Effect of antibody levels on the spread of disease in multiple infections

TL;DR

This work addresses how individual antibody levels interact with epidemic spread on complex networks by introducing a stochastic SIRS model in which antibody levels $A_i(t)$ evolve via an Ornstein-Uhlenbeck process and influence infection probability through a sigmoid function. The authors formulate a comprehensive system of stochastic differential equations for $S_i$, $I_i$, $R_i$, and $A_i$, and perform a mean-field reduction to analyze equilibrium points and a connectivity-dependent threshold; a threshold condition is derived as $\beta < \frac{\mu\bigl(1+e^{\alpha_p(\psi-\gamma_p)}\bigr)}{\langle k \rangle}$. They validate the model with numerical simulations on Watts-Strogatz and Barabási–Albert networks, showing that the antibody decay rate $\theta$ strongly affects epidemic spread and that network topology modulates sensitivity, with BA networks typically enabling faster and larger outbreaks. The results reveal Gaussian-like stationary distributions for both the infected population and final antibody levels, and they indicate that average final antibody levels are robust to individual-level antibody variation. Overall, the study provides insights into how antibody dynamics and network structure can inform epidemic prevention and control strategies, while highlighting limitations of the OU assumption and scope of simulations for future work.

Abstract

There are complex interactions between antibody levels and epidemic propagation, the antibody level of an individual influences the probability of infection, and the spread of the virus influences the antibody level of each individual. There exist some viruses that, in their natural state, cause antibody levels in an infected individual to gradually decay. When these antibody levels decay to a certain point, the individual can be reinfected, such as with COVID 19. To describe their interaction, we introduce a novel mathematical model that incorporates the presence of an antibody retention rate to investigate the infection patterns of individuals who survive multiple infections. The model is composed of a system of stochastic differential equations to derive the equilibrium point and threshold of the model and presents rich experimental results of numerical simulations to further elucidate the propagation properties of the model. We find that the antibody decay rate strongly affects the propagation process, and also that different network structures have different sensitivities to the antibody decay rate, and that changes in the antibody decay rate cause stronger changes in the propagation process in Barabasi Albert networks. Furthermore, we investigate the stationary distribution of the number of infection states and the final antibody levels, and find that they both satisfy the normal distribution, but the standard deviation is small in the Barabasi Albert network. Finally, we explore the effect of individual antibody differences and decay rates on the final population antibody levels, and uncover that individual antibody differences do not affect the final mean antibody levels. The study offers valuable insights for epidemic prevention and control in practical applications.

Figures (9)

• Figure 1: The propagation process. Where the left side is filled to represent the individual's antibody level. The turning point of the antibody level is represented by $\epsilon$, this value also represents a general antibody level regression value, indicating the state of an individual's antibody level after infection. The middle section of the diagram describes the probability and conditions of transmission, and the right provides a detailed explanation of the significance of each panel.
• Figure 2: Visual representation. Figure (a) displays the WS network with $N = 1000$, $k = 4$, and $p = 0.1$, along with the basic infection rate $\beta = 0.2$, disease recovery rate $\mu = 0.1$, and antibody level reversion value $\alpha = 0$, $\alpha_p = 3$, $\gamma_p = 1.2$. By varying the values of $\theta$ and $\sigma$ and allowing the propagation to continue for 50 time steps, we obtain the mean level of antibodies in the network. Figure (b) illustrates the changes in the infection probability $P_{infect}(i)$ with respect to the logarithm of $\gamma_p$ and $\alpha_p$.
• Figure 3: $P_{infect}(i)$ function diagram. The figure illustrates the relationship between the infection probability $P_{infect}(i)$ and antibody level $A_i(t)$ for different parameter values. In figure (a), where $\gamma_p = 1.5$ and $\beta = 0.3$, the effect of $\alpha_p$ on the curve slope is shown. It can be observed from the different curves that as $\alpha_p$ increases, the curve tends more and more towards an S-shape. On the other hand, figure (b) shows the effect of $\gamma_p$ on the turning point of the curve for fixed values of $\alpha_p = 3$ and $\beta = 0.3$. As $\gamma_p$ increases, the curve shifts left and right, as depicted by the different lines. It should be noted that the antibody level $A_i(t)$ is restricted to the range of 0 to 3.
• Figure 4: The propagation plots on the WS network. It depicts the spread of the virus under different $\theta$ parameters. The experiments were conducted on a WS network with $N = 1000$, an average degree of $k = 4$, a disconnected edge reconnection probability of $p = 0.1$, a basic infection rate of $\beta = 0.2$, a disease recovery rate of $\mu = 0.1$, and regression values of antibody levels $\alpha = 0$. The horizontal axis of the plots shows the number of time steps in a logarithmic scale, while the vertical axis shows the number of individuals in each state during propagation, averaged over 50 iterations. As shown in plots (a), and (b), the disease eventually disappears, while in plots (c), (d), (e), and (f), the transmission reaches a plateau. These results suggest that the behavior of the virus is affected by the value of $\theta$.
• Figure 5: The propagation on a BA network. where each subplot is labeled with its corresponding $\theta$ parameter value. The parameters used for the simulation are as follows: N = 1000 nodes in the BA network, with $k = 4$ as the network parameter, $\beta$ = 0.2 as the basic infection rate, $\mu$ = 0.1 as the disease recovery rate, and $\alpha$ = 0 as the regression values of antibody levels. The horizontal axis represents the number of time steps taken in a logarithmic scale, while the vertical axis indicates the number of individuals in each state during the propagation process. The data shown in the plot represents an average of 50 simulation runs. As observed, figure (a) shows that the disease eventually dies out, whereas figures (b)-(f) depict that the transmission eventually reaches a steady state.