Equivariant Hopf Bifurcation in a Time-Delayed Ring of Antigenic Variants

Let us begin by commenting that an in-depth description of time-delayed modification of Recker et al. [1] model can be found in [7, 8]. Here we simply give a very brief description, primarily for the express purpose of casting the model in the context of the analysis to come. The time-delayed modification of Recker et al. [1] is expressible in the form [7, 8] where the index j = 1, …, N separates the parasitised red blood cell population, denoted by Y_j, into N variants, each characterised by the unique major epitope of their displayed antigen (see [1, 7, 8] and references cited therein). The variables Z_j and W_j denote variant-specific and cross-reactive immune responses, respectively; ϕ is the intrinsic parasite growth rate, α and α^′ are the removal rates associated with specific and cross-reactive immune responses, respectively, β and β^′ are the proliferation rates of immune responses, μ and μ^′ are the decay rates of variant-specific and cross-reactive immune responses, and T_d is the discrete time delay of the IR. The coefficients c_jk of the connectivity matrix characterise cross-reactive intervariant interactions [1, 2, 7, 8, 11].

After normalisation and change of variables, [8] reduced system (1) to the following system: where $$ is a discrete time delay. The index j = 1, …, n < N separates the parasitised red blood cell population, denoted by y_j, into n < N variants. The variants j = n + 1, …, N are neglected in this reformulation of (1) [8]. As a consequence of this, the sum in (1) collapses to [8] Without going into specific details, it is important to point out that all the parameters in (2) are positive. Every variant in system (2) has the same n minor epitopes in common [7]. This point highlights a fundamental difference between the model studied in this work and the models studied in [3–5]. In essence, (2) represents a subsystem of (1). System (2) represents the interaction of malaria antigenic variants in the special case in which there are n minor epitopes characterised by n variants per epitope. The total number of variants in this case is given by The interaction of these different antigenic variants may be represented schematically as shown in Figure 1, from which it is evident that system (2) is endowed with some spatial symmetry, which we will attempt to describe in due course. We may gain some further insight about system (2) by analysing the structure of its associated adjacency matrix T, whose entries T_ij are identical to unity if the variants i and j have some minor epitopes in common; otherwise T_ij = 0 [4]. The matrix T is always symmetric [4]. In the case of a ring of nⁿ variants characterised by all-to-all coupling, as depicted in Figure 1, the corresponding nⁿ × nⁿ adjacency matrix is given by It is straightforward to construct the adjacency matrix T for an arbitrarily large number of minor epitopes [3]. In this paper, we focus on n minor epitopes, with n antigenic variants per epitope. By recourse to (5), we may express (2) in vectorial form as where y = (y₁, y₂, …, y_n) ^t, x = (x₁, x₂, …, x_n) ^t, w = (w₁, w₂, …, w_n) ^t, and $$. With appropriate initial conditions, it may be easily shown that system (6) is well posed [4]; that is, its solutions are nonnegative ∀t ≥ 0. Symmetry properties of (6) are encoded in the associated adjacency matrix T. In the present case, in which there are n minor epitopes with n variants per epitope, the dynamical system (6) is equivariant with respect to the symmetry group [12, 13] where $$ represents the symmetric group of all permutations in a network of n nodes with an all-to-all coupling and $$ is the cyclic group of order n, corresponding to rotations by 2π/n [4, 12, 13]. In particular, system (6) is equivariant under the action of the dihedral group $$, which is a 2nⁿ-dimensional symmetry group of an nⁿ-gon (see [3] for a pertinent brief outline of equivariance bifurcation theory). For the general dihedral group $$ of order 2nⁿ, whether nⁿ is even or odd is crucial as it demarcates two different choices as far as conjugacies of reflections are concerned (see page 128 of [3]).

System (2) has an infinitude of 3n-dimensional equilibria, of the generic form where We must comment at this point that the present study focusses entirely on the uniform equilibrium of (2). The linearisation of system (2) about the equilibrium E₀ is given by and yields the characteristic equation [7] where and where all the parameters are nonnegative. Equation (12) consists of the factors Δ₁(λ, τ)≔F₁(λ)F_ap(λ, τ) and Δ₂(λ, τ)≔F_s(λ, τ), with multiplicities n − 1 and 1, respectively. This type of factorisation of the characteristic equation is due to the presence of symmetry in (2). The continuous extension at λ = 0 is given by The equilibrium E₀ of (2) will be locally asymptotically stable if all of the roots, λ, of (12) have negative real part and unstable if at least one root has a positive real part.

First of all, we must note that F₁(λ) = 0 if, and only if, λ = −ab < 0. In other words, F₁(λ) = 0 always yields a real root of (12), λ = −ab < 0, of multiplicity n − 1. As a consequence of this observation, we see that an analysis of the distribution of roots of Δ₁(λ, τ) = 0 boils down to that of the distribution of roots of the factor F_ap(λ, τ) = 0. Let λ = ν + iω, $$ in the two factors, F_ap(λ, τ) and Δ₂(λ, τ), of (12), and separate into real and imaginary parts to obtain Δ_j(λ, τ) = R_j(ν, ω) + iI_j(ν, ω), with j = 1,2, where We begin our investigation by focussing on the factor Δ₁(λ, τ) of (12). It is of vital importance to establish conditions under which the factor F_ap(λ, τ) has purely imaginary roots. To this end, let ν = 0 in (15). This leads to the following simplifying relations: respectively. Solving for sin⁡(ωτ) and cos⁡(ωτ) in (16) and (17) yields respectively. Recalling that sin²⁡(ωτ) + cos²⁡(ωτ) = 1, squaring both sides of the equations in (18), adding, and rearranging give which always has at least one real root ω_∗ > 0, since Φ(0) = −c² < 0. In fact, the possible solutions of the quartic Φ(ω) = 0 are given by Since the frequency ω > 0, it follows that the only viable solution of (20) is ω₊. This implies that the characteristic equation (12) has (n − 1) pairs of purely imaginary roots ±iω₊. Similarly, (19) yields the octic polynomial where the coefficients A₀, …, A₈ are given in the Appendix. We now quickly review the local stability of E₀. We focus our attention on the factor Δ₁(λ, τ) of (12). Equation (20) has one real root, denoted by ω₊. Substitute ω₊ into (18) to obtain the sequence of critical time delays As a result, when $$, the characteristic equation (12) has (n − 1) pairs of purely imaginary roots; λ = ±iω₊. We define Let us shift our attention momentarily to the factor Δ₂(λ, τ) of (12). First of all, we note that (22) has at least one positive real root if A₀ < 0 and A₈ > 0. The proof of this fact is indeed elementary. Assume that (22) has eight simple real roots, denoted by ω_k, k = 1,2, …, 8. In fact, establishing conditions to guarantee the existence of such roots of (22) is far from trivial. For this reason, we will avoid delving into this subject in this paper. Suffice to say that the existence of such roots guarantees that a nondegenerate bifurcation occurs in the special case A₀ > 0, A₈ > 0. To establish that one of these roots is positive, we proceed in the following manner. Assume that A₈ ≠ 0, and define the function where $$, $$, $$, $$, $$, $$, and $$. It is important to stress that one of the roots of (22) is positive if at least one of the critical points of (25) is positive. The derivative function of (25) is Function (26) has at least one positive real root if, and only if, at least one of its critical points is positive. This happens if, and only if, A₂ < 0. Consequently, one of the roots of (22) is positive if, and only if, the condition A₂ < 0 is fulfilled. From (19), we compute the corresponding sequence of critical time delays: where j = 0,1, 2, … and k = 1,2, …, 8. When $$, it follows that (12) has a pair of purely imaginary roots, λ = ±iω_k. Finally, we define Assume that λ = λ(τ) in (12), and differentiate with respect to τ. For the simple root case, we recall from (12) that Δ₂(iω, τ) = 0. As a consequence of this, we obtain the following: where Using the computer algebra package MAPLE®, we deduce that the usual transversality condition corresponding to the factor Δ₂(λ, τ) is fulfilled if, and only if, By continuity, it follows that Re⁡[λ(τ)] becomes positive when $$ and the equilibrium E₀ of (2) becomes unstable. As a result, a simple root Hopf bifurcation occurs when τ passes through the critical time delay $$. Consider the equation Δ₁(λ, τ) = 0 when τ = 0; that is, Employing the well-known Routh-Hurwitz criterion and the fact that the parameters a, b, c, and n are strictly positive, it follows that all the roots of (32) have negative real part if, and only if, Similarly, for the equation Δ₂(λ, τ) = 0 with τ = 0, the Routh-Hurwitz criterion implies that all the roots of this equation have negative real part if, and only if,

The characteristic equation (12) can be written in the form By inspection, we can see that (35) has purely imaginary roots λ = ±iω of multiplicity n − 1 for parameters such that Δ₁(±iω, τ) = 0. That is, when The set of equations of (16) has four parameters, namely, (a, τ, c, n). If we fix three of the parameters, this gives two equations that may be solved for the critical value of the fourth parameter and the corresponding imaginary part of the eigenvalue ω_c. We consider a as the bifurcation parameter. All the results hold and are proved analogously, if any of the other parameters is used instead [16]. For convenience, we rewrite (16) in the form Taking the ratio of the two expressions in (37) yields an implicit equation for ω_c: Squaring and adding the expressions of (37) yield an equation for the corresponding critical value of a for Δ₁: Solving (39) for the parameter a gives where Let $$ denote the Banach space of continuous mappings from [−τ, 0] into $$ and endowed with the norm where |·| is the Euclidean norm on $$. Let x(t) be a solution of (2), and define x_t(θ) = x(t + θ), −τ ≤ θ ≤ 0. If the solution x(t) is continuous, then $$. We may now express (2) as the functional differential equation where $$ is defined by Analogously, the linearisation of (2) about E₀ is expressible as where the linear operator $$ is defined as where $$ and $$ are the n × n zero and identity matrices, respectively. The n × n matrices $$ and $$ are given by It is well known that a linear functional differential equation such as (45) generates a strongly continuous semigroup of linear operators with infinitesimal generator A(a) given by [17, 18]

The extension of the theory of equivariant Hopf bifurcation to functional differential equations was established in the series of papers [18, 21, 22]. Much of the development in this section is in the spirit of the work of [16, 19, 20] that independently studied equivariant Hopf bifurcation in a ring of n identical neurons characterised by signal transmission time delays. Defining the 3n × 1 column vector the linearisation (11) is expressible in the form where A and B are 3n × 3n matrices given by From the above consideration, we get that the characteristic matrix of the linearisation of (2) about E₀ is given by the 3n × 3n matrix and the associated characteristic equation is which reduces to (12). Mitchell and Carr [7, 8] have shown that the characteristic matrix of the linearisation of (2) about the equilibrium E₀ is given by the 3n × 3n block matrix of the form where D and E are given by It is evident that the characteristic matrix (54) is circulant, a fact to be exploited in the analysis to come. The corresponding characteristic equation is which can be shown to reduce to (12). With the apparatus developed above, we are now in a position to establish some lemmas [16] that will lead us to an equivariant Hopf bifurcation theorem.

As a consequence of the fact that the matrix (54) is circulant and to facilitate the analysis to follow, we set [16, 18–20] We note the following identities: We arrive at the following useful results: Following the footsteps of Lemma 3.1 of [16], we arrive at the following result.

In the characteristic equation (12), let λ = λ(τ) and differentiate with respect to τ to obtain from which it is straightforward to show that the usual transversality condition is fulfilled if, and only if, where and where relations (18) have been used to perform some simplifications in (66). The expression for $$ is given in the Appendix. Once again, following the footsteps of Lemma 3.2 of [16], we arrive at the following nonresonance condition.