Understanding original antigenic sin in influenza with a dynamical system

Abstract

Original antigenic sin is the phenomenon in which prior exposure to an antigen leads to a subsequent suboptimal immune response to a related antigen. Immune memory normally allows for an improved and rapid response to antigens previously seen and is the mechanism by which vaccination works. I here develop a dynamical system model of the mechanism of original antigenic sin in influenza, clarifying and explaining the detailed spin-glass treatment of original antigenic sin. The dynamical system describes the viral load, the quantities of healthy and infected epithelial cells, the concentrations of naïve and memory antibodies, and the affinities of naïve and memory antibodies. I give explicit correspondences between the microscopic variables of the spin-glass model and those of the present dynamical system model. The dynamical system model reproduces the phenomenon of original antigenic sin and describes how a competition between different types of B cells compromises the overall effect of immune response. I illustrate the competition between the naïve and the memory antibodies as a function of the antigenic distance between the initial and subsequent antigens. The suboptimal immune response caused by original antigenic sin is observed when the host is exposed to an antigen which has intermediate antigenic distance to a second antigen previously recognized by the host's immune system.

Figure 1. Time courses of the healthy cell concentration , the infected cell concentration , the dead cell concentration , the viral load , the concentrations of naïve and memory antibodies , respectively, and the naïve antibody affinity .

The memory antibody affinity is . The initial conditions are , , , , , and .

Figure 2. Time courses of the healthy cell concentration , the infected cell concentration , the dead cell concentration , the viral load , the concentrations of naïve and memory antibodies , respectively, and the naïve antibody affinity .

The memory antibody affinity is . The initial conditions are the same as those in Figure 1.

Figure 3. Trajectories of the maximum percentage of dead cells, the maximum viral load, the cumulative effects of naïve and memory antibodies defined by equations 9 and 10, respectively, and the final average antibody affinity defined by equation 11 with different memory antibody affinities .

The dashed horizontal lines in (a) and (b) are the maximum percentage of dead cells and the maximum viral load, respectively, at the lowest memory antibody affinity .

Figure 4. Trajectories of the concentration of dead cells and the viral load with different effects of immune response .

In each trajectory, , , , and is constant. When , viruses cannot be cleared at small values of , such as 0.1. The decay rates of both and increase with .

Figure 5. Trajectories of the effect of immune response, .

In each trajectory, , , . Each trajectory corresponds to one value of . When , viruses cannot be cleared at small values of , such as 0.1.

Figure 6. Sensitivity analysis of parameters and .

(a) and (b) The maximum percentages of dead cells and the average antibody affinities at different values of . (c) and (d) The maximum percentages of dead cells and the average antibody affinities at different values of . Initial conditions and parameters other than and are the same as those in Figure 3.

Figure 7. Sensitivity analysis of parameters and .

(a) and (b) The maximum percentages of dead cells and the average antibody affinities at different values of . (c) and (d) The maximum percentages of dead cells and the average antibody affinities at different values of . Initial conditions and parameters other than and are the same as those in Figure 3.