 100,000 times for each combination of vaccination rate and population level controls.

I consider the impact of the vaccination rate and population level controls on the total number of infections and hospitalisations. Figure 2 shows likelihood of the number of infections 28 days into an outbreak for V=0%, 30%, 60%, and 90% and no population level controls (*C*=1). The histograms indicate the likelihood of each number of infections and illustrate the randomness associated with the initial stages of an outbreak. For some simulations, random chance causes the outbreak to infect a small number of people, whereas for other simulations the outbreak can rapidly grow due to super-spreader events. For V=30%, there is a 10% chance that an outbreak will cause less than 98 infections after 28 days. However, there is also a 10% chance that an outbreak will cause more than 1340 infections. The potential spread of an outbreak is strongly dependent on the vaccination rate. For V=0%, there is a 50% chance than an outbreak will cause less than 2691 infections after 28 days compared to less than 75 infections for V=60%.

The number of infections is strongly dependent on the vaccination rate, with higher vaccination rates decreasing the number of infections by several orders of magnitude. For V=0%, the maximum number of infections after 28 days is 16,198 compared to 819 for V=60% and 101 for V=90%. The strong dependence of the number of infections on the vaccination rate is further visualised in Figure 3, which shows the median number of infections along with the 25% and 75% quartiles as a function of vaccination rate and population level controls. Increasing the vaccination rate can drastically decrease the number of infections and hence minimise the size of an outbreak.

**Figure 2**: Histograms showing the likelihood for a given number of infections 28 days into an outbreak for vaccination rates. (a) 0%, (b) 30%, (c) 60%, and (d) 90%. Red lines indicate the cumulative probability.

Figure 3 also shows the importance of population level controls, especially at low levels of vaccination. For V=20% (the approximate vaccination rate in New Zealand on 17 August 2021 when the Delta outbreak was first detected), the median number of infections after 28 days is 1,011 (542 and 1602 for 25% quantile and 75% quantile, respectively) at Level 1 but only 7 (3 and 17 and for 25% quantile and 75% quantile, respectively) at Level 4. This illustrates how, in the absence of high vaccination rates, population level controls are extremely important in limiting the growth of an outbreak. It is noted that New Zealand only shifted from Level 1 to Level 4 after a case was detected. At this point, the virus had been circulating undetected for approximately a week prior to this and had potentially already seeded 800 to 1000 cases in the community.[[23]] Furthermore, the effectiveness values used here for the population level controls were estimated for the original strain of COVID-19.[[3]] It is unclear if these values are appropriate for the more transmissible Delta variant[[10,24,25]] or if population level controls are less effective against Delta.

**Figure 3**: Number of infections as a function of vaccination rate and population level controls. (a,c) 14 days and (b,d) 28 days after an unvaccinated symptomatic individual is seeded into the community. Solid lines show the median of the 100,000 realisations while the shaded area shows the range between the 25% and 75% quantiles. Top plots (a,b) show the results on a linear scale while the bottom plots (c,d) show the results on a logarithmic scale. The vertical black dashed lines indicate vaccination rate for the total population that corresponds to the 90% target of the over 12 years old population (78.7%).

The results shown in Figures 2 and 3 illustrate how a vaccine can be extremely effective in preventing or limiting an outbreak even if the vaccine does not provide individuals with 100% protection from infection. Here, it is assumed that the Pfizer-BioNTech vaccine is 70% effective in preventing infection and, for breakthrough infections, 50% effective at preventing onward transmission.[[17]] Despite this imperfect protection, at a population level the vaccine drastically reduces the spread of the virus. For V=60%, the mean number of infections 28 days into the outbreak is only 3% of the mean number of infections for V=0%. For V=90%, the mean number of infections further decreases to only 0.3% compared to the unvaccinated scenario.

**Figure 4**: Histograms showing the likelihood for a given number of hospitalizations 28 days into an outbreak for C=1 and vaccination rates. (a) 0%, (b) 30%, (c) 60%, and (d) 90%. Red lines indicate the cumulative probability. (e) Number of hospitalisations 28 days into an outbreak as a function of vaccination rate showing the (solid line) median simulation result and (shaded area) 25th and 75th quantiles. Vertical black line indicates the vaccination rate of the total population that corresponds to 90% of the eligible (over 12 years old) population (78.7%).

The number of hospitalisations is also strongly dependent on the vaccination rate (Figure 4). For V=0%, there is a 50% chance of over 282 hospitalisations 28 days into the outbreak compared to 50% chance of less than six hospitalisations for V=60% and one hospitalisation for V=90%. The hospitalisation results presented in Figure 4 are calculated for *C*=1, which gives a worst-case scenario. Implementation of population level controls (ie Alert Level restrictions or vaccination certificates) will reduce the total number of hospitalisations at all vaccination rates but the general trend is unchanged; high vaccination rates drastically reduce the number of hospitalisations.

I assume that the vaccine is 87% effective against hospitalisation,[[22]] which results in a hospitalisation rate of 1.0% for fully vaccinated individuals. During the Delta outbreak, the hospitalisation rate for fully vaccinated individuals was 2.3% (as of 22 January 2022, prior to the emergence of Omicron in the community).[[21,26]] There are several possible reasons why the model presented here underpredicts the hospitalisation rate for fully vaccinated individuals. Firstly, the hospitalisation rate used in the model was estimated from data prior to the emergence of the more severe Delta variant.[[19]] Hospitalisation rates for vaccinated individuals can be twice as high for the Delta variant as for Alpha.[[27]] Secondly, I assume that the effectiveness of the vaccine is constant in throughout time, whereas recent studies have shown that vaccine effectiveness decreases with time (hence the current “booster” campaign).[[28,29,30]] Thirdly, the Ministry of Health reports the number of individuals in hospital who have COVID-19, rather than the number that are in hospital due to COVID-19,[[26]] and hence may over estimate the number of hospitalisations relative to what is modelled.

As the vaccination rate increases, there are more vaccinated individuals in the population, fewer unvaccinated individuals, and hence more breakthrough infections will occur . This is expected; the model assumes a 30% chance of breakthrough infections in vaccinated individuals.[[17]] As the number of vaccinated individuals increases, the number of cases in vaccinated individuals will increase, as shown in Figure 5a. For V=40%, there is an 83% chance that a new case will be unvaccinated and a 17% chance that they will be vaccinated. For V=80%, this switches to 46% unvaccinated and 54% vaccinated. It is important to note that, even though the number of infections in vaccinated individuals increases, it stays below the population proportion, which is shown by the dashed lines in Figure 5a. This indicates that, per population, infections are still more likely to occur in unvaccinated individuals. It should also be noted that infections in unvaccinated individuals are more likely to lead to onward transmission and hospitalisation than breakthrough infections.[[17,22]]

Figure 5b shows the probability that an infection occurs in a vaccinated or unvaccinated individual normalised by population. At all vaccination rates, an unvaccinated individual is more likely to be infected than vaccinated individual. Dividing the unvaccinated probability normalised by population by the vaccinated probability normalised by population gives the risk factor for unvaccinated individuals, which quantifies the likelihood of an unvaccinated individual getting infected compared to a vaccinated individual. This calculation shows that, even though the number of breakthrough infections increases with increasing vaccination rate, an unvaccinated individual is 3.3x more likely to be infected than a vaccinated individual for all vaccination rates.

**Figure 5**: Probability that an infected or hospitalized individual is (blue)vaccinated or (red) unvaccinated as a function of vaccination rate, as calculated from mean simulation result. (a) and (c) Probabilitythat an (a) infected or (c) hospitalised individual is vaccinated or unvaccinated. Dashed lines indicate proportion of population that are vaccinated or unvaccinated as a function of vaccination rate. (b) and (d) Probability normalised by population that an (b) infected or (d) hospitalised individual is vaccinated or unvaccinated.

I then perform the same analysis for hospitalisations. Figure 5c shows that the probability that a hospitalised case is vaccinated increases slower than the probability that an infected case is vaccinated. This shows that while the vaccine provides substantial protection against infection, it provides even greater protection against hospitalisation. At all vaccination rates, hospitalisations are significantly more likely to be unvaccinated than vaccinated (at V=80%, there is an 86% chance that a hospitalised case will be unvaccinated compared to 14% vaccinated). Figure 5d shows the probability of hospitalisation normalised by population. Unvaccinated individuals are 25x more at risk of hospitalisation than vaccinated individuals. This reinforces the need to vaccinate a large percent of the population to minimise hospitalisations and prevent strain on the healthcare system.

The results shown here are in broad agreement with a CDC study of 43,127 COVID-19 cases in Los Angeles County that showed that infection and hospitalisation rates among unvaccinated individuals were 4.9x and 29.2x, respectively, higher than for fully vaccinated individuals.[[31]] The risk factors calculated here may be lower due to underestimating the protection that the vaccine provides against infection or because I focus on the Delta variant, which transitioned to becoming the dominant variant during the study period of Griffin et al.[[31]]

The COVID-19 outbreak in New Zealand which began in August 2021, had resulted in the highest case counts experienced in New Zealand during the pandemic, prior to the arrival of Omicron in January 2022. Cases were centred in Auckland but spread to Northland and Waikato, with isolated cases spread around the rest of the country, including the South Island. The Government’s switch from an elimination to suppression strategy has emphasised the importance of vaccination in preventing COVID-19 from overwhelming the healthcare system. As restrictions eased, there was a need to understand how different vaccination rates will impact the initial stages of COVID-19 outbreak as cases become seeded in communities around New Zealand.

Here, I use a stochastic branching process model to examine the impact of vaccination rates on the initial spread of an outbreak. I show that increasing vaccination rates greatly decrease the number of infections (1.4% median number of infections 28 days into the outbreak for V=80% compared to V=20%), even if the Pfizer-BioNTech vaccine only provides individuals with imperfect protection (assumed to be 70% effective against breakthrough infection). This illustrates the effectiveness of the vaccine on a population level.

As the vaccination rate increases, the number of breakthrough infections and hospitalisations among vaccinated individuals will increase. This is expected and reflects the increased proportion of vaccinated individuals in the population. Unvaccinated individuals are 3.3x more likely to be infected and 25x more likely to be hospitalised than vaccinated individuals. The model results presented here agree with real-world data[[31]] and highlight how the Pfizer-BioNTech vaccine provides good protection against infection and extremely good protection against hospitalisation. This work illustrates the need for high vaccination rates to reduce infections and prevent the healthcare system from being overrun with COVID-19 patients.

The August 2021 COVID-19 outbreak in Auckland caused the New Zealand Government to transition from an elimination strategy to suppression, which relies heavily on high vaccination rates in the population. As restrictions ease and as COVID-19 spreads throughout New Zealand, there is a need to understand how different levels of vaccination will impact the initial stages of COVID-19 outbreaks that are seeded around the country.

A stochastic branching process model is used to simulate the initial spread of a COVID-19 outbreak for different vaccination rates.

High vaccination rates are effective at minimizing the number of infections and hospitalizations. Increasing vaccination rates from 20% (approximate value at the start of the August 2021 outbreak) to 80% (approximate proposed target) of the total population can reduce the median number of infections that occur within the first four weeks of an outbreak from 1011 to 14 (25th and 75th quantiles of 545–1602 and 2–32 for V=20% and V=80%, respectively). As the vaccination rate increases, the number of breakthrough infections (infections in fully vaccinated individuals) and hospitalisations of vaccinated individuals increases. Unvaccinated individuals, however, are 3.3x more likely to be infected with COVID-19 and 25x more likely to be hospitalised.

This work demonstrates the importance of vaccination in protecting individuals from COVID-19, preventing high caseloads, and minimising the number of hospitalisations and hence limiting the pressure on the healthcare system.

1) History of the COVID-19 Alert System. Unite Against COVID-19. [Accessed 2021 Nov 4]. Avaliable from https://covid19.govt.nz/alert-levels-and-updates/history-of-the-covid-19-alert-system/.

2) Grout L, Katar A, Ait Ouakrim D, et al. Failures of quarantine systems for preventing COVID-19 outbreaks in Australia and New Zealand. Med J Aust. 2021 Oct 4;215(7):320-324. doi: 10.5694/mja2.51240.

3) Plank MJ, Binny RN, Hendy SC, et al. A stochastic model for COVID-19 spread and the effects of alert level 4 in Aotearoa New Zealand. medRxiv. 2020. doi: 10.1101/2020.04.08.20058743.

4) Plank MJ, Hendy SC, Binny RN, Maclaren O. Modeling the August 2021 COVID-19 outbreak in New Zealand. Technical Report, Te Pūnaha Matatini. 2021.

5) Steyn N, Plank MJ, Binny RN, et al. A COVID-19 vaccination model for Aotearoa New Zealand, Technical Report, Te Pūnaha Matatini. 2021.

6) James A, Hendy SC, Plank MJ, Steyn N. Suppression and mitigation strategies for control of COVID-19 in New Zealand. medRxiv. 2020. doi: 10.1101/2020.03.26.20044677.

7) Davies NG, Klepac P, Liu Y, et al.; CMMID COVID-19 working group, Eggo RM. Age-dependent effects in the transmission and control of COVID-19 epidemics. Nat Med. 2020 Aug;26(8):1205-1211. doi: 10.1038/s41591-020-0962-9.

8) Hendy SC, Steyn N, James A, et al. Mathematical modelling to inform New Zealand’s COVID-19 response. J. R. Soc. N. Z. 2021 51(S1), S86-S106, doi: 10.1080/03036758.2021.1876111.

9) Endo A, Centre for the Mathematical Modelling of Infectious Diseases COVID-19 Working Group, Abbott S, Kucharski AJ, Funk S. Estimating the overdispersion in COVID-19 transmission using outbreak sizes outside China. *Wellcome open research* 5 2020.

10) Allen H, Vusirikala A, Flannagan J, et al.; COVID-19 Genomics UK (COG-UK Consortium). Household transmission of COVID-19 cases associated with SARS-CoV-2 delta variant (B.1.617.2): national case-control study. Lancet Reg Health Eur. 2021 Oct 28:100252. doi: 10.1016/j.lanepe.2021.100252.

11) Byambasuren O, Cardona M, Bell K, et al. Estimating the extent of asymptomatic COVID-19 and its potential for community transmission: systematic review and meta-analysis. JAMMI. 2020 Dec;5(4):223-34. doi: 10.3138/jammi-2020-0030

12) Lavezzo E, Franchin E, Ciavarella C, et al.; Imperial College COVID-19 Response Team, Brazzale AR, Toppo S, Trevisan M, Baldo V, Donnelly CA, Ferguson NM, Dorigatti I, Crisanti A; Imperial College COVID-19 Response Team. Suppression of a SARS-CoV-2 outbreak in the Italian municipality of Vo'. Nature. 2020 Aug;584(7821):425-429. doi: 10.1038/s41586-020-2488-1.

13) Pollán M, Pérez-Gómez B, Pastor-Barriuso R, et al; ENE-COVID Study Group. Prevalence of SARS-CoV-2 in Spain (ENE-COVID): a nationwide, population-based seroepidemiological study. Lancet. 2020 Aug 22;396(10250):535-544. doi: 10.1016/S0140-6736(20)31483-5.

14) Steyn N, Plank MJ, Hendy SC. Modelling to support a future COVID-19 strategy for Aotearoa New Zealand, Technical Report, Te Pūnaha Matatini. 2021.

15) Ministry of Health. https://www.health.govt.nz/our-work/diseases-and-conditions/covid-19-novel-coronavirus/covid-19-data-and-statistics/covid-19-vaccine-data. Accessed Feb 8, 2022.

16) Vattiato G, Maclaren O, Lustig A, et al. A preliminary assessment of the potential impact of the Omicon variant of SARS-CoV-2 in Aotearoa New Zealand. Covid-19 Modelling Aotearoa. 2022. https://www.covid19modelling.ac.nz/a-preliminary-assessment-of-the-potential-impact-of-the-omicron-variant/.

17) Scientific Pandemic Influenza Group of Modeling, SPI-M-O. Summary of further modelling of easing restrictions - Roadmap Step 4. Technical Report, Scientific Advisory Group for Emergencies. 2021. Available from https://www.gov.uk/government/publications/spi-m-o-summary-of-further-modelling-of-easing-restrictions-roadmap-step-4-9-june-2021.

18) Steyn, N, Binny RN, Hannah K, et al. Maori and Pacific people in New Zealand have a higher risk of hospitalization for COVID-19. New Zealand Medical Journal, 134(1538),28-43.

19) Verity R, Okell LC, Dorigatti I, et al. Estimates of the severity of coronavirus disease 2019: a model-based analysis. Lancet Infect Dis. 2020 Jun;20(6):669-677. doi: 10.1016/S1473-3099(20)30243-7.

20) COVID-19 Vaccine Data. Ministry of Health. [cited 2021 Nov 1]. Avaliable from https://www.health.govt.nz/our-work/diseases-and-conditions/covid-19-novel-coronavirus/covid-19-data-and-statistics/covid-19-vaccine-data.

21) Singh H. The Spinoff Covid-tracker: the live graphs that tell the story of delta in Aotearoa. 2021. [cited 2021 Nov 1]. Avaliable from https://thespinoff.co.nz/society/30-10-2021/the-spinoff-covid-tracker/.

22) Dagan N, Barda N, Kepten E, et al. BNT162b2 mRNA Covid-19 Vaccine in a Nationwide Mass Vaccination Setting. N Engl J Med. 2021 Apr 15;384(15):1412-1423. doi: 10.1056/NEJMoa2101765.

23) Derek Cheng. Big Read: Tracking the Covid-19 delta outbreak – how it arrived, simmered through lockdown, then spread beyond reach. The New Zealand Herald. 2021. [cited 2021 Nov 19]. Avaliable from https://www.nzherald.co.nz/nz/politics/big-read-tracking-the-covid-19-delta-outbreak-how-it-arrived-simmered-through-lockdown-then-spread-beyond-reach/UM5CJTAGMBFGA5X2ZGJ3XWBTY4/

24) Liu Y, Rocklöv J. The reproductive number of the Delta variant of SARS-CoV-2 is far higher compared to the ancestral SARS-CoV-2 virus. J Travel Med. 2021 Oct 11;28(7):taab124. doi: 10.1093/jtm/taab124.

25) Sonabend R, Whittles LK, Imai N, et al. Non-pharmaceutical interventions, vaccination and the Delta variant: epidemiological insights from modelling England's COVID-19 roadmap out of lockdown. medRxiv. 2021. doi: 10.1101/2021.08.17.21262164

26) Ministry of Health. https://www.health.govt.nz/our-work/diseases-and-conditions/covid-19-novel-coronavirus/covid-19-news-and-media-updates. Accessed Feb 8 2022.

27) Twohig KA, Nyberg T, Zaidi A, Thelwall S, et al. Hospital admission and emergency care attendance risk for SARS-CoV-2 delta (B.1.617.2) compared with alpha (B.1.1.7) variants of concern: a cohort study. The Lancet Infectious Diseases. 2021 22(1):P35-P42. doi: 10.1016/S1473-3099(21)00475-8.

28) Tartof SY, Slezak JE, Fischer H, et al. Effectiveness of mRNA BNT162b2 COVID-19 vaccine up to 6 months in a large integrated health system in the USA: a retrospective cohort study. Lancet. 2021 Oct 16;398(10309):1407-1416. doi: 10.1016/S0140-6736(21)02183-8.

29) Skowronski DM, Setayeshgar S, Febriana Y, et al. Two-dose SARS-CoV-2 vaccine effectiveness with mixed schedules and extended dosing intervals: test-negative design studies from British Columbia and Quebec, Canada. medRxiv, 2021, doi: 10.1101/2021.10.26.21265397

30) Andrews A, Tessier E, Stowe J, et al. Vaccine effectiveness and duration of protection of Comirnaty, Vaxzeria and Spikevax against mild and severe COVID-19 in the UK. medRxiv, 2021. doi: 10.1101/2021.09.15.21263583

31) Griffin JB, Haddix M, Danza P, et al. SARS-CoV-2 Infections and Hospitalizations Among Persons Aged ≥16 Years, by Vaccination Status - Los Angeles County, California, May 1-July 25, 2021. MMWR Morb Mortal Wkly Rep. 2021 Aug 27;70(34):1170-1176. doi: 10.15585/mmwr.mm7034e5.

contact nzmj@nzma.org.nz

In response to the COVID-19 global pandemic, New Zealand initially followed an elimination strategy that coupled tough lockdowns with strict border controls. The first COVID-19 case was reported on 28 February 2020 and on 19 March 2020 international borders were closed to all but New Zealand citizens and permanent residents.[[1]] On 25 March 2020, New Zealand moved to Alert Level 4 (Level 4 brings the toughest restrictions in the four-level alert system) with strict lockdown measures including the closure of educational and public facilities along with all non-essential businesses, stay-at-home orders, limits on travel and no gatherings allowed. From 10 April 2020, anyone entering the country had to undergo two weeks of managed isolation and quarantine (MIQ). These measures kept case numbers low with a total of 1504 cases before COVID-19 was declared eliminated in New Zealand on 8 June 2020.[[1]]

Mandatory quarantine of overseas arrivals has been broadly effective at keeping COVID-19 out of New Zealand. In the period up to 15 June 2021 there have been 10 border-related reincursions,[[2]] although these outbreaks were able to be quickly detected and successfully eliminated. On 17 August 2021 a COVID-19 case of the highly transmissible Delta variant, with no clear link to the border, was detected in Auckland. As a result, the entire country moved from Alert Level 1 (minimal restrictions) to Alert Level 4 (toughest restrictions). The lockdown measures and the implementation of an internal boundary around Auckland were largely successful in containing the outbreak to Auckland. However, a small number of cases leaked through the border, leading to community transmission in Northland and Waikato with a limited number of further cases detected around the country, including in the South Island. After 35 days at Alert Level 4 in Auckland, the Government began to ease restrictions and transition from an elimination to suppression strategy.

On 22 October 2021, the Government outlined the COVID-19 Protection Framework, which uses vaccination certificates along with public health measures to manage COVID-19 in the community. This system replaced the Alert Level framework when district health boards reached a vaccination target of 90% of the eligible population. As vaccination rates increased and restrictions eased, as expected, COVID-19 cases spread out of Auckland to other parts of the country (as was previously observed with prior outbreaks in Waikato and Northland). It is important to understand the impact of different vaccination rates on the growth of future outbreaks outside of Auckland along with the number of hospitalisations and stress on the healthcare system.

In this work, a stochastic branching process model[[3–5]] is used to simulate the initial stages of a COVID-19 outbreak within a community. Stochastic models are useful because they incorporate the randomness associated with the initial stages of an outbreak. For some simulations, COVID-19 will spread widely and form an extensive outbreak (eg Auckland in August 2021 where a returnee from Australia sparked an outbreak with >6000 cases at the time of writing), whereas for other simulations, despite the virus having a reproduction number larger than one, random chance will mean that COVID-19 does not spread far beyond the initial seed infection (eg Wellington in June 2021 when a COVID-19 infected traveller visited from Australia but did not infect anyone else). The stochastic model tracks each individual case and becomes computationally expensive for large case numbers. Therefore, while stochastic models are useful for simulating the initial stages of an outbreak, deterministic SEIR (Susceptible, Exposed, Infected, Recovered) models are frequently used for larger and longer-term population level studies of epidemics.[[5,6]]

Here, I use a stochastic model to study the how the number of infections and hospitalisations depend on the vaccination rate and population level controls. I calculate the likelihood that a new infected case or hospitalised individual is vaccinated and determine the relative risk of getting infected or hospitalised with COVID-19 between vaccinated and unvaccinated individuals.

The stochastic model presented here tracks the number of infections in the community and categorises individuals as symptomatic (clinical infections) or asymptomatic (subclinical infections). Each infected individual infects a random number of other individuals, *N*, drawn from a Poisson distribution (Figure 1).[[3]] For a symptomatic individual, the Poisson distribution is defined by *= RC* where *R* is the reproduction number and *C* is the effectiveness of population level controls (eg Level 1, 2, 3 or 4 in the Alert Level Framework or Green, Orange, or Red in the COVID-19 Protection Framework). For an asymptomatic individual, the Poisson distribution is defined by *= RC/2*, which assumes that asymptomatic individuals infect, on average, half as many people as symptomatic individuals.[[7]] This model accounts for “super-spreading” events through the tail of the Poisson distribution.[[8]] A symptomatic individual in an Alert Level 1 environment, for example, has an 8% chance of infecting more than 10 people. It is possible, however, that the spread of COVID-19 is more heterogeneous, potentially with up to 80% of COVID-19 infections caused by only 10% of cases.[[9]] Therefore, while multiple realisations of this model can be used to obtain an average perspective of the initial stages of an outbreak (which is the focus of this work), alternative models (and distributions) should be used to investigate super-spreading events and their impact on the evolution of an outbreak.

Population level controls include public health measures such as physical distancing, wearing of masks, closure of schools and non-essential businesses, and restrictions on gatherings and social activities. The effectiveness of population level controls are taken from Plank et al[[3]] as *C*=1 for Alert Level 1, *C*=0.72 for Alert Level 2, *C*=0.52 for Alert Level 3, and *C*=0.32 for Alert Level 4. These values were estimated for the initial variant of COVID-19, and population level controls may be less effective against the more transmissible Delta variant.[[10]] In addition, the model does not account for illegal gatherings or other non-compliance with restrictions.

The generation times between an individual becoming infected and infecting *N* other individuals are independently sampled from a Weibull distribution with *a*=5.57 and *b*=4.08 where* a* is the scale parameter and *b* is the shape parameter (mean=5.05 days and variance=1.94 days)[[8]] (Figure 1). The model assumes that 33% of new infections are asymptomatic (subclinical) with the remainder symptomatic (clinical).[[11–13]]

I consider a range of vaccination rates, *V*, from 0% to 90% of the total population (rather than the *eligible population*, which at the time of writing is the over 12 years old population). Unlike previous work by Steyn et al,[[5,14]] age is not accounted for in the model, either in the vaccination rollout where older individuals were initially prioritised (at the start of February 2022, two-dose vaccination rates are now relatively similar for all age groups over 12 years old, although older age groups have higher booster rates),[[15]] or in contact rates where younger people are likely to have more contacts, or in the susceptibility where older individuals are more likely to experience severe disease or death (individuals who are 65–69 years old are 19x more likely to be hospitalised than those who are 25–29).[[16]] Therefore, care should be taken when applying the model results across different age bands as there is considerable heterogeneity of risk with age.

The vaccination rate *V* is assumed to be constant throughout the simulated outbreak. Following Steyn et al,[[5,14]] the Pfizer-BioNTech vaccine, which is the only COVID-19 vaccine currently being widely administered in New Zealand, is assumed to be 70% effective against infection and 50% effective against transmission for breakthrough infections (infections in fully vaccinated individuals).[[17]] Throughout the simulations, the model tracks the total number of vaccinated and unvaccinated infections along with the number of symptomatic and asymptomatic cases. Infected individuals are assumed to be equally likely to interact with vaccinated and unvaccinated individuals, with probabilities based solely on the vaccination rate. This may lead to an underestimation of the spread of COVID-19 in unvaccinated communities, as unvaccinated individuals are more likely to have unvaccinated contacts. Other limitations of this model include not accounting for ethnicity, either in vaccination rates or differential risk factors for different ethnic groups[[18]] or socio-economic status. COVID-19 spreads rapidly through overcrowded households as well as posing a greater risk to those who do not have the economic resources to safely isolate or the ability to work-from-home.[[18]]

**Figure 1**:** **Probability distributions used in the stochastic model. (a) Number of infections caused by a symptomatic case and (b) by an asymptomatic case. The number of infections caused by a symptomatic or asymptomatic case is governed by a Poisson distribution and can only take integer values. (c) The generation time, which is the time between an individual getting infected and infecting others, is governed by a Weibull distribution.

The likelihood of hospitalisations is also modelled. Clinical infections are assumed to have a 7.8% probability of being hospitalised.[[3,19]] I note that this hospitalisation rate was estimated from data prior to the emergence of the more severe Delta variant[[24]] and may underestimate the severity of Delta. Nonetheless, the modelled hospitalisation rate agrees with the total number of cases hospitalised during the August 2021 Auckland outbreak (7.6% hospitalisation rate for all cases as of 30 October 2021).[[20,21]] Based on Dagan et al[[22]] who examined the effectiveness of the Pfizer-BioNTech vaccine against hospitalisation in Israel, it is assumed that the vaccine is 87% effective at preventing hospitalisations after two doses. The model only allows for cases to be fully vaccinated (defined as more than two weeks after the second dose of the two-dose Pfizer-BioNTech vaccine) or unvaccinated. It is assumed that vaccine effectiveness does not wane with time. The model does not include the additional complexity of individuals who are partially vaccinated, either by only receiving one dose of the vaccine or by being within two weeks after receiving the second dose (note that this simulation work was perform prior to the booster rollout and hence boosters are not considered). In addition, there is no lag time between becoming infected and becoming hospitalised. The model is only run for a short duration (28 days), and hence I do not simulate the likelihood of hospitalised individuals dying.

Note that the model presented here does not include any testing, contact tracing, or isolation of cases. Instead, I focus on the impact of vaccination rates, particularly on the early stages of an outbreak when cases may be circulating undetected. The reader is referred to Steyn et al[[5]] for a model that includes testing and estimates the number of infections at the time of detection of the outbreak for various vaccination rates and testing scenarios.

The simulations are seeded with one unvaccinated symptomatic individual at* t*=0 where *t* is the time in days. Simulations are run for 28 days with time steps of one day. The model tracks each infected individual, distinguishes between symptomatic cases (clinical infections) and asymptomatic cases (subclinical infections), tracks hospitalisation rates, and distinguishes between vaccinated and unvaccinated individuals. I consider vaccination rates between 0% and 90% of the total population in 5% increments (the vaccination rate referred to here is the total population rather than the eligible population, which is over 12 years old at the time of writing). The Government’s vaccination target of 90% of the eligible population corresponds to 78.7% of the total population.[[20,21]] For each vaccination rate, we consider four different population level controls based on New Zealand’s Alert Level system using the effectiveness values from Plank et al.[[3]] To get a representative sample of the possible outcomes for each scenario, we run the model 100,000 times for each combination of vaccination rate and population level controls.

I consider the impact of the vaccination rate and population level controls on the total number of infections and hospitalisations. Figure 2 shows likelihood of the number of infections 28 days into an outbreak for V=0%, 30%, 60%, and 90% and no population level controls (*C*=1). The histograms indicate the likelihood of each number of infections and illustrate the randomness associated with the initial stages of an outbreak. For some simulations, random chance causes the outbreak to infect a small number of people, whereas for other simulations the outbreak can rapidly grow due to super-spreader events. For V=30%, there is a 10% chance that an outbreak will cause less than 98 infections after 28 days. However, there is also a 10% chance that an outbreak will cause more than 1340 infections. The potential spread of an outbreak is strongly dependent on the vaccination rate. For V=0%, there is a 50% chance than an outbreak will cause less than 2691 infections after 28 days compared to less than 75 infections for V=60%.

The number of infections is strongly dependent on the vaccination rate, with higher vaccination rates decreasing the number of infections by several orders of magnitude. For V=0%, the maximum number of infections after 28 days is 16,198 compared to 819 for V=60% and 101 for V=90%. The strong dependence of the number of infections on the vaccination rate is further visualised in Figure 3, which shows the median number of infections along with the 25% and 75% quartiles as a function of vaccination rate and population level controls. Increasing the vaccination rate can drastically decrease the number of infections and hence minimise the size of an outbreak.

**Figure 2**: Histograms showing the likelihood for a given number of infections 28 days into an outbreak for vaccination rates. (a) 0%, (b) 30%, (c) 60%, and (d) 90%. Red lines indicate the cumulative probability.

Figure 3 also shows the importance of population level controls, especially at low levels of vaccination. For V=20% (the approximate vaccination rate in New Zealand on 17 August 2021 when the Delta outbreak was first detected), the median number of infections after 28 days is 1,011 (542 and 1602 for 25% quantile and 75% quantile, respectively) at Level 1 but only 7 (3 and 17 and for 25% quantile and 75% quantile, respectively) at Level 4. This illustrates how, in the absence of high vaccination rates, population level controls are extremely important in limiting the growth of an outbreak. It is noted that New Zealand only shifted from Level 1 to Level 4 after a case was detected. At this point, the virus had been circulating undetected for approximately a week prior to this and had potentially already seeded 800 to 1000 cases in the community.[[23]] Furthermore, the effectiveness values used here for the population level controls were estimated for the original strain of COVID-19.[[3]] It is unclear if these values are appropriate for the more transmissible Delta variant[[10,24,25]] or if population level controls are less effective against Delta.

**Figure 3**: Number of infections as a function of vaccination rate and population level controls. (a,c) 14 days and (b,d) 28 days after an unvaccinated symptomatic individual is seeded into the community. Solid lines show the median of the 100,000 realisations while the shaded area shows the range between the 25% and 75% quantiles. Top plots (a,b) show the results on a linear scale while the bottom plots (c,d) show the results on a logarithmic scale. The vertical black dashed lines indicate vaccination rate for the total population that corresponds to the 90% target of the over 12 years old population (78.7%).

The results shown in Figures 2 and 3 illustrate how a vaccine can be extremely effective in preventing or limiting an outbreak even if the vaccine does not provide individuals with 100% protection from infection. Here, it is assumed that the Pfizer-BioNTech vaccine is 70% effective in preventing infection and, for breakthrough infections, 50% effective at preventing onward transmission.[[17]] Despite this imperfect protection, at a population level the vaccine drastically reduces the spread of the virus. For V=60%, the mean number of infections 28 days into the outbreak is only 3% of the mean number of infections for V=0%. For V=90%, the mean number of infections further decreases to only 0.3% compared to the unvaccinated scenario.

**Figure 4**: Histograms showing the likelihood for a given number of hospitalizations 28 days into an outbreak for C=1 and vaccination rates. (a) 0%, (b) 30%, (c) 60%, and (d) 90%. Red lines indicate the cumulative probability. (e) Number of hospitalisations 28 days into an outbreak as a function of vaccination rate showing the (solid line) median simulation result and (shaded area) 25th and 75th quantiles. Vertical black line indicates the vaccination rate of the total population that corresponds to 90% of the eligible (over 12 years old) population (78.7%).

The number of hospitalisations is also strongly dependent on the vaccination rate (Figure 4). For V=0%, there is a 50% chance of over 282 hospitalisations 28 days into the outbreak compared to 50% chance of less than six hospitalisations for V=60% and one hospitalisation for V=90%. The hospitalisation results presented in Figure 4 are calculated for *C*=1, which gives a worst-case scenario. Implementation of population level controls (ie Alert Level restrictions or vaccination certificates) will reduce the total number of hospitalisations at all vaccination rates but the general trend is unchanged; high vaccination rates drastically reduce the number of hospitalisations.

I assume that the vaccine is 87% effective against hospitalisation,[[22]] which results in a hospitalisation rate of 1.0% for fully vaccinated individuals. During the Delta outbreak, the hospitalisation rate for fully vaccinated individuals was 2.3% (as of 22 January 2022, prior to the emergence of Omicron in the community).[[21,26]] There are several possible reasons why the model presented here underpredicts the hospitalisation rate for fully vaccinated individuals. Firstly, the hospitalisation rate used in the model was estimated from data prior to the emergence of the more severe Delta variant.[[19]] Hospitalisation rates for vaccinated individuals can be twice as high for the Delta variant as for Alpha.[[27]] Secondly, I assume that the effectiveness of the vaccine is constant in throughout time, whereas recent studies have shown that vaccine effectiveness decreases with time (hence the current “booster” campaign).[[28,29,30]] Thirdly, the Ministry of Health reports the number of individuals in hospital who have COVID-19, rather than the number that are in hospital due to COVID-19,[[26]] and hence may over estimate the number of hospitalisations relative to what is modelled.

As the vaccination rate increases, there are more vaccinated individuals in the population, fewer unvaccinated individuals, and hence more breakthrough infections will occur . This is expected; the model assumes a 30% chance of breakthrough infections in vaccinated individuals.[[17]] As the number of vaccinated individuals increases, the number of cases in vaccinated individuals will increase, as shown in Figure 5a. For V=40%, there is an 83% chance that a new case will be unvaccinated and a 17% chance that they will be vaccinated. For V=80%, this switches to 46% unvaccinated and 54% vaccinated. It is important to note that, even though the number of infections in vaccinated individuals increases, it stays below the population proportion, which is shown by the dashed lines in Figure 5a. This indicates that, per population, infections are still more likely to occur in unvaccinated individuals. It should also be noted that infections in unvaccinated individuals are more likely to lead to onward transmission and hospitalisation than breakthrough infections.[[17,22]]

Figure 5b shows the probability that an infection occurs in a vaccinated or unvaccinated individual normalised by population. At all vaccination rates, an unvaccinated individual is more likely to be infected than vaccinated individual. Dividing the unvaccinated probability normalised by population by the vaccinated probability normalised by population gives the risk factor for unvaccinated individuals, which quantifies the likelihood of an unvaccinated individual getting infected compared to a vaccinated individual. This calculation shows that, even though the number of breakthrough infections increases with increasing vaccination rate, an unvaccinated individual is 3.3x more likely to be infected than a vaccinated individual for all vaccination rates.

**Figure 5**: Probability that an infected or hospitalized individual is (blue)vaccinated or (red) unvaccinated as a function of vaccination rate, as calculated from mean simulation result. (a) and (c) Probabilitythat an (a) infected or (c) hospitalised individual is vaccinated or unvaccinated. Dashed lines indicate proportion of population that are vaccinated or unvaccinated as a function of vaccination rate. (b) and (d) Probability normalised by population that an (b) infected or (d) hospitalised individual is vaccinated or unvaccinated.

I then perform the same analysis for hospitalisations. Figure 5c shows that the probability that a hospitalised case is vaccinated increases slower than the probability that an infected case is vaccinated. This shows that while the vaccine provides substantial protection against infection, it provides even greater protection against hospitalisation. At all vaccination rates, hospitalisations are significantly more likely to be unvaccinated than vaccinated (at V=80%, there is an 86% chance that a hospitalised case will be unvaccinated compared to 14% vaccinated). Figure 5d shows the probability of hospitalisation normalised by population. Unvaccinated individuals are 25x more at risk of hospitalisation than vaccinated individuals. This reinforces the need to vaccinate a large percent of the population to minimise hospitalisations and prevent strain on the healthcare system.

The results shown here are in broad agreement with a CDC study of 43,127 COVID-19 cases in Los Angeles County that showed that infection and hospitalisation rates among unvaccinated individuals were 4.9x and 29.2x, respectively, higher than for fully vaccinated individuals.[[31]] The risk factors calculated here may be lower due to underestimating the protection that the vaccine provides against infection or because I focus on the Delta variant, which transitioned to becoming the dominant variant during the study period of Griffin et al.[[31]]

The COVID-19 outbreak in New Zealand which began in August 2021, had resulted in the highest case counts experienced in New Zealand during the pandemic, prior to the arrival of Omicron in January 2022. Cases were centred in Auckland but spread to Northland and Waikato, with isolated cases spread around the rest of the country, including the South Island. The Government’s switch from an elimination to suppression strategy has emphasised the importance of vaccination in preventing COVID-19 from overwhelming the healthcare system. As restrictions eased, there was a need to understand how different vaccination rates will impact the initial stages of COVID-19 outbreak as cases become seeded in communities around New Zealand.

Here, I use a stochastic branching process model to examine the impact of vaccination rates on the initial spread of an outbreak. I show that increasing vaccination rates greatly decrease the number of infections (1.4% median number of infections 28 days into the outbreak for V=80% compared to V=20%), even if the Pfizer-BioNTech vaccine only provides individuals with imperfect protection (assumed to be 70% effective against breakthrough infection). This illustrates the effectiveness of the vaccine on a population level.

As the vaccination rate increases, the number of breakthrough infections and hospitalisations among vaccinated individuals will increase. This is expected and reflects the increased proportion of vaccinated individuals in the population. Unvaccinated individuals are 3.3x more likely to be infected and 25x more likely to be hospitalised than vaccinated individuals. The model results presented here agree with real-world data[[31]] and highlight how the Pfizer-BioNTech vaccine provides good protection against infection and extremely good protection against hospitalisation. This work illustrates the need for high vaccination rates to reduce infections and prevent the healthcare system from being overrun with COVID-19 patients.

The August 2021 COVID-19 outbreak in Auckland caused the New Zealand Government to transition from an elimination strategy to suppression, which relies heavily on high vaccination rates in the population. As restrictions ease and as COVID-19 spreads throughout New Zealand, there is a need to understand how different levels of vaccination will impact the initial stages of COVID-19 outbreaks that are seeded around the country.

A stochastic branching process model is used to simulate the initial spread of a COVID-19 outbreak for different vaccination rates.

High vaccination rates are effective at minimizing the number of infections and hospitalizations. Increasing vaccination rates from 20% (approximate value at the start of the August 2021 outbreak) to 80% (approximate proposed target) of the total population can reduce the median number of infections that occur within the first four weeks of an outbreak from 1011 to 14 (25th and 75th quantiles of 545–1602 and 2–32 for V=20% and V=80%, respectively). As the vaccination rate increases, the number of breakthrough infections (infections in fully vaccinated individuals) and hospitalisations of vaccinated individuals increases. Unvaccinated individuals, however, are 3.3x more likely to be infected with COVID-19 and 25x more likely to be hospitalised.

This work demonstrates the importance of vaccination in protecting individuals from COVID-19, preventing high caseloads, and minimising the number of hospitalisations and hence limiting the pressure on the healthcare system.

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*N*, drawn from a Poisson distribution (Figure 1).[[3]] For a symptomatic individual, the Poisson distribution is defined by *= RC* where *R* is the reproduction number and *C* is the effectiveness of population level controls (eg Level 1, 2, 3 or 4 in the Alert Level Framework or Green, Orange, or Red in the COVID-19 Protection Framework). For an asymptomatic individual, the Poisson distribution is defined by *= RC/2*, which assumes that asymptomatic individuals infect, on average, half as many people as symptomatic individuals.[[7]] This model accounts for “super-spreading” events through the tail of the Poisson distribution.[[8]] A symptomatic individual in an Alert Level 1 environment, for example, has an 8% chance of infecting more than 10 people. It is possible, however, that the spread of COVID-19 is more heterogeneous, potentially with up to 80% of COVID-19 infections caused by only 10% of cases.[[9]] Therefore, while multiple realisations of this model can be used to obtain an average perspective of the initial stages of an outbreak (which is the focus of this work), alternative models (and distributions) should be used to investigate super-spreading events and their impact on the evolution of an outbreak.

*C*=1 for Alert Level 1, *C*=0.72 for Alert Level 2, *C*=0.52 for Alert Level 3, and *C*=0.32 for Alert Level 4. These values were estimated for the initial variant of COVID-19, and population level controls may be less effective against the more transmissible Delta variant.[[10]] In addition, the model does not account for illegal gatherings or other non-compliance with restrictions.

*N* other individuals are independently sampled from a Weibull distribution with *a*=5.57 and *b*=4.08 where* a* is the scale parameter and *b* is the shape parameter (mean=5.05 days and variance=1.94 days)[[8]] (Figure 1). The model assumes that 33% of new infections are asymptomatic (subclinical) with the remainder symptomatic (clinical).[[11–13]]

*V*, from 0% to 90% of the total population (rather than the *eligible population*, which at the time of writing is the over 12 years old population). Unlike previous work by Steyn et al,[[5,14]] age is not accounted for in the model, either in the vaccination rollout where older individuals were initially prioritised (at the start of February 2022, two-dose vaccination rates are now relatively similar for all age groups over 12 years old, although older age groups have higher booster rates),[[15]] or in contact rates where younger people are likely to have more contacts, or in the susceptibility where older individuals are more likely to experience severe disease or death (individuals who are 65–69 years old are 19x more likely to be hospitalised than those who are 25–29).[[16]] Therefore, care should be taken when applying the model results across different age bands as there is considerable heterogeneity of risk with age.

*V* is assumed to be constant throughout the simulated outbreak. Following Steyn et al,[[5,14]] the Pfizer-BioNTech vaccine, which is the only COVID-19 vaccine currently being widely administered in New Zealand, is assumed to be 70% effective against infection and 50% effective against transmission for breakthrough infections (infections in fully vaccinated individuals).[[17]] Throughout the simulations, the model tracks the total number of vaccinated and unvaccinated infections along with the number of symptomatic and asymptomatic cases. Infected individuals are assumed to be equally likely to interact with vaccinated and unvaccinated individuals, with probabilities based solely on the vaccination rate. This may lead to an underestimation of the spread of COVID-19 in unvaccinated communities, as unvaccinated individuals are more likely to have unvaccinated contacts. Other limitations of this model include not accounting for ethnicity, either in vaccination rates or differential risk factors for different ethnic groups[[18]] or socio-economic status. COVID-19 spreads rapidly through overcrowded households as well as posing a greater risk to those who do not have the economic resources to safely isolate or the ability to work-from-home.[[18]]

**Figure 1**:** **Probability distributions used in the stochastic model. (a) Number of infections caused by a symptomatic case and (b) by an asymptomatic case. The number of infections caused by a symptomatic or asymptomatic case is governed by a Poisson distribution and can only take integer values. (c) The generation time, which is the time between an individual getting infected and infecting others, is governed by a Weibull distribution.

* t*=0 where *t* is the time in days. Simulations are run for 28 days with time steps of one day. The model tracks each infected individual, distinguishes between symptomatic cases (clinical infections) and asymptomatic cases (subclinical infections), tracks hospitalisation rates, and distinguishes between vaccinated and unvaccinated individuals. I consider vaccination rates between 0% and 90% of the total population in 5% increments (the vaccination rate referred to here is the total population rather than the eligible population, which is over 12 years old at the time of writing). The Government’s vaccination target of 90% of the eligible population corresponds to 78.7% of the total population.[[20,21]] For each vaccination rate, we consider four different population level controls based on New Zealand’s Alert Level system using the effectiveness values from Plank et al.[[3]] To get a representative sample of the possible outcomes for each scenario, we run the model 100,000 times for each combination of vaccination rate and population level controls.

*C*=1). The histograms indicate the likelihood of each number of infections and illustrate the randomness associated with the initial stages of an outbreak. For some simulations, random chance causes the outbreak to infect a small number of people, whereas for other simulations the outbreak can rapidly grow due to super-spreader events. For V=30%, there is a 10% chance that an outbreak will cause less than 98 infections after 28 days. However, there is also a 10% chance that an outbreak will cause more than 1340 infections. The potential spread of an outbreak is strongly dependent on the vaccination rate. For V=0%, there is a 50% chance than an outbreak will cause less than 2691 infections after 28 days compared to less than 75 infections for V=60%.

**Figure 2**: Histograms showing the likelihood for a given number of infections 28 days into an outbreak for vaccination rates. (a) 0%, (b) 30%, (c) 60%, and (d) 90%. Red lines indicate the cumulative probability.

**Figure 3**: Number of infections as a function of vaccination rate and population level controls. (a,c) 14 days and (b,d) 28 days after an unvaccinated symptomatic individual is seeded into the community. Solid lines show the median of the 100,000 realisations while the shaded area shows the range between the 25% and 75% quantiles. Top plots (a,b) show the results on a linear scale while the bottom plots (c,d) show the results on a logarithmic scale. The vertical black dashed lines indicate vaccination rate for the total population that corresponds to the 90% target of the over 12 years old population (78.7%).

**Figure 4**: Histograms showing the likelihood for a given number of hospitalizations 28 days into an outbreak for C=1 and vaccination rates. (a) 0%, (b) 30%, (c) 60%, and (d) 90%. Red lines indicate the cumulative probability. (e) Number of hospitalisations 28 days into an outbreak as a function of vaccination rate showing the (solid line) median simulation result and (shaded area) 25th and 75th quantiles. Vertical black line indicates the vaccination rate of the total population that corresponds to 90% of the eligible (over 12 years old) population (78.7%).

*C*=1, which gives a worst-case scenario. Implementation of population level controls (ie Alert Level restrictions or vaccination certificates) will reduce the total number of hospitalisations at all vaccination rates but the general trend is unchanged; high vaccination rates drastically reduce the number of hospitalisations.

**Figure 5**: Probability that an infected or hospitalized individual is (blue)vaccinated or (red) unvaccinated as a function of vaccination rate, as calculated from mean simulation result. (a) and (c) Probabilitythat an (a) infected or (c) hospitalised individual is vaccinated or unvaccinated. Dashed lines indicate proportion of population that are vaccinated or unvaccinated as a function of vaccination rate. (b) and (d) Probability normalised by population that an (b) infected or (d) hospitalised individual is vaccinated or unvaccinated.

*Wellcome open research* 5 2020.

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