Abstract
Estimating the impact of vaccination and nonpharmaceutical interventions on COVID19 incidence is complicated by several factors, including successive emergence of SARSCoV2 variants of concern and changing population immunity from vaccination and infection. We develop an agestructured multistrain COVID19 transmission model and inference framework to estimate vaccination and nonpharmaceutical intervention impact accounting for these factors. We apply this framework to COVID19 waves in French Polynesia and estimate that the vaccination programme averted 34.8% (95% credible interval: 34.5–35.2%) of 223,000 symptomatic cases, 49.6% (48.7–50.5%) of 5830 hospitalisations and 64.2% (63.1–65.3%) of 1540 hospital deaths that would have occurred in a scenario without vaccination up to May 2022. We estimate the booster campaign contributed 4.5%, 1.9%, and 0.4% to overall reductions in cases, hospitalisations, and deaths. Our results suggest that removing lockdowns during the first two waves would have had nonlinear effects on incidence by altering accumulation of population immunity. Our estimates of vaccination and booster impact differ from those for other countries due to differences in age structure, previous exposure levels and timing of variant introduction relative to vaccination, emphasising the importance of detailed analysis that accounts for these factors.
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Introduction
Since late 2020, multiple new severe acute respiratory coronavirus 2 (SARSCoV2) variants have emerged and spread globally, of which the major variant groups (Alpha, Beta, Gamma, Delta, and Omicron) have shown substantially different levels of transmissibility, severity and/or immune escape. At the same time, first and seconddose vaccinations and booster doses against COVID19 have been rolled out in many countries around the world, drastically changing populationlevel immunity and reducing incidence of severe COVID19 outcomes^{1,2,3,4,5,6}. Many countries have experienced multiple waves from the same or different variants^{7,8,9}. In this context, estimating the impact of vaccination and nonpharmaceutical interventions (NPIs) on COVID19 incidence is challenging, because it is necessary to account for: different variant properties, a complicated and everchanging immune landscape from vaccination and previous infection, and the timing of variant emergence relative to vaccination rollout and previous epidemic waves. Most existing modelling analyses of vaccination impact have not explicitly accounted for different variant properties and the array of different levels and types of immunity that now exist^{1,2,3,4,10}, and thus may no longer offer the best available evidence. While frameworks for modelling multiple variants have been developed^{11,12,13,14,15}, most are country specific and not straightforwardly generalisable to other settings, or do not provide robust and flexible inference methodology for fitting to multiple data streams. Here we develop a framework that explicitly addresses these issues and apply it to COVID19 epidemic waves in French Polynesia.
As of mid 2023, French Polynesia had experienced five waves of COVID19 cases. The first, caused by the wildtype virus, started in August 2020 and peaked in early November 2020 (Fig. 1). Transmission then declined with the introduction of strict control measures, including a ban on gatherings in public places, mandatory mask wearing and a curfew, until cases reached very low levels again in February 2021. At this time a seroprevalence survey of 463 individuals on the main islands of Tahiti and Moorea was conducted to estimate the level of immunity in the population, and seroprevalence was estimated as 19.0% (95% confidence interval 15.5–22.9%). The low level of cases—driven by imports—was then maintained until mid2021 when the rollout of the 1st and 2nd vaccine doses occurred. Following the introduction of the Delta variant in June 2021, the country experienced a second larger and sharper wave of cases, hospitalisations and deaths, with cases peaking in midAugust 2021. A lockdown was implemented with the establishment of a curfew and confinement at home on the main island groups (the Windward and Leeward Islands) in August 2021, and cases declined quickly back to low levels in November 2021. A second seroprevalence survey of 673 individuals on Tahiti was performed in November and December 2021, in which seroprevalence from natural infection was estimated as 57.7% (95% confidence interval 53.8–61.4%)^{16}. The arrival of the Omicron BA.1/BA.2 variants in late December 2021 led to a relatively large third wave of cases, but fewer hospitalisations and deaths than in the previous waves, which coincided with the rollout of first booster doses. During the first trimester of 2022, incoming travellers were screened for infection at the border using PCR (polymerase chain reaction)/antigen tests. The third wave had largely subsided by April 2022. French Polynesia experienced a fourth wave of cases mainly caused by the Omicron BA.5 and BA.4 variants between June and September 2022^{16} and a fifth wave mainly caused by the BQ.1.1 Omicron subvariant in November and December 2022. During the third, fourth and fifth waves, no strong NPIs (curfews or case isolation) were implemented.
To understand how immunity and control measures shaped observed dynamics in French Polynesia, we fit an agestructured multistrain COVID19 transmission model to reported case, hospitalisation and death data up to May 2022, as well as data from the two aforementioned seroprevalence surveys. We then use the fitted model to estimate the impact of NPIs and vaccination on numbers of COVID19 cases, hospitalisations and deaths, and to estimate the immune status of the French Polynesian population.
Results
Model fit
The fit of the model to the overall numbers of confirmed cases, hospitalisations, and hospital deaths between July 2020 and May 2022 is shown in Fig. 2, and the fit of the model to the agestratified numbers of cases, hospitalisations and deaths is shown in Figures S5 and S4, and to the agestratified data from the seroprevalence surveys in Figure S6. The model reproduces the overall patterns in the data well, but underestimates hospital deaths among 60+ yearolds in the second wave. The estimated number of symptomatic cases over time corresponds closely to the numbers of confirmed cases during the three waves (Fig. 2), with an estimated reporting rate of 0.47 (95% credible interval (CI) 0.46–0.49), i.e. 47% of symptomatic cases having been reported.
Impact of NPIs
We estimate the counterfactual impact that the lockdowns during the first two COVID19 waves had on the numbers of symptomatic cases, hospitalisations and deaths in each wave and overall by simulating the model without the estimated reductions in the transmission rate corresponding to the lockdown periods in the first and second waves (Figure S10). We run 500 simulations with parameter values drawn from the posterior distribution of the parameters from the model fitting and compare the numbers of symptomatic cases, hospitalisations and hospital deaths to those in simulations with the estimated reductions in the transmission rate with lockdowns, to account for uncertainty in the estimated parameter values. This gives the results shown in Fig. 3 and Table S8.
The estimated overall numbers of symptomatic cases, hospitalisations and hospital deaths from fitting the model to the observed data up to May 2022 are 145,000 (95% CI 143,000–147,000), 2940 (95% CI 2810–3080) and 549 (95% CI 499–600) respectively. We estimated that removing the lockdowns in both the first and second waves would have had a nonlinear effect on dynamics, and—assuming everything else had remained the same—would have led to fewer hospitalisations and hospital deaths over the study period from July 2020 to May 2022 (45 (95% CI 42–134) and 193 (95% CI 158–231) fewer, respectively) but a slightly higher number of symptomatic cases (1800 (95% CI 1300–2200) more). This scenario assumes that patients hospitalised during the first wave would have been managed the same (in terms of treatment and intensive care unit (ICU) admission) had the number of hospitalisations in the first wave been nearly 75% higher. The nonlinear effect on overall incidence is due to the first wave of infections being much larger (with 27,600 (95% CI 25,700–29,900) more symptomatic cases, 860 (95% CI 760–990) more hospitalisations, and 105 (95% CI 85–129) more deaths), resulting in greater build up of immunity in the population prior to the introduction of the more severe Delta variant, and therefore a much smaller second wave of cases, hospitalisations and deaths (with 30,300 (95% CI 27,800–33,900) fewer cases, 940 (860–1040) fewer hospitalisations, and 299 (95% CI 263–338) fewer deaths). The impact on the third wave would have been relatively limited due to the effects on the first two waves approximately cancelling each other out in terms of cumulative infections, and the immune escape properties of the Omicron BA.1/BA.2 variants reducing the influence of immunity from previous infection. Overall hospitalisations and deaths would have decreased, despite the increase in overall cases, as the reduction in cases in the second wave would have been slightly greater than the increase in cases in the first wave and there are more hospitalisations and deaths per case in the second wave than the first due to the greater severity of the Delta variant.
We also considered the counterfactual impact that changing the timings of the lockdowns during the first two waves would have had on incidence in each wave and overall (Supplementary Information §2.3). We estimated that starting the first and second lockdowns 2 weeks earlier or later would have had relatively little impact on overall numbers of cases, hospitalisations and deaths due to a similar nonlinear cancellation effect between infections in the first and second waves as for removing the lockdowns (Figure S7).
Impact of vaccination
The counterfactual impact of vaccination on numbers of hospitalisations and deaths during each wave and overall was estimated by simulating the fitted model without any vaccination, and comparing the numbers of hospitalisations and deaths to those in simulations with the actual vaccination rollout (Fig. 3 and Table S8). The vaccination programme is estimated to have averted 77,500 (95% CI 77,20077,800) symptomatic cases, 2890 (95% CI 2730–3070) hospitalisations and 989 (95% CI 885–1088) hospital deaths overall, with nearly all of these being averted during the second and third waves, since vaccination did not start until midJanuary 2021 when the first wave had largely subsided. We also conducted a sensitivity analysis to determine the sensitivity of these estimates to uncertainty in the rates of waning of natural immunity and booster protection, crossimmunity to infection with Delta and Omicron from previous infection, and vaccine effectiveness (see Supplementary Information §§1.2 and 2.4 for details). The number of cases averted remains relatively constant across the range of parameter values considered, but the numbers of hospitalisations and deaths averted vary more significantly (from 2520 (95% CI 2380–2690) and 794 (95% CI 724–875) respectively under pessimistic assumptions about the parameters to 3630 (95% CI 3420–3860) and 1197 (95% CI 1080–1330) under optimistic assumptions).
Under our base case (‘central’) assumptions about the rate at which booster protection wanes, rate of waning of natural immunity, crossimmunity, and vaccine effectiveness, the booster campaign is estimated to have had a relatively small impact on the overall numbers of cases, hospitalisations and deaths, reducing them by 6800 (95% CI 6800–6900), 57 (95% CI 54–60) and 2 (95% CI 1–4) respectively (Fig. 3 and Table S8). However, these estimates are sensitive to uncertainty in these parameters. For pessimistic assumptions about the parameter values, the estimated reductions in cases, hospitalisations and deaths are 3300 (95% CI 3200–3300), 3 (95% CI 3–3), and 0 (95% CI 0–0) respectively, while for optimistic assumptions (including a less conservative assumption about the rate at which individuals lose all protection from boosters) they are 14,300 (95% CI 14,200–14,300), 163 (95% CI 153–173), and 9 (95% CI 6–15) respectively (see Supplementary Information §2.4 for further details).
Immune status of the population
The breakdown of the inferred immune status of the population over time and by age is shown in Fig. 4. The three waves of cases are visible where the proportion recovered from infection increases sharply in October 2020, August 2021, and February 2022. Based on the model, most infections in the Delta wave were among unvaccinated individuals without prior infection, while in the Omicron BA.1/BA.2 wave just under a half were among individuals with 2nd dose protection or waned 2nd dose protection, either with or without immunity from previous infection. The model also suggests that in May 2022 a very high proportion (78.9% (95% CI 78.5–79.5%)) of the population possessed either natural or hybrid (natural + vaccineinduced) immunity, and only 5.6% the population were fully susceptible. Table 1 shows the full breakdown of the estimated immune status of the population in May 2022. As expected, given prioritisation of older individuals in the vaccine and booster rollouts, the proportion of the population that had only natural immunity in May 2022 decreased with increasing age, from over 90% among 09yearolds to approximately 16% among 70+ yearolds, while the proportion with hybrid immunity from infection and a booster dose increased with age from 0% among 09 yearolds to over 20% among 40+ yearolds.
Discussion
Many existing modelling frameworks for estimating COVID19 vaccination and NPI impact^{1,2,3,4,10} do not explicitly account for the complicated COVID19 immune landscape that now exists, with different levels of protection against different outcomes from different variants due to varying vaccination and infection histories and variant properties (transmissibility, immune escape, and severity). Those that do^{11,13,14,15,17} tend to be country specific; reliant on detailed infection prevalence, hospitalisation and/or mobility data; and not readily transferable to estimate vaccination and NPI impact in settings without such data. We have developed an agestructured multistrain SARSCoV2 transmission model that addresses this gap, and used it to estimate the impact of vaccination and nonpharmaceutical interventions on incidence of cases and severe outcomes in the first three waves of COVID19 in French Polynesia. While our approach is similar to the ‘stacked’ SIRtype multistrain transmission model of different levels of immunity from vaccination and infection developed by LaJoie et al.^{12}, we also stratify by age and thus are able to model agedependent mixing and infer variation in immunity over time by age group, and we use a more robust and flexible framework for performing parameter inference across multiple data sources rather than just reported cases. We have estimated impact through comparison with counterfactual scenarios, including no lockdowns, earlier/later introduction of lockdowns, no vaccination and no boosters. Our results suggest that the first two vaccine doses had a large impact on incidence in the Delta and Omicron BA.1/BA.2 waves, averting over 75,000 symptomatic cases, nearly 2900 hospitalisations and nearly 1000 deaths up to May 2022 compared to a counterfactual scenario of no vaccination.
Unlike many other Pacific Island Countries (PICs), which succeeded in preventing or delaying community transmission of SARSCoV2 till 2021 or till vaccine rollouts had started, French Polynesia experienced largescale community transmission from relatively early in the pandemic in 2020^{18}. Despite successful control of the wildtype virus through various NPIs, and the vaccine rollout occurring in mid2021, the country suffered a very large wave of Delta infections and deaths in July–October 2021, with the highest COVID19 death rate of any of the PICs. After the Delta wave, however, the level of immunity in the population had risen sufficiently to limit the burden of subsequent outbreaks in terms of hospitalisations and deaths, and French Polynesia experienced lower death rates than other PICs^{19}. Nevertheless, the overall COVID19 death rate in French Polynesia was still much higher than that of other PICs (231 deaths per 100,000 people vs the next highest of 110 for New Caledonia^{20}). By providing a framework to quantify the impact of different interventions and evolution of population immunity in an island setting where local transmission becomes established, studies such as ours can offer valuable insight into intervention effectiveness and support planning of responses to future respiratory disease epidemics in such settings.
Although there have been global modelling analyses of vaccination impact at a country level^{1,10}, only one study has estimated vaccination impact in French Polynesia and our study is the first to estimate the impact in terms of different outcomes (cases, hospitalisations and deaths). A key difference between our study and these studies is that we fit our model to multiple direct data streams (hospital deaths, hospitalisations, reported cases and seroprevalence) rather than just estimates of excess deaths^{21,22}, which may make our estimates more robust. It does, however, lead to a much lower estimate of deaths averted through vaccination—Watson et al.^{1} estimated that 2120 (95% CI 1740–2570) deaths had been averted up to 8th December 2021—since the number of hospital deaths with recorded date of death (552 between July 2020 and May 2022) is much lower than the estimated number of excess deaths (920 (95% CI 790–1200) between December 2020 and December 2021^{22}). We chose to fit to hospital deaths rather than all deaths (hospital deaths + community deaths) since they are less sensitive to context bias as a data stream. We model variation in quality of patient care during the different waves by fitting a timedependent risk of death given hospitalisation. However, we do not account for changes in the risk of death in the community over the different waves, which may have been appreciable as there was considerable fear and distrust of hospitalisation during the Delta wave when hospitals reached capacity, and less distrust in healthcare in the first wave and less fear of severe outcomes during the Omicron BA.1/BA.2 wave. Given extensive followup of hospitalised cases it is likely that underreporting of hospital deaths in French Polynesia was not as high as elsewhere^{23}.
We estimated that the booster campaign had less of an impact on the BA.1/BA.2 wave than the first two doses had on the Delta wave, in both absolute and proportional terms, despite similar numbers of infections in the BA.1/BA.2 wave as in the Delta wave. Although this may seem surprising, e.g. when compared with the estimated impact of the booster rollout in the UK^{14}, the estimated small effect size is influenced by a combination of factors. These include the already high level of natural/hybrid immunity in the population from previous infection and/or vaccination (Fig. 4), the relatively low booster coverage during the wave ( < 35% of the overall population and only > 50% in individuals ≥50 years) (Fig. 5), and the lower severity of the BA.1/BA.2 variants.
Our results suggest that—all other things being equal—changing lockdown dates during the first two waves by two weeks would have had limited impact on overall numbers of cases, hospitalisations and deaths. This is because starting the first lockdown either earlier or later would have led to more infections prior to the second wave and thus been compensated for by a smaller second wave due to greater population immunity, and moving the second lockdown earlier or later would have had only a small impact on incidence due to its relatively limited estimated effect on the transmission rate (Figure S10). In addition, incidence in the Omicron BA.1/BA.2 wave would have been largely unaffected due to the limited net effect of changes in the lockdowns on incidence in the first two waves.
We also estimated the composition of immunity from infection and vaccination in the population in May 2022. We estimated that 94% of the population had some form of immunity, predominantly either natural or hybrid immunity (as opposed to only vaccineinduced immunity). From this we would expect that infection incidence (and therefore hospitalisation and death incidence) after May 2022 would have remained low without the advent of new variants with high levels of immune escape against Omicron BA.1/BA.2, which has been the case, with the Omicron BA.5/BA.4 and BQ.1.1 waves being relatively small in terms of detected cases, hospitalisations and deaths^{24,25}. Given that the rollout of 2nd booster doses took place between April and August 2022 we would expect incidence of infections and severe outcomes to remain low for some time if no new immune escape variants are introduced.
There are some limitations to the analysis we have presented (see Supplementary Information for further discussion). Since there is no social contact data available for French Polynesia, we have to rely on contact data from the literature for the agedependent contact rates used in the model and we choose to use data from France^{26} (adjusted to account for French Polynesia’s different population age structure). This is based on French Polynesia, as a French territory, having a societal structure more similar to that of France than most other countries and control measures implemented in French Polynesia during the pandemic being based on those in France.
We make the simplifying assumptions that mixing depends only on age and is otherwise homogeneous for the whole French Polynesian population, despite the fact that the population is spread over many islands in five archipelagos covering 2000km of ocean^{16}, and that the seroprevalence estimates from the main islands of Tahiti and Moorea are representative of seroprevalence on all the islands. However, the majority ( ~ 75%) of the population resides on Tahiti and Moorea (and ~ 69% on Tahiti), and most inhabited islands had frequent air connections with Tahiti during the pandemic, except during lockdowns, so these assumptions are not unreasonable. The suspension of interisland flights during lockdowns clearly would have had an impact on mixing of the population, but this is to some extent captured in the fitted values of the transmission rate parameters during the first and second epidemic waves. Ideally we would model transmission on the different archipelagos with a metapopulation model with the impact of lockdowns on interisland movement informed by mobility data, or focus the analysis on the Windward Islands, which include Tahiti and Moorea, as these were most affected by the epidemic, but no mobility data is available for French Polynesia and we do not have sufficient geolocation detail for cases.
We assume that if lockdowns had been removed and hospitalisations had increased by nearly 75% during the first wave, hospital capacity would not have been exceeded and hospitalised patients would have received the same quality of care. Based on our estimates, the peak incidence of hospitalisations in the first wave would have been slightly lower than that that occurred during the second wave (Figure S9), when the number of general hospitalised and ICU COVID19 cases at Centre Hospitalier de la Polynésie française (CHPF), the main hospital in French Polynesia where most COVID19 patients were treated, peaked at 246 and 48 respectively. Since it was possible to make 248 general beds available for hospitalised cases at CHPF and ICU bed capacity there had already been upgraded to 36 in August 2020 (with the army placed on standby to set up 10 more ICU beds if required), it is therefore not unreasonable to assume hospitals would have remained within capacity. Nevertheless, the increased pressure on hospital resources might have led to lower quality of care for hospitalised patients and hence poorer outcomes. The estimated reduction in overall hospitalisations and hospital deaths from removing lockdowns should thus be interpreted with caution. Further, given the heterogeneity in type, compliance and duration of lockdowns between countries, and the relative uniqueness of French Polynesia in terms of remoteness and population size, this result is unlikely to be generalisable across countries.
We may underestimate the impact of the vaccination programme as we estimate cases, hospitalisations and deaths averted from reported hospital deaths, which are considerably lower than estimates of allcause mortality and excess mortality^{22,27}, and do not account for increased death rates in the community when hospitals reached capacity during the Delta wave. The third wave was caused by a mixture of the Omicron BA.1 and BA.2 sublineages and there is evidence that the BA.2 variant is more transmissible than the BA.1 variant^{28,29,30,31} and can reinfect individuals previously infected with the BA.1 variant^{32}, but we do not distinguish between these subvariants when modelling the third wave, which may lead to some underestimation of the impact of the booster programme.
We assume initial vaccine effectiveness and rates of waning of immunity are the same for all ages. However, there is evidence that vaccine effectiveness against symptomatic infection is lower and wanes more quickly in older age groups (≥65 years) than in younger age groups, at least for the Delta variant^{33}, and of potential age differences in booster effectiveness against Omicron variants^{34}, which may introduce some bias into our estimates of vaccination and booster impact. We also assume waning rates are the same for different variants and infection outcomes, but data suggests waning is faster against the Omicron BA.1 variant than the Delta variant^{35} and that protection against severe outcomes wanes more slowly than that against infection^{33}. Further work is needed to determine the extent to which these differences affect vaccine impact estimates.
Nevertheless, the framework we have developed provides a means of estimating the impact of vaccination and NPIs on COVID19 incidence while accounting for the complex immune landscape that has developed over the course of the pandemic from myriad different infection and vaccination histories at an individual level. In particular, several different data streams can be incorporated in the inference to provide more robust estimates of key unobserved processes. The framework is sufficiently flexible that it could be used to model COVID19 dynamics and estimate vaccination and NPI impact for other countries, including those with less data available. As a minimum, COVID19/excess death or hospitalisation data, case or seroprevalence data, vaccination and booster coverage data, dates of major changes in restrictions, and broad date ranges for the introduction of different variants would be required to fit the model, but in such a scenario all parameters except for the timevarying transmission rate, variant introduction dates, and symptomatic case reporting rate would need to be fixed to avoid parameter identifiability issues. In settings without seroprevalence data, case or regular testing data would be required to infer infection levels, or a fixed infectionhospitalisation/infectionfatality rate would have to be assumed. For countries with no variant sequencing data, date ranges for the introduction of different variants would have to be based on variant introduction dates for countries in that region or estimates of global emergence dates of new variants. Caution would be required to only apply the model to countries for which COVID19 hospitalisation/death or excess death data was deemed to be reasonably complete to avoid biased estimates of impact. Nonetheless, the framework could still provide valuable insight in settings with different vaccine and booster availability and NPI levels.
Methods
Data
Multiple data streams are used in the fitting of the model. Anonymised line lists of confirmed cases and hospitalisations compiled by the Ministry of Health of French Polynesia with testing date and admission date, and date of death for those that died, and 10year age group were aggregated into agestratified time series of daily cases, hospitalisations and hospital deaths. Only 493 out of 74986 confirmed cases (0.66%) were missing their age group, so these cases were treated as unreported cases, since underreporting of cases is accounted for in the model fitting (see Confirmed cases). As testing dates were missing for a large number of cases early in the first wave and cases were numbered approximately sequentially by testing date in the surveillance system, we imputed the missing dates as being between the testing dates of the nearest numbered cases with recorded testing dates. Data from two serosurveys, the first conducted by Cellule Episurveillance COVID and the Health Department of French Polynesia in February 2021, the second by Institut Louis Malardé in NovemberDecember 2021, was also used. This data is described in detail in^{16} and summarised in Table 2. Briefly, in February 2021, 463 unvaccinated adults aged 18–88 years on the islands of Tahiti and Moorea were randomly selected and tested for antiSARSCoV2 immunoglobulin type G (IgG) antibodies with the Siemens SARSCoV2 IgG (sCOVG) test. Overall, 88 (19.0%, 95% confidence interval 15.5–22.9%) individuals had detectable IgG antibodies. In NovemberDecember 2021, 673 randomly selected individuals aged ≥18 years on Tahiti were tested for antibodies against the SARSCoV2 N antigen (i.e. for evidence of past infection) with the Roche Elecsys antiSARSCoV2 assay, and 388 (57.7%, 95% confidence interval 53.8–61.4%) were positive. For the purposes of the modelling, we assume that the seroprevalence in the 20–29 years age group in the model is the same as that in the 18–29 years age group in the data. We use data on the population of French Polynesia by year of age in 2020 from the UN World Population Prospects^{36} (for which the total population was estimated to be 280,904) aggregated into 10year age groups for the age group populations in the model.
We use data on daily numbers of first, second and booster doses administered by age group (12–17, 18–29, 30–39, 40–49, 50–59, 60–69, 70+ years) collected by the Ministry of Health of French Polynesia to determine the numbers of individuals moving between the different vaccination strata in the model. Since the model is stratified into 10year age groups, we split the doses in the 1829 years age group in the data into the 10–19 and 20–29 age groups in the model according to population proportion (the proportions of 18–29 yearolds that are 18–19 and 20–29 years old). Upon division by the population in each age group, this gives the vaccination coverage by age and dose shown in Fig. 5.
Model
We developed a deterministic agestructured multistrain SEIRtype model of COVID19 transmission with stratification by vaccination status (Fig. 6). The model is stratified into 8 age groups (0–9, 10–19, 20–29, 30–39, 40–49, 50–59, 60–69, 70+ years), and by 5 vaccination levels representing no vaccination, protection from 1 dose, protection from 2 doses, waned protection from the 2nd dose and protection from a booster dose.
In the model, susceptible individuals (S) enter an exposed state (E) upon infection with a particular variant, from where an agedependent proportion develop symptoms (I_{C}) after a presymptomatic infection period (I_{P}), while the rest progress to asymptomatic infection (I_{A}). Presymptomatic, symptomatic and asymptomatic individuals are all assumed to be infectious, though asymptomatic individuals less so. Most symptomatic individuals and all asymptomatic individuals recover naturally (R), but some symptomatic individuals develop severe disease (G or H) that can lead to hospitalisation. A proportion of these individuals die from the disease (D) while in hospital or at home, while the remainder recover following treatment. Infected individuals are assumed to cease being infectious upon recovery. Once recovered from infection individuals have immunity against reinfection with the same variant that wanes over time, but only partial immunity against infection with a different variant.
Individuals in the susceptible, exposed, presymptomatic, asymptomatic and recovered states can be vaccinated, providing them with increased levels of protection against infection, hospitalisation and death. The different vaccination strata and their associated levels of protection are shown in Tables 3 and S1.
The model is further stratified to account for different histories of infection with two different variants: the latest variant to have emerged and the previously dominant variant. Individuals can have been infected by only the previous variant, only the current variant, or the previous variant and the current variant (in either order), giving 4 possible infection histories. Once a new variant emerges the information stored in the strata for the two variants is combined into the stratum for the first variant, and the information for the new variant added to the second stratum. This simplification of the multistrain dynamics is to prevent the dimensionality of the model exploding as the number of variants and possible infection and vaccination histories increases, which would make the model prohibitively slow to fit.
Here we ignore transmission of the Alpha variant, as although Alpha was detected among travellers and a small number of local cases in early 2021 through variant screening (Table S5), transmission of Alpha remained localised and never became fully established. We therefore only explicitly model the introduction and spread of the Delta and Omicron variants. We also do not distinguish between the Omicron BA.1 and BA.2 sublineages, and model the introduction of Omicron and its sublineages as a single new variant.
Naturallyacquired immunity is assumed to wane slowly—individuals who have been infected are assumed to return to being susceptible to infection with the same variant after an exponentially distributed period with a mean of 6 years^{14}. Immunity between SARSCoV2 variants is assumed to be asymmetric, with infection with later variants conferring stronger protection against infection with earlier variants than vice versa (see Force of infection and Table S2 for details). Changes in populationlevel serological status with seroconversion and seroreversion following infection are modelled with a ‘parallel flow’.
Demographic processes such as birth, natural death and migration are ignored in the model (i.e. the population is assumed to remain constant in the absence of deaths from COVID19). These processes occur at a much slower rate than transmission processes and are therefore assumed to have a negligible impact on the transmission dynamics over the timescales modelled.
Vaccination
Details of the five vaccination strata in the model are shown in Table 3. Unvaccinated individuals move out of the first vaccination stratum (V_{1}) into the first vaccinated stratum (V_{2}) at a rate determined by the rollout of the 1st vaccine dose, with an assumed delay of 28 days for immunity from the 1st dose to develop. Likewise, movement into the 2nd dose vaccination stratum (V_{3}) is determined by the rollout of the 2nd dose, with a delay of 14 days for the dose to take full effect. Only nonsymptomatic and nonhospitalised individuals, i.e. individuals in the S, E, I_{A}, I_{P} and R states in the model, can be vaccinated. Protection from the 2nd vaccine dose is assumed to wane over an exponentially distributed period, with a mean duration of 6 months. Individuals whose protection wanes pass into a ‘waned’ vaccine stratum (V_{4}), with lower levels of protection. They either remain in this stratum or receive a booster vaccination and move into a ‘boosted’ vaccination stratum (V_{5}), with higher levels of protection. Individuals can only move between consecutive vaccine strata except when they are in the 2nd dose stratum (V_{3}), where they can receive their booster dose and move to the boosted stratum (V_{5}) before their protection has waned, skipping the waned 2nd dose stratum (V_{4}). Individuals in the 2nd dose and waned 2nd dose strata (V_{3} and V_{4}) are taken to be equally likely to receive a booster dose. Protection from the booster dose is assumed to wane slowly such that individuals eventually return to being fully susceptible.
In common with other transmission modelling studies^{14,37}, we model vaccine protection against five different outcomes:

1.
infection, with effectiveness e^{inf}

2.
symptomatic infection given infection, e^{sympt∣inf}

3.
severe disease given symptomatic infection, e^{SD∣sympt}

4.
death given severe disease, e^{death∣SD}

5.
onward transmission if infected, e^{ins}
Vaccine effectiveness against symptomatic infection, severe disease and death are conditional on previous outcomes and depend on overall vaccine effectiveness against infection, symptomatic infection, severe disease and death (e^{inf}, e^{sympt}, e^{SD} and e^{death}) as follows:
Estimates for e^{inf}, e^{sympt}, e^{SD} and e^{death} for different vaccination strata and variants taken from^{14} are provided in Table S1 (see^{14} for information on sources of these estimates), but we also vary these parameters in the sensitivity analysis (Table S7). As over 90% of the doses given in French Polynesia were of the PfizerBioNTech vaccine, we use effectiveness values for that vaccine for all doses given. We also make the simplifying assumption that vaccine effectiveness is the same across all age groups.
Waning immunity
The model accounts for waning of natural and vaccineinduced immunity as described in the previous sections. We assume that the waning rates of natural and vaccineinduced immunity are the same for all age groups and virus variants. When immunity from previous infection or booster vaccination wanes, individuals return to being fully susceptible, so immunity against different outcomes (infection, symptomatic infection, hospitalisation and death) is assumed to wane at the same rate. We note that this is a strong simplifying assumption as there is evidence to suggest that immunity against infection wanes more quickly than immunity against severe outcomes^{33}, and that immunity against infection and severe outcomes wanes faster for Omicron BA.1 than Delta^{35}. As there is no data that provides a direct measure of the rate of loss of all protection from vaccination, we use the rate of waning of protection against hospitalisation as a proxy for the rate at which individuals return to being fully susceptible following booster vaccination. Whilst a reasonable assumption, this may still be overly conservative, so we also conduct a sensitivity analysis with different values of the waning rate from the literature. Although we do not model variantspecific vaccine waning rates, we use estimates of the change in protection against hospitalisation over time following booster administration for Omicron BA.1^{35} for the booster waning rate in the analysis in the main text, since the booster campaign in French Polynesia coincided with the Omicron BA.1/BA.2 wave, and compare this to a lower waning rate assumed by Barnard et al.^{14} in their model with a similar structure (Table S6). See Supplementary Information §2.4 for results of the sensitivity analysis.
Parallel flow for serological status
So that we can fit to the data from the serosurveys we include a ‘parallel flow’ of compartments for serological status in addition to those for infection status and clinical progression (Fig. 6). We fit to the data on prevalence of seropositivity against the N antigen on the SARSCoV2 virus according to the Roche Elecsys antiSARSCoV2 assay, as this tests only for positivity resulting from infection. After a preconversion period (T_{pre}), individuals either seroconvert (T_{P}) with probability p_{P} or not (T_{N}). Those that do seroconvert eventually serorevert (to T_{N}) after an exponentially distributed time with mean 6.6 years^{38}.
Behaviour
The impact of lockdowns on transmission is described in the model through a timevarying transmission rate, with changepoints corresponding to major changes in restrictions in French Polynesia (Fig. 1). We make the simplifying assumption that adherence to these restrictions is the same across all age groups and vaccination strata, and regardless of infection history. Variation in careseeking behaviour with age is modelled through an agedependent probability of hospitalisation given symptomatic infection, where the relative risks of hospitalisation between age groups are based on data from France^{39} and we estimate the maximum probability of hospitalisation across all age groups to account for differences in careseeking and access to care between France and French Polynesia. The probability of hospitalisation varies across vaccination strata in the model due to the different levels of protection against severe disease with different levels of vaccination described above (see Vaccination), but we do not model any variation in careseeking behaviour with vaccination status beyond this. Vaccine and booster uptake by age and vaccination status in the model are determined by the data on the numbers of each dose received by age over time (Fig. 5), assuming that all individuals within each ageandvaccination stratum have an equal chance of being vaccinated (i.e. previous infection does not affect vaccine/boosterseeking) and can only receive successive vaccine doses (e.g. must have had the 2nd dose to receive a booster).
Model equations
Force of infection
The relative susceptibility to infection with variant j of a susceptible individual in age group i in vaccination stratum k is given by:
where \({e}_{ijk}^{inf}\) is the vaccine effectiveness against infection with variant j in vaccination stratum k ∈ {1, 2, 3, 4, 5} (see Table S1), and χ_{ij1} = 1, ∀ i, j (i.e. there is no protection in unvaccinated individuals). The index j denotes individuals’ infection histories, covering primary infection with one variant (j ∈ {1, 2}) and superinfection (infection with one variant followed by infection with another) (j ∈ {3, 4}) as follows:
We describe two periods of the epidemic with this setup, the first running up to 21st November 2021 and encompassing the wildtype and Delta waves, in which:
and the second, starting on 21st November 2021 shortly before the emergence of Omicron BA.1 and ending on 6th May 2022 and covering the Omicron BA.1/BA.2 wave, in which:
The relative infectiousness of an individual in age group i and vaccination stratum k infected with variant j compared with an unvaccinated individual infected with the wildtype virus is given by:
where ξ_{i,Wildtype,1} = 1, ∀ i, and σ_{j} is the relative transmissibility of variant j compared to the wildtype variant (and we assume σ_{1} = σ_{4} and σ_{2} = σ_{3}).
The infectiousnessweighted number of infectious individuals for variant j in age group i and vaccination stratum k on day t is given by
where θ_{A} is the relative infectiousness of an asymptomatic infected individual compared to a symptomatic individual in the same vaccination stratum infected with the same variant.
With these definitions, the force of infection on a susceptible individual in age group i and vaccination stratum k from variant j on day t is:
where \({m}_{i{i}^{{\prime} }}(t)=\beta (t){c}_{i{i}^{{\prime} }}\) is the timevarying persontoperson transmission rate from age group \({i}^{{\prime} }\) to age group i, composed of the timevarying transmission rate β(t) and the persontoperson contact matrix \({c}_{i{i}^{{\prime} }}\) between age groups. The contact matrix \({c}_{i{i}^{{\prime} }}\) was parameterised using estimates of contact rates \({d}_{l{l}^{{\prime} }}^{*}\) between 5year age groups for France from^{26}, where \({d}_{l{l}^{{\prime} }}^{*}\) is the mean number of contacts in age group \({l}^{{\prime} }\) an individual in age group l makes per day. Following^{26} and^{40}, these were corrected by the relative population densities of each age group of French Polynesia and France to account for differences in demography between the two countries:
where \({n}_{l}^{*}\) and n^{*} are the population of age group l and the total population for France and n_{l} and n are those for French Polynesia. They were then averaged over 10year age groups in the model and divided by the population in each age group to yield the persontoperson contact matrix \({c}_{i{i}^{{\prime} }}\):
Social contact data for France was used due to the absence of estimates for French Polynesia and the fact that French Polynesia is a French territory.
The total force of infection on a susceptible individual in age group i and vaccination stratum k is then the sum of the variantspecific forces of infection:
Crossimmunity between variants is modelled via partial immunity to infection with the other variant following infection with one variant, such that the force of infection on an individual recovered from infection with variant j in age group i and vaccination stratum k from the other variant is:
where η_{j} is the crossimmunity from infection with other variants against infection with variant j.
The timevarying transmission rate, β(t), represents temporal changes in the overall contact rates in the population due to changes in restrictions and behaviour. We assume that β(t) is piecewise linear with 5 changepoints corresponding to changes in alert levels and the imposition of islandwide restrictions such as curfews (Table 4 and Figure S10):
Seeding of variants
We seed each variant j at a daily rate of ω_{j}, over a period of ν_{j} days from time t_{j}. All seeding infections are from the S to E compartment in the 3039yearold age group and unvaccinated class.
For all variants, we seed at a rate of 10 infections per time step over one time step, i.e. ω_{j} = 40 day^{−1} and ν_{j} = 0.25 days for j ∈ {Wildtype, Delta, Omicron}. We fit the seeding dates t_{0} (which corresponds to the start date of the wildtype outbreak in 2020), t_{Delta}, and t_{Omicron} (see Table 4).
The daily seeding rate of variant j in age group i in vaccine stratum k, δ_{ijk}(t), is therefore:
Natural history parameters
Movement between model compartments is determined by parameters p_{X}, defining the probability of progressing to compartment X, and rate parameters γ_{X}, defining the time individuals stay in compartment X, which can vary with age group (i), variant (j) and vaccination status (k). Values of these parameters are given in Tables S2 and S3 and information on how they are calculated is given below.
There is now strong evidence that successive SARSCoV2 variants have had progressively shorter serial intervals^{41,42,43,44}. We therefore model this by reducing the mean durations of latent infection, asymptomatic infection, presymptomatic infection, and symptomatic infection (E, I_{P}, I_{C}, and I_{A}) of successive variants in line with percentage reductions in their serial intervals relative to the wildtype virus reported in the literature^{42} (Table S4).
The probability of developing symptoms given infection is
where \({{p}_{C}}_{i}\) is the agedependent probability of developing symptoms given infection for unvaccinated individuals.
The probability that an individual develops severe disease requiring hospitalisation given that they are symptomatically infected is
where \({{p}_{H}}_{i}\) is the agedependent probability of developing severe disease given symptomatic infection for unvaccinated individuals, \({{\pi }_{H}}_{j}\) is the variantdependent relative risk of severe disease, and \({{\eta }_{H}}_{j}\) is the crossimmunity from infection with other variants against hospitalisation with variant j. \({{p}_{H}}_{i}\) is defined as:
where \({{p}_{H}}_{max}\) is the maximum probability of hospitalisation across all age groups and \({{\psi }_{H}}_{i}\) is the agedependent relative risk of severe disease, such that \({{\psi }_{H}}_{i}=1\) for the group corresponding to the maximum. \({{\pi }_{H}}_{j}\) is parameterised as:
where π_{Delta/Wildtype} and π_{Omicron/Delta} are the relative risks of severe disease given infection for Delta compared to wildtype and Omicron compared to Delta, and we fit π_{Delta/Wildtype}.
The probability that a hospitalised individual will die is
where h(t) is the maximum probability of death given hospitalisation for unvaccinated individuals, \({{\psi }_{D}}_{i}\) is the agedependent relative risk of death for unvaccinated individuals (such that \({{\psi }_{D}}_{i}=1\) for the age group for which the probability of death is h(t)), and \({{\eta }_{D}}_{j}\) is the crossimmunity from infection with other variants against death from variant j. To allow for variation in the risk of death with changing quality of care and demand for hospital beds, we fit a piecewise linear form for h(t) with the following changepoints:
such that the probability of death given hospitalisation is constant during the first wave, changes with changing pressure on hospital beds in the Delta wave, and is constant after the Delta wave.
The probability that an individual dies in the community given that they have severe disease is
where p_{G} is the probability of death in the community given severe disease for unvaccinated individuals.
Compartmental model equations
The compartmental model is a deterministic approximation to a stochastic agestructured multistrain SEIRtype transmission model in which draws from random variables are replaced by their expected values (using the deterministic mode of the dust R package). This may have lower accuracy than an ODE formulation and solver, but we expect that the error is minimal based on the model fits. The model compartments are defined in Fig. 6. For completeness we provide the equations for the stochastic model here, and note that the stochastic version of the model can be fitted and run by setting the option deterministic < F in the code.
The compartments in the model are updated according to the following equations:
where \({{n}_{XY}}_{ijk}\) is the number of individuals in age group i and vaccination stratum k infected with variant j (if they are in an infection state) moving from state X to state Y at time t (and \({{n}_{XY}}_{ij0}={{n}_{XY}}_{ij5}\), and we have dropped the dependence on t from the notation for convenience); dt is the model time step, chosen to be 0.25 days; and \({{\mathbb{1}}}_{x}\) is the indicator function for condition x.
The flows between states are determined as follows:
where \({{n}_{XX}}_{ijk}\) is the number of individuals in age group i and vaccination stratum k infected with variant j (if they are in an infection state) who do not move from state X at time t. The fitted seeding dates t_{0}, t_{Delta}, and t_{Omicron} have continuous support, and the seeding process is handled within the discretisation to four update steps per day such that:
where
Model likelihood
The model likelihood is composed of the likelihoods for the different data streams that the model is fitted to, namely the agestratified time series of hospitalisations, hospital deaths and confirmed cases, and the agestratified seroprevalence data, as detailed below.
In the following, Y ~ Bin(n, p) denotes that Y follows a binomial distribution with n trials and success probability p, such that
and the mean and variance of Y are np and np(1 − p) respectively. Y ~ NegBin(m, κ) denotes that Y follows a negative binomial distribution with mean m and shape parameter κ, such that
where Γ(k) is the gamma function, and the variance of Y is m + m^{2}/κ.
Hospitalisations
We assume that the observed number of hospitalisations in each age group l at time t, Y_{hosp,l}(t), is distributed according to a negative binomial distribution
with mean
where the shape parameter κ_{hosp} determines the overdispersion in the observation process and thus accounts for noise in the underlying data, and we aggregate the four youngest age groups together due to low numbers of hospitalisations in these age groups such that l ∈ {0–39, 40–49, 50–59, 60–69, 70+} years. We fit the overdispersion parameter α_{hosp} = 1/κ_{hosp}. The contribution of the agestratified hospitalisation data to the likelihood is therefore:
Hospital deaths
The observed number of hospital deaths in each age group l ∈ {0–39, 40–49, 50–59, 60–69, 70+} years at time t is assumed to be distributed according to a negative binomial distribution:
with mean
and shape parameter κ_{death}. We fit the overdispersion parameter α_{death} = 1/κ_{death}. The contribution of the agestratified hospital death data to the likelihood is thus:
Confirmed cases
The daily number of confirmed cases in each age group i ∈ {0–9, 10–19, 20–29, 30–39, 40–49, 50–59, 60–69, 70+} years is assumed to arise as the noisy underreported observation of a hidden underlying Markov process
such that it follows a negative binomial distribution
with constant reporting factor ϕ_{cases} and shape parameter κ_{cases} = 1/α_{cases}, where α_{cases} is an overdispersion parameter that we fit. The corresponding likelihood contribution is
Seroprevalence
To fit the model to the agestratified data from the two serosurveys, we first calculate the number of seropositive and seronegative individuals in each age group over 20 yearsofage in the model (i.e assume the true serological status of all individuals is known):
We then compare the observed number of seropositive individuals in each age group in the serosurvey, \({{Y}_{P}}_{i}(t)\), with the number expected from the model based on the sample size Y_{test,i}(t) and the sensitivity p_{sens} and specificity p_{spec} of the serological assay:
where
is the apparent prevalence. The likelihood contribution of the serosurvey data is:
Full likelihood
The full likelihood is the product of the likelihoods for the hospitalisation, death, case and serosurvey data:
Prior distributions for fitted parameters
The prior distributions chosen for the fitted parameters are shown in Table 4. We use relatively informative gamma distributions for the transmission rate parameters β_{i} ~ Gamma(k, θ) (i = 1, 2, 3, 4, 5):
where Γ( ⋅ ) is the Gamma function, with shape parameter k = 4 and scale parameter θ = 0.005 to ensure that the basic reproduction number for the wildtype variant is in a sensible range. Targeted sequencing of samples from local cases and travellers to screen for new variants was performed from late December 2020 in French Polynesia (Table S5). Whilst this data is biased and so cannot be used to fit the variant proportions in the model, it can be used to constrain the introduction dates of the different variants. We use continuous uniform prior distributions for the introduction dates of the different variants, with the upper bounds of the distributions for Delta and Omicron BA.1 chosen to match the earliest date each variant was detected amongst local cases (since the variant cannot have been introduced into local circulation later than it was first detected), and the lower bounds chosen as 40 days and 12 days earlier respectively based on the earliest date each variant was detected amongst travellers and the much higher growth rate of the Omicron BA.1 variant (Tables 4 and S5). For the wildtype variant, we assume a lower bound of 39 days prior to the first reported hospitalisation and an upper bound of 9 days prior. We treat the introduction dates as continuous variables, and distribute the initial number of infections of that variant in proportion to how far between time steps the introduction date is. This helps to avoid mixing issues in the MCMC caused by treating the introduction date as a discrete variable. For the maximum probability of severe disease across all age groups and the symptomatic case reporting rate, we use completely uninformative priors, \({{p}_{H}}_{max},{\phi }_{cases} \sim \,{{\mbox{Beta}}}\,(1,1)\), where the density for X ~ Beta(a, b) is:
MCMC algorithm
We use the accelerated shaping and scaling adaptive Markov Chain Monte Carlo (MCMC) algorithm of Spencer^{45} to infer the values of the fitted parameters \({{{{{{{\boldsymbol{\theta }}}}}}}}=({{{{{{{\boldsymbol{\beta }}}}}}}},\,{t}_{0},\,{t}_{Delta},\,{t}_{Omicron},\,{{p}_{H}}_{max},\,{p_D}_{max,\,1},\,{p_D}_{max,2},\,{p_D}_{max,3},{\pi_H}_{Delta/Wildtype},{\phi }_{cases},\,{\alpha }_{cases},\,{\alpha }_{hosp},\,{\alpha }_{death})\), where β = (β_{1}, β_{2}, β_{3}, β_{4}, β_{5}). The algorithm adaptively shapes and scales the proposal matrix to achieve more efficient mixing. We refer the reader to^{45} for full details. The algorithm proceeds by repeating the following steps:

1.
At the ith iteration, draw new values of the fitted parameters from a multivariate normal proposal distribution
$${{{{{{{{\boldsymbol{\theta }}}}}}}}}_{i} \sim N({{{{{{{{\boldsymbol{\theta }}}}}}}}}_{i1},\,2.3{8}^{2}{c}_{i1}^{2}{{{{{{{{\boldsymbol{\Sigma }}}}}}}}}_{i1}/{n}_{{{{{{{{\boldsymbol{\theta }}}}}}}}})$$(87)where Σ_{i−1} is the running estimate of the covariance matrix of the posterior distribution, n_{θ} is the dimension of the posterior density, and c_{i−1} is a scaling parameter that is tuned to achieve a desired acceptance rate (see Step 4).

2.
Accept θ_{i} with probability:
$$\alpha ({{{{{{{{\boldsymbol{\theta }}}}}}}}}_{i},{{{{{{{{\boldsymbol{\theta }}}}}}}}}_{i1})=\min \left(1,\frac{L({{{{{{{{\boldsymbol{\theta }}}}}}}}}_{i})P({{{{{{{{\boldsymbol{\theta }}}}}}}}}_{i})}{L({{{{{{{{\boldsymbol{\theta }}}}}}}}}_{i1})P({{{{{{{{\boldsymbol{\theta }}}}}}}}}_{i1})}\right)$$(88)where P(θ) is the prior density of θ.

3.
Calculate the running mean and covariance as: if i = 1:
$${\overline{{{{{{{{\boldsymbol{\theta }}}}}}}}}}_{1}=\frac{1}{2}\mathop{\sum }\limits_{j=0}^{1}{{{{{{{{\boldsymbol{\theta }}}}}}}}}_{j}$$(89)$${{{{{{{{\boldsymbol{\Sigma }}}}}}}}}_{1}=\frac{1}{{i}_{0}+{n}_{{{{{{{{\boldsymbol{\theta }}}}}}}}}+3}\left(({i}_{0}+{n}_{{{{{{{{\boldsymbol{\theta }}}}}}}}}+1){{{{{{{{\boldsymbol{\Sigma }}}}}}}}}_{0}+\mathop{\sum }\limits_{j=0}^{1}{{{{{{{{\boldsymbol{\theta }}}}}}}}}_{j}{{{{{{{{\boldsymbol{\theta }}}}}}}}}_{j}^{T}2{\overline{{{{{{{{\boldsymbol{\theta }}}}}}}}}}_{1}{\overline{{{{{{{{\boldsymbol{\theta }}}}}}}}}}_{1}^{T}\right)$$(90)if f(i) = f(i − 1) + 1, where \(f(i)=\lfloor \frac{i}{2}\rfloor\):
$${\overline{{{{{{{{\boldsymbol{\theta }}}}}}}}}}_{i}={\overline{{{{{{{{\boldsymbol{\theta }}}}}}}}}}_{i1}+\frac{1}{if(i)+1}({{{{{{{{\boldsymbol{\theta }}}}}}}}}_{i}{{{{{{{{\boldsymbol{\theta }}}}}}}}}_{f(i)1})$$(91)$${{{{{{{{\boldsymbol{\Sigma }}}}}}}}}_{i}= {{{{{{{{\boldsymbol{\Sigma }}}}}}}}}_{i1}+\frac{1}{if(i)+{i}_{0}+{n}_{{{{{{{{\boldsymbol{\theta }}}}}}}}}+2}\left({{{{{{{{\boldsymbol{\theta }}}}}}}}}_{i}{{{{{{{{\boldsymbol{\theta }}}}}}}}}_{i}^{T}{{{{{{{{\boldsymbol{\theta }}}}}}}}}_{f(i)1}{{{{{{{{\boldsymbol{\theta }}}}}}}}}_{f(i)1}^{T}\right.\\ \left. \, (if(i)+1)({\overline{{{{{{{{\boldsymbol{\theta }}}}}}}}}}_{i1}{\overline{{{{{{{{\boldsymbol{\theta }}}}}}}}}}_{i1}^{T}{\overline{{{{{{{{\boldsymbol{\theta }}}}}}}}}}_{i}{\overline{{{{{{{{\boldsymbol{\theta }}}}}}}}}}_{i}^{T})\right)$$(92)such that the new observation replaces the oldest, and if f(i) = f(i − 1):
$${\overline{{{{{{{{\boldsymbol{\theta }}}}}}}}}}_{i}=\frac{1}{if(i)+1}((if(i)){\overline{{{{{{{{\boldsymbol{\theta }}}}}}}}}}_{i1}+{{{{{{{{\boldsymbol{\theta }}}}}}}}}_{i})$$(93)$${{{{{{{{\boldsymbol{\Sigma }}}}}}}}}_{i}= \frac{1}{if(i)+{i}_{0}+{n}_{{{{{{{{\boldsymbol{\theta }}}}}}}}}+2}\left((if(i)+{i}_{0}+{n}_{{{{{{{{\boldsymbol{\theta }}}}}}}}}+1){{{{{{{{\boldsymbol{\Sigma }}}}}}}}}_{i1}+{{{{{{{{\boldsymbol{\theta }}}}}}}}}_{i}{{{{{{{{\boldsymbol{\theta }}}}}}}}}_{i}^{T}\right.\\ \left. \, (if(i)){\overline{{{{{{{{\boldsymbol{\theta }}}}}}}}}}_{i1}{\overline{{{{{{{{\boldsymbol{\theta }}}}}}}}}}_{i1}^{T}(if(i)+1){\overline{{{{{{{{\boldsymbol{\theta }}}}}}}}}}_{i}{\overline{{{{{{{{\boldsymbol{\theta }}}}}}}}}}_{i}^{T}\left.\right)\right)$$(94)such that a new observation is included, where i_{0} is a constant that determines the rate at which the influence of Σ_{0} on Σ_{i} decreases.

4.
Update the covariance scaling parameter c_{i}:
$${c}_{i}=\max \left({c}_{min},\,{c}_{i1}\exp \left(\frac{\delta }{{i}_{start}+i}(\alpha ({{{{{{{{\boldsymbol{\theta }}}}}}}}}_{i},\,{{{{{{{{\boldsymbol{\theta }}}}}}}}}_{i1})a)\right)\right)$$(95)where
$$\delta=\left(1\frac{1}{{n}_{{{{{{{{\boldsymbol{\theta }}}}}}}}}}\right)\frac{\sqrt{2\pi }\exp ({A}^{2}/2)}{2A}+\frac{1}{{n}_{{{{{{{{\boldsymbol{\theta }}}}}}}}}a(1a)}$$(96)$$A={{{\Phi }}}^{1}(a/2)$$(97)$${i}_{start}=\frac{5}{a(1a)}$$(98)with Φ( ⋅ ) the cumulative distribution function of the standard normal distribution, c_{min} is a minimum value for the scaling parameter (to prevent the proposal matrix being shrunk too much, which can lead to very slow mixing), and a is the target acceptance rate.

5.
If \( \log ({c}_{i})\log ({c}_{start}) > \log (3)\), restart the tuning of the scaling parameter from its current value:
$${c}_{start}\mapsto {c}_{i}$$(99)$${i}_{start}\mapsto \frac{5}{a(1a)}i.$$(100)
We run 4 chains of the above algorithm from different initial parameter values with i_{0} = 100, c_{0} = c_{start} = 1, c_{min} = 1, and a target acceptance rate of a = 0.234 for 50,000 iterations. We thin the chains by a factor of 10, then discard the first 4000 iterations of each thinned chain as burnin, and combine the remaining iterations to form a sample of size 4000. We assess convergence of the MCMC chains by visual assessment of the trace plots, and calculating the maximum GelmanRubin statistic and minimum effective sample size across all the parameters for the thinned combined sample (4000 iterations).
Reporting summary
Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.
Data availability
All data used in the analysis are available online on Zenodo at https://doi.org/10.5281/zenodo.8320333 and on GitHub at https://github.com/LloydChapman/covid_multi_strain.
Code availability
The code used in this analysis was developed in R version 4.1.0 and uses the odin, odin.dust, dust and mcstate R packages for simulating discretetime stochastic processes^{46,47,48,49}. The model structure is similar to that of the COVID19 transmission model in the sircovid R package^{50}, and some of the code from this package is reused. All code used in the analysis is available online at https://github.com/LloydChapman/covid_multi_strain.
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Acknowledgements
We would like to thank Richard FitzJohn of the MRC Centre for Global Infectious Disease Analaysis at Imperial College for help with the R package mcstate and Rosanna Barnard of the London School of Hygiene and Tropical Medicine for helpful discussions. LACC, ESK and AJK were supported by funding from the National Institute for Health and Care Research (NIHR) Health Protection Research Unit in Modelling and Health Economics, a partnership between the UK Health Security Agency (UKHSA), Imperial College London, and the London School of Hygiene & Tropical Medicine (NIHR200908). LACC and AJK were also supported by funding from the Wellcome Trust (206250/Z/17/Z). The views expressed are those of the authors and not necessarily those of the UK Department of Health and Social Care, NIHR, UKHSA, or Wellcome Trust.
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L.A.C.C., V.M.C.L., and A.J.K. conceived the study. L.A.C.C. wrote the computer code, conducted the analysis and wrote the first draft of the manuscript. M.A., N.M., and H.P.M. processed the data. T.W.R., E.S.K., J.A.L. provided support with the development of the code. L.A.C.C., M.A., H.P.M., V.M.C.L., A.J.K. interpreted the results and critically revised the manuscript. All authors read and approved the final version of the manuscript.
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Secondary data analysis of routinely collected COVID19 data from French Polynesia was approved by the London School of Hygiene and Tropical Medicine Observational Research Ethics Committee (ref 28129).
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Chapman, L.A.C., Aubry, M., Maset, N. et al. Impact of vaccinations, boosters and lockdowns on COVID19 waves in French Polynesia. Nat Commun 14, 7330 (2023). https://doi.org/10.1038/s4146702343002x
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DOI: https://doi.org/10.1038/s4146702343002x
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