Epidemiological and evolutionary considerations of SARSCoV2 vaccine dosing regimes
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Abstract
In the face of vaccine dose shortages and logistical challenges, various deployment strategies are being proposed to increase population immunity levels to SARSCoV2. Two critical issues arise: how will the timing of delivery of the second dose affect both infection dynamics and prospects for the evolution of viral immune escape via a buildup of partially immune individuals. Both hinge on the robustness of the immune response elicited by a single dose, compared to natural and twodose immunity. Building on an existing immunoepidemiological model, we find that in the shortterm, focusing on one dose generally decreases infections, but longerterm outcomes depend on this relative immune robustness. We then explore three scenarios of selection and find that a onedose policy may increase the potential for antigenic evolution under certain conditions of partial population immunity. We highlight the critical need to test viral loads and quantify immune responses after one vaccine dose, and to ramp up vaccination efforts throughout the world.
As the severe acute respiratory syndrome coronavirus 2 (SARSCoV2) betacoronavirus (βCoV) pandemic continues, the deployment of safe and effective vaccines presents a key intervention for mitigating disease severity and spread and eventually relaxing nonpharmaceutical interventions (NPIs). At the time of writing, eleven vaccines have been approved by at least one country (1). We focus mainly on the vaccines from Pfizer/BioNTech, Moderna, and Oxford/AstraZeneca. The first two elicit adaptive immunity against SARSCoV2 in response to the introduction of messenger ribonucleic acid (mRNA) molecules that encode the spike protein of SARSCoV2 (2), and appear to offer greater than 95% (Pfizer/BioNTech (3), approved in 60 countries) and 94% (Moderna (2), approved in 38 countries) protection against symptomatic coronavirus disease 2019 (COVID19). Both of these mRNA vaccines were tested in clinical trials according to a twodose regime with dose spacing of 21 and 28 days for the Pfizer/BioNTech and Moderna platforms, respectively. The Oxford/AstraZeneca vaccine uses a nonreplicating adenovirus vector, and has also been tested in clinical trials according to a twodose regime with a target 28day interdose period (although for logistical reasons some trial participants received their second dose after a delay of at least 12 weeks). Clinical trials indicated 62%–90% efficacy for this vaccine according to the specific dose administered (4). While we base our parameter choices and modeling assumptions on these three vaccines, our results are generalizable across platforms.
As these vaccines have been distributed internationally, several countries including the UK (5) and Canada (6) have chosen to delay the second dose in an effort to increase the number of individuals receiving at least one or in response to logistical constraints (7). Although a number of participants dropped out after a single dose of the vaccine in the Pfizer/BioNTech and Moderna trials, these studies were not designed to assess vaccine efficacy under such circumstances, and Pfizer has stated that there is no evidence that vaccine protection from a single dose extends beyond 21 days (5), although other data paint a more optimistic picture (8, 9). The Oxford/AstraZeneca clinical trials did include different dose spacings, and limited evidence suggests that longer intervals (two to three months) did not affect and may even have improved vaccine efficacy (4, 5). Ultimately, the consequences of deviating from manufacturerprescribed dosing regimes at the population scale remain unknown, but will hinge on immune responses.
While there has been significant progress in quantifying host immune responses following infection (10–12), substantial uncertainty regarding the strength and duration of both natural and vaccinal SARSCoV2 immunity remains. Previous work suggests that these factors will play a central role in shaping the future dynamics of COVID19 cases (13). Future cases also create an environment for the selection of novel variants [e.g., (14–16)]. Of particular concern is the possibility of antigenic drift [e.g., for influenza (17), and (18) for the seasonal human coronavirus 229E] via immune escape from natural or vaccinal immunity. For example, immune escape might be especially important if vaccinal immunity elicited after the complete twodose regime is highly protective whereas a single vaccine dose provides less effective immunity. Consequently, the longer term epidemiological and evolutionary implications of these different SARSCoV2 vaccine dosing regimes are not yet clear; the immediate need for effective mass vaccination makes understanding them critical to inform policy (19).
Here, we explore these epidemiological and evolutionary considerations with an extension of a
recent immunoepidemiological model for SARSCoV2 dynamics (13), depicted schematically in Fig. 1.
Without vaccination, our model reduces to the SusceptibleInfectedRecovered(Susceptible) [SIR(S)]
model (13, 20), where individual immunity after recovery from primary infection may
eventually wane at rate δ, leading to potentially reduced susceptibility to secondary
infections, denoted by the fraction ϵ relative to a baseline level of unity. This parameter
ϵ is thus related to the (transmissionblocking) strength of immunity, and titrates between
the SIR (lifetime immunity, ϵ = 0) and SIRS (hosts regain complete susceptibility, ϵ =
1) paradigms. Quantifying ϵ is challenging because it requires measuring reinfection rates
after the waning of immunity. Some studies have made significant progress in this direction (11, 12); however, uncertainties remain, particularly related to quantifying the
average duration of immunity 1/δ. In this model extension (Fig.
1 and Materials and methods) we incorporate two vaccinated classes;
V_{1} accounts for individuals who have received one dose of a SARSCoV2
vaccine and V_{2} tracks individuals who have received two doses. In the
short term, we assume that both dosing options decrease susceptibility by fractions
(one dose) and
$\left(1{\u03f5}_{{V}_{2}}\right)$ (two doses), inferred from the clinical trial data (though the nature
of the infecting variant may influence this); we also assume that I_{V}
tracks infection following vaccination. We allow for vaccinal immunity to wane at separate rates
[ρ_{1} (one dose) and ρ_{2} (two doses)], moving individuals to the
partially susceptible immune classes
and
${S}_{{S}_{2}}$ characterized by (possibly different) levels of immune protection
ϵ_{1} and ϵ_{2}. Infection following waned onedose or twodose
vaccinal immunity is tracked by the immune classes
and
${I}_{{S}_{2}}$, respectively. We consider a continuous spectrum for the interdose
period
, with an infinite value corresponding to a “onedose
strategy”, and model the rate of administration of the first dose ν as an increasing
function of the interdose period (Fig. 1 and Materials and
methods) to reflect the increase in available doses due to a delayed second dose. Thus, dosing
regimes with longer interdose periods allow for higher coverage with the first dose.
We begin by projecting the epidemiological impacts of the different dosing regimes on mediumterm temporal dynamics of COVID19 cases. We then examine the potential evolutionary consequences of dosing regime by calculating a timedependent relative net viral adaptation rate (17). This term is related to the strength of natural and vaccinal immunity (either via inducing selection through immune pressure or suppressing viral replication) as well as the sizes of classes of individuals experiencing infections after immune waning.
Epidemiological impacts
As a base case, we consider a high latitude European or North American city with initial conditions that qualitatively correspond to early 2021 (see supplementary materials and figs. S5 and S6 for other scenarios, e.g., a high initial attack rate or almost full susceptibility), in addition to a seasonal transmission rate (21) with NPIs (see Materials and methods). Note that given immunological and future control uncertainties, we are aiming to project qualitatively rather than formulate quantitative predictions for particular locations. The UK and Canadian policy is for a delayed second dose; they are not aiming for an “exclusively” onedose policy. However, we explore the onedose strategy as an extreme case for the “twodose” vaccines; it also encompasses a pessimistic situation of waning public confidence in vaccination and individuals’ own decisions to forgo the second dose. Finally, this onedose policy could capture vaccines which only require a single dose, e.g., the Johnson & Johnson vaccine.
In Fig. 2, we present potential scenarios for mediumterm SARSCoV2 infection and immunity dynamics contingent upon vaccine dosing regimes. We start by assuming that vaccination occurs at a constant rate, and assume a relatively optimistic maximum rate of administration of the first dose of ν_{0} = 2% of the population per week (see supplementary materials for other scenarios). Figure 2A and Fig. 2B correspond, respectively, to scenarios with weaker (and shorter) and stronger (and longer) natural and vaccinal adaptive immune responses. Thus, the former represents a scenario with higher secondary susceptible density than the latter. In each panel, the top and bottom sections consider poor and robust onedose vaccinal immunity, respectively. The leftmost column represents a onedose vaccine policy (captured in the model by infinite dose spacing), with dose spacing decreasing to 4 weeks in the rightmost column (i.e., a strict twodose policy with doses separated by the clinical trial window corresponding to Moderna’s recommendations for their vaccine, hereafter referred to as the “recommended twodose strategy”).
As expected, we find that broader deployment of widelyspaced doses is beneficial. Specifically, a onedose strategy (or a longer interdose period) may lead to a substantially reduced “first” epidemic peak of cases after the initiation of vaccination (compare the leftmost top panels of Fig. 2, A and B, with the no vaccination scenarios in fig. S1, A and B). This result applies even if immunity conferred by one vaccine dose is shorter and weaker than that following twodoses (top panels of Fig. 2, A and B). However under these conditions of imperfect immunity, an exclusively onedose strategy then leads to an earlier subsequent peak due to the accumulation of partially susceptible individuals. When the rate of administration of the first dose is very high (fig. S4, ν_{0} = 5% per week), this subsequent infection peak may be larger than that expected in the scenario with no vaccination. In general, the accumulation of partially susceptible individuals with waned onedose vaccinal immunity can be mitigated by implementing a twodose strategy and decreasing the time between doses. Thus, in situations of a less effective first dose where the second dose is delayed, it is important to ensure individuals eventually do obtain their second dose.
In line with intuition, longer and stronger immunity elicited after a single dose heightens the benefits of a onedose strategy or of delaying the second dose (compare the top and bottom leftmost panels of Fig. 2, A and B). Additionally, the protective effects of adopting these strategies instead of the twodose regime are maintained in the mediumterm, with decreased burden in all future peaks. This is further summarized in Fig. 3A via the cumulative number of total and severe cases (right and left panels, respectively) over approximately four years from the time of vaccine initiation, normalized by the burdens with no vaccination; these ratios are plotted as a function of the interdose period and the one to twodose immune response ratio x_{e} (see figure caption for details). When the immune response conferred by a single dose is close to the robustness following two doses, total case numbers (Fig. 3A, right panel) can be substantially reduced by delaying the second dose. However, for smaller values of x_{e}, larger interdose periods are associated with more cases. The reduction in the cumulative burden of severe cases is even more sizeable (Fig. 3A, left panel) due to the assumed reduction in the fraction of severe cases for partially immune individuals. When vaccination rates are substantially lower (fig. S2, ν_{0} = 0.1% per week; and fig. S3, ν_{0} = 1% per week), the benefits of a single dose strategy diminish even for an effective first dose, as an insufficient proportion of the population are immunized. The short term effect of the vaccine on case numbers is sensitive to when it is introduced in the dynamical cycle (figs. S7 and S8), highlighting the critical interplay between the force of infection and the level of population immunity (see supplementary materials for further details).
Vaccines will be central to efforts to attain community immunity (22), and thus prevent local spread due to case importation. We therefore analytically calculated the first vaccine dose administration rate for a given interdose spacing required for community immunity in our model (see supplementary materials). In the long term, however, individuals whose one or twodose immunity has waned will likely be able to be vaccinated again before infection in countries with adequate supplies; we therefore incorporated revaccination of these individuals into the extended model and computed an analogous minimal vaccination rate which we plot in Fig. 3B. We find that as the interdose period grows, this minimal rate depends increasingly on the degree of reduction in susceptibility after the waning of onedose vaccinal immunity ϵ_{1} (Fig. 3B and see fig. S13 for other parameter choices). Vaccine refusal (23) may also impact the attainment of community immunity through vaccinal immunity in the longerterm (see supplementary materials).
Evolutionary impacts
The recent emergence of numerous SARSCoV2 variants in still relatively susceptible populations underline the virus’s evolutionary potential (24–26). We focus here on the longer term potential for immune escape from natural or vaccinal immunity (17). For immune escape variants to spread within a population, they must first arise via mutation, and then there must be substantial selection pressure in their favor. We expect the greatest opportunity for variants to arise in (and spread from) hosts with the highest viral loads, likely those with the least immunity. On the other hand, we expect the greatest selection for escape where immunity is strongest. Previous research on the phylodynamic interaction between viral epidemiology and evolution (based on seasonal influenza) predicts that partially immune individuals (permitting intermediate levels of selection and transmission) could maximize levels of escape (17) (Fig. 4A). Under this model, we would project that different categories of secondarily infected people (after waning of natural immunity or immunity conferred from one or two doses of vaccine) would be key potential contributors to viral immune escape.
In Fig. 4, we consider three potential evolutionary scenarios, exploring different assumptions regarding viral abundance and withinhost selection for the various immune classes. In all scenarios, we assume for simplicity that immunity elicited after two doses of the vaccine is equivalent to that elicited after natural infection. We also assume that transmission rises with viral abundance in hosts (17). In Scenario I (black borders on circles, top panel of Fig. 4A), we assume that infections of all classes of partially susceptible individuals lead to strong selective pressures and low viral abundance (a marker of low transmission), and thus low rates of adaptation, with only slightly reduced immune pressure for infections after a waned single vaccine dose relative to natural infection or two doses. Scenario II (blue borders on circles, middle panel of Fig. 4A), considers a situation where natural and twodose vaccinal immunity again lead to low viral abundance, but onedose vaccinal immunity is associated with intermediate immune pressure that results in substantially higher rates of viral adaptation. Finally, in Scenario III (purple borders on circles, bottom panel of Fig. 4A), adaptive immune responses following waned natural, one dose, and two dose vaccinal immunity all lead to similar intermediate levels of immune pressure and high rates of viral adaptation. In all cases, we assume for tractability that viral immune escape is not correlated with clinical severity (27).
The relative potential viral adaptation rates [see (17) for more details] corresponding to each scenario are presented in the top rows of Fig. 4, B and C. This relative rate is estimated as the sum of the sizes of the infection classes following waned immunity (i.e., I_{S} after S_{S},
${I}_{{S}_{1}}$after
${S}_{{S}_{1}}$, and
${I}_{{S}_{2}}$after
${S}_{{S}_{2}}$) weighted by the infection classspecific net viral adaptation rate assigned in each scenario. Therefore, this quantity reflects a weightaveraged potential rate for viral adaptation perindividual perinfection. The corresponding immune and susceptibility classes are plotted in the middle and bottom rows, respectively, according to the color scheme defined in Fig. 1A. The weaker immunity scenario of Fig. 2A is considered, with Fig. 4B and Fig. 4C corresponding, respectively, to the situations of a weaker and more robust single vaccine dose relative to two doses. The leftmost column corresponds to a one dose strategy, an interdose period of
$\frac{1}{\omega}=24$weeks is assumed in the middle column, and the rightmost column assumes a two dose strategy with doses separated by the clinical trial window of
$\frac{1}{\omega}=4$weeks.
Different assumptions regarding the strength and duration of adaptive immune responses to vaccines and natural infections alter projections for the proportions of individuals in the partially susceptible immune classes over time. When one dose vaccinal immunity is poor, a onedose strategy results in the rapid accumulation of partially susceptible
${S}_{{S}_{1}}$individuals (Fig. 4B, bottom row) and a greater infection burden. (Note, this
${S}_{{S}_{1}}$immune class is highlighted in orange for visibility in Figs. 1, 2, and 4.) When the assumed individual rates of evolutionary adaptation arising from these infection classes are high (Scenarios II and III), we find that a onedose strategy could lead to substantially higher relative rates of adaptation. This effect can be mitigated by implementing a twodose strategy even with a longer interdose period than the recommended duration, echoing our epidemiological findings.
A single dose strategy of a strongly immunizing vaccine reduces infection rates, resulting in lower relative rates of adaptation when a one dose strategy is used; however the resulting large fraction of
${S}_{{S}_{1}}$individuals may still lead to evolutionary pressure, particularly when the potential viral adaptation rate associated with
${I}_{{S}_{1}}$infections is large. A twodose strategy mitigates this effect, but the corresponding reduction in vaccinated individuals increases the infection burden from other classes. Thus, to avoid these potentially pessimistic evolutionary outcomes, our results highlight the importance of rapid vaccine deployment. More broadly, our results further underline the importance of equitable, global vaccination (28, 29): immune escape anywhere will quickly spread.
Impact of increasing vaccination through time
In the supplementary materials (figs. S10 to S12), we explore the implications of ramping up vaccine deployment through two approaches. First, we examine a simple increase in the rate of administration of the first dose and unchanged dosing regimes (fig. S10). Qualitatively, these results are largely analogous to our previous results, and reflect the benefits of increasing population immunity through an increase in vaccination deployment.
However, as vaccines become more widely available, policies on dosing regimes may change. The second approach we consider is a timely shift to a twodose policy with recommended interdose spacing as vaccine deployment capacity increases (figs. S11 and S12). Initially delaying (or omitting) the second dose decreases the first epidemic peak after the initiation of vaccination. Such a reduction in first peak size would also reduce secondary infections, and thus potentially immune escape in most cases (i.e., an evolutionary advantage). Subsequently, the switch to a manufacturertimed vaccine dosage regime mitigates the potential mediumterm disadvantages of delaying (or omitting) the second dose that may arise if immunity conferred from a single dose is relatively poor, including the accumulation of partially susceptible
${S}_{{S}_{1}}$individuals whose onedose vaccinal immunity has waned. These contrasts highlight the importance of datadriven policies that undergo constant reevaluation as vaccination progresses.
Caveats
Our immunoepidemiological model makes several assumptions. While heterogeneities (superspreading, age, space, etc.) (30–33) are important for the quantitative prediction of SARSCoV2 dynamics, we previously found that these do not qualitatively affect our results (13). Nevertheless, we again briefly explore the epidemiological consequences of heterogeneities in transmission and vaccine coverage in the supplementary materials. We have also assumed that the robustness of immune responses following the second dose is independent of the interdose period, yet it is possible that delaying the second dose may actually enhance adaptive immune responses (34). Detailed clinical evaluation of adaptive immune responses after one and two vaccine doses with different interdose spacing is an important direction for future work.
Additionally, we have assumed highly simplified scenarios for NPIs. The chosen scenario was selected to qualitatively capture current estimates of SARSCoV2 prevalence and seropositivity in large cities. However, these values vary substantially between locations, a notable example being recent estimates of a large infection rate in Manaus, Brazil, during the first wave (35), or countries having almost no infections due to the successful implementation of NPIs (36–38). We have examined these scenarios in the supplementary materials (figs. S5 and S6). The qualitative projections of our model are sensitive to the composition of infection and immune classes at the onset of vaccination (including, therefore, the assumption of dramatically higher seropositivity levels, i.e., the sum of the S_{S} and R classes). We further explore this in the supplementary materials through the initiation of vaccination at different times in the dynamic cycle (figs. S7 and S8). Thorough explorations of various NPIs, seasonal transmission rate patterns, vaccine deployment rates, dosing regimes, and clinical burdens can be investigated for broad ranges of epidemiological and immunological parameters with the online interactive application, available at (39).
Finally, we have explored the simplest evolutionary model, which can only give a general indication of the potential for evolution under different scenarios. Including more complex evolutionary models (40, 41) into our framework is thus another important area for future work. Population heterogeneities likely have complex impacts on viral evolution. First, heterogeneities in immune responses and transmission (e.g., chronically infected hosts that shed virus for extended periods (42), or focused versus polyclonal responses) may have important impacts on the accumulation of genetic diversity and the strength of selection pressures, and hence on evolutionary potential [e.g., for influenza, see (43)]. Second, there are complex evolutionary implications of disease severity minimization by vaccination (27, 44). Third, superspreading and contact structure could influence the rate of spread of novel variants through a population (45). Additionally, increases in viral avidity to the human ACE2 receptor might generate multiple benefits for the virus in terms of enhanced transmission and immune escape (46). Finally, genetic processes such as clonal interference, epistasis, and recombination also add substantial complexity to evolutionary dynamics [e.g., (17, 47, 48)]. Further model refinements should also include these details for increased accuracy. A full list of caveats is presented in the supplementary materials.
Conclusion
The deployment of SARSCoV2 vaccines in the coming months will strongly shape postpandemic epidemiological trajectories and characteristics of accumulated population immunity. Dosing regimes should seek to navigate existing immunological and epidemiological tradeoffs between individuals and populations. Using simple models, we have shown that different regimes may have crucial epidemiological and evolutionary impacts, resulting in a wide range of potential outcomes in the medium term. Our work also lays the foundation for a number of future considerations related to vaccine deployment during ongoing epidemics, especially preparing against future pandemics.
In line with intuition, spreading single doses in emergency settings (i.e., rising infections) is beneficial in the short term and reduces prevalence. Furthermore, we find that if immunity following a single dose is robust, then delaying the second dose is also optimal from an epidemiological perspective in the longer term. On the other hand, if onedose vaccinal immunity is weak, the outcome could be more pessimistic; specifically, a vaccine strategy with a very long interdose period could lead to marginal shortterm benefits (a decrease in the shortterm burden) at the cost of a higher infection burden in the long term and substantially more potential for viral evolution. These negative longer term effects may be alleviated by the eventual administration of a second dose, even if it is moderately delayed. With additional knowledge of the relative strength and duration of onedose vaccinal immunity and corresponding, clinicallyinformed policies related to dosing regimes, pessimistic scenarios may be avoided. For context, at the time of writing, the UK for example has been particularly successful in rolling out vaccination to a large population with a wide spacing between doses (49). Our model illustrates that, ultimately, the long term impacts of this strategy, especially in terms of transmission and immune escape, will depend on the duration and strength of onedose vaccinal immunity. Recent experience of weaker vaccinal immunity against the B.1.351 strain (50) underlines the importance of both detecting novel strains and titrating the strength of natural and vaccinal immunity against them.
In places where vaccine deployment is delayed and vaccination rates are low, our results stress the subsequent negative epidemiological and evolutionary impacts that may emerge. Particularly since these consequences (e.g., the evolution of new variants) could emerge as global problems, this underlines the urgent need for global equity in vaccine distribution and deployment (28, 29).
Current uncertainties surrounding the strength and duration of adaptive immunity in response to natural infection or vaccination lead to very broad ranges for the possible outcomes of various dosing regimes. Nevertheless, ongoing elevated COVID19 case numbers stresses the rapid need for effective, mass vaccine deployment. Overall, our work emphasizes that the impact of vaccine dosing regimes are strongly dependent on the relative robustness of immunity conferred by a single dose. It is therefore imperative to determine the strength and duration of clinical protection and transmissionblocking immunity through careful clinical evaluations (including, for instance, randomized control trials of dose intervals and regular testing of viral loads in vaccinated individuals, their contacts, and those who have recovered from natural infections) in order to enforce sound public policies. More broadly, our results underscore the importance of exploring the phylodynamic interaction of pathogen dynamics and evolution, from within host to global scales, for SARSCoV2, influenza, and other important pathogens (40, 41, 47, 48, 51, 52).
Materials and methods
Model formulation
We extend the model of (13) to examine different vaccination strategies. The additional compartments are as follows: V_{i} denotes individuals vaccinated with i doses and are thus immune;
${S}_{{S}_{i}}$denotes individuals whose complete idose immunity has waned and are now partially susceptible again;
${I}_{{S}_{i}}$denotes individuals who were in
${S}_{{S}_{i}}$and have now been infected again; I_{V} denotes individuals for whom the vaccine did not prevent infection.
The extended model contains several new parameters:
$\frac{1}{{\rho}_{i}}$is the average duration of vaccinal immunity V_{i};
$\frac{1}{\omega}$is the average interdose period;
${\u03f5}_{{V}_{i}}$is the decrease in susceptibility following vaccination with dose i; ϵ_{i} is the decrease in susceptibility following waning of idose immunity; α_{i} is the relative infectiousness of individuals in
${I}_{{S}_{i}}$; and α_{V} is the relative infectiousness of individuals in I_{V}. To allow for heterogeneity in vaccinal immune responses and potentially cumulative effects of natural and vaccinal immunity, we take c to be the fraction of previouslyinfected partially susceptible individuals (S_{S}) for whom one dose of the vaccine gives equivalent immunity to twodoses for fully susceptible individuals (S_{P}). Finally, x_{i} is the fraction of individuals in
${S}_{{S}_{i}}$that are revaccinated, and (1p_{i}) is the fraction of individuals in
${S}_{{S}_{i}}$for whom readministration of the “first dose” provides equivalent immune protection to two doses (i.e., they transition to the V_{2} class). The full set of equations governing the transitions between these infection and immunity classes is then given by
$$\frac{d{S}_{P}}{dt}=\mu \beta {S}_{P}\left[{I}_{P}+\alpha {I}_{S}+{\alpha}_{V}{I}_{V}+{\alpha}_{1}{I}_{{S}_{1}}+{\alpha}_{2}{I}_{{S}_{2}}\right]\left({S}_{\text{vax}}\nu +\mu \right){S}_{P}$$
(1a)
$\frac{d{I}_{P}}{dt}=\beta {S}_{P}\left[{I}_{P}+\alpha {I}_{S}+{\alpha}_{V}{I}_{V}+{\alpha}_{1}{I}_{{S}_{1}}+{\alpha}_{2}{I}_{{S}_{2}}\right]\left(\gamma +\mu \right){I}_{P}$(1b)
$\frac{dR}{dt}=\gamma \left({I}_{P}+{I}_{s}+{I}_{V}+{I}_{{S}_{1}}+{I}_{{S}_{2}}\right)\left(\delta +\mu \right)R$(1c)
$$\frac{d{S}_{S}}{dt}=\delta R\u03f5\beta {S}_{S}\left[{I}_{P}+\alpha {I}_{S}+{\alpha}_{V}{I}_{V}+{\alpha}_{1}{I}_{{S}_{1}}+{\alpha}_{2}{I}_{{S}_{2}}\right]\left({S}_{\text{vax}}\nu +\mu \right){S}_{S}$$(1d)
$\frac{d{I}_{S}}{dt}=\u03f5\beta {S}_{S}\left[{I}_{P}+\alpha {I}_{S}+{\alpha}_{V}{I}_{V}+{\alpha}_{1}{I}_{{S}_{1}}+{\alpha}_{2}{I}_{{S}_{2}}\right]\left(\gamma +\mu \right){I}_{S}$(1e)
$$\begin{array}{l}\frac{d{V}_{1}}{dt}={s}_{\text{vax}}\nu {S}_{P}+c{s}_{\text{vax}}\nu {S}_{S}+{x}_{1}{P}_{1}{s}_{\text{vax}}\nu {S}_{{S}_{1}}+{x}_{2}{P}_{2}{s}_{\text{vax}}\nu {S}_{{S}_{2}}\\ {\u03f5}_{{V}_{1}}\beta {V}_{1}\left[{I}_{P}+\alpha {I}_{S}+{\alpha}_{V}{I}_{V}+{\alpha}_{1}{I}_{{S}_{1}}+{\alpha}_{2}{I}_{{S}_{2}}\right]\left(\omega +{\rho}_{1}+\mu \right){V}_{1}\end{array}$$(1f)
$$\begin{array}{l}\frac{d{V}_{2}}{dt}=\left(1c\right){s}_{\text{vax}}\nu {S}_{S}+{x}_{1}\left(1{p}_{1}\right){s}_{\text{vax}}\nu {S}_{{S}_{1}}+{x}_{2}\left(1{p}_{2}\right){s}_{\text{vax}}\nu {S}_{{S}_{2}}+\omega {V}_{1}\\ {\u03f5}_{{V}_{2}}\beta {V}_{2}\left[{I}_{P}+\alpha {I}_{S}+{\alpha}_{V}{I}_{V}+{\alpha}_{1}{I}_{{S}_{1}}+{\alpha}_{2}{I}_{{S}_{2}}\right]\left({\rho}_{2}+\mu \right){V}_{2}\end{array}$$(1g)
$\frac{d{I}_{V}}{dt}=\beta \left({\u03f5}_{{V}_{1}}{V}_{1}+{\u03f5}_{{V}_{2}}{V}_{2}\right)\left[{I}_{P}+\alpha {I}_{S}+{\alpha}_{V}{I}_{V}+{\alpha}_{1}{I}_{{S}_{1}}+{\alpha}_{2}{I}_{{S}_{2}}\right]\left(\gamma \mu \right){I}_{V}$(1h)
$\frac{d{S}_{{S}_{1}}}{dt}={P}_{1}{V}_{1}{\u03f5}_{1}\beta {S}_{{S}_{1}}\left[{I}_{P}+\alpha {I}_{S}+{\alpha}_{V}{I}_{V}+{\alpha}_{1}{I}_{{S}_{1}}+{\alpha}_{2}{I}_{{S}_{2}}\right]\left({s}_{\text{vax}}{x}_{1}\nu +\mu \right){S}_{{S}_{1}}$(1i)
$\frac{d{S}_{{S}_{2}}}{dt}={P}_{2}{V}_{2}{\u03f5}_{2}\beta {S}_{{S}_{2}}\left[{I}_{P}+\alpha {I}_{S}+{\alpha}_{V}{I}_{V}+{\alpha}_{1}{I}_{{S}_{1}}+{\alpha}_{2}{I}_{{S}_{2}}\right]\left({s}_{\text{vax}}{x}_{2}\nu +\mu \right){S}_{{S}_{2}}$(1j)
$$\frac{d{I}_{{S}_{1}}}{dt}={\u03f5}_{1}\beta {S}_{{S}_{1}}\left[{I}_{P}+\alpha {I}_{S}+{\alpha}_{V}{I}_{V}+{\alpha}_{1}{I}_{{S}_{1}}+{\alpha}_{2}{I}_{{S}_{2}}\right]\left(\gamma +\mu \right){I}_{{S}_{1}}$$(1k)
$\frac{d{I}_{{S}_{2}}}{dt}={\u03f5}_{2}\beta {S}_{{S}_{2}}\left[{I}_{P}+\alpha {I}_{S}+{\alpha}_{V}{I}_{V}+{\alpha}_{1}{I}_{{S}_{1}}+{\alpha}_{2}{I}_{{S}_{2}}\right]\left(\gamma +\mu \right){I}_{{S}_{2}}$(1l)
For all simulations, we take μ = 0.02 y^{−1} corresponding to a yearly crude birth rate of 20 per 1000 people. Additionally, we take the infectious period to be 1/γ = 5 days, consistent with the modeling in (13, 21, 53) and the estimation of a serial interval of 5.1 days for COVID19 in (54), and assume that c = 0.5. We take the relative transmissibility of infections to be α = α_{V} = α_{1}= α_{2} = 1, and therefore only modulate the relative susceptibility to disease ϵ. For the initial conditions of all simulations, we take I_{P} = 1 × 10^{−9} and assume the remainder of the population is in the fully susceptible class. The values of the remaining parameters used in the various simulations are specified throughout the main text.
Determination of seasonal reproduction numbers
In order to reflect observed seasonal variation in transmission rates for respiratory infections arising from related coronaviruses (21), influenza (21) and respiratory syncytial virus (55), we base seasonal reproduction numbers in this work on those in (13), which were calculated in (21) based on the climate of New York City. Other seasonal patterns can be explored using the interactive online application. In all simulations, we modify these values to force a mean value for the basic reproduction number of
${\overline{R}}_{0}=\u3008{R}_{0}\left(t\right)\u3009=2.3$by multiplying the climatederived time series R_{0,}_{c}(t) by 2.3 and dividing by its average value, i.e.
${R}_{0}\left(t\right)={R}_{0,c}\left(t\right)\frac{2.3}{{\overline{R}}_{0,c}}$
Modeling of nonpharmaceutical interventions (NPIs)
In all simulations, we enforce periods of NPI adoption (arising from behaviors and policies such as lock downs, maskwearing, and social distancing) in which the transmission rate is reduced from its seasonal value described in the previous section. In particular, we assume that NPIs are adopted between weeks 8 and 47 following the pandemic onset resulting in the transmission rate being reduced to 45% of its seasonal value. Between weeks 48 and 79, we assume that the transmission rate is to 30% higher than the previous time interval (reflecting an overall reduction to 45(1.3) = 58.5% of the original transmission rate), due to either behavioral changes following the introduction of the vaccine or the emergence of more transmissible strains. Finally, we assume that NPIs are completely relaxed beyond week 80.
Linking vaccination rate to interdose period
We consider an exponential relationship between the rate of administration of the first vaccination dose ν[ω] and the interdose period
$\frac{1}{\omega}$. We assume that this rate is maximized at ν_{0} when no second dose occurs (i.e., ω = 0, an infinite interdose period), and that when the first and second doses are spaced by the clinically recommended interdose period L_{opt}
$\left({\omega}_{\text{opt}}=\frac{1}{{L}_{\text{opt}}}\right)$, the rate of administration of the first dose is one half of its maximum value. Thus,
$\nu \left[\omega \right]={2}^{{L}_{\text{opt}}\omega}{\nu}_{0}$.
References and Notes
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 ↵We qualitatively describe the fraction of population that have received vaccines by assuming that vaccination occurs at random in the population. Specifically, suppose vaccination occurs at time τ = 0. The rate of change of the fraction not vaccinated follows
$\frac{dN}{d\tau}=\nu N$, N(0) = 1, giving N(τ) = e^{−}^{v}^{τ} so that the fraction vaccinated with one dose Y(τ ) = 1 – N(τ) = 1 – e^{−}^{v}^{τ}. The fractions of those vaccinated with one but not two doses follows
$\frac{d{W}_{1}}{d\tau}=\nu N\omega {W}_{1}$, giving
${W}_{1}\left(\tau \right)=\frac{\nu \left({e}^{\mathrm{\nu \tau}}{e}^{\mathrm{\omega \tau}}\right)}{\omega \nu}$. Then, the fraction vaccinated with two doses is
${W}_{2}\left(\tau \right)=Y\left(\tau \right){W}_{1}\left(\tau \right)=1\frac{\omega}{\omega \nu}{e}^{\mathrm{\nu \tau}}+\frac{\nu}{\omega \nu}{e}^{\mathrm{\omega \tau}}$.
 ↵We calculate the total number of cases at any time point as
${I}_{T}={I}_{P}+{I}_{S}+{I}_{V}+{I}_{{S}_{1}}+{I}_{{S}_{2}}$. Similarly, the number of severe cases is given by
${I}_{T,\text{sev}}={x}_{\text{sev},p}{I}_{P}+{x}_{\text{sev},s}{I}_{S}+{x}_{\text{sev},V}{I}_{V}+{x}_{\text{sev},1}{I}_{{S}_{1}}+{x}_{\text{sev},2}{I}_{{S}_{2}}$. Cumulative case numbers for a give time period are calculated through
$\gamma {\displaystyle \sum {I}_{T}}$(total cases) and
$\gamma {\displaystyle \sum {I}_{T,\text{sev}}}$(severe cases), where the summation occurs over all time steps.
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R. M. Anderson, R. M. May, Infectious Diseases of Humans: Dynamics and Control (Oxford Univ. Press, 1992).
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