Shaping the phylogenetic tree of influenza by cross-immunity

Abstract

Cross-immunity among related strains can account for the selection producing the slender phylogenetic tree of influenza A and B in humans. Using a model of seasonal influenza epidemics with drift (Andreasen, 2003. Dynamics of annual influenza A epidemics with immuno-selection. J. Math. Biol. 46, 504–536), and assuming that two mutants arrive in the host population sequentially, we determine the threshold condition for the establishment of the second mutant in the presence of partial cross-protection caused by the first mutant and their common ancestors. For fixed levels of cross-protection, the chance that the second mutant establishes increases with $\rho$ the basic reproduction ratio and some temporary immunity may be necessary to explain the slenderness of flu's phylogenetic tree. In the presence of moderate levels of temporary immunity, an asymmetric situation can arise in the season after the two mutants were introduced and established: if the offspring of the new mutant arrives before the offspring of the resident type, then the mutant-line may produce a massive epidemic suppressing the original lineage. However, if the original lineage arrives first then both strains may establish and the phylogenetic tree may bifurcate.

Introduction

The phylogenetic trees of influenza genes are long and narrow with short side branches and rare bifurcations where two trunks coexist over several years (Buonagurio et al., 1986, Cox and Subbarao, 2000, Fitch et al., 1991, Fitch et al., 1997, Hay et al., 2001). We here show how cross-immunity among genetically related viral variants can allow one mutant to establish while suppressing other equally viable mutants of the same ancestor, a mechanism that will give rise to a narrow tree with short side branches.

The slender phylogenetic tree of influenza A contrasts with the phylogenetic trees that have been observed for influenza in birds and for genes of other infectious diseases (Kawaoka et al., 1998, Frey et al., 1998, Rambaut et al., 2001). From a theoretical view-point a virus species can be seen as a quasi-species, i.e. as a cloud of viral types kept together in a balance between mutation and selection (Eigen and Schuster, 1979, Eigen, 1993, Abu-Raddad and Ferguson, 2004). For such a system we would expect a constant rate of divergence in non-structural genes suggesting a more branched structure of the phylogenetic tree than that observed for the flu. The mechanism thought to be responsible for the shape of the flu tree is natural selection induced by the cross-immunity among related viral strains. Such cross-immunity constantly inhibits branching by natural selection against mutants that are related to previously successful variants allowing only one lineage to spread in the population (Buonagurio et al., 1986).

Influenza A is an example of a virus undergoing antigenic change at an intermediate time scale, longer than the duration of an infection event, yet shorter than host life span. Thus during an infection, a single antigenic strain of the pathogen colonizes a host, but through its life the same host may be infected several times by antigenically different variants. The selection processes in the viral population giving rise to antigenic change and to the exclusion of multiple lineages are therefore determined by the immunity structure of the host population (Grenfell et al., 2004).

The most important antigen of influenza A and B is the hemagglutinin surface-molecule (HA), although antibodies are formed in response to many other sites, most noticeably the neuraminidase antigen (NA). Due to its configuration, antibodies are not formed to the functionally active part of the HA-molecule, instead antibodies are formed to five non-functional epitopes allowing for significant diversity in flu antigens. The antigenic variation in influenza virus is caused by two distinct processes. In a process known as virus drift, point mutations in the gene coding for HA give rise to new virus variants (strains) with gradually changing antigenic properties. Immunity obtained from infection with a specific strain of influenza confers permanent immunity to that particular strain and a partial protection against related strains. In general, the level of cross-protection decreases with the number of amino acids by which the HA-gene of the two strains differ and hence with the distance between the two strains in Hamming space. Smith et al. (2002) suggest that 3–4 amino-acid substitutions must occur in the virus before there is an appreciable chance of reinfection of the same host and in general cross-immunity seems to protect against the mutations that accumulate over a few years (Smith et al., 2002, Cox and Subbarao, 2000, Potter et al., 1977, Larson et al., 1978). Recently Smith et al. (2004) have found that the antigenic variation in influenza A/H3N2 can be represented in a two-dimensional space with most of the variation occurring along a single axis.

In addition to antigenic drift antigens also change in distinct shifts where reassortment with avian influenza replaces whole segments of the viral surface-structure introducing a new subtype. Such shift events occur at irregular intervals on the order of decades and they are usually associated with the disappearance of the old subtype. Thus, our focus will be on an intermediate time scale describing the period between two shifts.

Previous models of influenza drift have focused on the epidemiological consequences of drift by assuming that mutations occur along a one-dimensional axis representing the main trunk of the phylogenetic tree and that the drift-mutation is constant over time (Pease, 1987, Inaba, 2001, Thieme and Yang, 2002, Girvan et al., 2002, Andreasen, 2003). They have modeled the speed of the drift along the axis (Andreasen et al., 1996, Gog and Grenfell, 2002, Lin et al., 2003) or the mutation rate required for viral drift (Boni et al., 2004). Because of the multiple strains and the complexities of the population-based herd-immunity, an account for the transmission dynamics and mutation process sufficiently detailed to reproduce the drift-like behavior seems to require individual-based computer simulations (Ferguson et al., 2003, Tria et al., 2005). To avoid the complexities of such models we will not attempt to include all processes involved in influenza drift but rather study branching as a “perturbation” of the normal drift process. Most of the analytical drift-models do not allow for the introduction of additional mutants because the immunological structure of the virus population is modeled in a way that links to the time-progression so we shall base our model on that of Andreasen (2003).