A Novel Mathematical Model That Predicts the Protection Time of SARS-CoV-2 Antibodies

Abstract

Infectious diseases such as SARS-CoV-2 pose a considerable threat to public health. Constructing a reliable mathematical model helps us quantitatively explain the kinetic characteristics of antibody-virus interactions. A novel and robust model is developed to integrate antibody dynamics with virus dynamics based on a comprehensive understanding of immunology principles. This model explicitly formulizes the pernicious effect of the antibody, together with a positive feedback stimulation of the virus-antibody complex on the antibody regeneration. Besides providing quantitative insights into antibody and virus dynamics, it demonstrates good adaptivity in recapturing the virus-antibody interaction. It is proposed that the environmental antigenic substances help maintain the memory cell level and the corresponding neutralizing antibodies secreted by those memory cells. A broader application is also visualized in predicting the antibody protection time caused by a natural infection. Suitable binding antibodies and the presence of massive environmental antigenic substances would prolong the protection time against breakthrough infection. The model also displays excellent fitness and provides good explanations for antibody selection, antibody interference, and self-reinfection. It helps elucidate how our immune system efficiently develops neutralizing antibodies with good binding kinetics. It provides a reasonable explanation for the lower SARS-CoV-2 mortality in the population that was vaccinated with other vaccines. It is inferred that the best strategy for prolonging the vaccine protection time is not repeated inoculation but a directed induction of fast-binding antibodies. Eventually, this model will inform the future construction of an optimal mathematical model and help us fight against those infectious diseases.

Keywords: SARS-CoV-2; antibody dynamics; mathematical modeling; protection time; vaccine.

Figure 1

A simple diagram of host-virus interaction.

Figure 2

The role of environmental antigens in immune response.

Figure 3

Antibody and virus dynamics modeling using different p(0) value. p(0) represents the concentration of environmental antigens. The virus-antibody dynamics are modeled under different environmental antigen concentrations (A) and different environmental antigen attributes (B). As shown in (A), the antibody decay rate is significantly slower (shown in solid yellow curve) when there is a large amount of environmental antigen-like substances. The parameter set we used is: x(0) = 0, y(0) = 100, z(0) = 10, p(0) = 1000 or p(0) = 1,000,000, $k_1=0.1, k_2$ = 1 × 10⁻⁵, $k_{-2}$ = 1 × 10⁻¹⁴, $k_3$ = 1, $k_4$ = 2, $k_5$ = 0.02, $k_6$ = 0.02, $k_7$ = 1 × 10⁻⁸, $k_{-7}$ = 1 × 10⁻¹⁴. As shown in (B), the antibody decay rate is significantly slower (shown in solid yellow curve) when $k_8$ sets a large value which corresponds to a better binding kinetics between environmental antigen-like stuff and its corresponding antibody. The parameter set we used is: x(0) = 0, y(0) = 100, z(0) = 10, p(0) = 1,000,000, $k_1=0.1, k_2$ = 1 × 10⁻⁵, $k_{-2}$= 1 × 10⁻¹⁴, $k_3$ = 1, $k_4$ = 2, $k_5$ = 0.02, $k_6$ = 0.02, $k_7$ = 1 × 10⁻⁸ or $k_7$ = 1.8 × 10⁻⁸, $k_{-7}$ = 1 × 10⁻¹⁴.

Figure 4

Antibody dynamics modeling with different $k_7$ values. (A) One scenario where environmental antigen-like substances do not trigger antibody growth. As shown in (A), the antibody does not engage proliferation due to the presence of environment antigen-like molecules. The parameter set we used is: x(0) = 0, y(0) = 100, z(0) = 0, p(0) = 1,000,000, $k_1=0.1, k_2$ = 1 × 10⁻⁵, $k_{-2}$= 1 × 10⁻¹⁴, $k_3$ = 1, $k_4$ = 2, $k_5$ = 0.02, $k_6$ = 0.02, $k_7$ = 1 × 10⁻⁸, $k_{-7}$ = 1 × 10⁻¹⁴. (B) One scenario where environmental antigen-like substances do trigger antibody proliferation. As shown in (B), the antibody does engage proliferation due to the presence of environment antigen-like molecules. The parameter set we used is: x(0) = 0, y(0) = 100, z(0) = 0, p(0) = 1,000,000, $k_1=0.1, k_2$= 1 × 10⁻⁵, $k_{-2}$= 1 × 10⁻¹⁴, $k_3$ = 1, $k_4$ = 2, $k_5$ = 0.02, $k_6$ = 0.02, ${\mathrm{k}}_7$ = 1 × 10⁻⁷, $k_{-7}$ = 1 × 10⁻¹⁴. The antibodies might significantly increase when the environmental antigenic substances bind strongly with them. This always induces allergic reactions.

Figure 5

Different immune behaviors toward variants with different replication activities. The parameter set we used is: x(0) = 0, y(0) = 100, z(0) = 10, p(0) = 1000, $k_1=0.06 or 0.1, k_2$ = 1 × 10⁻⁵, $k_{-2}$ = 1 × 10⁻¹⁴, $k_3$ = 1, $k_4$ = 2, $k_5$ = 0.02, $k_6$ = 0.02, $k_7$ = 1 × 10⁻⁸, $k_{-7}$ = 1 × 10⁻¹⁴. As shown in this figure, the antibody response is milder when the host is infected by a less toxic strain (smaller $k_1$ value 0.06). The peak viral load is also less compared to its high toxic counterpart (bigger $k_1$ value 0.1). This indicates that even for the same virus infection, the immune response would vary greatly due to the differences in viral replication capacity of different variants.

Figure 6

Dynamics of different antibodies with different kinetic attributes. The parameter sets we used are: x(0) = 0, y(0) = 1, z(0) = 1, w = 1, $k_1=0.1, k_2$ = 1 × 10⁻⁵, $k_{-2}$= 1 × 10⁻¹⁴, $k_3$ = 1, $k_4$ = 2, $k_5$ = 0.02, $k_6$ = 0.02, for antibody 1;$k_2$ = 9 × 10⁻⁶, $k_{-2}$ = 9 × 10⁻¹⁵ for antibody 2; $k_2$ = 8 × 10⁻⁶, $k_{-2}$= 8 × 10⁻¹⁵ for antibody 3; $k_2$ = 7 × 10⁻⁶, $k_{-2}$ = 7 × 10⁻¹⁵ for antibody 4; $k_2$ = 6 × 10⁻⁶, $k_{-2}$ = 6 × 10⁻¹⁵ for antibody 5. It is demonstrated in this figure that the faster-binding antibodies engage amplification at a greater magnitude. In this way, the immune system selects those specific neutralizing antibodies.

Figure 7

High concentrations of weakly binding antibodies can provide effective protection. A plot of the inhibitory capacity of a specific concentration of weakly binding antibodies against infection is presented. Two types of antibodies are presented in this figure: antibody 1 has a strong binding capacity ($K{1}_{\mathrm{on}}$ = 1 × 10⁻⁵) while antibody 2 has a relatively weak binding capacity ($K{2}_{\mathrm{on}}$ = 5 × 10⁻⁶). Two scenarios are simulated: both antibodies have low initial concentrations in case 1; weakly binding antibody has a high initial level in case 2. The parameter sets we used are: $x_1$(0) = $x_2$(0) = 0, $y_1$(0) = $y_2$(0) = 1, z(0) = 1, $w_1$ = $w_2$ = 1, $K{1}_{\mathrm{on}}$ = 1 × 10⁻⁵, $K{1}_{\mathrm{off}}$= 1 × 10⁻¹⁴, $K{2}_{\mathrm{on}}$ = 5 × 10⁻⁶, $K{2}_{\mathrm{off}}$ = 1 × 10⁻¹⁴, $k_3$ = 1, $k_4$ = 2, $k_5$ = $k_6$ = 0.02, $k_1$ = 0.1 for case 1; $y_2$(0) = 100 for case 2. It can be seen that the peak viral load and antibody level are significantly lower in case 2, which corresponds to a milder immune response. It indicates that the elevated weakly binding antibodies could also provide protection against severe infection.

Figure 8

An illustration of protection time calculation. The parameter set we used is: x(0) = 0, y(0) = 100, z(0) = 10, p(0) = 10,000, $k_1=0.1, k_2$ = 1 × 10⁻⁷, $k_{-2}$ = 1 × 10⁻¹⁴, $k_3$ = 1, $k_4$ = 2, $k_5$ = 0.02, $k_6$ = 0.02, $k_7$ = 1 × 10⁻⁸, $k_{-7}$ = 1 × 10⁻¹⁴. Incubation time is calculated as the time interval between virus entrance and the production of antibodies. The viruses would engage a proliferation earlier than the antibodies. The patient is still asymptomatic even though the virus has reached a high level. Symptoms such as fever would appear when the antibody–virus complexes reach beyond a specific threshold. The second infection is marked with a green arrow. It can be seen in this figure that a second infection could occur when the IgG level drops below a certain level. It does not necessarily require a zero IgG level when a breakthrough infection happens.

Figure 9

Protection time calculation when the antibody has a specific binding kinetic constant $k_2$. It can be inferred from this figure that a fast-binding neutralizing antibody would provide a longer protection time when we compared (A) with (C). The protection time against severe infection can also be prolonged, given the faster binding kinetics when comparing (B) to (D). (A) Protection time calculation when the antibody has a weak binding kinetic constant $k_2$ = 1 × 10⁻⁶. The parameter set we used is: x(0) = 0, y(0) = 100, z(0) = 10, q(0) = 1 × 10⁶, $k_1=0.1, k_2$ = 1 × 10⁻⁶, $k_{-2}$ = 1 × 10⁻¹⁴, $k_3$ = 1, $k_4$ = 2, $k_5$ = 0.02, $k_6$ = 0.02, $k_7$ = 1 × 10⁻⁹, $k_{-7}$ = 1 × 10⁻¹⁴. (B) Maximal virus load at different infection points when the antibody has a binding kinetic constant $k_1$ = 1 × 10⁻⁶. The parameter set used is the same as (A). (C) Protection time calculation when the antibody has a strong binding kinetic constant $k_2$ = 1 × 10⁻⁵. The parameter set we used is: x(0) = 0, y(0) = 100, z(0) = 10, q(0) = 1 × 10⁶, $k_1=0.1, k_2$ = 1 × 10⁻⁵, $k_{-2}$= 1 × 10⁻¹⁴, $k_3$ = 1, $k_4$ = 2, $k_5$ = 0.02, $k_6$ = 0.02, $k_7$ = 1 × 10⁻⁹, $k_{-7}$ = 1 × 10⁻¹⁴. (D) Maximal virus load at different infection points when the antibody has a binding kinetic constant $k_2$ = 1 × 10⁻⁵. The parameter set used is the same as (C).

Figure 10

(A) the dynamic behavior of antibodies in the overall population through time. (The blue zone around mean curve stands for 95% confidence interval). It can be seen that the IgG level would significantly decline after reaching a peak level. However, its degradation does not follow a simple mathematical formula. Its descent rate would gradually decline and be maintained at a relatively stable level after 200 days. (B) The protection performance of antibodies in the overall population through time. It can be seen in this figure that the protection efficiency of induced neutralizing antibodies would be maintained at a relatively high level in the first 100 days. Its protection efficiency would engage a rapid decline after the first three months.

Figure 11

(A) Self-reinfection scenario. The parameter sets we used are: x(0) = 0, y(0) = 1000, z(0) = 1, w = 1000, $k_1=0.1, k_2$ = 1 × 10⁻⁵, $k_{-2}$ = 1 × 10⁻¹⁴, $k_3$ = 1, $k_4$ = 2, $k_5$ = 0.02, $k_6$ = 0.02. Reinfections are represented as repeated waves in this figure. It indicates that the viruses could re-proliferate when the antibodies cannot completely eliminate them. The viruses start to proliferate when the antibodies drop to a certain level, leading to self-reinfection in this case. (B) Scenario of chronic infection. The parameter sets we used are: x(0) = 0, y(0) = 1000, z(0) = 1, w = 1000, $k_1$ = 0.1, $k_2$ = 3 × 10⁻⁵, $k_{-2}$ = 1 × 10⁻⁵, $k_3$ = 1, $k_4$ = 2, $k_5$ = 0.02, $k_6$ = 0.02. In this case, pathogens would not be eliminated but maintained at a relatively stable level, forming a chronic infection. The low concentration of pathogenic antigens only provides a limited driving force for promoting antibody reproduction.