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# Modeling how antibody responses may determine the efficacy of COVID-19 vaccines

## Abstract

Predicting the efficacy of COVID-19 vaccines would aid vaccine development and usage strategies, which is of importance given their limited supplies. Here we develop a multiscale mathematical model that proposes mechanistic links between COVID-19 vaccine efficacies and the neutralizing antibody (NAb) responses they elicit. We hypothesized that the collection of all NAbs would constitute a shape space and that responses of individuals are random samples from this space. We constructed the shape space by analyzing reported in vitro dose–response curves of ~80 NAbs. Sampling NAb subsets from the space, we recapitulated the responses of convalescent patients. We assumed that vaccination would elicit similar NAb responses. We developed a model of within-host SARS-CoV-2 dynamics, applied it to virtual patient populations and, invoking the NAb responses above, predicted vaccine efficacies. Our predictions quantitatively captured the efficacies from clinical trials. Our study thus suggests plausible mechanistic underpinnings of COVID-19 vaccines and generates testable hypotheses for establishing them.

## Main

The coronavirus disease 2019 (COVID-19) pandemic has triggered the fastest vaccine development efforts so far, as well as urgent global vaccination rollouts. Yet, limited vaccine supplies are hindering our ability to fight the pandemic1,2. Enormous efforts are ongoing to develop new vaccines. As of 14 October 2021, there were 332 vaccine candidates under development, of which 113 were in clinical testing3. The ability to predict vaccine efficacies may expedite vaccine development by helping to shortlist promising candidates and/or minimize the subsequent reliance on expensive and time-consuming clinical trials for assessing their efficacies4. Simultaneously, it may help identify optimal vaccination strategies5,6. For example, although several approved vaccines are administered in two doses separated by a few weeks, it would be useful to know the protection conferred by a single dose, by lower dosages or by doses separated by longer intervals, especially in less vulnerable populations, so as to ease demand7,8,9.

Approved COVID-19 vaccines have shown remarkable but varying efficacies in clinical trials, reducing the incidence of ‘symptomatic’ infections with the wild-type severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) strain by 62–96% (refs. 7,10,11,12,13,14,15,16). The protection has been argued to be due to neutralizing antibodies (NAbs) elicited by the vaccines. Cellular immunity appears to play a less important role, especially after the recommended doses of the vaccine are administered10,11,17,18,19,20,21,22. The protection due to NAbs is consistent with independent observations, in which higher levels of pre-existing NAbs were correlated with protection and lower risk of infection, respectively, in an early outbreak in a fishery vessel23 and in a longitudinal study of healthcare workers24. Protection from seasonal coronaviruses too has been associated with pre-existing NAbs25. Strong statistical correlations have been identified between COVID-19 vaccine efficacies and the NAb responses they elicit17,18. An important question that arises is how the NAb responses confer the protection observed. An understanding of the dependence of the level of protection on the NAb titers and their neutralization efficiencies is lacking.

In this Article, we develop a theoretical framework that describes and integrates several key phenomena associated with SARS-CoV-2 infection and vaccination, across length and time scales spanning the range from the workings of individual NAbs to the responses of clinical cohorts to vaccination. The framework employs testable, mechanistic hypotheses and quantitatively predicts the population-level protection conferred by vaccines as a function of the NAb titers they elicit.

## Results

### Construction of the SARS-CoV-2 NAb landscape

A major challenge to describing the effects of vaccination is the diversity of the NAb responses elicited. The NAb response to primary SARS-CoV-2 infection in unvaccinated individuals is diverse, spanning >1,000-fold variation in Ab titers and in vitro neutralization efficiencies across individuals26,27. NAb titers following vaccination have been found to be comparable to those from convalescent patients10,11,28. No formalism exists to predict this diversity or its effects on protection. We addressed this challenge by adapting the classic idea of ‘shape space’, which has aided quantification of the immune repertoire29, for characterizing NAbs. Accordingly, we sought features (also termed shape parameters) of the NAbs that would predict their neutralization efficiencies. Numerous studies have isolated individual NAbs from patients and assessed their neutralization efficiencies in vitro. We compiled dose–response curves (DRCs) of ~80 NAbs26,27,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44 thus isolated and fit them using the median-effect equation45 (Methods, Supplementary Fig. 1 and Supplementary Data 1). The equation fit the data well (Fig. 1a and Supplementary Fig. 2), indicating that two parameters, the 50% inhibitory concentration (IC50) and the slope (m) of the DRC, were sufficient to characterize the neutralization efficiency of individual NAbs (Fig. 1a and Supplementary Data 1). The best-fit IC50 and m varied widely across NAbs (Fig. 1b). IC50 values ranged from ~10−3 to ~140 µg ml−1 (Fig. 1b), in close agreement with reported estimates (Supplementary Fig. 3a and Supplementary Data 1). The values of m, the importance of which has been recognized with HIV-1 and hepatitis C45,46 but has not typically been reported for SARS-CoV-2, ranged from ~0.2 to 2 (Fig. 1c). This variability in IC50 and m was not restricted to a particular pseudotyped virus construct or backbone (Supplementary Fig. 3b,c), cell line (Supplementary Fig. 3d,e) or assay conditions, which could vary across studies (Supplementary Fig. 3f,g). The variability was thus not attributable to these potential confounding factors and appeared to be intrinsic to the NAbs, indicating the spectrum of NAbs elicited. Furthermore, akin to HIV-1 antibodies45, the variations in IC50 and m appeared independent. For example, the NAbs BD-361 and REGN10954 had similar IC50 values (both ~0.04 µg ml−1) but vastly different m (~0.7 and ~1.5, respectively), whereas the NAbs CC12.3 and 515-5 had different IC50 values (~0.02 µg ml−1 and 1.6 µg ml−1, respectively) but similar m (both ~1). IC50 and m were thus not only sufficient but also necessary for quantifying the neutralization efficiencies of NAbs. We therefore employed IC50 and m as the shape parameters. Plotting the NAbs on an IC50m plot, we identified the NAb shape space (Fig. 2), which, because of its two-dimensional nature, we termed the ‘landscape of SARS-CoV-2 NAbs’.

The landscape contains potent NAbs, with low IC50 and high m, as well as weak NAbs, with the opposite traits. To compare the NAbs, we employed the instantaneous inhibitory potential (IIP), a composite metric of IC50 and m (refs. 45,46,47). IIPD represents the log10 decline in viral load in a single round infection assay due to the NAb present at concentration D. Thus, the higher the IIPD, the more potent is the NAb at that concentration. NAbs displayed a wide distribution of IIP100 values: six NAbs had the highest IIP100 values, >5, and seven had the least, ≤1 (D = 100 µg ml−1) (Fig. 2b and Supplementary Data 1). This distribution demonstrated further the wide spectrum of neutralization efficiencies of NAbs. Using another NAb concentration did not affect this distribution substantially (Supplementary Fig. 4).

### Prediction of the NAb responses of convalescent patients

The landscape established bounds on the neutralization efficiencies of the NAbs elicited. We reasoned next that the diversity of the NAb responses across individuals would arise from the way NAbs are sampled from the landscape. Although a large number of NAbs can be isolated from individuals, studies of convalescent patient plasma26,27,48,49,50 as well as on NAb epitope profiling51 have argued that the NAb response of an individual can be attributed to a small subset of five to ten distinct NAbs. Furthermore, although some epitopes on the SARS-CoV-2 spike protein, S, are targeted more than others by NAbs, the collection of NAbs produced differs substantially across individuals52. We therefore assumed that the response elicited by an infected individual would be a small, random subset of the landscape. We analyzed DRCs of NAbs isolated from individual patients and found that they indeed constituted such subsets (Supplementary Fig. 5). Accordingly, we sampled random combinations of ten NAbs each, each combination representing the response of an individual. Our results were robust to an increase in the number of NAbs sampled beyond ten (Supplementary Fig. 6). We let NAb concentrations vary across individuals, to mimic the observed variation of the NAb titers48,49,50. We quantified the neutralization efficiency of the NAb response by simulating standard plasma dilution assays (Methods and Fig. 3a). Experimental plasma dilution assay results follow an inverse sigmoidal pattern and are characterized by NT50, the dilution at which the neutralization efficiency of the plasma decreases by 50% (Fig. 3b). In our simulations, we let the NAbs exhibit Bliss independence or Loewe additivity, the former representing NAbs targeting distinct, non-occluding epitopes and the latter the same or occluding epitopes53. Our simulations recapitulated the dilution curves associated with patient plasma (Fig. 3b). Furthermore, the values of NT50 we predicted were in close agreement with experimental observations49 (Fig. 3c). (Note that immunoglobulin-G (IgG) targeting the receptor binding domain of the SARS-CoV-2 spike protein, reported in the experiments in Fig. 3c, do not represent all NAbs. They are, however, a major component of the NAbs. The dilution assays in Fig. 3b do include all NAbs, indicating that our predictions capture the measured NT50 values reliably.) The latter data were described better by Bliss independence at low NAb titers and Loewe additivity at high titers. This is expected, because, at low titers, the NAbs are unlikely to interact with each other and would thus follow Bliss independence, whereas at high titers, they may compete for binding sites on S or occlude each other and exhibit Loewe additivity53. At any NAb titer, there existed substantial variation in NT50, attributed to the random combinations of NAbs sampled. The variation, however, was outweighed by the overall rise of NT50 with the NAb titer, consistent with patient data (Fig. 3c). For example, the geometric mean NT50 computed using Loewe additivity was 7.2 at an IgG titer of 0.1 µg ml−1 and 455.1 at 10 µg ml−1. Sampling from the NAb landscape thus successfully recapitulated patient responses. Note that no adjustable parameters were involved in this comparison. We were thus able to describe the diversity of the NAb responses elicited across patients. Armed with this description, we examined next the protection accorded by vaccines in clinical trials.

### Prediction of COVID-19 vaccine efficacies

Following vaccination, NAb titers rise and are expected to remain stable (or decay slowly) over weeks to months54, protecting individuals exposed to the virus during this period. Individuals were assumed to be protected if they did not report symptomatic infection. Loss of protection involved symptoms and a positive result on a nucleic acid amplification test10,11. Protection with NAbs is not expected to be sterilizing, as suggested by animal studies21,55; NAbs help suppress virus load and facilitate more rapid clearance of the infection. We assumed that the severity of the symptoms would be proportional to the virus load. If the peak is sufficiently suppressed, no symptoms may result, as is the case with naturally asymptomatic infections56. Here we assumed that an individual would be detected as symptomatically infected if the viral load rose above a threshold.

To estimate the peak viral load, we developed a mathematical model (Fig. 4a) of the early time course of the infection, where the viral load typically rises, attains a peak and declines57. To ascertain its ability to describe these dynamics in vivo, we fit the model to longitudinal viral load data from individual patients from three cohorts57,58,59. The model fit the data well (Fig. 4b and Supplementary Figs. 7 and 8) and yielded parameter estimates quantifying inter-patient variability in the within-host dynamics (Supplementary Tables 2 and 3). We applied the model to describe the effect of vaccination (Methods). We assumed that NAbs generated following vaccination would exist at the start of infection. Although NAbs may perform many functions, including stimulating the cellular adaptive response56,60, direct evidence of these functions in SARS-CoV-2-infected humans is yet to be gathered56. (The vaccines may stimulate other immune arms independently of the NAbs, which, as we argue above, is expected to make limited contributions to vaccine efficacy against the wild-type SARS-CoV-2 strain10,11,17,18.) We therefore focused on their ability to neutralize free viruses, effectively reducing viral infectivity. We recall that protection due to NAbs is not expected to be sterilizing21,55. The reduced infectivity is also consistent with in vitro pseudovirus neutralization assays, which measure the drop in infectivity with increasing NAb concentrations. Our model captures these in vitro assays quantitatively (Fig. 3). The greater the reduction in infectivity, the lower the peak viral load in vivo (Fig. 4c, curves with efficacy ε > 0). This assumption of the activity of NAbs is thus also consistent with the inverse correlation between the peak viral load and NAb titers elicited by vaccination in mouse models21 and in macaques61.

Substantial de novo NAb production post-infection typically occurs after the peak in virus load56 and has been argued to have limited impact on viral clearance62. We therefore considered pre-existing NAbs as responsible for protection and assumed their titers not to vary substantially during the course of the infection, given the typically short course of the infection and the much longer durability of the NAb response to vaccination54. We let the pre-existing NAbs be drawn as random subsets from the landscape, as we did above. We assumed that the NAbs neutralized free viruses with an efficiency that we estimated using Loewe additivity. NAb titers in the lung airways are expected to be similar to those in the blood given the close coupling between the lungs and the circulatory system56. The efficiency of NAbs in vivo can differ from that in vitro63, which we took into account. We simulated a virtual patient population of 3,500 individuals, similar to the number of individuals infected in the placebo arms of clinical trials. The individuals all had distinct viral dynamics parameters drawn from previously known and/or estimated ranges (Supplementary Tables 1 and 2), to mimic inter-patient variability in addition to the variability arising from NAb sampling from the landscape. Our model predicted wide variability in the peak viral load (Fig. 4d). At low pre-existing NAb concentrations (0.01 µg ml−1), indicative of the scenario without vaccination, the predicted peak viral load ranged from ~103 to 109 copies per milliliter, consistent with the range in symptomatic individuals64. The peaks declined as NAb titers increased. Following clinical trials, we set the limit of detection to ~102 copies per milliliter (ref. 65). (Note that, even if symptoms were to arise with lower viral loads in some individuals, such individuals would not be diagnosed as infected because of assay limitations.) The fraction of individuals with peaks below detection would indicate the level of protection due to the vaccine.

To quantify the mean level of protection and test it against data from clinical trials, we used viral dynamics parameters representative of detectable infections66,67 (Supplementary Tables 1 and 2) and simulated the dynamics in a cohort of 10,000 infected individuals. Vaccination studies report the NT50 values of the NAb responses elicited and the associated mean protection level, or efficacy7,10,12,68,69,70 (Supplementary Table 4). We binned individuals into narrow NT50 bands and calculated the mean protection and 95% confidence interval (CI) in each band. We found that the mean protection was low for an NT50  of ~1. It increased in a sigmoidal manner to 50% at an NT50 of ~15 and asymptotically reached 100% at an NT50 of ~200 (Supplementary Fig. 9).

To compare these predictions with clinical data across trials, we normalized the data using the NT50 values from convalescent patients reported by the respective trials (Supplementary Table 4). This ensured that assay variations across studies did not confound our comparisons. We accordingly also normalized our predictions of NT50 with those corresponding to convalescent patients captured by our model (Methods, Fig. 3 and Supplementary Fig. 10). For the set of parameters employed, the data for all the eight approved vaccines we considered fell on this ‘protection curve’ (Fig. 5). Thus, for example, a single dose of the vaccine Ad26.COV2.S elicited NAbs with a scaled NT50 of 0.43 and accorded 66.1% protection. Following two doses of the vaccine BNT162b2, the corresponding values were 3.84% and 94%, respectively. These values were captured accurately by our model predictions. This agreement extended to data from phase III trials of the other vaccines we considered as well (Fig. 5).

We tested the robustness of these predictions to model parameter variations. Using local (Supplementary Fig. 11a) and global sensitivity analysis (Supplementary Fig. 11b), we found that our predictions of vaccine efficacy were robust to parameter variations so long as the parameter values were consistent with patient data (Supplementary Fig. 11c–e). One parameter not involved in the patient data fitting was ω, the assumed ratio of the IC50 values of the NAbs in vivo and in vitro, because NAb responses, given their limited role, were not part of our model of unvaccinated individuals62. The protection curve was sensitive to ω (Supplementary Fig. 11f). Recent studies71,72 have estimated ω for a few NAbs of SARS-CoV-2 and found it to be in the range ~5–40 (Supplementary Table 1), similar to estimates for other viruses63,73. In the above predictions we set ω to ~10, which lies in the above range and yielded the protection curve that captured clinical data. Future studies may provide independent estimates of the model parameters, limiting these uncertainties and sharpening the mechanistic links between COVID-19 vaccine efficacies and NAb responses assumed in our formalism.

## Discussion

Our formalism for predicting COVID-19 vaccine efficacies as a function of the NAb responses they elicit required the description and integration of several key phenomena, varying over many length and time scales. These include (1) the neutralization potential of individual NAbs, (2) the diversity of the NAb response within and across individuals, (3) the relationship between NAb titers in individuals and their neutralization potential, (4) the within-host dynamics of disease progression, (5) the influence of vaccination on the within-host dynamics and (6) the variability of the latter influence across a patient population in clinical trials. At each step, we established quantitative connects with experimental data, rendering our approach rigorous.

Identifying correlates of the protection offered by COVID-19 vaccines has been challenging74. Several recent studies have pointed to NAb responses as strong correlates of vaccine-mediated protection17,18,19,20. Our study independently arrived at the correlation between vaccine efficacy and NT50 and generated testable hypotheses of its mechanistic origins. Briefly, our study assumed that, in response to vaccination, different individuals in a population produce NAbs that are well represented by random samples from the NAb landscape. The neutralization potential of the NAbs elicited, quantified by NT50, accordingly varies across individuals in a dose-dependent manner. The presence of the NAbs, in our model, caused a reduction in the peak viral load following infection, also in a dose-dependent manner. A sufficient reduction in the peak viral load was assumed to prevent symptoms and manifest as protection conferred by vaccination, potentially giving rise to the correlation of vaccine efficacy with NT50. Thus, our study suggests a plausible mechanism that can be tested experimentally for how multiple NAbs generated within an individual following vaccination provide protection by being members of a shape space. Our model proposes a mechanism to reduce the net neutralization effect of these antibodies into a single metric. Further, our formalism deduced a relationship between NT50 and the concentration of NAbs in plasma, so that the latter may also be used to estimate vaccine efficacies.

Our formalism employs several assumptions and hypotheses underlying the protection conferred by NAbs elicited by vaccination. First, NAbs are assumed to be generated randomly from the NAb landscape. Although this appears plausible (Supplementary Fig. 5), it remains to be established. Furthermore, that the NAb landscape in convalescent patients and following vaccination is similar needs to be ascertained. Second, assay variations in the estimates of NAb characteristics (IC50 and m) are assumed not significant. To verify this, estimates of IC50 and m could be standardized across studies, for example, by normalizing them using the IC50 and m values generated in each study for the same ‘reference’ NAb and by using the same viral backbone and cell line. Third, our model assumed that protection following vaccination is predominantly due to NAbs and that other immune arms play a secondary role. Although the extent to which COVID-19 vaccines trigger other immune arms is being investigated20, the high NAb titers generated after the boost support this assumption. Fourth, NAbs are assumed to act primarily by neutralizing virus and thus reducing viral infectivity and the peak viral load in a dose-dependent manner. It is possible that NAbs trigger other immune arms, via Fc-mediated mechanisms20. Innate immune responses and CD8 T cells may also reduce the peak viral load56,75. The contributions of these mechanisms to vaccine efficacy remain to be ascertained. Fifth, the severity of symptoms is assumed to depend on the viral load. In clinical trials of vaccine efficacy, individuals are diagnosed after symptom onset. If the minimum viral load for symptom onset, which is unknown, is greater than the assay limit of 100 copies per milliliter, fewer people would display symptoms than would test positive on a nucleic acid test if all individuals were frequently monitored. Vaccine efficacy trials thus report protection against symptoms, whereas our model predicts protection against positivity in a nucleic acid test. The latter might be slightly lower than the former. Sixth, our within-host model assumes that innate immune responses would render target cells refractory to infection. Although this has been observed with other viruses76, and has been used in models of SARS-CoV-2 dynamics77, it remains to be demonstrated explicitly in vivo. Seventh, although we assessed the robustness of our model to parameter variations, several of the parameter values employed were based on previous studies or from other viral infection settings. For example, we assumed that the ratio of the neutralization efficiency of NAbs in vitro and in vivo, ω, was a constant. We set its value, based on estimates for a few NAbs, to one that helped recapitulate clinical data. However, ω might vary across NAbs. Knowledge of these parameter values would make our model more robust and improve its predictive ability. Challenge studies on macaques post vaccination could help test these assumptions.

Finally, we note that our study did not consider viral mutations. With five to ten NAbs active, viral escape from NAb responses is expected to be unlikely51,78. With the new circulating mutant strains7, however, the NAb landscape may have to be reconstructed. Future studies may report DRCs of NAbs against the new strains, facilitating such reconstruction. Our formalism could then be applied to predict the efficacies of vaccines against the new strains.

## Methods

### Data of DRCs

We considered data from studies that reported in vitro DRCs of NAbs using SARS-CoV-2 pseudotyped virions26,27,30,31,32,34,35,36,37,38,39,40,41,42,43,44. We included early studies in our analysis so that the NAbs considered were unlikely to be against mutant virus strains. The assays estimated the fraction of infection events unaffected by the NAbs as a function of the NAb concentration (Fig. 1 and Supplementary Fig. 2). We extracted the data using Engauge Digitizer 12.1 and ensured consistency with reported details, such as dilution levels used.

### Analysis of DRCs

We used the median-effect equation (equation (1)) to analyze the data:

$${\log _{10}\left( {\frac{{f_{\rm{a}}}}{{f_{\rm{u}}}}} \right)} = {{m}{\log _{10}}\left( {\frac{D}{{{\rm{IC}}_{50}}}} \right)}$$
(1)

where $${f_{\rm{u}}}$$ and $${f_{\rm{a}}}$$ are the fraction of infection events unaffected and affected, respectively, by the NAbs in a single round of infection, D is the NAb concentration, IC50 is the half-maximal inhibitory concentration and m is the slope. Data were fitted using the tool REGRESS in MATLAB R2017b. Data points with 1% < fu < 99% were considered for parameter estimation. We fit the data using equation (1), with m and $${- {m}{\log _{10}}({\rm{IC}}_{50})}$$ as adjustable parameters, and obtained estimates of IC50 and m for each NAb. We then computed $${\rm{IIP}}_{100} = {{\log _{10}}\left( {{1} + {\left( {\frac{{100}}{{{\rm{IC}}_{50}}}} \right)^m}} \right)}$$. We did not include NAbs for which the fits were not satisfactory (R2 ≤ 0.8; Supplementary Data 1), possibly arising from large uncertainties in the data. The details of the NAbs and parameter estimates are presented in Supplementary Data 1.

### In silico simulation of plasma dilution assays

We simulated plasma dilution experiments as follows. We assumed that the plasma contained N NAbs in equimolar concentrations sampled from the landscape (Fig. 2a). The reciprocal plasma dilution curve was predicted assuming Loewe additivity (equation (2)) or Bliss independence (equation (3)) between the different NAbs79,80,81 using

$${\mathop {\sum}\limits_{i = 1}^{N} {\frac{{{D_i}/\gamma }}{{{{\rm{IC}}_{50_i}}\left( {\frac{1}{{\varepsilon _{\rm{L}}}} - {1}} \right)^{ - 1/{m_i}}}}}} = {1}$$
(2)
$${\varepsilon _{\rm{B}}} = {{1} - {\mathop {\prod}\limits_{i = 1}^{N}} {\frac{{\left( {{\rm{IC}}_{50_i}} \right)^{m_i}}}{{\left( {{\rm{IC}}_{50_i}} \right)^{m_i} + \left( {{D_i}/\gamma } \right)^{m_i}}}}}$$
(3)

where γ is the plasma dilution factor, εL and εB are the fractions of infection events affected by the plasma in a single round of infection estimated using Loewe additivity and Bliss independence, respectively, Di is the concentration of the ith NAb in the plasma before dilution, and IC50 is its half-maximal inhibitory concentration and mi its slope, with $${{i} \in \left\{ {1,\,2,\,...,\,N} \right\}}$$. We let N = 10 in our simulations26. We estimated the value of γ at which ε = 0.5 as the corresponding NT50. We chose Di as D0/N, and varied D0 between 0.1 and 100 µg ml−1 (D0 is the total NAb concentration).

We repeated these simulations 100 times at different NAb titers, with each simulation representative of an individual patient. We compared the resulting predictions at D0 = 30 µg ml−1 with observations from three patients (Fig. 3b)82, which we also digitized. The equation $${f_{\rm{u}}} = {\frac{{(\gamma )^{n}}}{{{(\gamma )^{n}} + {({\rm{NT}}_{50})^n}}}}$$ was fit to the observations from three patients, merely to ascertain the shapes of the curves and their similarity to those predicted by our calculations. Here, n is the Hill coefficient, γ is the plasma dilution and NT50 is the half-maximal inhibitory plasma neutralizing titer. We also compared predictions of the dependence of NT50 on NAb titers with experimental observations (Fig. 3c)49.

### Model of SARS-CoV-2 dynamics

To predict the protection conferred by vaccines, we developed a mathematical model of within-host SARS-CoV-2 infection post vaccination. We adapted previous models58,67,77,83 by focusing on early dynamics, required to accurately predict the reduction in the peak viral load due to pre-existing NAbs. The following equations described the resulting infection dynamics in vaccinated individuals exposed to the virus:

$${\frac{{{\rm{d}}T}}{{{\rm{d}}t}}} = {- \beta ({1} - \varepsilon ){VT} - {\rho _X}{XT}}$$
(4)
$${\frac{{{\rm{d}}R}}{{{\rm{d}}t}}} = {\rho _XXT}$$
(5)
$${\frac{{{\rm{d}}I_1}}{{{\rm{d}}t}}} = {\beta ({1} - \varepsilon ){VT} - {k}{I_1}}$$
(6)
$${\frac{{{\rm{d}}{I_2}}}{{{\rm{d}}t}}} = {{k}{I_1} - \delta {I_2}}$$
(7)
$${\frac{{{\rm{d}}V}}{{{\rm{d}}t}}} = {{p}{I_2} - {cV}}$$
(8)
$${\frac{{{\rm{d}}X}}{{{\rm{d}}t}}} = {\frac{{{\sigma _X}{I_2}{(1 - X)}}}{{{\phi _X} + {I_2}}} - {d_X}{X}}$$
(9)

Here, uninfected target cells, T, are infected by SARS-CoV-2 virions, V, with second-order rate constant β, producing infected cells in eclipse phase, I1. Cells I1 convert to productively infected cells, I2, with a rate constant k. Cells I2 produce virions at rate p per cell and are lost with rate constant δ. This transition from target cells to infected cells in the eclipse phase and then the productive phase has been employed in previous models58,77. The virions are cleared with rate constant c. The activation of the innate immune response, quantified phenomenologically using X, is assumed to be a saturable function of I2, with maximal rate σX and half-maximal activation parameter ϕX. If I2 is not limiting, X would rise at rate σX at low X and cease to rise as X approaches 1. This form for the innate immune response has been proposed previously84. X converts uninfected cells to an infection-refractory state, R, at a per capita rate ρX, and decays with rate constant dX. The presence of the refractory population has also been proposed in previous models77,85,86. The pre-existing NAbs are drawn as random subsets from the landscape (Fig. 2a) to block new infections with an efficacy ε, which is a function of NAb titers and computed using

$${\mathop {\sum}\limits_{i = 1}^{N}} {\frac{{{D_i}/\gamma }}{{\omega {\rm{IC}}_{50_i}\left( {\frac{1}{\varepsilon } - {1}} \right)^{ - 1/m_i}}}} = {1}$$

where ω accounts for the difference between in vitro and in vivo IC50 values63,71. The neutralizing activity of NAbs follows from our observations above (Fig. 3), where this activity recapitulated plasma dilution experiments.

### Fits to longitudinal patient data

We considered viral load data from non-vaccinated individuals in three cohorts. One dataset57 has patients with mild symptoms. The second dataset59 has measurements in the viral expansion phase. We chose nine individuals from the latter dataset who had frequent viral load measurements. The third dataset58 has data from hospitalized patients classified by their age (above and below 65 years). We simultaneously fit our viral dynamics model (equations (4)–(9)) to viral loads from all three datasets using a population-based fitting approach via nonlinear mixed effects models (Supplementary Section 1).

### Prediction of vaccine efficacy

#### Vaccine efficacy

To examine the variation in peak viral loads with NAb titers (Fig. 4d), we predicted the viral dynamics of 3,500 infected individuals, each individual with different pre-existing NAbs sampled from the landscape as well as different values of viral dynamics parameters (Supplementary Tables 1 and 2). To obtain the protection curve (Fig. 5), we sampled antibody concentrations from a uniform distribution ranging between 0.05 and 100 μg ml−1, mimicking the range seen experimentally, and simulated viral dynamics in 10,000 virtual individuals, as we describe above, and obtained peak viral loads. Furthermore, for each individual, we also simulated plasma dilution assays and estimated NT50. We then binned individuals into narrow ranges of NT50 values. In each bin, we estimated the fraction of individuals with peak viral load below 100 copies per milliliter. This fraction yielded the protection curve. The error bars (95% CI) were obtained using the Clopper–Pearson method and following the protocols used in clinical trials10,87,88.

#### Convalescent NT50

Our model captured the experimentally observed dependence of NT50 on NAb titers in 15 convalescent patients (Fig. 3c)49. NT50 as a function of time post symptom onset has also been measured in the study described in ref. 49. Following vaccine immunogenicity studies28,89, we randomly sampled one timepoint per patient at least 20 days after initial symptoms and obtained a geometric mean NT50 of 565 (Supplementary Fig. 10). We used this value to normalize the NT50 predicted by our model.

#### Parameter sensitivity

The available longitudinal patient data did not allow estimation of all the parameters in our model. We thus fixed several parameters (ρX, σX, ΦX, k, c and dX) and initial conditions (T(0) and I2(0)) to values from previous studies and estimated the rest (β, p and δ) from fits to patient data (Methods). Overall, the good fits (Fig. 4b and Supplementary Fig. 7) indicate that the parameters recapitulate the patient viral load data. Yet, the choice of values of the ‘fixed’ parameters could introduce uncertainties in the overall parameter estimates. We therefore tested the sensitivity of model predictions of vaccine efficacy to fixed model parameter values and initial conditions by increasing or decreasing the parameter values by twofold, one at a time (Supplementary Fig. 11a). We also performed a global sensitivity analysis and estimated the partial rank correlation coefficients of each parameter (Supplementary Fig. 11b). We found that the vaccine efficacy predictions were most sensitive to the viral clearance rate, c. We therefore varied c, refitted the model to patient data using the same adjustable parameters, and repeated our vaccine efficacy predictions. The protection curves were close to the one described above (Supplementary Fig. 11c). We also obtained protection curves using two different iterations after fitting using Monolix (Supplementary Fig. 11d) and when fixed parameter values were varied over a defined range or held constant (Supplementary Fig. 11e). Finally, we tested the sensitivity of the protection curve to the IC50 values of the NAbs in vivo by varying ω (Supplementary Fig. 11f).

## Data availability

All data supporting the findings of this study are available within the paper and its Supplementary Information files. Source data are provided with this paper.

## Code availability

The codes are written in MATLAB and are available on GitHub (https://github.com/PraneshPadmanabhan/COVID-19-vaccine-efficacies) and on Zenodo90.

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## Acknowledgements

This work was supported by the DBT/Wellcome Trust India Alliance Senior Fellowship IA/S/14/1/501307 to N.M.D.

## Author information

Authors

### Contributions

P.P. provided conceptualization, carried out investigations and formal analysis, contributed to writing the original draft and reviewed and edited the manuscript. R.D. performed formal analysis and reviewed and edited the manuscript. N.M.D. provided conceptualization, contributed to writing the original draft and reviewed and edited the manuscript.

### Corresponding authors

Correspondence to Pranesh Padmanabhan or Narendra M. Dixit.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature Computational Science thanks Joshua T. Schiffer and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Primary Handling Editor: Ananya Rastogi, in collaboration with the Nature Computational Science team.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Supplementary information

### Supplementary Information

Supplementary Section 1, Figs. 1–11 and Tables 1–4.

### Supplementary Data 1

Best-fit estimates of dose–response curve parameters of NAbs.

## Source data

### Source Data Fig. 1

Statistical source data.

### Source Data Fig. 2

Statistical source data.

### Source Data Fig. 3

Statistical source data.

### Source Data Fig. 4

Statistical source data.

### Source Data Fig. 5

Statistical source data.

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Padmanabhan, P., Desikan, R. & Dixit, N.M. Modeling how antibody responses may determine the efficacy of COVID-19 vaccines. Nat Comput Sci 2, 123–131 (2022). https://doi.org/10.1038/s43588-022-00198-0

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• ### Correlates of protection via modeling

• Joshua T. Schiffer

Nature Computational Science (2022)

• ### Mathematical model predicts efficacy of COVID-19 vaccines

Nature India (2022)