A speed limit on serial strain replacement from original antigenic sin

Figures

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  Figure 1:. Cross-protection and blunting from a single memory in 2-dimensional antigenic space. A strain-specific memory s→ at the center protects against infection by nearby strains according to Eq. 1. The distance over which protection drops by a factor of 1/e is r; the blunting distance is b; hatched regions represent excluded regions in the (A) every- and (B) any-epitope models. Only infections with strains outside these regions would generate new memory. Here, b=1.7 and r=1.25.

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  Figure 2:. Successive strain replacement as traveling waves in antigenic space. (A) Strain replacement is modeled as a localized infection distribution moving along a one-dimensional trajectory in d-dimensional antigenic space with constant velocity v→. (B) The memory distribution is an exponentially decaying distribution with wavefront at x1=vt, localized in orthogonal directions. (C) The axes aligned with the direction of motion of the infection and memory distributions are not the same as the epitope-aligned axes. The localized infection peak is visualized as a point at the location of the wavefront.

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  Figure 3:. Blunting neighborhoods about reference point p→ for two blunting models, shown in 2d antigenic space. If an individual has memory to a strain in a blunting neighborhood of a given reference point and is later infected by a different strain in that neighborhood, no new memory will be created. (A) Blunting neighborhoods for epitope 1 (yellow) and epitope 2 (blue) about p→ in the every-epitope model. (B) Blunting neighborhood about p→ in the any-epitope model.

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  Figure 4:. Protection is order-dependent with blunting. (A) Infection history of hosts h1 and h2. (B) Memory profiles and protection distributions of hosts h1 and h2. Here, r=b2.

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  Figure 5:. Immune blind spots arise when b is large compared to r. Given a memory at the origin, the “allowed strains” are examples of the closest possible strains that could be added to the memory profile. The protection is necessarily low at the immunity blind spot strains regardless of infection history in (A) the every-epitope model and (B) the any-epitope model as long as br is sufficiently large (and, in (A), the distance from the memory strain to the immune blind spot is greater than both the blunting distance and the cross-protection distance). (C, D) show the probability of infection at the blind spot points considered here for (C) the every-epitope model and (D) the any-epitope model. Vertical dotted lines indicate approximate values of b/r above which infection is essentially guaranteed (decreasing with dimension in (C), dimension-independent in (D)).

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  Figure 6:. Transmission limit implied by the OAS constraint. (A) Strain replacement imposes a maximum value on R0. (B) Maximum value for R0 as a function of br for different values of θ*.

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  Figure 7:. Dynamical limits implied by the OAS constraint. (A) Maximum speed of evolution v (antigenic distance per infection), which depends on the diffusion constant D and the blunting distance modulated by the projection onto the fastest (slowest) evolving epitope for the every (any) epitope model (left), and the time for the infection peak to travel the cross-protection distance as a function of br for different values of r and θ* at saturating value of vmax (right). (B) Minimum standing diversity σ (antigenic distance) as a function of the diffusion coefficient and blunting distance modulated by the projection onto the fastest (slowest) evolving epitope for the every (any) epitope model (left), and the minimum standing diversity in units of cross-protection distance as a function of br for different values of r and θ*. Parameter combinations lying above or close to the black solid line at σmin/r=1 cannot lead to serial strain replacement, as the traveling wave solution requires σ/r≪1. (C) Minimum time to most recent common ancestor, tMRCA (measured in duration of infection), as a function of the diffusion coefficient D and the blunting distance modulated by the projection onto the…

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  Figure 8:. The slowest (fastest) epitope determines the integration bounds in the every- (any-) epitope model. (A, B) The every-epitope model. (A) The blue stripe is the blunting neighborhood that maximizes the number of memories being integrated over. The interval of x1 that lies in the blue region has length b/cosθ1. The projection of the velocity vector onto the epitope 1 direction is less than the projection onto the epitope 2 direction, confirming that θ1 is the angle corresponding to the slowest evolving epitope. (B) The yellow stripe is the blunting neighborhood that maximizes the number of memories being integrated over. The interval of x1 that lies in the yellow region has length b/cosθ2. (C, D) The any-epitope model. The green regions are the blunting neighborhoods that maximize the number of memories being integrated over. The length of the x1 interval that lies in the green region equals b/cosθ2 in (C), and b/cosθ1 in (D).

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  Figure 9:. Dynamical quantities are monotonic as a function of the fitness gradient s. A) The wave velocity is monotonically increasing, and (B, C) the infection standard deviation σ as well as the time to most recent common ancestor tMRCA are monotonically decreasing.