COVID-19 Adaptive Humoral Immunity Models: Weakly Neutralizing Versus Antibody-Disease Enhancement Scenarios.

The interplay between the virus, infected cells and immune responses to SARS-CoV-2 is still under debate. By extending the basic model of viral dynamics, we propose here a formal approach to describe neutralisation versus weak (or non-)neutralisation scenarios and compare them with the possible effects of antibody-dependent enhancement (ADE). The theoretical model is consistent with the data available in the literature; we show that both weakly neutralising antibodies and ADE can result in final viral clearance or disease progression, but that the immunodynamics are different in each case. As a significant proportion of the world's population is already naturally immune or vaccinated, we also discuss the implications for secondary infections after vaccination or in the presence of immune system dysfunctions.

Background

SARS-CoV-2 is a new virus of the coronavirus family, responsible for the ongoing COVID-19 pandemic. To date, there are more than 300 million cases and over five million deaths worldwide (John’s Hopkins University 2000). SARS-CoV-2 is the third severe beta-coronavirus to emerge in the last 20 years, after SARS-CoV-1 and MERS-CoV. Hence the growing need for effective drugs and/or vaccines, not only in the immediate future but also in anticipation of a subsequent coronavirus resurgence.

However, the promising initial successes of antiviral treatments have also raised the possibility of negative side-effects. With regard to vaccines, an autoimmune disease (which lead to the temporary suspension of clinical trials) occurred during the AstraZeneca vaccine trial (9 September 2020); this context has demonstrated the importance of understanding qualitatively and quantitatively the immune response to primary infection and challenges (vaccines fall into both categories). In particular, relevant mathematical models of immune dynamics may be of interest to understand and predict the complicated behavior often observed.

We focus here on humoral adaptive immunity (antibody-mediated immunity) and refer to future works for an extension to the cellular and/or innate immune system. For clinical reasons and also for the understanding of those studying vaccines, antibody responses are of paramount importance. However, the neutralizing abilities of antibodies are still under discussion, especially as weak or non-neutralizing antibodies can promote infection through a process called antibody-dependent enhancement (hereafter abbreviated ‘ADE’) (Taylor et al. 2015; Iwasaki and Yang 2020; Yip et al. 2014; Jaume et al. 2011), see also the online supplementary information (Danchin et al. 2020). Therefore, here we studied both primary and secondary COVID-19.

To summarize, we propose a mathematical model of the immune response and virus dynamics that includes the possibility of weakly neutralizing antibodies and / or ADE and discuss its implications. At the time of writing the second version of the manuscript (January 2022) a significant part of the world’s population is either vaccinated or naturally immunized and the consequences of reinfection events are a major source of uncertainty concerning the evolution of the pandemic. This situation naturally calls for scientific investigation.

Methods

Mathematical Model

We present below the viral and immune response model. It is a compartmental model similar to those used to describe the epidemic propagation, see Kermack and McKendrick (1927), Diekmann et al. (2000), Hethcote (2000), Ng et al. (2003) for a general introduction and Faraz et al. (2020), Drożdżal et al. (2021), Liu et al. (2020), Danchin and Turinici (2021), Dolbeault and Turinici (2020) and Danchin et al. (2021) for COVID-19 specific works.

The viral-host interaction (excluding the immune response) is called the basic model of virus dynamics. It has been extensively validated both theoretically and experimentally, see Nowak and May (2000, Eq. (3.1), p. 18) and Wodarz (2007, Eqs. (2.3)–(2.4), p. 26) and references therein. See also Louzoun (2007), Castro et al. (2016) and Eftimie et al. (2016) for general overviews of mathematical immunology.

The model involves several classes: that of the target cells, denoted T, the infected cells, denoted I, the free virus denoted V and the antibodies denoted A. The model is illustrated in Fig. 1.

Fig. 1

Graphical illustration of the flow in the model (1)–(5)

Target cells T, which in our case are the epithelial cells with ACE2 receptors located, for instance in the respiratory tracts including lungs, nasal and trachea/bronchial tissues, are produced at a rate and die at rate . The parameters and define tissue dynamics in the absence of infection, see also “Model Without a Virus, Nor Immune Response” section in “Appendix 3”. When these susceptible cells meet free virus particles V, they become infected at a rate . Furthermore, target cells can also become infected via ADE if virus entry is mediated by antibodies. The parameter represents the rate of ADE infection route which is the result of a three-species interaction: T, A and V.

Infected (initially target) cells, denoted I, die at a rate . Note that this death rate will often be larger than the death rate of uninfected cells because viruses cause cell damage and cell death, Wodarz (2007) and Nowak and May (2000). Infected cells produce new virus particles at a rate , and the free virus particles which have been released from infected cells decay at a rate c called the clearance rate.

Free virions are neutralized by antibodies A, which can block virus entry into cells but also facilitate phagocytosis, at a rate b. Finally, the antibodies can be stimulated by the free virus with a production rate a while declining at a rate of (see for instance Wodarz 2007, eq. (9.4), p.126)). Note that alternative proposals for the antibody dynamics exist, see e.g. André and Gandon André et al. (2006) who assume that immune response, once started, grows at a constant rate while Pawlek et al. (2016) design a more complex model that takes into account the macrophage activation. The complete model reads (all constants are positive):Several hypotheses in this model need to be further documented. The first one in that all infected cells including ADE infected cells support viral replication and can produce virus. However, to date, it is still unclear whether ADE infected cells can support viral replication in vivo, Yip et al. (2014) and Jaume et al. (2011). Here we choose not to distinguish between virus productive and non productive infected cells to keep the model simple (see however the comments in “Appendix 4”). For the same reason, we do not discriminate between neutralizing, weakly neutralizing or non-neutralizing antibodies but consider all as members of the same class, the antibodies neutralizing capacity will therefore be the average of the neutralizing power and the average is described by the parameter b; on the other hand the ADE magnitude will be monitored by parameter . These parameters are the most important part of the immune response and the object of our study.

Stability of Equilibria and Further Considerations

We operate under the assumptions that all parameters are positive and furthermore the following two assumptions hold (see “Appendix 3” for details):where we define as in Nowak and May (2000, Eq. (6.2), p. 53):Note that (7) implies in particular which is a standard condition for such models. We will further denote

Stability of the Equilibrium without ADE

With these definitions we can give the main theoretical properties of the model depending on the presence or not of the ADE term.

Proposition 1

The model (1)–(4) without ADE i.e., (that is ) has a single stable equilibrium given by:

Proof

The proof of the stability of the equilibrium (10) is technical and is given in full detail in “Model: Virus and Immune Response but No Enhancement” section in “Appendix 3”.

Stability of the Equilibrium with ADE

We investigate now the full model having a non-null ADE term

Proposition 2

The model (1)–(4) has three equilibria:

the trivial equilibrium which is unstable;

the immmunosuppression equilibrium, also unstable, given by :

and a third equilibrium characterized as follows:

the antibody level is the unique positive solution of the following second order equation in the unknown A:

the other quantities are:

The following affirmations hold true concerning this third equilibrium

when is small enough the equilibrium is stable;

when is large enough the equilibrium is stable;

however there exist choices of parameters (in particular values of ) for which this equilibrium is unstable.

The proof is presented in “Appendix 3”.

Dynamical Aspects

The equilibrium analysis in the previous sections does not yet tell the full story of the evolution of the system (1)–(4). Depending on the parameters, a common behavior is the following: initially A will increase as response to V being above threshold the increase of A will drive both I and V to zero. Such a dynamics is stable over a long period and in practice I and V will keep small values for a time long enough to ensure virus clearance (when V is small enough, due to the random nature of the events, V will disappear).

Taking I and V to be constant equal to zero, the new evolution is:Note that equations for I and V are missing because if the initial states are then for all . This evolution drives T to and A to zero. If however during the slow decay of A a challenge is presented in the form of a virus load a new infection will start and V and I will rise again.

In conclusion, the stable equilibrium (12)–(13) is not necessarily reached in practice. The precise dynamics depends crucially on the parameters b and , see main text for details.

Results

Theoretical Results

We refer the reader to Sect. 3 for the rigorous statements concerning the theoretical properties of the model (1)–(5). Several situations may occur, but in summary the absence of ADE (i.e., ) insures stable equilibrium while intermediate values (neither too small not too large) may provide examples of unstable equilibria; moreover, stochastic events prevent the stable equilibrium state to be reached in practice, cf. Sect. 3.3. The parameters b and are shown to be the most important for the viral-host-antibody dynamics.

Empirical Results: Initial Infection

Taking into account the available data from the literature and the methodology in “Appendix 1” we run a numerical procedure to fit the model parameters to reproduce at best the viral load data in Fig. 2 (left) and obtained the values in Table 1. The numerical simulation for a primary infection corresponding to these parameters is shown in Fig. 3.

Fig. 2

Left Clinically observed typical variation of SARS-CoV-2 viral load in nasopharyngeal swab normalised using cell quantification. Data taken from Lescure et al. (2020, Fig. 3, p. 703, patient 4). Right Typical time variations for IgG. Data taken from Padoan et al. (2020, Fig. 2, p. 1085). Note that the antibody data is a mean over several days and corresponds to a different patient cohort

Table 1

Baseline parameters use in numerical simulations of the model (1)–(4)

μ	Λ	ω	β_0	β_1	δ	c	b
9.66	9.66×10^6	59.74	1.28×10^-6	0	16.22	1.45	0.52
a	σ	I(0)	V(0)	A(0)
9.15×10^-7	0.02	372.11	994.84	1.17

Fig. 3

Numerical simulation of the first infection without ADE for model (1)–(4) and parameters in Table 1. A good fit for the viral load from Fig. 2 is obtained (data is truncated below the value 1). On the contrary the fit is not as good for A(t) because data does not correspond to the same patient (joint V(t)/A(t) data was not available)

There is a 20% fall of target cells which either become infected or naturally die. The viral load peaks around 4-5 days after symptoms onset at copies/ml. While SARS-CoV-1 viral load, as MERS-CoV, peaked around 10 days after symptoms onset, most studies agree that SARS-CoV-2 viral load peaks sooner, around 5 days, Lescure et al. (2020) and Zou et al. (2020). Concerning antibodies, they increase sharply until week 2 then slower until a month after infection and start to decrease within 2–3 months (Long et al. 2020; Seow et al. 2020). Qualitative agreement is observed with clinically observed variations variations of viral load and partially with antibodies concentration depicted in Fig. 2 (see references in the figure).

Note that, although we expect agreement between V(t) and the viral load evolution in Fig. 2 (left) (which corresponds to a precise, real patient) the antibody data from Padoan et al. (2020, Fig. 2, p. 1085) does not correspond to the same patient (data unavailable) but is a mean value over several days and patients (not always the same). Each individual is likely to have his own immuno-kinetic parameters: the parameters of the individual that may fit the A(t) data from Fig. 2 (right) are not the same as the parameters that fit the data in left side of the same figure.

The equilibrium state (10) when (no ADE present) is reached after 2 years for all variables in Fig. 3. However, viral load and infected cells reach a minimum within several weeks post-infection before increasing and oscillating toward equilibrium state (10) (simulations not shown here). Therefore, if the virus load is very small close to the minimum, all other variables decrease towards 0 and the infection has vanished. The equilibrium state (10) is stable but not reached in practice as the patient is cured.

Empirical Results: Secondary Infection, Variants, Vaccination

We focus on a scenario where the immuno-kinetic parameters such as the neutralizing efficacy (b) or the ADE parameter () change; the causes can be multiple: a primary infection with a different variant, vaccination, or some immune evolution (aging being an example). In all cases we investigate the infection, called challenge, that takes place with a different set of b of parameters than in Table 1.

Variation of the Neutralizing Capacity b

When there is no ADE, decreasing the neutralizing capacity of antibodies (parameter b) leads on the one hand to a higher viral load peak but on the other hand to higher antibodies concentrations. The less neutralizing the antibodies are, the more abundant they are in order to have the infection always cleared. The simulations results are presented in Fig. 4. Infection resolution is obtained with little target cell destruction for larger values of b. On the contrary, low values of b will lead to significant increase of the antibody number and simultaneous decay of target cells, both largely pejorative for the patient.

Fig. 4

Numerical simulation of the model (1)–(4) and parameters in Table 1. Only the neutralizing capacity b is changed around the nominal value . Lower value of the neutralizing capacity b (solid blue line) leads to lower target cell count and higher antibody levels. See Fig. 6 for simulation over a longer time span

In the cases where the viral load reaches low values the infection stops before converging to the theoretical equilibrium.

Fig. 6

Simulation in Fig. 4 for a longer time span

Presence of ADE ()

We investigated in Figs. 5 and 6 the possibility of the ADE mechanism present (), for a range of possible parameter values. We plot all variables upon challenge with the same neutralizing capacity for antibodies. A higher ADE parameter leads to more destroyed target cells, more infected cells, more viral load and more antibodies. However the antibodies concentration is restricted by an upper limit (see Fig. 5 and compare with theoretical insights in the proof of point 3b of Proposition 2 in “Appendix 3”). Therefore after some threshold value, a higher ADE parameter cannot be compensated by more antibodies.

Fig. 5

The effect of the ADE parameter [the model (1)–(4)]. The secondary (challenge) infection has fixed neutralizing antibodies capacity b but several possible (ADE) parameter values; for all other parameters we use the nominal values given in Table 1. See Fig. 7 for simulation over a longer time span

For example, unlike , if the viral load directly stabilizes to its equilibrium state (13), without reaching a minimum close to 0 while oscillating (simulation not shown here). In this case, the infection wins (leading to respiratory function disruption and possibly patient death). Large values of lead to significant (possibly total) destruction of target cells.

Fig. 7

Simulation in 5 for a longer time span

Discussion

We studied the immune response to infection and reinfection with SARS-CoV-2 using a numerical model; the model can also take into account the possible presence of ADE, either during the first infection or during a challenge (secondary infection or reinfection with a different phenotype, after vaccination, etc.).

As there is no clear evidence to date that ADE occurs in severe COVID-19 patients, we assume that ADE only occurs in the case of challenge.

We started from a classical virus-host dynamic (Wodarz 2007; Tang et al. 2020) and modified it by adding the parameter to account for a possible ADE mechanism. In order to keep the model at its lowest complexity, we do not distinguish between ADE triggering and neutralising antibodies.

We performed a theoretical study of our system by calculating steady states and stability with and without ADE. We showed that stochastic events can also play a role and prevent the stable steady state from being reached in practice; we identified the parameters b (neutralising capacity) and (presence of ADE) as crucial for the dynamics of our system.

Next, we calibrated our parameter values to match the reference viral load from the literature (Lescure et al. 2020; Seow et al. 2020) and obtained good results.

We then studied a secondary infection (or an infection following vaccination or another immune event) which may have different immunodynamic parameters. We saw that without ADE, the low neutralising capacity of antibodies was systematically compensated by higher concentrations of antibodies leading to viral clearance. On the other hand, the addition of an ADE was not always associated with viral clearance but possibly with high target cell killing. Simulations and equilibrium analysis showed that the antibody concentration had an upper limit that prevented a higher ADE from being compensated by an unlimited amount of antibody. Therefore, ADE should be taken into consideration as a serious risk in understanding the disease, treatment and vaccine development and programming.

On the other hand, we have shown that the results are sensitive to the capacity of neutralizing antibodies (the b parameter); it should be noted that a decrease in this parameter can occur in several situations, e.g. due to degradation of the immune function, malfunctioning of the antibody immunodominance mechanism that ends up selecting too many weakly neutralizing antibodies or due to poorly calibrated therapeutic interventions. Regardless of the cause, such a decrease in neutralizing capacity is likely to imply a substantial deterioration of the outcome.

In summary, our results seem to confirm that the presence of ADE correlates mainly with significant target cell destruction, whereas the loss of neutralizing capacity correlates with both a higher number of antibodies (leading to inflammation) and a higher target cell destruction.

Limitations and Future Work

Like any other model, our model of course has several limitations. First, we assumed that all infected cells supported viral replication, including ADE-infected cells. Concerning SARS-CoV-2, the questions of ADE is still under debate, but for SARS-CoV-1 in vitro ADE evidence suggested abortive viral replication in ADE infected cells. Therefore, if we changed the model (1)–(4) to include this distinction, the steady state would change and ADE could be compensated for. Similarly, we did not distinguish between old and new antibodies secreted upon challenge. This would involve more parameters and change the equilibrium levels but not intrinsically change the behavior of the variables. The dynamic antibody model can also be modified to include, for example, constant antibody production after a threshold or more specific effects (André et al. 2006; Pawelek et al. 2016). As far as parameter are concerned, we did not have enough usable data to train our model and better fit the parameters. Finally, a single model can hardly account for the extreme variability in clinical outcomes of COVID-19, see Callaway et al. (2020); some studies proposed that part of this variability comes from genetics, see e.g., Ellinghaus et al. (2020) where genetic information from about 4,000 people from Italy and Spain was correlated with COVID-19 severity. This may lead to a variability of our model parameters in the form of random variables.

To date (January 2022), billions of people have been vaccinated, at least by one injection, and over half a billion have been infected. Fortunately, despite the spread of the highly contagious omicron variant, it appears that morbidity and mortality are declining. This implies that, for the time being, the most disastrous consequences of the phenomena included in this model are not being observed. However, it must be emphasized that human polymorphism, viral polymorphism and highly variable environmental conditions, as well as the considerable variety of vaccination protocols, mean that there may be isolates where ADE or the other immune responses we have explored could be significant. It is therefore particularly important to monitor variations in morbidity and mortality around the world so that a rapid response can be implemented if there is a local increase. Finally, the types of vaccines used are very different. For those based on well-established technologies, we do not foresee any consequences other than those discussed in this work, except perhaps in terms of vaccination protocols (time between primary and booster injections. In contrast, the use of vaccines based on indirect antigen production (adenovirus or synthetic RNA-based vaccines) requires specific encapsulation of the active ingredient in a variety of capsules or cassettes. These containers can, by themselves, be immunogenic. The consequence would be that after several immunisations, patients would develop a response against the vaccine, rendering it ineffective against the disease. We have not considered this possibility in our work.

Another limitation of this work is that in a secondary challenge there would be presumably already present memory cells allowing for a faster antibody response, thus increasing the value of A(0) relatively to a primary infection. This circumstance was not considered in this work.

The more science sheds light on the full picture of SARS-CoV-2, the more complex and precise details our model can go into. In the meantime, the main take-home message is that, with parameters consistent with the available clinical data, neutralizing capacity and ADE mechanisms can play an important immunological role in the outcome of primary and secondary infection.