Dynamics of influenza A drift: the linear three-strain model

Abstract

We analyze an epidemiological model consisting of a linear chain of three cocirculating influenza A strains that provide hosts exposed to a given strain with partial immune cross-protection against other strains. In the extreme case where infection with the middle strain prevents further infections from the other two strains, we reduce the model to a six-dimensional kernel capable of showing self-sustaining oscillations at relatively high levels of cross-protection. Dimensional reduction has been accomplished by a transformation of variables that preserves the eigenvalue responsible for the transition from damped oscillations to limit cycle solutions.

Introduction

The simultaneous circulation of several antigenic variants of the same pathogen can give rise to complex dynamics, including sustained oscillations and chaos in disease prevalence when the pathogen confers sufficiently strong cross-protection against related strains to recovering hosts [1]. Previous efforts to illustrate how periodic dynamics can be sustained in disease systems have necessarily invoked a complex of factors; in this paper we demonstrate that in a system with three interacting strains, herd-immunity alone can support sustained oscillations. The simplest such case is the `linear three-strain model', in which one of the strains is of an intermediate type in the sense that it confers partial cross-protection to the two other strains while these two strains induce no reciprocal cross-reaction.

The cocirculation of cross-reacting pathogen types is of interest from both ecological and evolutionary viewpoints. In a few diseases, such as influenza A and canine pavovirus, new antigenic variants arise continuously thus affecting significantly the epidemiology of the disease. Other pathogens display antigenic variation that seems to play no role in the epidemiology of these diseases [2], though the evolutionary forces that maintain this variation remain important to study. Our focus in this paper will be on the influenza A virus, which has been particularly well studied.

Influenza A is an RNA virus of the Orthomyxoviridae family [3]. Its persistence in many vertebrate species appears to be linked to its high degree of genetic plasticity [4], [5]. Two processes cause the rapid evolution of the glycoprotein molecules, hemagglutinin (HA) and neuraminidase (NA), on the surface of the virus: antigenic drift and genetic shift [6], [7]. In antigenic drift, point mutations in HA and NA gradually change the aminoacid composition of antigenic sites. These mutants or strains are responsible for annual or biennial epidemics affecting tens of millions of people worldwide. In genetic shift, gene reassortment in the negatively charged segments of the nuclear RNA gives rise to a new virus subtype with a different set of antibody binding sites (epitopes) in the HA and NA molecules [8]. This virus shift is responsible for the pandemics of 1918 (Spanish flu, H₁N₁ subtype), 1957 (Asian flu, H₂N₂ subtype) and 1968 (Hong Kong flu, H₃N₂ subtype). Although before the Russian flu of 1977 only one subtype was present at any one time, since then both subtypes H₁N₁ and H₃N₂ have been cocirculating worldwide [9], [10]. There is still no clear understanding of why this change has occurred [11], [12], [13].

Cross-immunity between different subtypes of influenza A is weak or difficult to detect [14], [15], [16], [17]. This is not so with drift variants of the same subtype. Functionally related strains show partial cross-reaction to the antibodies produced in the host against a previous virus strain [18], [19], [20]. The degree of cross-reaction between two strains can be identified serologically from hemagglutination inhibition with antibodies reacting fully with one of the strains [21]. Partially cross-reacting strains are functionally related, and zero cross-reactivity means no cross-protection with the antibodies produced in the host against a previous virus strain.

Theoretical studies of cross-immunity in disease transmission dynamics are relatively few. By keeping track of hosts infected with each single strain, one is naturally led to consider extensions of the well-known continuous-time SIR models to describe the outcomes of selection on a group of strains [22], [23]. Models incorporating partial cross-immunity between two strains have been shown to maintain sustained oscillations when age-specific mortalities are included [24], [25], or when disease is transmitted by a vector [26]. These oscillations become weakly damped without some sort of delay [27], [28].

In a previous paper [29] we analyzed an epidemiological model of influenza A drift that included life-long partial cross-protection among neighboring strains. When strains are placed on a one-dimensional lattice with periodic boundary conditions and partial cross-protection to nearest neighbors, it is found that at relatively high levels of cross-protection sustained oscillations are possible only if the number of cocirculating strains exceeds three. In this paper we show that it is also possible to have a Hopf bifurcation to a periodic solution even when only three strains, in a linear chain configuration, are cocirculating. Therefore, the presence of oscillations does not rely on the cyclic nature of the immunity structure as in [29], but may occur when immunity is organized linearly, mimicking the immunity structure that arises under antigenic drift.

It has recently been shown that the evolutionary tree of the hemagglutinin gene has the shape of a cactus tree with the main trunk of surviving genes evolving significantly faster than the short lateral branches of non-surviving genes [30]. To some this is a sign of positive Darwinian evolution [31], [32], while others take a more cautionary view [33]. In this paper we study the dynamics of virus drift for the surviving lineage when the total virus population is divided in subgroups with varying degrees of cross-protection among strains.

In the first part we outline the linear three-strain model, using an index set notation to describe population subgroups with a particular history of infection. In the second part we study a submodel that exhibits sustained oscillations. This submodel has the property that individuals infected with the middle strain are immune to further infections by the other two symmetrical strains. By taking advantage of the remaining symmetry imposed on this submodel we further reduce it to a six-dimensional system. This is so far the simplest system we have been able to find that has self-sustaining oscillations.