A mathematical model simulating the adaptive immune response in various vaccines and vaccination strategies

Vaccination plays a pivotal role in upholding human health, and its impact is closely intertwined with population growth and increased life expectancy^1,2,3. Throughout history, numerous infectious diseases of great significance, such as smallpox, plague, and cholera, have been eradicated or effectively controlled through vaccination efforts^4,5,6.

In recent years, significant advancements have been made in vaccine technologies, including adenoviral^7,8, mRNA^9,10, and attenuated vaccines^11,12. Adenoviral and mRNA vaccines have been extensively utilized in the prevention of SARS-CoV-2¹³. Quantitative research using mathematical models to study the interactions between vaccines and hosts can provide a better understanding of the mechanisms by which vaccines stimulate antibody responses, offering theoretical guidance for scientific vaccination strategies and dosing regimens. The application of mathematical modeling in studying host-pathogen interactions also provided substantial theoretical guidance for the prevention and treatment of infectious diseases^{14,15,16,17,18}. Moreover, mathematical modeling has found promising applications in the field of vaccines. For instance, with the consideration of pre-existing immunity, Rajat Desikan et al. developed vaccine models to anticipate guidelines for updating vaccines against evolving pathogens like SARS-CoV-2 and influenza¹⁹. Cristina Leon et al. successfully simulated the innate and adaptive immune responses of hosts to COVID-19 infection or vaccination through the utilization of a mathematical model²⁰. Indrajit Ghosh utilized a mathematical model to investigate the efficacy of antiviral drugs and vaccination in the dynamics of SARS-CoV-2 infection²¹.

However, to our knowledge, no model has been developed that comprehensively simulates antibody generation following administration of different types of vaccines. Drawing from a comprehensive review of the aforementioned research, we have advanced the prevailing antibody kinetics model²² by integrating a novel element: vaccination. Our primary focus is to scrutinize the specific activation mechanisms of vaccines on host adaptive immune responses. This manuscript presents a meticulous examination of our research findings, commencing with a systematic overview of our model and its distinguishing characteristics vis-à-vis other models. In this context, we establish a lucid demarcation of various parameters within the model, ascribing them to factors such as viral pathogenicity²³, clinical symptom severity²⁴, and antigen-specific T-cell immunogenicity²⁵.

Subsequently, employing our refined model, we extensively evaluate the efficacy of distinct vaccines and varied administration strategies. Furthermore, utilizing the developed model as a foundation, we propose four fundamental strategies to inform vaccine design: enhancement of antigen-specific T-cell immunogenicity, targeted elicitation of high-affinity antibodies, attenuation of IgG decay rate, and reduction of peak levels in vaccine antigen-antibody complexes. Our model provides a comprehensive and quantitative elucidation of the modulatory effects induced by diverse antigenic substances on adaptive humoral immunity. As a result, it offers invaluable theoretical insights for future endeavors in both mathematical modeling and experimental investigations in this field.

An overview of the immnodynamic model

Prior to delving into the specific mathematical equations, we provide a macroscopic overview of our model to enhance readers’ comprehension of this mathematical framework. Our model can be concisely represented by the aforementioned flowchart, which comprises five core components and thirteen significant reactions. Specifically, Reaction 1 denotes the binding of B cells producing IgM with antigenic substances, resulting in the formation of antigen-antibody complexes. Concurrently, these antigen-antibody complexes interact with Th cells, eliciting immunological responses from Th cells. The antigenic substances implicated in this reaction may encompass protein constituents found in inactivated vaccines, live viruses, or those translated from mRNA vaccines. Reaction 2 signifies the recognition and swift elimination of IgM-antigen complexes by the immune system, potentially involving various immune cells such as Natural Killer (NK) cells.

Reaction 3 encompasses the proliferative influence exerted by Th cells on adjacent B cells, a process that is frequently overlooked in prevailing mathematical models. We explicitly incorporate this positive feedback effect into our model through Reaction 3. When B cells expressing specific antibodies bind to antigens, the antigens are recognized, engulfed, and subsequently lysed by B cells, leading to the generation of corresponding peptide fragments. These peptide fragments possess distinct Th cell immune stimulatory properties. The interaction between B cells and Th cells is facilitated through the binding of B cell antibodies to Th cell binding sites, which may involve intermediary molecules such as CD8. The lysed peptide fragments are presented to Th cells, triggering alterations in their signal transduction pathways and subsequent secretion of diverse cytokines. These cytokines foster the proliferation of Th cells themselves and neighboring cells. As a consequence, B cells recognizing the pertinent antigen, along with their corresponding Th cells, can undergo rapid and substantial proliferation within a compressed timeframe^26,27. Hence, we posit that the regeneration of B cells, or more precisely, the regeneration of antibodies, emanates from the presence of antigen-antibody complexes. Consequently, the rate of regeneration is directly proportional to the concentration of antigen-antibody complexes.

Reaction 4 embodies the intricate process by which IgM undergoes transformation into IgG. The generation of IgG-producing B cells involves two distinct sources: one arises from the conversion of IgM molecules of the same isotype, while the other emerges through the proliferation of B cells themselves, a process denoted as Process 10. Unfortunately, prevailing models often overlook this transformation and fail to consider the interconnectedness between IgM and IgG. This oversight significantly undermines the accuracy and reliability of these models. For instance, when studying dengue virus infection, dynamic simulations involving IgM are invariably required. However, models that do account for IgM frequently isolate it from IgG for separate analyses, a practice that evidently lacks veracity. In the subsequent results, we will demonstrate the profound implications of the IgM-IgG relationship, particularly in the context of selecting optimal vaccination strategies. It is noteworthy that many vaccines incorporate booster doses precisely due to this relationship. IgM represents the initial class of antibodies within the human body, boasting an antibody repertoire significantly larger than that of IgG. For viruses to which we have never been exposed, our IgM antibody repertoire comprises antibodies with robust binding affinity, whereas the IgG antibody repertoire may lack antibodies of comparable strength. Consequently, in the face of such infections or upon receiving the initial dose of such vaccines, neutralizing antibodies primarily stem from IgM, while IgG production occurs subsequently, initially arising from the transformation of IgM.

Reaction 5 signifies the specific binding of self-antigenic substances to IgM antibodies. Our previous theory on antibody kinetics has substantiated the indispensable role of environmental or self-antigens in antibody preservation, a principle applicable to both IgM and IgG antibodies. Reaction 6 elucidates the clearance process of IgM-self-antigen complexes, while Reaction 7 illustrates the stimulating effect of these complexes on the regeneration of IgM antibodies. Incorporating the role of self-antigens in antibody preservation constitutes another pivotal characteristic of our model. Neglecting the presence of self-antigenic substances would result in degradation, leading to the gradual loss of the initial antigenic stimulus and subsequent decline of all antibodies. Ultimately, this would lead to the eradication of antibodies. Therefore, it is imperative to consider self-antigens in order to explain the phenomenon of long-term antibody persistence and the provision of sustained protection, which numerous existing models fail to address. Consequently, we introduce the concept of self-antigenic substances. In fact, self-antigens play an exceedingly crucial role in the maturation, differentiation, and clonal selection of immune cells. We have conducted more comprehensive research on this subject in a previous article²⁸. The same principle applies to the maintenance of IgG antibodies through self-antigens, which is represented by Reactions 11, 12, and 13. As for IgG, the interaction between antigens and IgG follows the same type as that between IgM and antigens, and it is delineated by Reactions 8, 9, and 10.

A mathematical modeling of adaptive immune response in different vaccine types

According to Fig. 1, we have listed the reactions involved and presented them in Table 1. Tables 2, 3 represent the selection of parameters and the values of initial variables, respectively. The antigen replication reaction is represented as reaction 14. All degradation processes are represented as reaction 15–19.

Model schematic. 13 core reactions and 5 components are illustrated in the model. Those 5 components include virus antigen, self-antigen, Immunoglobulin M (IgM)-producing B-cell, IgG-producing B-cell, and helper T-cell. Each reaction is represented in green line with arrow indicating the reaction direction.

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Corresponding ordinary differential equations (ODEs) can be derived as follows:

The above equations were solved numerically using the ode15s function in MATLAB³². It is important to note that the concentration of self-antigen-like substances is considered constant, as they can be rapidly replenished from the environment. Therefore, the sixth term in the equation is set to 0.

For inactivated vaccines, such as inactivated primary vaccines or inactivated recombinant vaccines like adenovirus vector vaccines, the viruses are unable to replicate. Consequently, the value of k₁₄ is set to 0. Similarly, for mRNA vaccines, the expressed protein antigens cannot undergo self-amplification, resulting in k₁₄ also being equal to 0.

However, in the case of attenuated vaccines, this value is non-zero. Unlike regular viruses, attenuated vaccines, such as those based on defective viruses, exhibit significantly reduced replication activity. Thus, the value of k₁₄ is significantly smaller compared to regular viral infections.

For mRNA vaccines, mRNA molecules can be transcribed into antigen-like substances. Therefore, the initial concentration of antigen-like substances is set to 0, while the initial mRNA concentration is assigned a higher value. Conversely, for inactivated and attenuated vaccines, the initial mRNA concentration is assumed to be 0. Although attenuated viruses still rely on mRNA for replication, we have omitted the consideration of mRNA in the model. The ultimate outcome of virus replication is the production of replicated antigens.

Another notable distinction between inactivated vaccines and mildly pathogenic virus infections lies in the difference in initial antigen concentration. Inactivated vaccines, due to the inability of antigens to self-replicate, require a higher injection concentration to achieve a desirable immune response. Conversely, there is no dosage threshold for vaccines against mildly pathogenic virus infections. As a result, the initial invading viral antigen concentration can be relatively low, yet still elicit an effective immune response.

Parameter estimation based on clinical data of adenovirus-based COVID-19 vaccines

Strictly speaking, adenovirus-based COVID-19 vaccines do not belong to the traditional category of inactivated vaccines. However, since the antigens expressed by these vaccines also possess non-self-proliferating capabilities, we define them as a subtype of traditional vaccines in our simulation, and consider them as a form of inactivated vaccine. In order to enhance the reliability and applicability of our model, after presenting the theoretical framework, we further utilized adenovirus-based COVID-19 vaccines to fit the model parameters. We employed two sets of data for parameter fitting: the change in antibody titers for the 18–55 age group after two normal doses, and the change in antibody titers for the 56–69 age group after two normal doses³³. Peripheral blood samples were collected for immunological analysis at four key time points: prevaccine baseline (time point 1), 28 days after the first dose (time point 2) which is also the time for the second dose, 42 days after the first dose (time point 3), and 56 days after the first dose (time point 4). All fits to clinical data using our model were performed in Matlab using non-linear mixed-effects models (Eq. 10). This is the difference on Log10 scales between the model output (\(\:{\stackrel{-}{y}}^{ij}\)), and the experimental measurement (\(\:{y}^{ij}\)). n is number of the data points per replicate, m is the number of replicates. Since IgG concentrations vary over many orders of magnitude over time, we log-transformed these quantities during fitting.

An excessive number of parameter settings can diminish the reliability of parameter estimation, leading to reduced parameter identifiability. Due to the limited data available for fitting, we have employed the following simplifications to achieve more accurate estimations for all parameters: in our model, IgM and IgG have the same isotype, hence k1 = k8 and k−1 = k−8. k2 = k6 = k9 = k12, k3 = k7 = k10 = k13. Furthermore, due to the extremely small value of the dissociation constant k−1, sensitivity analysis showed that the output was not very sensitive to this parameter. Therefore, the parameters we fitted were k1, k2, k3, and k16, which may exhibit significant inter-individual differences. From a physical mechanism perspective, the parameters k11 and k−11, which represent the properties of environmental antigenic substances, have very weak impact on the short-term trend of IgG concentration changes. Additionally, since clinical data only involves changes in IgG, parameters such as the decay rate of IgM (k15), which cannot directly influence the fitting data, were not included in our parameter fitting. k1 represents the affinity between different antibodies and antigens, which is an important factor that can vary among individuals. k2 represents the clearance time of antigen-antibody complexes. If NK cells have high activity, then the clearance of these complexes will be faster. k₃ represents the ability of antigen-antibody complexes to stimulate antibody regeneration, which is related to the activity of germinal centers. k₁₆ represents the decay rate of IgG in the body. In the model, the initial value of x₂ is the vaccine dose, and since there is no mRNA, the initial value of x₉ is 0. Through this simplification, the number of parameters to be estimated has been reduced from 17 to 4, with the fitting results provided in Table 5. For the antibody kinetics following mRNA vaccination in individual subjects, we introduced two parameters related to mRNA properties, decreasing the total number of parameters from 19 to 6. The fitting results for this model are presented in Table 6.

Parameter estimation based on clinical data of mRNA vaccine

In order to enhance the reliability and applicability of our model, we further utilized clinical data from mRNA vaccines to fit the model parameters. The dataset consisted of IgG antibody data from 39 individuals who received two doses of the BNT162b2 vaccine. Originally, the dataset included information from 44 participants, but 5 of them had missing data³⁴. We selected the 39 individuals with complete data (IgG concentrations at 4 time points) for parameter fitting. Blood samples were collected at four key time points: prevaccine baseline (time point 1), 2 weeks after the first dose (time point 2), the day of the second dose (time point 3), and 1 week after the second dose (time point 4).

All fits to the clinical data using our model were also performed in Matlab fmincon function using non-linear mixed-effects models (Eq. 10). Due to the wide range of IgG concentrations over time, we log-transformed these quantities during fitting to account for the magnitude differences.

Unlike Adenovirus-Based COVID-19 Vaccines, where the key mechanism is the immune response induced by the antigens expressed by the vaccine, mRNA vaccines rely on the translation of mRNA into antigens to initiate immune responses. Therefore, the initial value of x₂ is 0, indicating no mRNA present initially, while the initial value of x₉ represents the vaccine dosage. Additionally, in the parameter fitting process, we introduced two additional parameters, k₁₈ and k₁₉. A full list of estimated parameters is listed in Table 4.

We performed parameter fitting for each individual to obtain their individual parameters, and the analysis of these parameters allows us to obtain the distribution of population-level parameters. The same method can be used to fit parameters for attenuated vaccines. However, as there is currently no attenuated vaccine available for COVID-19, no such fitting work was carried out in this study.

This study utilized de-identified clinical data obtained from previously published studies^33,34. The original studies had obtained appropriate ethical approval from their respective Institutional Review Boards (IRBs) and had obtained informed consent from participants. No new data collection or human participant involvement was conducted for the purposes of this study.

Kinetic modeling of antibody response following inactivated vaccine administration

Inactivated vaccines represent a conventional approach in vaccine production. The term “inactivated vaccines” encompasses both the traditional method of rendering original pathogens non-infectious, as well as other approaches utilizing non-replicating forms of antigens for antigen delivery. This definition also extends to genetically modified inactivated vaccines produced through gene recombinant techniques. However, conventional inactivated vaccines encounter notable challenges in the development of RNA virus vaccines. Instability of RNA viruses during replication cycles poses a significant obstacle, leading to an increased presence of defective viral genomes (DVGs) with each passage. Eventually, this results in the complete loss of viral replication activity³⁵. Nonetheless, genetically engineered vaccines have been industrially developed to combat RNA virus infections, such as the AstraZeneca vaccine for COVID-19³⁶. Optimal efficacy for most inactivated vaccines often necessitates booster doses, and our simulation aims to elucidate the underlying mechanisms involved.

As depicted in Fig. 2, the concentration changes subsequent to the initial dose are indicated by solid lines, while dashed lines represent the concentration changes of different substances following the second dose. Sustaining elevated levels of IgG within the body holds paramount importance in preventing reinfection owing to its notably slower decay rate in comparison to IgM. It is evident that the initial rise in IgG levels following the first dose is limited (as denoted by the green solid line), whereas there is a substantial increase in IgM levels (as indicated by the red solid line). This phenomenon primarily arises due to the initial scarcity of IgG within the body at the time of vaccination. Consequently, a small quantity of IgG is derived from the conversion of IgM, accompanied by the self-amplification of IgG through antigen binding. Simultaneously, IgM undergoes rapid proliferation, but experiences swift decay once antigen-like substances become depleted. The proliferation of IgM ceases when it is no longer stimulated by antigens, leading to a rapid decline. During this process, a fraction of IgM continues to transform into IgG.

At the 50th time unit, administering the second dose leads to a notably distinct pattern. IgG experiences a rapid surge, with its concentration swiftly escalating to a higher level (> 2e⁵). Correspondingly, IgM levels also rise (as indicated by the red dashed line). This phenomenon arises from the presence of a certain level of IgG attained after the initial vaccination. The substantial increase in IgG content during the second dose primarily stems from the proliferation process of IgG itself, stimulated by antigens, as elucidated in Reaction 10 outlined in the methods section. By administering booster doses, IgG levels can attain a considerable threshold, thereby prolonging the protective effect and duration beyond that achieved by a single dose. Hence, multiple-dose administration strategies are commonly employed for vaccines³⁷. Notably, Fig. 2 not only describes the kinetic changes of antibodies but also specifically depicts the dynamics of IgG-antigen and IgM-antigen complexes. These antigen-antibody complexes play a crucial role in the immune response. They not only participate directly in antibody regeneration through feedback regulation but also serve as a direct indicator of the intensity of patient symptoms. Symptoms resulting from viral infections or vaccine administration, such as fever, are positively correlated with the concentration of antigen-antibody complexes rather than the concentrations of viruses or antibodies alone. Therefore, when assessing vaccine side effects, it is indispensable to consider changes in the concentration of antigen-antibody complexes induced by the vaccine. Figure 2 illustrates that the blue curve represents the concentration changes of IgM-antigen complexes, with their peaks exceeding 2e⁵ in both doses, while the concentration of IgG-antigen complexes also exhibits a noticeable increase after the second dose (as denoted by the purple dashed line). Additionally, inactivated vaccines may have a potential drawback when antigen structures undergo changes after treatment with high temperatures or chemical reagents. Such alterations can lead to modifications in antigenic determinants, akin to the effect of antigenic drift, potentially resulting in a significant decline in vaccine efficacy^38,39.

Antibody dynamics after 2 doses of inactivated vaccines. First dose is injected at the initial time unit, second dose is injected at 50th time unit. Both injection dosages of antigen substances are 10⁶.

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The results of parameter fitting using Adenovirus-Based COVID-19 vaccines in different age groups are presented in Fig. 3. Figure 3 demonstrates that the experimental data exhibit a strong concordance with the simulation results for both younger and older age groups. The data indicate that a booster vaccination strategy significantly enhances IgG levels. Furthermore, antibody concentrations in the younger cohort are marginally elevated compared to those in the older cohort. Distinct differences in curve trajectories between the two age groups are evident and can be further elucidated through the parameter fitting results presented below.

(a) IgG level fit to the clinical data for the 18–55 (two doses) age group. (b) IgG level fit to the clinical data for the 56–69 (two doses) age group.

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From Table 5, it can be observed that the value of k₁ for older recipients is significantly higher than that of younger individuals. This indicates that antibodies exhibited stronger antigen affinity in older individuals, which aligns with biological principles. This is because older individuals have a greater abundance of antibodies in their bodies, resulting in significantly higher antigen affinity or neutralization capacity of IgG following subsequent vaccination compared to younger individuals. However, younger infected individuals exhibit higher germinal center activity, indicating a stronger stimulus for antibody regeneration, as reflected by a larger value of k₃. This ensures rapid proliferation of antibodies.

This result also highlights the differential strategies employed by different age groups during the viral infection process. Due to fewer exposures to pathogens, younger individuals have a lower concentration or probability of high-affinity binding antibodies in their bodies. However, due to the enhanced activity of Th cells, the stimulation of B cells in germinal centers leads to a strong capacity for antibody regeneration, thereby generating a large quantity of protective antibodies. In contrast, older individuals, despite having less regenerative capacity in germinal centers compared to younger individuals, possess a higher concentration of high-affinity binding antibodies due to historical exposure to various pathogens. Although their proliferation rate is slower, their antibodies have higher initial values and better binding activity, enabling them to combat microbial infections through this mechanism.

Kinetic modeling of antibody response following mRNA vaccine administration

With the increasing comprehension of immunological mechanisms and the advancements in mRNA preparation and packaging technologies, significant progress has been made in the development of mRNA vaccines. Pfizer’s BNT162b2 and Moderna’s mRNA-1273 serve as notable examples that have exhibited superior efficacy compared to traditional vaccines in clinical trials^40,41. The merits of mRNA vaccines stem from their ability to maintain the original antigenic structure due to the absence of inactivation processes, thereby mitigating the risk of antigenic drift⁴². Adverse reactions associated with mRNA vaccines are often less pronounced, owing to the gradual process of antibody synthesis following mRNA administration, which results in lower concentrations of antigen-antibody complexes observed in the model. The concentration changes of IgM-antigen complexes are specifically depicted by the blue curve in Fig. 4. It is evident from Fig. 4 that the peak concentrations of IgM-antigen complexes induced by the two doses are significantly smaller when compared to those elicited by traditional inactivated vaccines. Analogous to inactivated vaccines, the IgG levels after two doses of mRNA vaccines exhibit substantial elevation as compared to those following a single dose, signifying that multiple doses remain the optimal vaccination strategy for mRNA vaccines. It is important to note that both the final IgG concentration of inactivated vaccines and mRNA vaccines are closely correlated with the vaccination dosage. Excessive vaccination dosages can give rise to an excessive accumulation of antigen-antibody complexes, resulting in severe vaccine side effects. Insufficient vaccination dosages are inadequate to induce a sufficient IgG concentration, leading to decreased efficacy of protection and shortened duration of immunity. The determination of the optimal vaccination dosage can be achieved more scientifically through mathematical models. However, it should be emphasized that both inactivated vaccines and mRNA vaccines necessitate a specific vaccination dosage, which poses short-term production challenges and constraints on their widespread implementation. For highly contagious respiratory viruses, surpassing the natural infection rate through vaccination is crucial for disease prevention, which imparts limitations on the future development of vaccines^43,44. Additionally, due to mRNA’s rapid degradation rate in comparison to proteins, mRNA vaccines encounter the drawback of storage difficulties, which introduces inconveniences during their utilization⁴⁵. Another significant concern is the potential occurrence of myocarditis. To enable the entry of mRNA into cells for the translation of corresponding antigenic substances, carriers are utilized. These carriers, distinct from the spike protein of the original virus, have the capacity to indiscriminately infect all cells, including non-epithelial tissues and cells with minimal expression of ACE2 receptors. Consequently, there exists the possibility of infecting cardiomyocytes, and during the process of eliminating infected cells subsequent to antibody production, damage to cardiomyocytes can occur⁴⁶.

Antibody dynamics after 2 doses of mRNA vaccines. First dose is injected at the initial time unit, second dose is injected at 50th time unit. Both injection dosages of mRNA are 10⁶.

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As the clinical data for mRNA vaccines are divided into different individuals, parameter fitting can be performed separately for each individual. The parameter fitting results for all 39 individuals are presented in figures in the supplementary materials, and the individual parameter fitting results are summarized in supplementary Table 1. These 39 individuals are divided into two groups, with 10 individuals having previously been infected with COVID-19 before receiving the vaccine (individuals 5, 15, 17, 19, 20, 22, 29, 32, 35, and 42) and the remaining 29 individuals having not been infected before vaccination. The parameter fitting results for both groups are presented in Table 6.

As shown in Table 6, it is displayed that the k₁ value for vaccinated individuals with a history of infection is significantly greater than that for uninfected vaccine recipients, indicating that their antibodies exhibit stronger antigen affinity. This is consistent with immunological principles, as the initial infection process selects for high binding antibodies. Therefore, after subsequent vaccination, their IgG antigen affinity or neutralization ability is significantly higher than that of uninfected individuals. k₂ and k₃ values are also significantly increased, which is consistent with immunological principles, as the initial infection brings about an increase in T-cell immunity, which manifests in two ways. The increase in the number and activity of helper T cells will enhance the capacity of germinal centers to produce antibodies, leading to an increase in the k₃ value. Meanwhile, the increase in cytotoxic T cells will lead to an increase in the clearance rate of antigen-antibody complexes, resulting in an increase in the k₂ value.

The most distinctive feature of our model is that it can explicitly represent the properties of antibodies, i.e., the antigen-binding activity k₁. By fitting data from different individuals, we can identify which individuals produce antibodies with stronger binding affinity, and these antibodies can be further sequenced by experimental researchers for the production of ultra-potent spectral neutralizing antibodies. Moreover, based on these fitted parameters, we can calculate the changes in IgG concentration for different individuals in the future, as well as the trend in IgG concentration changes for the population. Furthermore, we can conduct further dose optimization and vaccination timing optimization to achieve optimal antibody maintenance.

Kinetic modeling of antibody response following attentuated vaccine administration

The concept of utilizing attenuated vaccines has recently emerged as a potential immunization strategy. Attenuated vaccines involve the administration of viruses with reduced replicability to stimulate antibody-mediated immunity in hosts. Recent advancements in our understanding of viruses, particularly defective viruses, have shed light on the existence of various genotypes within a single viral strain that exhibit decreased replication activity and milder symptoms^47,48,49. In the context of COVID-19, many asymptomatic cases have been attributed to infections with defective viruses. Consequently, some scholars have proposed and implemented the use of attenuated viruses as natural vaccines⁵⁰. Different approaches are employed to reduce viral activity, such as introducing rare codons in the host to hinder translation efficiency^51,52 or utilizing defective viruses^50,53. The characteristics of the antibody response elicited by attenuated vaccines are depicted in Fig. 5. From Fig. 5, it is evident that the administration of attenuated vaccines induces a substantial increase in IgG levels (represented by the solid green line), surpassing the effects of mRNA and inactivated vaccines after two doses. Importantly, the immune response triggered by attenuated vaccines maintains a balanced intensity, as indicated by relatively low levels of antigen-antibody complexes. The peak concentration of IgG-antigen complexes (represented by the dashed purple line) is approximately 2.5e⁵, while the peak concentration of IgM-antigen complexes (represented by the dashed blue line) is around 1.5e⁵. The advantages of attenuated vaccines can be summarized as follows: Firstly, a high level of IgG can be achieved without the need for multiple doses. Unlike traditional vaccine regimens, the antibody response induced by attenuated vaccines exhibits kinetics similar to those observed during natural viral infections. This is attributed to the low viral inoculum, which allows for sufficient time for viral replication and the subsequent conversion from IgM to IgG. As a result, a significant increase in IgG levels can often be achieved with a single low-dose infection caused by attenuated viruses. Secondly, attenuated vaccines require minimal vaccine dosage. Due to the inherent replicability of the virus, only a small amount of attenuated virus is needed to stimulate an adequate level of antibodies, and the final antibody levels are not significantly influenced by the vaccine dosage. This greatly reduces production costs and usage requirements. Thirdly, attenuated vaccines exhibit good transmissibility. Since attenuated vaccines consist of live viruses, they possess similar infectivity to the original virus. Consequently, unvaccinated individuals can acquire the virus from vaccinated individuals, further accelerating the achievement of herd immunity. Lastly, attenuated vaccines closely resemble the original virus, containing antigenic epitopes that more accurately represent the original virus and a diverse range of antigenic epitopes. This results in a superior immune response. Although mRNA vaccines have demonstrated promising antigenic determinants, their expression primarily focuses on specific antigens rather than the entire virus. Specific antigens often exist in a free state, exposing fewer antigenic epitopes that are absent in their natural conformation. Antibodies induced by such epitopes are evidently insufficient to provide protection against real infections. However, it should be noted that the development of attenuated vaccines is still in its early stages, and these vaccines may pose significant risks to individuals with compromised immune function. The design of attenuated vaccines requires precise control over viral replicability, as excessive replication activity can lead to severe side effects, while insufficient replication activity may fail to stimulate an adequate concentration of protective antibodies.

Antibody dynamics after attenuated virus vaccination. Only one injection is implemented at the initial time with an attenuated live virus.

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The impact of viral inoculum dose on the ratio of IgM to IgG

The correlation between IgM and IgG levels has been extensively investigated in the context of dengue fever virus, where initial attention was directed to this phenomenon. Primary dengue infection is associated with a considerable increase in IgM levels, while the rise in IgG concentrations is not prominent^54,55. However, during secondary dengue infection, a substantial increase in IgG levels is observed, similar to the response seen after secondary vaccination. In contrast, for respiratory infectious diseases such as COVID-19, a significant elevation in IgG levels can generally be detected after the initial infection. The differences in these observations can be attributed to variations in viral inoculum dose. A high viral inoculum dose, akin to that of inactivated vaccine administration, promptly enhances IgM levels but due to the absence of an initial IgG reservoir, the initial IgG levels are negligible. Conversion from IgM to IgG is reliant on the inefficient process of isotype switching, resulting in limited IgG elevation during the initial infection. However, this scenario changes during secondary infections, as elucidated in Sect. 3.2. In the case of bloodborne infectious diseases like dengue fever, where the viral inoculum dose is higher, the dynamics of IgM and IgG resemble those of vaccine administration. Conversely, for respiratory infectious diseases like COVID-19, where the viral inoculum dose is low, ample time is provided for IgG conversion, extending the incubation period and leading to a higher IgG/IgM ratio post-infection⁵⁶. As demonstrated in Fig. 6, an increase in viral inoculum dose from 1 to 100, compared to Fig. 5, results in a shorter incubation period and a significant rise in induced IgM levels (solid red line) along with a marked decrease in IgG levels (solid green line), leading to a highly significant reduction in the IgG/IgM ratio.

Dosage effect on IgM/IgG ratio. Only one injection is implemented at the initial time with larger number of attenuated live viruses (100 in this case).

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Calculation of the protection time brought by vaccination

The term “herd immunity” was a common concept discussed prior to the widespread administration of COVID-19 vaccines. The attainment of herd immunity is achievable by vaccinating a proportion of the population equivalent to 1–1/R₀, where R₀ refers to the basic reproduction number of the virus, thus resulting in the fundamental eradication of the infectious disease. This approach has been successful in eliminating highly transmissible diseases like smallpox. However, combating COVID-19 has proven to be a more complex issue as it has become increasingly recognized that antibodies can undergo decay and result in time-limited protection, in addition to viral mutation effects. Consequently, the theory of herd immunity may not be applicable for certain infectious diseases, since repeated infections become unavoidable for the majority of the population. Our previous research²² provided a detailed explanation of the reasons why certain vaccines, such as those for smallpox and mumps, provide lifelong protection, while others, such as the hepatitis B vaccine, offer protection for over a decade, and vaccines like the COVID-19 and influenza vaccines confer shorter-term protection, typically within a year.

The study of the duration of vaccine-induced protection is analogous to the study of the duration of naturally acquired immunity, both of which require the calculation of critical threshold levels of IgG. However, it should be noted that the critical thresholds differ for different antibody isotypes, with high-affinity antibodies having much lower thresholds compared to low-affinity antibodies. Once the host’s antibody kinetic parameters have been determined, our model can be used to calculate the duration of vaccine-induced protection. Our calculation methodology involves simulating viral invasion at different time points by setting the viral quantity at a specific time to 1. We then observe the dynamic changes in the virus and antibody populations. If significant viral proliferation and peak concentrations of antigen-antibody complexes occur during subsequent time points, it indicates an infection. When the concentration is relatively low, as shown in Fig. 7a, the infection may be asymptomatic or mild. When the concentration is higher, as shown in Fig. 7b, it corresponds to symptomatic infection. Figure 7a,b represent calculated diagrams illustrating the duration of protection for all infections and the duration of protection specifically against symptomatic infections following vaccination, respectively.

In Fig. 7a, the actual viral invasion occurs at 200 time units, with viral proliferation and antibody elevation observed in the red region on the right after a long incubation period. At this point, the concentration of IgG-antigen complexes increases only slightly, which can be considered a case of asymptomatic infection. The critical time unit, marked as the 200th time unit, is the threshold before which viral infection barely leads to any viral proliferation due to rapid neutralization by high concentrations of IgG. Beyond this critical point, the virus demonstrates varying degrees of proliferation, and the later the invasion occurs, the lower the IgG concentration, resulting in more pronounced viral proliferation. As shown in Fig. 7b, when viral invasion occurs at the 400th time unit, significant viral proliferation, antibody elevation, and antigen-antibody complex elevation are observed. The concentration of antigen-antibody complexes exceeds 1e⁵, which can be considered the critical concentration for symptomatic infection. Therefore, viral invasions occurring before the 400th time unit do not lead to symptomatic infections, while those occurring thereafter consistently result in symptomatic infections. It can be observed that the duration of protection against symptomatic infections conferred by the vaccine is significantly longer than the duration of protection against all infections.

An interesting phenomenon is that after vaccination or natural infection, to prevent the recurrence of severe infections, moderate exposure to the virus is advisable rather than achieving complete self-protection. Early exposure to the virus without significant symptoms, as shown in Fig. 7a, can lead to a re-elevation of IgG antibody levels, thus providing more durable subsequent protection. Complete avoidance of virus exposure may result in more pronounced clinical symptoms upon encountering the virus at a later stage, as demonstrated in Fig. 7b. The later the viral invasion occurs, the higher the peak concentration of antigen-antibody complexes, leading to more pronounced symptoms. For this reason, long periods of excessive self-protection, such as habitual mask-wearing, are not recommended.

(a) An illustration of protection time calculation toward asymptomatic infection. First dose is injected at the initial time unit, second dose is injected at 50th time unit. Both injection dosages of antigen substances are 10⁶. Single live virus invaded at 200th time unit. The protection duration and subsequent infection curve are both marked in this figure. (b) An illustration of protection time calculation toward symptomatic infection. First dose is injected at the initial time unit, second dose is injected at 50th time unit. Both injection dosages of antigen substances are 10⁶. Single live virus invaded at 400th time unit. The protection duration and subsequent infection curve are both marked in this figure.

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Four suggestions in vaccine design

In the preceding sections, we expounded upon diverse mathematical models pertaining to vaccine administration. Drawing upon these models, we now proffer four recommendations for prospective vaccine design. Although the systematic overview and theoretical summarization of these approaches are yet to be accomplished, numerous scientists have endeavored to undertake and implement them.

Enhancing the T-cell immunogenicity of the antigen

The T-cell immunogenicity of antigens is a critical factor in eliciting antibody production, yet it is often overlooked in vaccine design. Indeed, the T-cell immunogenicity of antigens provides the basis for host recognition of self and foreign components. When antibodies strongly bind to self-antigenic substances due to the low T-cell immunogenicity of self-antigens, they cannot undergo extensive proliferation with the aid of T cells. Consequently, they are rapidly eliminated by the immune system, forming the basis for clonal deletion²⁸. The T-cell immunogenicity of antigens arises from the peptide sequences derived from antigen degradation, called primary sequences. Through bioinformatics techniques, scientists can now quantitatively analyze the T-cell immunogenicity of antigens^57,58,59,60. Pathogenic microorganisms capable of causing acute infections, such as SARS-CoV-2, have primary sequences of antigens with highly potent T-cell immunogenicity. This is reflected in our model by larger values of k₃ and k₁₀. For the development of vaccines against this type of pathogenic microorganisms, the immunogenicity of helper T-cells is often not the primary consideration.

Nevertheless, in the case of chronic infections such as HIV, where the T-cell immunogenicity of the antigen is low, it is necessary to moderately enhance the T-cell immunogenicity of the antigen. In immunization, increasing the T-cell immunogenicity of the antigen, indicated by the values of k₃ and k₁₀, is crucial for boosting the level of neutralizing antibodies. As shown in Fig. 8a, the IgG antibody level (denoted by the purple dashed line) after secondary immunization with an antigen exhibiting strong T-cell immunogenicity is significantly higher than that achieved with an antigen displaying weak T-cell immunogenicity (denoted by the red dashed line). Figure 8b presents two commonly used methods to enhance the T-cell immunogenicity of antigens, both of which have been widely applied in practice. The prerequisite for these methods is to not disrupt the antigenic epitopes of the antigen. The first method involves molecular engineering, where antigens are artificially modified without altering their antigenic epitopes. This is achieved by introducing point mutations in internal or non-epitope regions to alter their primary sequences and maximize T-cell immunogenicity. With the advent of computational protein design technologies, this method has been used to reduce or increase the T-cell immunogenicity of target antigens^61,62. Another approach is grafting the original antigen onto other proteins to enhance its T-cell immunogenicity. Many vaccines use other viral vectors for production, inadvertently leading to the fusion of the target antigen with other protein components. Composite antigens generated through fusion often exhibit stronger T-cell immunogenicity and greatly enhance the induction of neutralizing antibodies. Encouraging results from the clinical trial of the HIV vaccine sv144 demonstrated that increasing the T-cell immunogenicity of antigens has the potential to overcome the challenges of HIV vaccines^63,64. The RV144 trial, a randomized, double-blind phase 3 efficacy trial, employed a recombinant canarypox vector vaccine, ALVAC-HIV (vCP1521), expressing Env (clade E), group-specific antigen (Gag) (clade B), and protease (Pro) (clade B), along with an alum-adjuvanted AIDSVAX B/E and a bivalent HIV glycoprotein 120 (gp120) subunit vaccine. One inherent risk of this method is that the fusion antigens may introduce additional antigenic epitopes, thereby posing risks of inducing antibody responses against non-target antigenic epitopes.

(a) Effect of T-cell immunogenicity on IgG induction. k₃ and k₁₀ are assigned to a smaller number (1.5) for the low T-cell immunogenicity group. They are assigned to a larger number (2.5) for its strong T-cell immunogenicity counterpart. (b) An illustration of approaches in improving Th-cell immonogenity. Two stragties are presented : rational design of antigen with conserved epitope structure and the design of polymer antigen with strong T-cell immonogenity segments.

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Directing the induction of high-affinity neutralizing antibodies

The generation of high-affinity antibodies against targeted antigens represents a major challenge in vaccine development. A critical bottleneck in this process is the paucity of template antibodies in the IgM repertoire that display strong binding to vaccine antigens. This scarcity may be attributed to the high similarity between the antigenic epitopes of the vaccine antigen and self-antigens, resulting in the loss of highly binding antibodies to the target antigen due to clonal deletion. In the context of chronic infections such as HIV, only a small fraction of individuals are able to produce neutralizing antibodies^65,66. To increase the production levels and probability of generating neutralizing antibodies, it is necessary to direct their induction through a targeted approach, as depicted in Fig. 9a. In a seminal study, Jardine et al. employed computational protein design techniques to induce mice to generate high concentrations of neutralizing antibodies against the classic HIV antigen, gp120⁶⁷. This involved obtaining the crystal structure of the antibody-antigen complex and using computational protein design methods to introduce point mutations in the epitope region of the antigen to enhance antibody binding affinity. The mutated gp120 was then used as the vaccine for the first immunization, followed by a second immunization with the original vaccine. This approach yielded superior results compared to traditional two-dose immunization approaches. The simulated process of this method is depicted in Fig. 9b. During the simulation, two competitive antibody isotypes were utilized—one with strong binding affinity as a neutralizing antibody and another with weak binding affinity as a non-neutralizing antibody. At the outset, the level of neutralizing antibodies was extremely low at 1e⁻⁵, but exhibited high binding capability (k₁ = 1e⁻⁵), while the initial level of non-neutralizing antibodies was higher at 1e³ but with lower binding affinity (k₁ = 1e^−-6). Through the employment of the directed induction approach, the binding affinity between antigen and antibody was enhanced to 5e^−-5 due to the altered antigen in the first immunization. However, in the second immunization, when the original antigen was reintroduced, the binding affinity of the target antibodies reverted back to 1e^−-5. As shown in Fig. 9b, the IgG levels induced by the new strategy (represented by the red dashed line) were significantly higher than those induced by the traditional method (represented by the yellow dashed line).

An alternative and more direct strategy for stimulating the production of neutralizing antibodies involves the augmentation of initial antibody levels. This approach entails the utilization of gene editing techniques to introduce the genes encoding potent neutralizing antibodies into B cells. Subsequently, these genetically modified B cells are administered to the host in conjunction with vaccine administration, resulting in a rapid proliferation of neutralizing antibodies^68,69,70. This methodology bears resemblance to the use of convalescent blood from individuals who have recovered from specific acute infectious diseases for therapeutic purposes. However, it is crucial to recognize that this approach carries inherent risks, as the antibodies generated are not naturally derived from the host and have not undergone clonal selection and deletion processes. Consequently, there exists a potential for robust immune reactions against host tissues.

(a) An illustration of induced neutralizing antibody production by mutated antigen. The mutated antigen is used as the vaccine in the first immunization. It was designed to increase the binding affinity with targeted neutralizing antibody. (b) Comparison between induced antibody production and traditional vaccination strategy. First dose is injected at the initial time unit, second dose is injected at 100th time unit. Both injection dosages of antigen substances are 10⁶.

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Reduce the decay rate of IgG

Recurrent viral infections in certain individuals often result from discrepant decay rates observed among distinct antibodies, alongside variations in immunocompetence attributed to factors such as age and overall health. The diverse decay rates experienced under the complex circumstances of self-antigenic stimulation deviate from conventional direct decay rates of antibodies. Prior investigations have identified the crucial role played by self-antigenic components in sustaining IgG levels. The presence of self-antigenic moieties impedes a simple exponential relationship governing antibody decay, leading to extended protective periods demonstrated by select antibodies. The paramount importance of self-antigens in maintaining antibody concentration is exemplified in Fig. 10. Case 1 presents the original profile of antibody fluctuations (indicated by the red dashed line). Enhancing the initial concentration of self-antigenic material (Case 2; increasing from 1e⁵ to 1e⁶) effectively decelerates the decline rate of antibodies (depicted by the yellow dashed line). Similarly, amplifying the binding affinity between antibodies and self-antigens (Case 3; increasing from 1e⁻⁸ to 5e⁻⁸) substantially retards antibody decay (shown as the blue dashed line). It is noteworthy that while manipulating self-antigenic components remains beyond our control, we can regulate antibody attributes. Each antibody variant corresponds to a unique self-antigenic moiety. Some antibodies exhibit robust self-antigenic moieties, which maintain relatively high concentrations and confer prolonged protection against secondary infections, underscoring their significance. Extending vaccine-induced protection durations entails not only enhancing antibody neutralization capacity but also inducing targeted slow-decay-rate antibody responses. It is critical to acknowledge that repetitive antigenic exposure merely amplifies existing antibody levels, without imparting alterations to antibody type and attributes. As a result, individuals experiencing recurrent infections continue to exhibit antibodies characterized by rapid decay rates, resulting in substantially abbreviated protective cycles relative to their counterparts. This realization emphasizes that repetitive vaccination does not necessarily represent the optimal strategy.

IgG dynamics in different self-antigen scenarios. First dose is injected at the initial time unit, second dose is injected at 50th time unit. Both injection dosages of antigen substances are 10⁶. It can be seen that all IgG would decline after the peak but with different decay speeds.

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Reduce the adverse effects of vaccine

All vaccines exhibit varying degrees of adverse reactions, and we shall refrain from delving into the specific adverse effects induced by different vaccines in this particular context. Instead, our focus is to introduce the concept that, within our model, adverse reactions can be evaluated based on the alterations in the concentrations of antigen-antibody complexes. Vaccines operate by stimulating the production of IgG, which inevitably leads to the formation of complexes with antigenic substances during the process of antibody synthesis. While these adverse reactions are unavoidable, the extent of adverse effects resulting from vaccines of different types and administration methods, at the same level of antibody induction, displays significant variations. Through systematic comparisons among inactivated vaccines, mRNA vaccines, and attenuated vaccines, we can draw preliminary conclusions. For individuals with normal immune function, when achieving equivalent levels of antibody induction, mRNA vaccines and attenuated vaccines demonstrate notably lower adverse effects compared to inactivated vaccines. In other words, at the same level of adverse reactions, antibodies induced by mRNA vaccines and attenuated vaccines are significantly higher than those induced by inactivated vaccines. This observation also elucidates why mRNA vaccines exhibit superior preventive capabilities against COVID-19 when compared to traditional vaccines. Furthermore, our model allows for the quantitative assessment of such adverse reactions, thereby facilitating better control over vaccine dosages. In the case of mRNA vaccines and inactivated vaccines, adverse reactions demonstrate a substantial positive correlation with the administered dosage. Our model provides a theoretical framework for optimizing vaccination strategies in future in-silico research.

The investigation of host-virus interactions is crucial for comprehending the therapeutic mechanisms of infectious diseases and provides valuable guidance for vaccine development. The establishment of a rational mathematical model permits quantitative exploration of the dynamic alterations in host-virus interactions, thereby facilitating rational vaccine design. Building upon our prior model of host-virus interactions, we have extended and refined this model to more accurately simulate the dynamics of antibody changes under diverse vaccination conditions. To this end, we have differentiated antibodies into IgM and IgG subclasses and incorporated the process of IgM to IgG conversion.

Utilizing this model, we have conducted an analysis of the dynamics of antibodies within the organism following distinct vaccine administrations. Specifically, in the case of inactivated vaccines, which encompass protein-based inactivated vaccines generated via exogenous vectors, the levels of IgG exhibit a substantial increase subsequent to secondary doses. This phenomenon elucidates the significance of employing sequential vaccination strategies. The primary vaccine dose solely triggers the initiation of IgG production but fails to elevate IgG levels to a higher range. This trajectory bears resemblance to the antibody alterations observed during the initial infection of blood-borne infectious diseases, such as the dengue fever virus. Sequential vaccination also assumes critical importance for mRNA vaccines. In comparison to conventional inactivated vaccines, mRNA vaccines excel in preserving the original antigenic epitopes. Moreover, owing to the gradual release of antigens facilitated by mRNA vaccines, they frequently achieve superior effects in enhancing antibody responses while concurrently maintaining antigen-antibody complexes at a relatively lower level. Consequently, this results in a reduced incidence of side effects and heightened safety. We have further deliberated on the prospects associated with attenuated vaccines. We posit that attenuated vaccines harbor broader developmental potential when juxtaposed with traditional inactivated vaccines and mRNA vaccines. They obviate the need for specific dosages and can significantly elevate IgG levels without necessitating secondary or multiple doses. Additional merits intrinsic to attenuated vaccines, including their contagiousness and minimal side effects, position them as a promising candidate for future vaccine development endeavors.

Drawing upon this model, we have formulated four recommendations for the advancement of future vaccine development: enhancing the T-cell immunogenicity of antigens, selectively eliciting neutralizing antibodies, selectively inducing antibodies characterized by protracted decay rates, and mitigating vaccine-associated adverse effects. These principles emanate from our inferential deductions grounded in the model and have been substantiated through a multitude of empirical applications.

The development of an HIV vaccine has always posed significant challenges. From a theoretical perspective, we identify two core issues. The first issue pertains to the inherent low T-cell immunogenicity of natural antigens, which impairs the effective elevation of antibody levels following vaccine administration. The second issue concerns the initial low binding activity levels of antibodies in the antibody library, resulting in an insufficient increase of neutralizing antibodies after vaccination. To thwart HIV infection, we require antibodies with higher binding affinity capable of effectively neutralizing invading viruses. Once the viruses infiltrate cells, complete clearance becomes arduous, as it is closely linked to HIV’s infection of immune cells and the low T-cell immunogenicity. Thus, to prevent infection, we often necessitate antibodies with stronger binding affinity, commonly referred to as neutralizing antibodies. Protein engineering constitutes a powerful tool capable of addressing both of these bottlenecks, particularly with the rapid progression of computational protein design techniques in recent years. We can augment the immunogenicity of vaccines by engrafting other highly immunogenic proteins and modifying the core sequences of antigen proteins without altering the antigenic epitopes, thereby heightening their T-cell immunogenicity. Some pioneering work even entails modifying host autologous proteins to stimulate antibody production, inducing immune responses against self-tissues. This work can also furnish valuable insights for future cancer immunotherapy. Through computer-aided protein design, we can design antigens that bind more efficiently to potential neutralizing antibodies, typically attained via epitope mutations, selectively augmenting their binding affinity. Of course, this technique may necessitate experimental approaches such as protein-directed evolution. Utilizing engineered antigens as the primary immunogenic substances for the initial vaccination, followed by secondary vaccination utilizing the original antigens, can induce higher levels of neutralizing antibodies. By judiciously integrating these two methods, we may achieve greater breakthroughs in HIV vaccine research.

According to our model, there is substantial variability in the decay effects of antibodies among individuals due to their inherent properties. Certain antibodies possess robust self-immunogenicity and can persist at high levels, providing long-lasting protection. Conversely, other antibodies exhibit lower self-immunogenicity and experience faster decay rates in the absence of stimulation from self-antigen substances, resulting in an increased susceptibility to recurrent infections. This discrepancy may be attributed, in part, to the composition of an individual’s innate repertoire. The original antibody composition is influenced by factors such as the scarcity of IgMs in the innate repertoire, as discussed in Subsection 3.6. The limited size of the V(D)J DNA sequences, which serve as the source for these IgMs on chromosome 14⁷¹, plays a role in this phenomenon. Specifically, the genes encoding the heavy chain of IgMs are located on chromosome 14, while the alternative light chain loci (kappa and lambda) are found on chromosomes 2 and 22, respectively. The process of DNA recombination in B lymphocytes in the bone marrow generates a diverse array of immunoglobulins through random recombination of different DNA segments within the V(D)J complex^72,73,74. This composition is hereditary, and its heterogeneity is primarily a result of evolutionary selection pressures exerted by exposure to lethal infectious agents. Although the current model does not account for this heterogeneity, future iterations incorporating genetic regulation of the immune system⁷⁵ could address this limitation. In addition to individual variations, age and health factors also influence the ability to stimulate antibody regeneration by T cells. Healthy individuals tend to generate more antibodies under the same conditions of self-antigen substance, resulting in a slower decay rate of antibodies. Consequently, individuals with compromised immune systems are more susceptible to recurrent infections, such as COVID-19. Our model also suggests that repetitive administration of the same vaccine does not significantly alter the original antibody composition. Therefore, repetitive vaccination is not considered a viable long-term solution for preventing infections. Future vaccine development, including efforts related to COVID-19, should focus on selectively inducing high-binding and slow-decay antibodies. Furthermore, our model provides quantitative insights into the concentration of antigen-antibody complexes, which can serve as an indicator of vaccine side effects, including the severity of symptoms following natural infection. Our research demonstrates that different vaccines may elicit significant variations in the induction of IgG at equivalent levels. Thus, the selection of vaccines with minimal side effects is crucial in the context of future vaccine development.

In summary, by utilizing computer protein design and bioinformatics, we can selectively alter the primary sequence of antigens to increase their immunogenicity and enhance their ability to generate better antibodies without altering the antigenic determinants. Additionally, as described in Fig. 9, we can induce high-affinity antibodies through a stepwise approach. Furthermore, since different antibodies have varying decay periods, when selecting highly active antibodies in experiments, we need to consider their decay rates and select excellent antibodies with slower decay rates. Finally, we need to evaluate the side effects of different vaccines. In brief, the peak concentration of the vaccine antigen-antibody complex induced by vaccines that directly inject antigen-like substances is often higher than that induced by vaccines that induce antigen production, such as mRNA vaccines and attenuated vaccines. Therefore, they are also more likely to cause greater side effects.

Finally, our theoretical research provides a basis for future investigations concerning the development of vaccines through mathematical modeling. However, it is crucial to acknowledge a fundamental premise: all models, by their very nature, are subject to fallibility, though they possess varying degrees of utility. The immune system is remarkably intricate and heterogeneous, yet we have allocated limited attention to aspects of innate immunity, such as the production of interferon regulated by a complex genetic network⁷⁵, primarily directing our scientific inquiries towards humoral immunity. Additionally, we have overlooked the significance of somatic hypermutation on the generation of neutralizing antibodies and the gradual loss of antibody binding capacity following infection²⁸. It is vital to recognize the inherent uncertainties associated with our model, and its refinement demands ongoing validation through additional experimental investigations and clinical data.