Decisive conditions for strategic vaccination against SARS-CoV-2

After the initial identification of the novel severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) in Wuhan, China in December 2019, the virus quickly reached pandemic proportions and caused major public health and economic problems worldwide.¹ The disease associated with SARS-CoV-2 infections was termed coronavirus disease 2019 (COVID-19). As of August 28, 2021, the number of confirmed COVID-19 cases exceeded $216$ million, and more than $4.5$ million COVID-19 deaths in at least 221 countries and territories were reported.² Large differences between excess deaths and reported COVID-19 deaths across different countries suggest that the actual death toll associated with COVID-19 is even higher.³ 

With the start and continuation of vaccination campaigns against SARS-CoV-2 in many countries,⁴ millions of people will receive partial and full immunization in the coming months. The mRNA vaccines BNT162b2 (BioNTech–Pfizer) and mRNA-1273 (Moderna) have been approved in several countries.⁵ When administered as a prime-boost regimen, these vaccines have a reported protective efficacy of 95%⁶ and 94.1%,⁷ respectively. An effectiveness evaluation of the BioNTech–Pfizer vaccine shows that it may offer about 50% protection against SARS-CoV-2 infections about 2–3 weeks after receiving the first dose.⁸ The adenovirus-based vaccine ChAdOx1 (Oxford–AstraZeneca) is being used in the UK, EU, and other countries with a reported single-dose efficacy between 62%–79%.^9,10 Estimates of the vaccine effectiveness of prime-boost regimens against symptomatic disease are reported to be 88% and 80% for Alpha and Delta variants,¹¹ respectively. Studies on the effectiveness of prime-boost vaccines against hospitalization^12,13 suggest values of 92% and 94% for Alpha and Delta strains, respectively.

Taken together, the majority of currently available SARS-CoV-2 vaccines elicit strong immune responses against all studied variants when administered as prime-boost regimens. However, given the current distribution and production constraints, it may take months until the production of COVID-19 vaccines can meet the actual global demand. Similar to vaccination campaigns in previous disease outbreaks, it may, therefore, be a favorable alternative to administer a single vaccination dose to twice as many people. In 2016, a single-dose vaccination campaign against cholera was implemented in Zambia because of the insufficient number of vaccination doses that were available to complete a standard two-dose campaign.¹⁴ Other vaccines, such as the oral cholera vaccines that require two doses, are highly effective after a single dose, but their protection is short lived compared to that obtained with prime-boost vaccination.^15,16

While there are countries that adopted a prime-boost protocol with an inter-dosing interval of about 1 month for COVID-19 vaccinations,¹⁷ some countries implemented vaccination protocols with longer inter-dosing intervals.¹⁸ For example, the Canadian Ministry of Health describes in Ref. 19 that they follow the National Advisory Committee on Immunization in recommending an interval of 4 months between the first and second dose to “increase the number of individuals benefiting from the first dose of vaccine.”

Despite clear advantages of such prime-first vaccination campaigns, including faster immunization of a larger number of people and lower vaccine-distribution infrastructure requirements and costs, any deviation from the immunologically favorable prime-boost protocol may negatively affect the level of vaccination-induced immunity. An analysis of blood samples from COVID-19 patients suggests that T cells play an important role in the long-term defense against SARS-CoV-2²⁰ since antibody concentrations were found to decay faster than those of T cells that respond to SARS-CoV-2 epitopes. Clinical trial results²¹ on the COVID-19 vaccine BNT162b1 show that the vaccination-induced $CD4+$ and $CD8+$ T-cell responses are significantly reduced if no boost shot was administered, indicating that booster doses are important for T-cell-mediated immunity against SARS-CoV-2. In the same study, antibody concentrations in patients who received prime-boost regimes were found to be about 5–20 times higher than those observed in patients who only received a single vaccination dose, highlighting the need for boosting. Similar observations were made for type-1 inactivated poliovirus vaccine (IPV), for which clinical trial results²² suggest that boost injections are needed to increase the level of neutralizing antibodies. However, for type-2 and -3 IPV, the first vaccination dose already elicits a neutralizing antibody response. In addition, single-dose vaccination may provide already a substantial degree of protection against infection, as confirmed in studies for BioNTech–Pfizer⁸ and Oxford–AstraZeneca²³ vaccines. Still, the mechanisms underlying vaccination-induced humoral (antibody-mediated) and cell-mediated immune responses against SARS-CoV-2 are not well understood, and data on immunity waning are scarce.^24,25

Here, we study population dynamics of SARS-CoV-2 that account for vaccine-induced protection levels, immunity waning, and other immunological factors to analyze under which epidemiological conditions prime-first vaccination is favorable over prime-boost vaccination. The advantages and disadvantages of these alternative vaccination protocols are subject to debate in many countries, including the US,^26,27 the UK,^28,29 and Germany,¹⁸ as they are fearing the spread of more contagious and more deadly SARS-CoV-2 mutants,³⁰ such as the Delta and Gamma variants, and the risk of collapsing health care systems.³¹ 

The current controversy around prime-first and prime-boost vaccination strategies raises two connected questions, which we address in this paper: How do shortages in vaccine supplies and uncertainties in epidemiological parameters alter the possible advantage of prime-first over prime-boost vaccination? And how do possible differences in vaccine efficacy and loss of vaccine-induced immunity affect the decision boundary separating prime-first and prime-boost vaccination regimes in a high-dimensional parameter space? By combining methods from epidemiological modeling, statistical mechanics,^32,33 and decision tree learning, we explore position, extent, and sensitivity of the decision boundary and provide a characterization of discriminative criteria,³⁴ sufficiently simple and immediately accessible to decision makers.

Different vaccination campaigns may lead to different proportions of infected, recovered, and deceased individuals at a given time. We study the differences between prime-first (Fig. 1) and prime-boost campaigns by accounting for a vaccination-induced reduction in transmissibility in a susceptible–exposed–infected–recovered–deceased (SEIRD)-based model³⁵ (see  Appendices A– C and Fig. 1). To quantify the effect of prime-first and prime-boost vaccination protocols on the overall disease-induced fatality, we use two fatality measures. The first measure is based on fatality rates, and the second one is based on cumulative deaths. Specifically, let $d1$ and $d2$ be the maximum (daily) changes in the total number of deaths within the time horizon of about 10 months (⁠$T=300$ days) for prime-first and prime-boost protocols, respectively. In addition, we use $D1$ and $D2$ to denote the corresponding cumulative numbers of disease-induced deaths. As a measure of the relative difference between $d1$ and $d2$⁠, we use the relative fatality change (RFC-$δ$⁠),

As a cumulative measure, we study the relative change in the cumulative number of disease-induced deaths (RFC-$Δ$⁠),

defined within the same time horizon as RFC-$δ$⁠.

A positive sign of RFC-$Δ$ and RFC-$δ$⁠, respectively, indicates that the cumulative number and the maximum daily number of disease-induced deaths is larger under a prime-boost vaccination strategy than under a prime-first vaccination campaign. A negative sign of RFC-$Δ$ and RFC-$δ$ indicates the opposite. In  Appendix D, we show that and how the two measures are correlated.

Current vaccination campaigns prioritize health care workers and vulnerable groups (e.g., elderly people with comorbidities) with a high risk of infection, leading to variations in vaccination rates. Further heterogeneity in model parameters may arise from infection rates that differ between age groups because of different degrees of susceptibility to infection³⁶ and different mobility characteristics. Our model accounts for these variations in epidemiological parameters through a large degree of parameterization. Nine different infection rates describe contacts between (susceptible and infectious) unvaccinated, single-dose vaccinated, and prime-boost vaccinated individuals. This large degree of parameterization can effectively account for possible correlations between age group, transmissibility, and mobility. We, therefore, choose not to incorporate demographic compartmentalization in our model.³³ In the following sections and in  Appendices E,  F,  G, and  H, we study the effects of age-stratification, natural immunity waning, and effects from parameter constraints in separate scenarios.

To provide mechanistic insight into the population-level differences between prime-first and prime-boost vaccination campaigns, we study how RFC-$δ$ and RFC-$Δ$ are impacted by epidemiological parameters and epidemic state. As a function of two parameters, green domains as shown in Fig. 2 indicate preference for prime-first campaigns, while red regions indicate that prime-boost is favored over prime-first vaccination. The parameter ranges follow the existing literature or are chosen sufficiently broad to cover uncertainties. Empirical data³⁷ suggest an estimated range of the basic reproduction number $R0∈[1,4]$ for the wild-type virus strain. Other SARS-CoV-2 variants such as the Delta strain may be outside this range. The following analysis and the data presented in Fig. 2 suggest that our main results also hold for larger values of $R0$⁠.

Differences in the waning rates $η1$ and $η2$ are not known at the present time, not even conclusive estimates,³⁸ while clinical trials are still ongoing. The characteristic time scale of waning immunity for prime-boost vaccinated individuals associated with a waning rate $η2=3×10−3day−1$ is about 1 year. Thus, we sample a broad parameter range, $η1−η2∈[10−4,10−1]day−1$ with $η2=3×10−3day−1$⁠, which includes waning time scales that were reported earlier for SARS-CoV.²⁴ The corresponding sampling interval for $η1$ includes waning time scales between 10 days and 1 year. For sampling the initial disease prevalence $I(0)$⁠, we use the interval $[10−4,10−1]$⁠. This interval includes up to 10% infected individuals but may lie outside prevalence estimates that were reported for some places with a very high prevalence such as Manaus, Brazil³⁹ August 2020 and New York City, USA.⁴⁰ For the range of the maximum vaccination rate $νmax$⁠, we use $[0,10−1]day−1$⁠, which we inferred from current vaccination-campaign data.⁴ 

We assume that the transmission rates $β1$ and $β2$ are proportional to the vaccine efficacies after single-dose and prime-boost vaccination, respectively. Thus, we identify the relative efficacy for single-dose immunization (RE) with the ratio $β2/β1$⁠. Values close to one are favorable for prime-first campaigns, while a low RE disfavors prime-first.

In order to analyze the effect of RE on the effectiveness of prime-first and prime-boost vaccination campaigns, we sample $β2/β1$ from the interval $[10−4,1]day−1$⁠. We choose this rather broad range to account for the lack of reliable data, in particular, regarding new variants of SARS-CoV-2 and possible adverse effects in vaccine protection.⁴¹ Parameters that are held constant in our simulations are listed in Table I.

The vaccination-campaign-preference diagrams (Fig. 2) suggest that prime-first vaccination campaigns are associated with a smaller death toll compared to prime-boost campaigns for a wide range of $R0$⁠, maximum vaccination rates, epidemic states, and relative efficacy ratios (green-shaded regions in Fig. 2).

As the main result of our study, we identify a two-parameter threshold combination that separates vaccination-campaign preferences (dashed black lines in Fig. 2). For a sufficiently small waning-rate difference $η1−η2≲0.02day−1$ and a sufficiently low maximum vaccination rate $νmax≲0.02day−1$⁠, we observe that prime-first vaccination outperforms prime-boost vaccination in all projections where parameters are held constant as specified in Table I. In the projections involving $η2−η1$⁠, prime-boost preference is observed if immunity wanes significantly faster for prime-vaccinated individuals than for prime-boost vaccinated individuals.

All projections in Fig. 2, therefore, suggest that prime-boost vaccination should only be favored for $νmax≳0.02day−1$⁠, which largely exceeds SARS-CoV-2 immunization rates worldwide.⁴ 

How a relatively low single-dose efficacy affects the preference for each campaign is shown in Figs. 2(k)–2(m). In Figs. 2(k) and 2(l), we assume a transmission reduction of only 10% after \hbox{single-dose immunization, $β1=0.9β$⁠, together with a 40% reduction in fatality,^11,12 $f⋆=0.6×10−2=0.6f$⁠, and all other parameters as in Table I.

The low single-dose efficacy domain is characterized by the occurrence of additional prime-boost preference regions in the parameter space [red-shaded regions in Figs. 2(k) and 2(l)]. However, even if the fatality rates of prime-first and prime-boost differ substantially, $f⋆⋆/f⋆≲0.8$⁠, only for low values of the relative prime-first efficacy $RE=β2/β1≲0.1$⁠, preference for prime-boost is observed [Fig. 2(k), shown range $0≤β2/β1≤0.1$]. Given this range and current data on SARS-CoV-2,^8,11,12,23 the diagram suggests preference for prime-first vaccination campaigns. Regarding the waning-rate difference, $η1−η2$⁠, a low single-dose efficacy does not suggest a threshold lower than $0.017day−1$ for prime-first preference [Fig. 2(l)].

Figure 2(m) shows the dependence of RFC-$δ$ and RFC-$Δ$ on $β1/β$ and $R0$⁠. Values of $β1/β≈1$ indicate a very low single-dose efficacy, whereas $β1/β≈0$ indicates an unrealistically high efficacy. For large $R0$ and very low single-dose efficacies, $β1/β≳0.8$⁠, prime-boost is preferred over prime-first [Fig. 2(m)]. These parameters are, however, unlikely to be characteristic of SARS-CoV-2,^8,23 recent data on the Delta variant included.¹² 

Finally, the waning-rate threshold $η1−η2=0.017day−1$ robustly separates prime-first and prime-boost preference regions for varying natural immunity-waning rates and empirical vaccination time series data (see  Appendices I,  F, and  G). The waning-rate threshold below which preference for prime-first is observed depends only weakly on the initial infection prevalence: it slightly increases as $I(0)$ decreases, $η1−η2≲0.010$ to $0.017day−1$ for $I(0)=10−5−10−2$ (see  Appendix J).

The presented campaign-preference diagrams are two-dimensional projections of a 25-dimensional parameter space, with the majority of parameters kept arbitrarily fixed (Table I). Due to this clear limitation, we further examine if the prime-first vaccination scheme is supported by other mathematical methods.

Our previous results suggest a pronounced preference for prime-first vaccination for a wide range of key epidemiological parameters. To further substantiate this conclusion, we performed Monte Carlo sampling of the entire 25-dimensional parameter space (see  Appendix E). For the analyzed high-dimensional parameter space, our results support that preference for prime-boost occurs significantly less frequently than samples indicating an advantage of prime-first vaccination.

The relative frequencies of samples for which prime-boost vaccination outperforms prime-first vaccination, characterized by RFC-$δ<0$ and RFC-$Δ<0$⁠, are estimated as 7.9% [standard error (SE): 0.2%] and 23.2% (SE: 0.4%); see orange bars in Fig. 3(a). For waning-rate differences $η1−η2≤0.056day−1$ and vaccination rates $νmax≤0.047day−1$⁠, we find that the proportions of prime-boost-preference samples are 7.0% (SE: 0.2%) for RFC-$δ<0$ and 15.4% (SE: 0.3%) for RFC-$Δ<0$ [beige bars in Fig. 3(a)]. Further restricting the parameter space using the condition $η1−η2<0.017day−1$ (dashed black lines in Fig. 2) and reported vaccination rates $νmax<0.013day−1$⁴ lead to proportions of prime-boost-preference samples of 8.5% (SE: 0.2%) for RFC-$δ<0$ and 6.9% (SE: 0.2%) for RFC-$Δ<0$ [blue bars in Fig. 3(a)]. The corresponding distributions of the differences in the maximum change of fatalities, $d2−d1$⁠, and in the cumulative numbers of deaths, $D2−D1$⁠, are shown in Figs. 3(b) and 3(c). As indicated by the mean relative frequencies of prime-boost-preference samples in Fig. 3(a), the majority of samples satisfy RFC-$δ>0$ (i.e., $d2−d1>0$⁠) and RFC-$Δ>0$ (i.e., $D2−D1>0$⁠).

Based on the results shown in Figs. 3(a)–3(c), we conclude that constraining the studied parameter space by lowering $νmax$ and $η1−η2$ results in a substantially enhanced preference for prime-first in terms of reduced excess deaths, RFC-$Δ$⁠. We also conclude that the preference of prime-first vaccination campaigns with respect to the fatality measure RFC-$δ$ is almost independent of the chosen parameter restrictions, which is indicated by the observed narrow range between 7% and 9% in Fig. 3(a). This supports the robustness of our results.

The discriminative power of the thresholds associated with the parameters $η1−η2$ and $νmax$ is also supported by random-sampling results for different risk groups and in situations with natural immunity waning, as seen in Figs. 3(d)–3(f). As detailed in  Appendix H, simulations were performed for two age groups⁴² with infection fatality ratios (IFRs) of less than 0.1% and between 0.1% and 1%, respectively. We observe slightly smaller proportions of prime-boost preference samples in the older age group than in the younger age group.

As another independent method for determining decisive conditions for strategic vaccination campaigns, we performed binary decision tree learning with repeated stratified cross-validation.^43,44 This technique has proven useful to extract the most discriminative features in high-dimensional data. Our analysis suggests that $νmax$ and $η1−η2$ are the most discriminative parameters within the 25-dimensional parameter space (see  Appendix K). For the samples that we generated according to the distributions listed in Table I (orange lines and markers in Fig. 3) and the constraints $νmax≤0.047day−1$ and $η1−η2≤0.056day−1$⁠, about 70% of vaccination preferences of simulated scenarios are correctly predicted (see  Appendix K for accuracy scores and details). Additionally, constraining the parameter space with the thresholds that we used in the previous paragraph (beige and blue lines and markers in Fig. 3) results in prime-first preference for 93% of the parameter-space volume. This suggests that for realistic vaccination rates, the vaccination-dose-dependent immunity-waning rate difference is the only highly discriminative factor.

Effective vaccination protocols are crucial to achieve a high immunization coverage, especially if vaccination supplies are limited. The ongoing debate on the most effective way of distributing prime-boost regimens against SARS-CoV-2 has been sparked by arguments suggesting that, from an epidemiological perspective, prime-first vaccination protocols may be more effective than immediate prime-boost administration if supply cannot meet demand.^{18,26,28,45,46} For many COVID-19 vaccines, prime-boost protocols are considered immunologically efficient due to their ability to elicit strong and long-term humoral and cellular immune responses.²¹ Immunologically efficient vaccination protocols, however, may be not epidemiologically favorable,⁴⁷ in particular, for exponentially increasing infection numbers and vaccine doses shortages on times scales of months. By combining epidemiological modeling, methods from statistical mechanics, and decision tree learning, we have studied the effect of relevant immunological and epidemiological parameters (e.g., vaccine efficacy and immunity waning) on a possible advantage of prime-first over prime-boost vaccination. We have identified and analyzed decision boundaries separating the parameter regimes in which one or the other vaccination protocol is preferable. Our results suggest that prime-first campaigns are associated with a lower death toll compared to prime-boost vaccination campaigns even for relatively high vaccination rates and more surprisingly for low single-dose efficacies, which is in contrast to the existing literature.^8,23,45,48

A related study⁴⁵ compares prime-first and prime-boost vaccination campaigns against SARS-CoV-2, without accounting for immunity waning. This study reports that single-dose vaccination campaigns make optimal use of resources in the short term, given a sufficiently large single-dose efficacy that they identify as the main discriminative factor. In contrast, our study calls attention to immunity waning and vaccination rates as the highly discriminative factors, while we find that vaccine efficacies are less discriminative.

Previous works^15,48 consistently emphasize that due to the complexity of underlying models and limitations from available data, a vaccine campaign recommendation can only be given once the precision in all key epidemiological parameters, such as field efficacy data and the level of protection obtained after a single dose, becomes sufficiently high. Laubenbacher et al.⁴⁹ highlight the need for further data collection and model integration in infectious disease modeling, which are important steps to better estimate immunity-waning rates and vaccine effectiveness.^8,15,48

Saad-Roy et al.⁵⁰ focus on the long-term effects of waning and evolutionary immune response in a highly parameterized model. Certain scenarios they analyze suggest that prime-first campaigns may be favorable for some time scales but not for others, depending on a combination of parameters, waning rates included. In contrast, for the critical time scale of months, we provide a simple decision criterion by showing that the waning-rate difference is the main discriminative threshold.

Preference for prime-first vaccination is not unexpected. During the initial inter-dose interval, both vaccination strategies are identical since, regardless of the chosen strategy, booster doses are not yet administered. In the subsequent time interval, twice as many susceptible individuals can be immunized by following a prime-first protocol compared to prime-boost campaigns. This means that about 50% of individuals who could have been vaccinated will actually remain unvaccinated in a prime-boost campaign. Let us refer to this unvaccinated group as group $A$ and denote by group $B$ the group of individuals that receive two vaccination doses in a prime-boost scenario. One can assume that the infection rates of individuals in group $A$ are larger than those of individuals in group $B$⁠, who benefit from a more effective immune response. As a result, higher transmission in group $A$ is the expected dominating differential adverse effect. As expected for effective prime-boost vaccines, one may assume that the prime-boost infection rate, $β2$⁠, and the fatality rate, $f⋆⋆$⁠, are significantly smaller than their counterparts in prime-first campaigns. Thus, the effective transmission rate for group $A$ and $B$ combined is dominated by group $A$’s rate but not critically dependent on $β2$⁠, which intuitively explains why the relative efficacy ratio $RE=β2/β1$ is not a highly discriminative factor.

For very low single-dose efficacies or very high single-dose disease-induced fatality rates, the single-dose efficacy $β1/β$⁠, the relative prime-first efficacy $RE$⁠, and $R0$ may be discriminative, depending on the circumstances. Current data on SARS-CoV-2 vaccination campaigns,^8,23 however, suggest that those parameter combinations are unlikely to occur. Furthermore, if immunity wanes substantially faster after the first dose than after the additional booster dose, prime-boost vaccination may become favorable over prime-first, depending on the value of $R0$⁠. Unvaccinated and susceptible individuals should also receive both vaccination doses if a few percent of a jurisdiction’s total population can be vaccinated daily. However, even for the relatively large vaccination rate of about $∼1%$ per day, as realized in Israel,⁴ our analyses suggest that prime-first vaccination is still favorable over prime-boost campaigns.

A recent study in single-dose vaccinated SARS-CoV-2 patients infected with Beta or Delta variants showed neutralizing antibody concentrations well below the limit of detection.⁵¹ While antibody titers correlate with protection against severe disease,⁵² they are only a single component of the intricate immune response mechanisms and are not a necessary condition for effectiveness of vaccines against symptomatic disease or hospitalization. In fact, recent effectiveness estimates suggest that the first dose of BioNTech–Pfizer and Oxford–AstraZeneca is about 78% and 75% effective against hospitalization after an infection with the Alpha and Delta variants, respectively.^12,13 These data strongly support our conclusions given the high correlation between hospitalization and fatality.

The effectiveness estimates for Alpha and Delta variants as reported in Refs. 12 and 13 are as low as 31% for single-dose immunization when compared to the values of about 80% that are reported for prime-boost vaccinated individuals. While efficacies below 50% may strongly suggest that the delay between the first and second dose should be kept limited, our study finds that prime-first vaccination should be considered if the primary health objective is minimizing hospitalizations and fatalities. This means that vaccination campaigns that deviate from the recommended immunization protocol are particularly relevant in countries where new variants, such as the Delta and Gamma strains, are highly prevalent.^19,53,54

In summary, our results contrast the existing literature^15,45,48,50 in the sense that not all key epidemiological data are required to be collected to identify most effective vaccination protocols. On the contrary, our analysis suggests that, even for a large degree of uncertainty in key epidemiological data, prime-first vaccination is robustly preferred over prime-boost vaccination—if the waning-rate difference between prime-first and prime-boost is sufficiently small. For epidemiological parameters that describe the spread of COVID-19, we found this threshold to be in the narrow range of 0.01–0.02 $day−1$⁠. Unfortunately, to date, there are no reliable data available on waning time scales.^24,38 While some estimates suggest that no significant waning occurs for Oxford–AstraZeneca in the first 90 days after receiving the first dose,²³ recent estimates from Israel may suggest substantial immunity waning regarding protection from infection after 6 months for BioNTech–Pfizer prime-boost regimens.⁵⁵ For a characteristic prime-boost immunity-waning time scale of about 6 months, the waning immunity threshold condition suggests that prime-first protocols are more effective than prime-boost protocols if immunity waning of prime-vaccinated individuals is not expected to occur already within the first two months. Once vaccination-dependent waning rates can be estimated from data²⁵ and adverse immunological effects can be assessed or excluded, our criterion may become highly valuable for decision makers in countries facing vaccine shortages.

Although clinical studies of the approved SARS-CoV-2 vaccines may suggest that these vaccines are safe and effective, only little is known about their possible long-term adverse effects.⁵⁶ Clearly, for the comparison of different vaccination strategies, we assume that negative long-term effects are negligible. In addition, we do not consider harm measures covering non-hospitalized symptomatic cases. Adverse effects and different levels of protection may be incorporated in models that account for different subgroups.⁵⁷ 

To conclude, while current vaccine supply does not meet global demand, especially in low- and middle-income countries, and new variants of concern may reduce the effectiveness of currently available vaccines,⁵⁸ it is desirable to provide decision makers with transparent tools that support them in assessing different vaccination protocols. This study may be of help to healthcare officials and decision makers since, in contrast to the existing literature, our combination of tools results in unexpectedly robust and highly decisive criteria. More generally, the presented framework establishes how epidemiologically efficient vaccine dosing strategies^59,60 can be integrated into effective pandemic control plans.

L. B. acknowledges financial support from the SNF (No. P2EZP2_191888), the National Institutes of Health (NIH) (No. R01HL146552), and the Army Research Office (ARO) (No. W911NF-18-1-0345). Parts of the simulations were performed on the ETH Euler cluster.

The authors declare no competing interests.

The data that support the findings of this study are openly available in GitHub at https://github.com/lubo93/vaccination, Ref. 67.

We adapt the SEIRD model^32,35 to account for immunity waning and a vaccination-induced reduction in transmissibility [Fig. 1]. The fractions of susceptible, exposed, infected, recovered, and deceased individuals at time $t$ are denoted by $S(t)$⁠, $E(t)$⁠, $I(t)$⁠, $R(t)$⁠, and $D(t)$ respectively. Moreover, we denote the fractions of prime and prime-boost vaccinated susceptible individuals by $S⋆(t)$ and $S⋆⋆(t)$⁠, respectively. With rate $ν1$⁠, susceptible individuals receive their first vaccination dose, and with rate $ν2$⁠, prime-vaccinated susceptible individuals receive their booster doses. The time dependence in the vaccination rates reflects temporal variations in the availability of vaccination doses, as explained below. The corresponding fractions of vaccinated exposed and infected individuals are denoted by $E⋆(t)$ and $E⋆⋆(t)$ and $I⋆(t)$ and $I⋆⋆(t)$⁠. We use three constant rates $η1$⁠, $η2$⁠, $η3$ to model immunity waning (i.e., transitions from $S⋆$⁠, $S⋆⋆$⁠, and $R$ to $S$⁠). Characteristic time scales of waning immunity,²⁴ defined by the inverse of the corresponding rates, are much longer than those associated with entering and leaving exposed and infected compartments; therefore, we do not explicitly model waning immunity in these compartments. For long time horizons, additional birth and death processes may be employed to model birth and age-related death.

The resulting dynamics of the susceptible and exposed classes is described by the following rate equations:

The maximum proportion of susceptible individuals that can be prime and prime-boost vaccinated is $S(t)$ and $S⋆(t)$⁠, respectively. Based on vaccination data from Israel (Fig. 4), we assume linearly increasing immunization over time in our model and use the vaccination rates

and

where $μ1=νmax$ and $μ2=0$ for prime-first vaccination and $μ1=μ2=νmax/2$ for prime-boost vaccination. Here, $H[x]$ denotes the Heaviside step function, which is zero for $x<0$ and one for $x≥0$⁠. The function $H[t−td]$ describes the delay $td$ of about 2–3 weeks⁵ (Fig. 4) between prime and booster doses. Up to time $td$⁠, susceptible individuals get vaccinated with rate $μ1+μ2$⁠. If no susceptible individuals are left, prime-vaccinated individuals get vaccinated with rate $μ1$ too, leading to the term $μ1(1−H[S(t)])H[S⋆(t)]H[t−td]$ in Eq. (A3). In our model, only susceptible individuals are vaccinated. This can be justified by the assumption that susceptible individuals outnumber those in other disease states.

Exposed individuals transition to an infected state at rates $σ$⁠, $σ1$⁠, and $σ2$⁠. The evolution of the infected, recovered, and deceased compartments is described by

Only ten of equations (A1) and (A4) are independent since we employ the normalization condition $S+S⋆+S⋆⋆+E+E⋆+E⋆⋆+I+I⋆+I⋆⋆+R+D=1$⁠. Different transmissibilities $β,β⋆,β⋆⋆$⁠, $β1,β1⋆,β1⋆⋆$⁠, and $β2,β2⋆,β2⋆⋆$ describe interactions between susceptible and infected individuals with different immunity levels.

For each infected compartment $I$⁠, $I⋆$⁠, and $I⋆⋆$⁠, we calculate the infection fatality ratios (IFRs)³ by dividing the associated cumulative number of deaths by the total number of infections in the unvaccinated, prime-vaccinated, and prime-boost-vaccinated compartments, respectively. The IFR of the unvaccinated pool of individuals is

For constant $γ,f$⁠, we obtain

As the number of infected individuals approaches zero for long time horizons [i.e., $limt→∞I(t)=0$], the IFR satisfies $limt→∞IFR(t)=f$⁠. Similarly, $limt→∞IFR⋆(t)=f⋆$ and $limt→∞IFR⋆⋆(t)=f⋆⋆$ if $γ⋆,f⋆$ and $γ⋆⋆,f⋆⋆$ are time-independent.

Due to ergodicity breaking effects from multiplicative noise,⁶¹ deterministic models tend to overestimate infection and fatality. However, it is realistic to assume that the effects from noise are not discriminative as they do not differ for either vaccination campaign.

After all, the immunological intricacies of SARS-CoV-2 remain largely unknown, and there is no single commonly accepted epidemiological standard model. At the same time, we anticipate more reliable data on immunity waning and other immunological effects in the coming months. Our framework is transparent and flexible enough to change or augment the (already high) degree of parameterization, or compartmentalization, if warranted.

We calculate the basic reproduction number $R0$ of the epidemic model (A1) and (A4) using the next-generation matrix method.⁶⁴ As a first step, we rewrite the rate equations (A1) and (A4) of the infected compartments in a matrix form,

where $x=(E,E⋆,E⋆⋆,I,I⋆,I⋆⋆)⊤$⁠, $y=(S,S⋆,S⋆⋆,R,D)$⁠, $F$ represents the vector of new infections, and $V$ describes all remaining transitions. We thus find for the corresponding Jacobians of $F$ and $V$ at the disease-free equilibrium,

and

The basic reproduction number $R0$⁠, the expected number of infections generated by an infectious individual in an otherwise completely susceptible population, is the spectral radius of the next-generation matrix $FV−1$⁠.⁶⁴ Finding $R0$ for the general system (B2)

involves the analytically cumbersome task of finding roots of a cubic equation, which can be avoided by using numerical methods (e.g., the power method). For very effective vaccines, however, one may assume that the transmissibility of prime-boost vaccinated individuals is much lower than the transmissibility of unvaccinated individuals; that is, $β⋆⋆≪β$⁠, $β1⋆⋆≪β$⁠, and $β2⋆⋆≪β$⁠. In this approximation, we obtain

For $S(0)=1$ and $S⋆(0)=0$⁠, the basic reproduction number is

To solve Eqs. (A1) and (A4) numerically, we use the Dormand–Prince method⁶⁵ with a maximum time step of $10−1$ and simulate the evolution of different epidemics in the time interval $[0,T]$⁠, where $T=300$ days. For the simulation results that we show in Fig. 2, we set $I(0)=10−2$ and $S(0)=1−I(0)$⁠. If model parameters are held constant in Fig. 2, we use the parameters that are listed in Table I. Transmissibilities of infection events that involve at least one vaccinated individual are smaller than or equal to the baseline transmissibility $β$ as long as mobility and distancing characteristics of vaccinated individuals do not differ significantly from those who are unvaccinated. In our model, this means that $β≥β⋆≥β⋆⋆$⁠, $β≥β1≥β2$⁠, $β1≥β1⋆≥β1⋆⋆$⁠, and $β2≥β2⋆≥β2⋆⋆$ (equality holds for very ineffective vaccines). As vaccination campaigns and vaccine effectiveness analyses are ongoing, we used estimates for $β⋆$⁠, $β⋆⋆$⁠, $β1$⁠, $β1⋆$⁠, $β1⋆⋆$⁠, $β2$⁠, $β2⋆$⁠, and $β2⋆⋆$ as reported in Table I. We also model the effect of small incidence rates and broader parameter ranges in a random-sampling analysis as reported in the following sections.

There are two more constraints that our model parameters have to satisfy to describe the impact of vaccination campaigns on disease transmission. First, the fatality ratio in the unvaccinated compartment is larger than the fatality ratios in the vaccinated compartments (i.e., $f≥f⋆≥f⋆⋆$⁠). We assume that differences in $f⋆$ and $f⋆⋆$ are negligible. Second, the waning rate in the prime-boost vaccinated compartment is smaller than the waning rate in the prime-vaccinated compartment (i.e., $η2≤η1$⁠).

The fatality measures RFC-$δ$ and RFC-$Δ$ as defined in Eqs. (1) and (2) are correlated, [Figs. 8(a)–8(c), $R=0.94$ (a), $R=0.68$ (b), and $R=0.68$ (c); corresponding p-values are smaller than machine precision]. The correlation observed for the threshold combination $νmax≤0.013day−1$ and $η1−η2≤0.017day−1$ confirms the discriminative power and robustness of our results regarding the choice of both fatality measures.

The parameter distributions that we use in our random sampling and decision tree analysis are summarized in Table II. We generate two datasets with $N=50000$ samples each and analyze the influence of different combinations of model parameters and initial conditions on RFC-$δ(d1,d2)$ [Eq. (1)] and RFC-$Δ(D1,D2)$ [Eq. (2)].