Explicit modeling of antibody levels for infectious disease simulations in the context of SARS-CoV-2

Summary Measurable levels of immunoglobulin G antibodies develop after infections with and vaccinations against severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). These antibody levels are dynamic: due to waning, antibody levels will drop over time. During the COVID-19 pandemic, multiple models predicting infection dynamics were used by policymakers to support the planning of public health policies. Explicitly integrating antibody and waning effects into the models is crucial for reliable calculations of individual infection risk. However, only few approaches have been suggested that explicitly treat these effects. This paper presents a methodology that explicitly models antibody levels and the resulting protection against infection for individuals within an agent-based model. The model was developed in response to the complexity of different immunization sequences and types and is based on neutralization titer studies. This approach allows complex population studies with explicit antibody and waning effects. We demonstrate the usefulness of our model in two use cases.


Model overview
We propose a generalized antibody model.We assume generalized antibody levels, N ab;v , against all virus variants v of interest.These generalized antibody levels are initially zero.The first immunization event, be this a vaccination or an infection, sets all N ab;v to initial levels, which need to be calibrated.From then on, all levels follow an exponential decay, until another immunization event pushes them up again (the principle is shown in Figure 1).
So far, this follows pre-existing models in the literature.The novelty is that we plug this generalized antibody model into a dose-response model for agent-based infection dynamics.Mathematically, the dose-response model uses an infection probability of where d v is the viral dose of the virus variant v, and Q v is a calibration parameter, which depends on the transmissibility of the virus variant under consideration.In order to calculate the reduced infection probability of immunized persons, we integrate generalized antibody levels in the model as follows: Clearly, for N v = 0 the model returns to its previous form, and N v = 1 has the same effect as dividing the dose by 2.

Calibration
Our model uses data from studies regarding vaccine effectiveness, VE, as input (see also Figure 2).These studies compare persons with a certain immunization history, i.e., vaccination(s) or infection(s), with persons without that; VE is typically given as a function of time after the last of these immunization events.At the beginning of a pandemic, such studies are comparatively straightforward to conduct since only one virus variant is in circulation, and many immunologically naive persons are available for comparison.Later in the pandemic, it becomes harder to study VE as people have increasingly divergent immunization histories, which leads to inhomogeneous immunity profiles; additionally, there are fewer immunologically naive persons to compare with.Our model interpolates the results of vaccination effectiveness studies, which allows us to generalize these to arbitrary sequences of immunization events.Additionally, it is possible to extend immunity profiles, i.e., a synthetic person's N v values, to a new virus variant by using the lab-based neutralization titer measurements.Specifically, if a neutralization titer of a new virus variant w is reduced by a factor of a against an older virus variant v, all new N w values are obtained as N w = a$N v :

Use cases
In the following, we show two use cases of our model (details about STAR Methods and data can be found in Sec.Method details).All results refer to the city of Cologne in Germany.
1. We show that the model is able to calculate population-wide immunization statistics at a given time, even for large populations, i.e., people living in a large city.This is the basis for more complex use cases, such as the following one.Our model is validated by the fact that its outputs closely match the observed data.
2. We show how the model can be used to gain insight into individual population groups and how they are protected against different virus strains.The model can be used as a data source to develop strategies, such as vaccination campaigns, and can compensate for missing data.

Use case 1: Population-wide immunization statistics
The model presented in this paper can be implemented as an extension of our agent-based model (ABM). 10In this way, we can calculate infection dynamics that are also (but not only) dependent on immune protection.We show in the following section how the model can be used to calculate the population-wide immunity at a given point in time.This allows us to evaluate the model in a real-world scenario by comparing it to available data for time periods in which data are available.Moreover, since the strategies of policymakers often depend on the existing immunity in the population, this is a relevant parameter.
In Table 1, we show how our model result compares to observed data. 11The observed data contain information on the percentage of study participants who tested positive for antibodies against the S antigen and for antibodies against the N antigen.Antibodies against the N antigen indicate that the participant had an infection in the past.Antibodies against the S antigen indicate that the participant has been vaccinated or infected in the past.Therefore, we compare the two quantities with the percentage of the population in the model that has had at least one infection and at least one infection or vaccination, respectively.The observed data stem from measurements across Germany in the period from June 2022 until September 2022; the model data refer to Cologne and July 2022.As the observed data are available in this way, the comparison is made for different age groups.The age group < 18 is not shown because we do not have the observational data to compare it to.
When comparing the observed data in Table 1 to the model results, one finds the following.(1) The proportion of participants testing positive for antibodies against the S antigen fits very well with the model results.According to the data, in the adult population, the proportion of those testing positive is 96%; in the model, approximately 97% had at least one infection or vaccination.(2) The proportion of those with at least one infection in the model fits the data less well, with a value of 44% in the data vs. 52% in the model.The deviations at this point could presumably be explained by the fact that the Cologne model is being compared here with data from all of Germany or that the proportion of undetected infections in the model is somewhat too high.The model is compared to reported case numbers in reality, and in order to model the population-wide immunity correctly, it thus needs to make assumptions about under-reporting, i.e., infections that occurred in reality but were not reported.In addition, the study that we compared our results to could be associated with some inaccuracies; e.g., because the measurements have a detection limit, fewer infections are found than have actually occurred.(3) In terms of the proportion with at least one infection, both the model and the data show that this proportion decreases with increasing age.However, the spread in the data between the different age groups is greater in reality than in the model.It is worth noting that our model does not currently account for the age distribution within households, which could contribute to the discrepancy.Additionally, the mobile phone data we utilized to simulate behavioral changes throughout the pandemic are not age dependent, potentially overlooking the higher cautionary behavior exhibited by older individuals in reality.This absence of age-dependent behavioral patterns might explain the observed deviation in our model's representation.
In addition to the data shown in Table 1, there are further surveys 12,13 that attempted to determine what proportion of the population has had at least one infection.We only found data from summer 2022, but the available studies show results similar to our model: by then, about 35%-50% of the German population had been infected with COVID-19 at least once.Since the proportion of vaccinated individuals is higher in the studies than that in the general population, this value must be interpreted as a lower limit and can only be used as a rough guide.For this reason, and because the study results vary somewhat, we believe our model results are acceptable.
The data shown in Table 1 allow a comparison to the model within the different age groups.In order to be able to compare the development over time, we use a publication from the RKI (Robert Koch-Institut), which has compiled various studies on seroprevalence in Germany.The nationwide results are shown in Figure 3 (red) and are compared with our model (blue).The model points show the proportion of the population with at least one infection or vaccination.
In general, it becomes clear that the model and the data match quite well here.At the beginning of 2021, there is a low seroprevalence in the population in both cases, which then increases significantly, especially in summer and fall 2021, and finally reaches values above 90%.

Use case 2: Variant-specific protection of sub-groups
In general, we assume in our calculations that there is no immune protection at the beginning of the pandemic and that each infection or vaccination increases protection, where protection is defined in the same way as vaccine effectiveness, i.e., a reduced probability to become infected compared with immunologically naive persons.The exact methodology is explained in Sec.Method details.In the following, we depict how the population is protected against infection from different virus variants according to the model presented in this paper.These are simulation results that are difficult to validate against real data because the real-world data are not available with this level of accuracy.The results must therefore be interpreted with some caution but offer considerable added value precisely because they cannot be collected in any other way.However, the results from the first use case show that where there are data to compare to, the model fits the data well when it comes to general immunity in the population.
Figure 4 shows the population-wide protection against infection over time averaged over all age groups.The gray area shows that there is a large spread in immune responses; some individuals are subsequently very well protected, while others have almost no protection.This can The observed data refer to Germany and the period from June to September 2022; the model refers to Cologne and July 2022.
partially be explained by the fact that some individuals are unvaccinated (blue dots), while others are vaccinated (red dots) or boostered (more than two vaccinations, green dots).The model results clearly show that vaccinated individuals are better protected than unvaccinated individuals, and missing vaccinations are not compensated for by infections.Thus, unvaccinated individuals do not achieve the same protection through infections as vaccinated individuals.
We calculate the protection for the different variants (left to right).Individuals do not have protection against any variants at the beginning of the pandemic and do not acquire significant immunity throughout 2020.This is because only a small fraction of the population was infected in 2020 and vaccinations were not yet available.Relevant immune protection is achieved by mid-2021 because vaccinations became available for the entire adult population.Beginning in July 2021, a significant decline in immune protection through waning is clearly visible.In winter 2021/2022, we see another protection increase when a third round of vaccinations (boosters) was administered to large segments of the population.The significant increase in immunity in the group of vaccinated persons (red dots) at the beginning of 2022 can be explained on the one hand by infections and on the other hand by the statistical effect that the group becomes smaller and smaller due to booster vaccinations.Due to the vaccination interval between the first vaccination and the booster vaccination, the proportion of persons in the group of vaccinated persons, for whom the vaccination took place a long time ago, is reduced more and more in the course of the booster campaign.
In addition, the different facets of Figure 4 show the impact of immune escape variants: in general, protection against infection with Alpha is significantly higher than that against Delta, and protection against Delta is significantly higher than that against either Omicron variant.
Figure 5 shows how protection varies across age groups.It is clear that the mean protection between the age groups differs significantly.On average, children acquired less protection than adults, which can be explained by low vaccination rates in these age groups.Evidently, according to the model, the lower vaccination rates are not compensated for by infections.According to the officially reported numbers, the group of children under 5 years of age is almost entirely unvaccinated, which results in a low level of immune protection.It also becomes  apparent that the different age groups were vaccinated at different times during the vaccination campaign.The elderly over 60 were vaccinated very early, so immune protection was also built up early.However, due to the early vaccination, there is already a significant decline in vaccination protection in the summer of 2021.In the younger adults, a similar but less pronounced effect is seen; the effect is barely visible for individuals under 18.Because vaccinations were no longer administered strictly by age during the winter 2021/2022 booster campaign, it can be speculated that younger individuals had a shorter interval between 2nd and 3rd vaccination than older individuals.
The simulation results show that protection varies significantly depending on the age group, the variant, the time point, and the number of administered vaccinations.In particular, there is little protection in young children; hence, according to our model, potential vaccines for this age group could have a significant effect.In addition, it is clear that a vaccine adapted to the new variants would be helpful for all ages since the mean protection in July of 2022 in all age groups is only about 50% or less.

DISCUSSION
This study describes a methodology that explicitly models an individual's antibody levels and the resulting immunity to infection, using an ABM as an example framework.The incorporation of this strategy into other frameworks can facilitate the execution of complex population studies taking into account the dynamics of antibodies and their waning effects.During infectious disease outbreaks, such as the COVID-19 pandemic, the formulation of effective public health policies and interventions requires an accurate estimation of population protection and individual infection risk.
In the described use cases we emphasize the significance of incorporating antibody and waning effects into models that predict infection dynamics.If this is taken into account, a more accurate estimation of the individual risk of infection and population protection can be reached.The use cases also demonstrate the ability to infer immune protection for both individuals and populations based on variables including time, age, vaccination status, and virus variant.As a result, an enhanced understanding of immunity dynamics among disparate subpopulations can provide guidance for targeted vaccination initiatives, resource allocation, and the development of more effective public health strategies.Our model's adaptability to other infectious diseases presents an opportunity to improve responses to future pandemics and outbreaks and increases its applicability and potential influence.Nonetheless, our model has limitations and inherent assumptions: except for the initial immunization administered to an individual, in most cases no distinction is made between the effects of vaccinations and infections (see Sec. Method details for details).Furthermore, it is assumed that the half-life of antibodies remains the same regardless of whether they originated from infections or vaccinations.Further, we assume a correlation between antibody levels and protection against infection.This is a key aspect of many epidemiological models, including the one described in this study.This assumption is based on the understanding that the presence of specific antibodies, such as IgG antibodies, can help neutralize the virus, thereby reducing the risk of infection or the severity of the disease.For SARS-CoV-2, evidence from various studies has shown this kind of positive relationship between antibody levels and protection against infection. 5,15,16Research indicates that individuals with higher levels of neutralizing antibodies are generally less likely to become infected or experience severe COVID-19 symptoms.However, it is important to note that the relationship between antibody levels and protection might be non-linear.Multiple other factors, such as cellular immunity or the presence of immunological memory, might also play a significant role in determining an individual's level of protection.
Overall, we believe that these considerations do not limit the applicability of the presented model, as the given empirical evidence from the presented use cases demonstrated.With the increasing availability of research on SARS-CoV-2, COVID-19, and other infectious diseases, it will be possible to refine and expand these assumptions to further improve the precision and applicability of our model.

. Protection against infection according to model for different variants and age groups
The color coding is as follows: blue: unvaccinated, red: vaccinated, green: boostered, black: mean (whole age group).
In the field of infectious disease epidemiology, the methodology presented in this paper offers a valuable method for modeling antibody dynamics and protection against infection.This will help inform more effective public health policies and interventions, ultimately benefiting global efforts to combat and manage the spread of infectious diseases.

Summary & conclusion
We have presented an approach on how to model the variant-specific neutralizing effect of antibodies and how to convert it into a protection against infection.The presented use cases demonstrate that the model produces valid results that match the observed historical data in Germany very well.Further, we have shown how this approach can be used within an agent-based modeling framework to allow computation of infection dynamics.In the (current) situation of high population immunity, considering immune protection is essential for achieving realistic simulation results.
Our model estimates that in summer 2022 there was still a significant difference in immune protection between unvaccinated and vaccinated individuals.According to the model, the lack of vaccination was not fully compensated for by infections.This effect also becomes clear when looking at the age groups: according to the model, children had a significantly lower protection against infections than adults.In addition, the model allows quantification of the protection against the immune escape variants.These results suggest that the protection against the Omicron variants is significantly lower than that against the original (wild-type) variant.This matches the available data.
The necessary model parameters have either been taken from the available literature or are based on calibration to available data.This process necessarily includes modeling choices.Given the solid agreement between our model results and the available data, we are confident that sensible parameters and fitting parameter values have been identified.This is also confirmed by simulation results that have been achieved by using the presented antibody model in conjunction with our own ABM (see, for example, Mueller et al. 17 and Mueller et al. 18 ).These results demonstrate once more that the ABM with the presented antibody-model extension is able to soundly replicate many important parameters, such as case numbers, R values, and hospitalizations.This is a significant improvement based on the explicit antibody model for each agent.
To the best of our knowledge-and based on our literature review-no other currently available model allows both (1) the integration of antibody levels as a proxy for protection against infection and (2) the modeling of individual immunization histories.While a small number of models implemented one of these, we have not encountered any that implements both.In consequence, our approach could help others to integrate any permutation of immunization events, as well as waning, into their COVID-19 models.

Limitations of the study
The proposed model, as it is described in this paper, is well suited for the presented application scenarios and can also be adapted for future use cases.However, limitations and possible improvements exist and will be briefly described in the following.
As described in detail in the STAR Methods section, our model uses a uniform factor of 15 to increase antibodies for all vaccinations and infections, if this is not the first immunization event.That is, for this parameter, we do not distinguish between the different virus variants or vaccines.In principle, that factor could be different for each virus or vaccine variant, and/or for each prior immunization sequence.Differentiating between all these would yield many free parameters, which would be difficult or impossible to calibrate.This would go against the purpose of the model to be used for predictive simulations during the acute phase of a pandemic.For this, certain simplifications are necessary.The simplification here was to use the factor of 15 from data for the most typical case, which was a booster vaccination after the initial vaccination, and then to assume that the same factor applies for all other cases.Clearly, it would be possible during a pandemic to systematically test for other cases, for example, for a vaccination after an infection, or an infection after a vaccination, and from that to decide whether the values need to be differentiated or not.For the COVID pandemic, we were not aware of such studies.
Furthermore, when it comes to waning, we do not distinguish between infection and vaccination, but assume the same half-life.This is on the one hand due to the fact that we have not found any data that assume the opposite, and it on the other hand once more avoids that the number of model parameters increases significantly.Where possible, we have generally tried to reduce the number of model parameters.
Adjustments to the model would be straightforward since the entire source code 19 is publicly available.

STAR+METHODS
Detailed methods are provided in the online version of this paper and include the following: Individual may pass through the following infection stages: susceptible, exposed, infectious, and recovered (SEIR).The associated GitHub repository has last been committed to in January 2022, and their paper, 24 which was published in July 2021, mentions that they will incorporate waning efficacy, but no more recent information or model description could be found.Their additional methods preprint, 25 which does not explicitly mention Covasim but is noted on their project website, served as the basis for our antibody model and is discussed in   variant.This is then integrated into their integro-differential equations which describe the transitions between compartments.In the appendix of their recent publication 36  September 2021 does mention neither antibodies nor waning, describes their agent-based model (not mentioning the influence of multiple vaccine jabs/infections), and focuses on finding an optimal lockdown strategy for Poland.In said preprint, they define the probability of infection as p infection = 1 À expð À aIÞ.Here, a is the transmission coefficient and I is the total infectivity defined via I = P c w c I c , where the sum is over the different infection contexts (household, workplace, preschool, school, university, large university, street, and travel), w c is the time-dependent contact rate of context c, and I c is the time-dependent context infectivity depending on the number of (a) symptomatic and all agents at context c, the fraction of symptomatic agents who do not self-isolate, and a curbing parameter for the infectivity of the asymptomatic agents.Consequently, no protection acquired from infection/vaccination is included.

Background
We used the models of Cohen et al. 25 and Cromer et al., 46 with details for the latter in Khoury et al., 47 as starting points for the process of integrating antibodies into our agent-based model.Both deal with the connection between neutralizing antibodies with protection against infection.They both postulate a logistic model of type for vaccine effectiveness, where N ab is the measured antibody level, N 50 is the antibody level at which VE is 50%, and b determines the slope at N ab = N 50 .Translated into relative risk, which we here call immFac, this can be rearranged to where N is a strain-specific relative antibody level and is defined as N : = N ab =N 50 (see Sec. Modelling the antibody level for more explanation).N is unit-less and would need to be multiplied with N 50 to be expressed in laboratory units.Note that N is time-dependent, as antibodies decrease over time, and increase when an infection or vaccination occurs (see Sec. Modelling the antibody level).The value for b is chosen through calibration (see also Sec.Calibration).The equation shows that a relative antibody level of 0 leads to an immunity factor of 1, i.e. a VE of 0%.An antibody level of 1 leads to an immunity factor of 0.5, i.e. a VE of 50%.An antibody level above 1 corresponds to an immunity factor below 0.5, i.e. a VE higher than 50%.

Integration with a dose-response model
Our agent-based model 10 uses the following well-established dose-response model to calculate the probability of infection: [48][49][50] where d is the viral dose, and Q is a calibration parameter, which depends on the transmissibility of the virus under consideration.
The open question was how to include immFac in the above dose-response infection model; since most simulations use a compartmental approach, they do not need to resolve this issue.A possible form, p immunized inf = immFac$p not À immunized inf , would imply full protection for people with high antibody levels, even in virus rich environments. 51This does not seem plausible, given that the virus eventually overcomes the antibodies if the ratio of virus to antibodies is large enough.
As a consequence, we put immFac into the exponent: Note that this has the consequence that in a virus-limited environment, where dose d is small, immFac becomes a risk reduction: This linear approximation in a virus-limited environment follows from 1 À expðÀ xÞzx for x R 0 and sufficiently small.That is, a model that was originally developed for a macroscopic situation is now used at a more microscopic level.The epidemiological risk reduction would come out as an average over many exposures with different values of d.
Equation 2 shows how antibodies reduce the likelihood of becoming infected (reduced susceptibility).However, we also included the fact that individuals with antibodies have reduced probability to transmit the virus (reduced infectivity).Thus, when an unvaccinated agent has contact with a vaccinated agent, the unvaccinated agent indirectly benefits from the vaccinated agent's antibodies because the probability of infection is reduced.If both agents are vaccinated, the probability is further reduced.This is in accordance with findings by Eyre et al. 52 In our model, the infectivity is reduced according to the same principle as explained above, but to a lesser extent.The effect of the antibodies on infectivity is 25% of the effect they have on susceptibility.Thus, if an agent has a 50% reduced probability of infection due to their antibodies, the probability of transmission is reduced by only 12.5%.

Modeling the antibody level
In the next step, the relative antibody levels (N in Equation 2) are modelled.For every simulated day and agent, the model updates the agent's relative antibody level with respect to each SARS-CoV-2 strain.A relative antibody level of 0 corresponds to no protection, while a relative antibody level of 1 corresponds to 50% protection (see Sec. Background).At the beginning of the simulation, all agents are initialised with a relative antibody level of 0. Immunization events (vaccinations and infections) increase an agent's relative antibodies.On days on which no immunization event occurs, the antibody levels follow an exponential decay curve, NðtÞ = N 0 $2 À t=t 0:5 ; (Equation 4) where NðtÞ is the antibody level on day t after the most recent immunization event, N 0 is the antibody level immediately after the most recent immunization event and t 0:5 is the half-life, which is calibrated as 60 days, see Sec.Calibration.The value of 60 days is on the lower end of what can be found in the literature 46,53,54 ).
The general principle of the model is exemplified in Figure 4.The left figure shows how the antibody level of an agent develops over time.The spikes in relative antibodies correspond, in this illustrative example, to a SARS-CoV-2 infection, an mRNA vaccination, and an infection with the Delta variant.On days without an immunization event, the waning becomes apparent.In addition, it becomes clear that we distinguish between the different virus variants.As a result, this means that agents are less protected against the immune escape variants after vaccination.The right plot shows how we translate the antibodies into protection.See Sec. Background and Sec.Integration with a doseresponse model for details.

Initial immunization
As noted above, we assume that initially (at the beginning of the pandemic) all agents have an antibody level of 0. The first immunization event generates a strain-dependent initial antibody level, which is shown in Table S1.The agent's antibodies have varying neutralizing effects against different SARS-CoV-2 strains.Thus, we model that an agent has a different relative antibody level per strain.As shown in Table S1, an infection with Delta provides more relative antibodies against a reinfection with Delta than against an infection with an Omicron variant.Similarly, the vaccinations were designed to protect against the wild-type and Alpha variants; thus, the vaccinations provide more relative antibodies against these strains than against later variants.Table S1 is based on studies that examined protection against (symptomatic) infection and on various studies that measured antibody titers after vaccination or infection. 55,56Here, protection obtained through vaccination with the mRNA vaccines developed by Moderna (mRNA-1273), and by BioNTech-Pfizer (BNT162b2) are summarized under 'mRNA', while the vector vaccines developed by AstraZeneca (ChAdOx1-S) and Johnson & Johnson (Ad26.COV2.S) are summarized under 'vector'.In consequence, we do not distinguish between vaccine brands, but only between vaccine types.
The starting point for Table S1 was protection after vaccination with an mRNA vaccine against the wild-type, the Alpha, the Delta and the Omicron BA.1 variant (marked with + in Table S1).][23] To match these studies, the corresponding initial antibody values in Table S1 were calibrated.In the same step, the half-life of 60 days from Equation 4 was estimated (for the calibration process, see Sec.Calibration and for the conversion between vaccine effectiveness and neutralizing antibodies, see Equation 6. In the next step, we used measurements from Roessler et al. 55,56 to populate the other entries.For example, the second row of Table S1 represents the relative antibodies versus various strains resulting from a vector vaccination.For Alpha, 55 measure a neutralizing effect of approximately 700 after mRNA vaccination and approximately 210 after vector vaccination (we obtained these values from Figure 1 in Roessler et al; 55 ).We used this ratio to calculate the relative antibodies against Alpha after vector vaccination: 29:2$210=700 = 8:76.The remaining entries in the table were filled following the same logic.
The measurements by Roessler et al. 55,56 and others show that there is virtually no neutralizing effect if the initial immunization event is an Omicron infection, so we assume a very low value (0.01) here.We do not use 0, as it is to be expected that at least a small protection is present in the case of repeated infections.
We did not have accurate measurements for Omicron BA.2 and BA.5 at the time of the study; thus, we calibrated the immune escapes using our agent-based model.Here, we take the values for BA.1 from Table S1 and divide them by a factor.The factor was calibrated so that our model correctly replicates the infection dynamics, in particular the initial growth of BA.2 and BA.5, respectively.

Agent heterogeneity
To account for the fact that immune response towards vaccinations or infections varies across the population, we assign an immuneRes-ponseMultiplier to each agent.The lowest possible immuneResponseMultiplier is 0.1, which is an attempt to adequately depict the immunocompromised population; the maximum multiplier is 10.0.Table S1 presents the initial antibodies for an individual with an average response to immunization events (immuneResponseMultiplier = 1.0); for low and high responders, the antibodies shown in the table are multiplied by an agent's immuneResponseMultiplier to calculate the antibodies gained in response to an immunization event.A log-normal distribution of immuneResponseMultiplier with a m of 0.0 (corresponding to a median of 1.0) and s of 3.0 is applied to the population.

Subsequent immunizations
If the agent is subject to an additional immunization event, their antibody levels against each strain will be multiplied by a factor of 15, regardless of whether a vaccination or infection occurs. 57The maximum antibody level that an agent can have is 150 (which corresponds to a protection of nearly 100%).If the multiplication by 15 still leads to a lower protection than indicated in Table S1, then the value from Table S1 is used insead.This means that, at minimum, the initial antibody level from Table S1 is always reached.can calculate the antibodies separately by vaccination status.For the analysis, we use one data point per month to save computing time during the analysis and to keep the figures clear.However, the actual simulation runs on each day, so that a day-by-day view would be possible.
Antibody models for epidemiological predictive modeling of COVID-19: A literature search The previous section described our approach for modelling antibody levels in an epidemiological context.In this section, we present an overview of similar approaches that exist in the literature, as of July 2022.We compiled a list of all models that have been listed in one of the following resources: (a) the Covid-19 Forecast Hub (From the community sub-section, as of 07/21/22.)( 59), (b) the European Covid-19 Forecast Hub (From the community sub-section, as of 07/21/22.)( 60,61 ), and (c) the European Covid-19 Scenario Hub (From the models sub-section, as of 08/31/22.)( 62 ).The final list contains 90 models.Additionally, to the 86 models from the three resources, we also included four more models that we found through a PubMed literature search.The full list can be found in Appendix A. To get the relevant information for the individual models, we went to the respective websites, and analyzed connected publications and available source codes (e.g. from GitHub).We were in particular interested in models that either (1) related antibodies and protection against infection and integrated this into their model or (2)  acknowledged and integrated into their model the waning of protection against infection (after vaccination and/or infection).
From this literature review, we conclude that, apart from Covasim, 24,25 whose influence on our model is discussed in Sec. Background and which can be found as model # 1 in Appendix A, none of the reviewed models explicitly integrates antibody levels as part of their infection sub-model.This is also due to the fact that many models focus on the prediction of hospitalization numbers and thus do not need to explicitly model individual antibody levels.However, some approaches seem to have integrated some kind of vaccination or antibody sub-model, but no detailed description was found.This includes (see Appendix A for details): IHME, UC3M-EpiGraph, ECDC-CM_ONE, and SIMID-SCM.

Figure 2 .
Figure 2. Calibration results.Dots were taken from the literature, 15,16,20-23 lines are the fitted curves On day 210 the agent receive a booster dose, which increases their level of protection.

Figure 1 .
Figure 1.Exemplary immunization history The agent gets infected with the wild type on day 50, receives the mRNA vaccine on day 200, and gets infected with the Delta variant on day 600.Left: Neutralizing antibody levels, Right: Resulting protection against infection (Protection is computed as follows: 1 À immFac = VE).

Figure 3 .
Figure 3.Comparison of seroprevalence in the model (blue) and observed data 14 (red) over time The observed data refer to all of Germany, the model to Cologne.

Figure 4 .
Figure 4. Protection against infection according to the model for different variantsThe color coding is as follows: blue: unvaccinated, red: vaccinated, green: boostered, black: mean protection (whole population), gray area: 10th to 90th percentile.The red dots (vaccinated) do not include boostered individuals, meaning that every person is part of only one group.Reading example: for the Delta variant (2nd plot) it becomes apparent that unvaccinated (blue) have a significantly lower protection that vaccinated (red) or boostered (green) by July 2022.

Figure 5
Figure 5. Protection against infection according to model for different variants and age groupsThe color coding is as follows: blue: unvaccinated, red: vaccinated, green: boostered, black: mean (whole age group).

Sec
Agent-based model developed by MRC Center for Global Infectious Disease Analysis hosted at Imperial College, London.Model documentation, as part of the associated GitHub repository, 26 has last been updated in February 2021.No mentions of vaccines, waning or antibodies.3 CoSim N/A Lit.Search Expanded SEIR model containing 27 compartments (as of December 2021). 27COVID-19-related metrics are computed on German federal state level.Checking both the model description on their website as well as the FAQs for their simulator (which have last been updated in December 2021), no information on waning or antibodies could be A stochastic, discrete-time, individual-based transmission model of infections and disease dynamics.In their paper, 28 dating from December 2021, the authors note that they did not consider waning immunity in this study.But, in the supplemental information, they model the probability of transmission as pðtransmissionÞ = b$n i ðtÞ$4 1 $sðtÞ$ð1 À m s Þ, where n 1 ðtÞ ˛½0; 1 denotes the viral load of the infectious individual and t denotes they days following infection.4 1 denotes the infectivity factor of the SARS-CoV-2 variant with which the infectious individual is infected, sðtÞ denotes the season factor at date t, and m s denotes the immunity of the susceptible individual.Here, m s = 83% for recovered individuals (independent of disease severity, risk group or age) and m s = 80% for vaccinated individuals.GitHub repository has last been committed to in January 2022.Hence, we could not determine whether or not they are still continuing their work and if they have by now integrated waning immunity into their model.(Continued on next page) from the COVID-19 Forecast hub are ranked according to their performance over the previous four weeks and then ensembled and weighted iteratively to achieve a combined forecast.
Blog post dating from December 2021,35 which describes their model update in detail.Their new model is a system of integrodifferential equations.(1) Individuals are placed in different compartments based on variant by which they were more recently infected and round of vaccination they most recently got / One compartment for each combination of vaccinations and infections (at the time of the blog post they are considering 24 compartments); (2) Time since last vaccination and/or infection is tracked; (3) Protection from infection and vaccine interact multiplicatively.In other words, if ε is the protection (against a variant) acquired from vaccination and 4 is the protection acquired from infection, then a person's risk of infection is ð1 À εÞð1 À 4Þ times the risk of a naive individual.Hence, an individual's risk of infection depends on the variant and time of their last infection, the brand and time of their last vaccination, and the variant they're currently confronted with.They estimate the average protection in a particular susceptible compartment to a specific

Table 1 .
11mparison of population-wide immunity according to observed data from11(1st and 3rd row) and our model (2nd and 4th row)

TABLE
YES, NO, N/A) Explicit Modeling of immunisation events (distinguish between infection, immunisation etc.) AND/OR of waning AND/ OR of a relation between antibodies and protection against infection.If the model description solely mentions any of the criterion above, but provides no details, we still set EMI = N/A.If the same model was, under a slightly different name, part of multiple of our sources, we put down all names separated by a ''/''., which were discovered through (a) the Covid-19 Forecast Hub, (b) the European Covid-19 Forecast Hub, (c) the European Covid-19 Scenario Hub, or (d) a literature search d RESOURCE AVAILABILITY B Lead contact B Materials availability B Data and code availability d EXPERIMENTAL MODEL AND STUDY PARTICIPANT DETAILS d METHOD DETAILS