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The generation of influenza outbreaks by a network of host immune responses against a limited set of antigenic types

Communicated by Robert May, University of Oxford, Oxford, United Kingdom, March 8, 2007 (received for review November 16, 2006)
Article Figures & SI
Figures
Data supplements
Recker et al. 10.1073/pnas.0702154104.
Supporting Information
Files in this Data Supplement:
SI Figure 5
SI Figure 6
SI Table 1
SI Table 2
SI Table 3
SI Table 4
SI Methods
SI Figure 5Fig. 5. Bifurcation diagram for z_{ax} demonstrating the transition from NSS to CSS as the level of crossimmunity, g, increases, clearly showing the four distinct regions of (A) a state of equilibrium, (B and C) periodic and quasiperiodic oscillations, respectively, and (D) chaotic fluctuations.
SI Figure 6Fig. 6. Time courses and phase plots for three different values of g demonstrating the transition from periodic to chaotic oscillations, g = 0.59, g = 0.61, and g = 0.69 (left to right). The top row shows the time series for the proportion of the population that have been exposed to type ax, z_{ax}; the middle row shows the rate of change in that proportion, dz_{ax}/dt; and the bottom row shows the phase plots (y_{ax}, dy_{ax}/dt) for periodic, quasiperiodic, and chaotic behavior, respectively.
Table 2. Haplotype distribution among the isolates used in Table 1
Haplotype
Count
NSNTARKQKSDSLQRVSG
15
IDDGAGNKESDSLRRQNG
15
TGDKAGNKESDSLQRIND
13
TGDTAGKKESDSLQRVSG
11
TGDKAGNKESDSLQRING
10
NSNTARKKESDSLQRVSG
9
IDDKAGNKESDSLRRLNG
9
IDSGAGNEEIENLRRLTG
9
IDDKAGNKESDSLRRQNG
9
TGDTAGKKESDSLQRVSD
8
TNDKAGKKESDSLRKLNG
8
TGDTAGKKESDSLQRISD
7
IDDGAGNKESDSLRRLNG
7
IDSGAGNEESENLRRLTG
6
IGSGAGNKESENLRRLTG
6
IGSGAGNEESENLRRLTG
5
TGDTAGKKESVSLQRVSG
5
TGDKAGKKESDSLQRIND
5
TGDKAGKKESDSLQRING
5
TGDTAGKKESDSLQRISG
4
IDNDAGKEESESLRRLTG
4
TGNTAGKKESDSLQRISD
3
TGDTARKKESDSLHRVSD
3
NSNTARKKESDSLQRISG
3
NSNTARKKEIESLQRVSG
3
INDKAGNKDSDSLRRLNG
3
IGSGAGNGESENLRRLTG
3
IGSGAGNEEIENLRRLTG
3
IGSGAGNEESENLRKLTG
3
TDDGAGNKESDSLRRLNG
3
IDSDAGKEESESLRRLTG
3
IDSDAGKKESESLRRLTG
3
IDSEAGKEEIESLRRLTG
3
IDSEAGKEDIESLRRLTG
3
TGDKSGNKESDSLQRIND
2
TSNTAGKKESDRLQRISD
2
TSNTAGKKESDSLQRVRD
2
NSNTARKKESDSLQRVSD
2
NSNTAGKKEIESLQRVSG
2
INDKAGNKESDSLRRQNG
2
IDDGAGNKESVSLRRQNG
2
IGSGTGNEESENLRRLTG
2
IGNGAGNEEIENLRRLTG
2
IDSGAGNKEVENLRRLTG
2
IDSGTGNEESENLRRLTG
2
IDSGAGNKESENLRRLTG
2
TGSGAGNEESENLRRLTG
2
IDDKTGNKESDSLRRLNG
2
IDDKSGNKESDSLRRLNG
2
IDDKAGKKESDSLRGLNG
2
IDDKAGKKESDSLRRLNG
2
IDDKAGKKESDSLRRQNG
2
TGDTAGKKESDSIQRISG
2
IDSEAGKKESESLRRLTG
2
IDSEAGKEDSESLRRLTG
2
INDKAGKKESDSLRRLNG
2
TGDKAGNKESDSLRRING
2
TGDKAGNKESDRIQRING
2
TGNTAGKKESDSIQRVSG
1
TGNTAGKKESDSLQRVSD
1
TGDTAGKKESVSLQRISD
1
TGDTAGKKEIESLQRVSD
1
TGDTAGKKESDSIQRISD
1
TGDTAGKKESDSLQRIGD
1
TGDTAGNKESDSLQRVND
1
TGDTAGKKSDSLQRISG
1
IGDTAGKKESDSLQRISG
1
TSNTAGKKESDSLQRIKD
1
TSNTAGKKESDSLQRVKD
1
NSNTAGKKESDSIQRISG
1
NSNTAGKKESDSLQRISG
1
NSNTAGKKESDSIQRVSG
1
NSNTAGKKESDSLQRVSG
1
NSNTAGKKESDRIQRVSG
1
NSNTARKKESDRIQRVSG
1
NSNTARKKKSDSLQRISG
1
NSNTARKQKSDSLQRISG
1
NSNTAGKKESDSLQRVSD
1
NSNTARKKESDRLQRISD
1
NSNTARKKESDSLQRISD
1
NSNTASKQKSDSIQRISG
1
NSNTAGKQKSDSLQRVSG
1
INDKAGNKDSDSLRRQNG
1
IGSGSGNEECENLRRLTG
1
IGNGAGNEESENLRRLTG
1
IGSGSGNEESEKLRRLTG
1
IGSGAGNKESENLRRQTG
1
IDSGAGNVESEKLRRLTG
1
IDGGSGNEESENLRRLTG
1
IDSGSGNEESENLRRLTG
1
IDSGSGNEESEKLRRLTG
1
IDSGAGNEESEKLRRLTG
1
TGSGAGNEESVNLRRLTG
1
IGSGAGNEESEKLRRLTG
1
IGSGAGKEESENLRRLTG
1
IDDKAGNSKSDSLRRQNG
1
IDDKAGNKESDRLRRQNG
1
IDDKAGNKESDSLRRQTG
1
IDDTAGKKESDSIRRING
1
IDDTSGNKKSDSLRGLNG
1
VDDKSGNKESDSLRRLNG
1
IDDEAGNKESDSLRRQNG
1
IDDKAGNKESDILRRQNG
1
IDDKSGNEESDSLRRLNG
1
IDDKAGKKESDSLRKLNG
1
IDDGAGNKESDSLQRQNG
1
IDDGAGNKESDRLRRQNG
1
IDDGAGNKESDRLRRLNG
1
IDDGSGNKESDSLRRLNG
1
IDDGAGKKESDSLRRQNG
1
TDDGAGNKESVSLRRQNG
1
IDDKTGNQKSDSLRRQNG
1
IDDKTGNQKSDILRRLNG
1
IDDKTGNKESDSLRGQNG
1
IDDKAGNKESDSLRGQNG
1
IDSGAGNKESENLRRQTG
1
IDSDAGKEEIERLRRLTG
1
TGDTAGKKESDSLQRLSG
1
TGDTAGKKEGDSLQRVSG
1
NSNTARKKESVSLQRVSG
1
TNDKAGKKESDSLRTLNG
1
INDKTGNKESDSIRRLNG
1
INDKAGNKESVSLRRQNG
1
TNNKTGNKESDRLRKLNG
1
TNDKSGNKESDSLRRLND
1
IDSEAGKEEIENLRRLTG
1
IDSDAGKEESENLRRLTG
1
IDSGAGKEEIENLRRLTG
1
IDDGAGNEESDSLRRQNG
1
IDNDAGKEEIESLRRLTG
1
TSDKAGNKESDSLQRING
1
IDNGAGKEESESLRRLTG
1
IDDGAGNEKSDSLRRQNG
1
IDSGAGNEESDSLRRLNG
1
IDSGAGNEESDSLRRQNG
1
IDSDAGKEEIESLRRLTG
1
TGDTAGKKESDRIQRVSG
1
TGDGAGNKESDSLQRING
1
TGDTAGKKESDRLQRISD
1
TGDTAGKKESVSLQRISG
1
TDSEAGKEDSESLRRLTG
1
IDSEAGKKEIESLRRLTG
1
IDSETGKEESESLRRLTG
1
TDSDAGKEEIESLRRLTG
1
TSNTAGKKEIDSIQRVSD
1
TSNTAGKKESDSIQRVSD
1
TDSEAGKEDIESLRRLTG
1
IDSEAGNEESENLRRQTG
1
TNDKSGNKESDSLRKLNG
1
TNNKSGNKESDSLRKLNG
1
TNDKAGKKESVSLRKQNG
1
TNDKAGKKESDSLRKQNG
1
INDKAGNKDSDTLRRLNG
1
TGDKAGNKEIDRLQRING
1
TGDKSGNKESDSLQRING
1
TGNKAGNKESDSLQRING
1
TGDKAGNKESDSLQRITD
1
TGDKAGKKESDSIQRIND
1
Table 3. Distribution of antigenic types at the 15 time points corresponding to the successive epidemic peaks shown in Fig. 3A
Antigenic type
Time point
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
00000
00001
16
00010
24
00011
23
00100
00101
00110
23
00111
01000
01001
22
01010
25
01011
01100
01101
22
01110
3
25
01111
10000
25
10001
10010
10
2
10011
10100
1
3
10101
25
10110
2
9
3
10111
11000
25
12
11001
11010
24
11011
11100
1
3
11101
25
11110
22
11111
Antigenic types are designated by alleles (0,1) at five loci. At each time point, 25 isolates were randomly sampled from the model population.
SI Methods
Model Behavior. The transition between NSS (no strain structure) and CSS (cyclical or chaotic strain structure) can be illustrated as a bifurcation diagram (SI Fig. 5) or a series of time courses and phase plane plots (SI Fig. 6) for a twolocus system with three alleles at one locus and four alleles at the other. The bifurcation diagram in SI Fig. 5 indicates four regions of distinct dynamical behavior. At low levels of crossimmunity, g, all types exist at the same stable frequency (naturally subject to stochastic fluctuation). This equilibrium point becomes unstable as the level of crossinhibition increases, resulting in periodic oscillations that are characterized by successive epidemic outbreaks of types least inhibited by the previous epidemic types. In the case of a twolocus model with three and four alleles at each respective locus, this would take on the form of ax ® by ® cz ® aw ® bx ® cy... Increasing g even further results first in quasiperiodic cycles and then chaotic oscillations, where the sequence of succeeding epidemic types does not follow a predictable order but is still characterized by types with little or no overlap in their antigenic repertoire (high antigenic distance between two succeeding epidemic types). If crossimmunity was to be increased to very high levels, a further bifurcation point can be found that leads the dynamics back into a state of equilibrium with only one antigenic type prevailing and other types being either driven to extinction or residing at a subdominant level (not shown here).
SI Fig. 6 also shows this transition from periodic to chaotic behavior, via a state of quasiperiodicity, for one antigenic type, here ax. The top row depicts the time course of the proportion of the population that have been exposed to type ax, z_{ax}, and the middle row shows the progression of its derivative over time. The phase plane diagrams in the bottom row of SI Fig. 6 shows a clear distinction between the three different dynamical behaviors, particularly the quasiperiodic (Center) and chaotic oscillations (Right).
Amino Acid Usage at 18 Positively Selected Codons of the Influenza HA Gene. SI Table 1 enumerates the influenza isolates encoding particular amino acids at the 18 sites identified by Bush et al. (1) as being under positive selection. The gene sequence data were obtained from Bush et al. (1) and Fitch et al. (2). We found that 14 of 18 codons contained just two or three different amino acids at >1% frequency, suggesting strong evolutionary constraint. We also analyzed the haplotype distribution among the above isolates (as shown in SI Table 2). Interestingly, ≈50% of isolates can be explained by only 25 possible patterns. This suggests that there are restrictions on the possible combinations of amino acids, given that many of the singletons might represent transient slightly deleterious variants (or may be attributed to sequencing error). Thus the antigenic space may be constrained to a much higher degree than suggested even by the restrictions on amino acid usage shown in SI Table 1.
Change in Antigenic Types over Time. SI Table 3 shows the distribution of antigenic types at the 15 time points corresponding to the successive epidemic peaks shown in Fig. 4a. Antigenic types are designated by alleles (0,1) at five loci. At each time point, 25 isolates were randomly sampled from the model population. In SI Table 4, we construct a map of their antigenic relationships as a surrogate for a hemagglutination inhibition (HI) or neutralization assay data matrix. As mentioned in the text, to replicate the structure of the HI data matrix used in ref. 3, antigenic distances between isolates from nonadjacent time points were replaced by censored distances. SI Table 5 represents HI data from Vincent et al. (4) on subtype 1 swine influenza, which we believe to echo the patterns shown in SI Table 4.
The limited locus structure of our model is supported by a recent paper (5) showing two vaccine strains (A/Panama/2007/99 and A/Wyoming/03/2003) from that differed in 16 aa in the HA, could be rendered antigenically equivalent (as determined by HI and microneutralization assays using ferret postinfection antisera) after substitutions of only 2 of these 16 aa. However, these data are also compatible with the antigenic drift model. Studies of this sort conducted over longer time scales will help discriminate between the two models.
1. Bush RM, Bender CA, Subbarao K, Cox NJ, Fitch WM (1999) Science 286:19211925.
2. Fitch WM, Bush RM, Bender CA, Cox NJ (1997) Proc Natl Acad Sci USA 94:77127728.
3. Smith DJ, Lapedes AS, de Jong JC, Bestebroer TM, Rimmelzwaan GF, Osterhaus AD, Fouchier RA (2004) Science 305:371375.
4. Vincent AL, Lager KM, Ma W, Lekcharoensuk P, Gramer MR, Loiacono C, Richt JA (2006) Vet Microbiol 118:212222.
5. Jin H Zhou H, Lin H, Chan W, Adhikary L, Mahmood K, Lee MS, Kemble G (2005) Virology 336:113119.
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