Abstract

We consider an intrahost malaria model allowing for antigenic variation within a single species. The host’s immune response is compartmentalised into reactions to major and minor epitopes. We investigate the dynamics of the model, paying particular attention to bifurcation and stability of the uniform nonzero endemic equilibrium. We establish conditions for the existence of an equivariant Hopf bifurcation in a ring of antigenic variants, characterised by time delay.

1. Preliminaries

An intrahost mathematical model of Plasmodium falciparum, a species of parasites that cause malaria in humans, is considered. The central achievement of the model, first proposed by [1], is its ability to replicate the phenomenon of antigenic variation, which is a mechanism employed by the parasite in order to evade detection by the host’s immune system. In addition, the proposed model incorporates the effects of immune response (IR) mounted by the human host. Such a model has been the subject of a number of previous studies (see [2–11], e.g.). In particular, [7, 8] introduce the idea of a delayed IR, leading to a mathematical model comprised of a set of coupled nonlinear delay differential equations (DDEs), where it is assumed that the IR time delay is discrete. Specifically, there is a time delay between changes in the parasite load and the production of immune effectors [8]. The authors then proceed to show that a range of interesting dynamics (synchronous and asynchronous oscillations) result as a consequence of the (small) time delay. The current paper is a further development of the model studied in [7, 8]. The distinction of our work is that we focus on the effects of symmetry on the dynamics of the model and that our time delay is not constrained to be small. In particular, we establish sufficient and necessary conditions for the existence of an equivariant Hopf bifurcation. We must state at the onset that the current study is similar in spirit to the recent work of [3–5]. However, there are fundamental differences in what we do with the model, as explained below. First of all, the study conducted in [3] concerns the Recker et al. [1] model, with no time delays in the host’s immune response. Following in the footsteps of the monumental work of [12, 13], the author then uses elements of equivariant bifurcation theory to study the effects of symmetry on the dynamic interactions of the host and the pathogens. In the work of [4], the authors attack the problem of symmetry-breaking in system (1). They establish the existence of a fully symmetric steady state of (1) and then employ ideas of equivariant bifurcation theory [12, 13] to study the dynamics of this steady state. Essentially, the authors investigate the effects of immune response time delay on the symmetric dynamics of (1). They do so by employing the technique of isotypic decomposition [12, 13] to reduce the stability problem to a simple transcendental equation for the eigenvalues [4]. In [5], the authors employ the groupoid formalism developed in [14, 15] to study the dynamics of cross-reactivity from antigenic variation and establish a synchrony-breaking Hopf bifurcation emanating from a nontrivial synchronous equilibrium of system (1). To the best of our knowledge, the problem of equivariant Hopf bifurcation in the time-delayed modification of the Recker et al. [1] model (2) or (1) has never been addressed before in the literature.

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 separates the parasitised red blood cell population, denoted by , into variants, each characterised by the unique major epitope of their displayed antigen (see [1, 7, 8] and references cited therein). The variables and denote variant-specific and cross-reactive immune responses, respectively; is the intrinsic parasite growth rate, and are the removal rates associated with specific and cross-reactive immune responses, respectively, and are the proliferation rates of immune responses, and are the decay rates of variant-specific and cross-reactive immune responses, and is the discrete time delay of the IR. The coefficients 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 separates the parasitised red blood cell population, denoted by , into variants. The variants 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 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 minor epitopes characterised by variants per epitope. The total number of variants in this case is given byThe 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 , whose entries are identical to unity if the variants and have some minor epitopes in common; otherwise [4]. The matrix is always symmetric [4]. In the case of a ring of variants characterised by all-to-all coupling, as depicted in Figure 1, the corresponding adjacency matrix is given byIt is straightforward to construct the adjacency matrix for an arbitrarily large number of minor epitopes [3]. In this paper, we focus on minor epitopes, with antigenic variants per epitope. By recourse to (5), we may express (2) in vectorial form aswhere , , , and . With appropriate initial conditions, it may be easily shown that system (6) is well posed [4]; that is, its solutions are nonnegative . Symmetry properties of (6) are encoded in the associated adjacency matrix . In the present case, in which there are minor epitopes with 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 nodes with an all-to-all coupling and is the cyclic group of order , corresponding to rotations by [4, 12, 13]. In particular, system (6) is equivariant under the action of the dihedral group , which is a -dimensional symmetry group of an -gon (see [3] for a pertinent brief outline of equivariance bifurcation theory). For the general dihedral group of order , whether 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]).

Figure 1 

Interaction of antigenic variants in the case of minor epitopes with variants per epitope. Every variant in the ring will be connected to every other variant in much the same way that variant is connected. For clarity and to avoid cluttering the diagram, we have only shown the full network connections of variant .

System (2) has an infinitude of -dimensional equilibria, of the generic formwhereWe must comment at this point that the present study focusses entirely on the uniform equilibriumof (2). The linearisation of system (2) about the equilibrium is given byand yields the characteristic equation [7]whereand where all the parameters are nonnegative. Equation (12) consists of the factors and , with multiplicities and , respectively. This type of factorisation of the characteristic equation is due to the presence of symmetry in (2). The continuous extension at is given byThe equilibrium of (2) will be locally asymptotically stable if all of the roots, , 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 if, and only if, . In other words, always yields a real root of (12), , of multiplicity . As a consequence of this observation, we see that an analysis of the distribution of roots of boils down to that of the distribution of roots of the factor . Let , in the two factors, and , of (12), and separate into real and imaginary parts to obtain , with , whereWe begin our investigation by focussing on the factor of (12). It is of vital importance to establish conditions under which the factor has purely imaginary roots. To this end, let in (15). This leads to the following simplifying relations:respectively. Solving for and in (16) and (17) yieldsrespectively. Recalling that , squaring both sides of the equations in (18), adding, and rearranging givewhich always has at least one real root , since . In fact, the possible solutions of the quartic are given bySince the frequency , it follows that the only viable solution of (20) is . This implies that the characteristic equation (12) has pairs of purely imaginary roots . Similarly, (19) yields the octic polynomialwhere the coefficients are given in the Appendix. We now quickly review the local stability of . We focus our attention on the factor of (12). Equation (20) has one real root, denoted by . Substitute into (18) to obtain the sequence of critical time delaysAs a result, when , the characteristic equation (12) has pairs of purely imaginary roots; . We defineLet us shift our attention momentarily to the factor of (12). First of all, we note that (22) has at least one positive real root if and . The proof of this fact is indeed elementary. Assume that (22) has eight simple real roots, denoted by , . 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 , . To establish that one of these roots is positive, we proceed in the following manner. Assume that , and define the functionwhere , , , , , , 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) isFunction (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, . Consequently, one of the roots of (22) is positive if, and only if, the condition is fulfilled. From (19), we compute the corresponding sequence of critical time delays:where and . When , it follows that (12) has a pair of purely imaginary roots, . Finally, we defineAssume that in (12), and differentiate with respect to . For the simple root case, we recall from (12) that . As a consequence of this, we obtain the following:whereUsing the computer algebra package MAPLE®, we deduce that the usual transversality condition corresponding to the factor is fulfilled if, and only if,By continuity, it follows that becomes positive when and the equilibrium of (2) becomes unstable. As a result, a simple root Hopf bifurcation occurs when passes through the critical time delay . Consider the equation when ; that is,Employing the well-known Routh-Hurwitz criterion and the fact that the parameters , and are strictly positive, it follows that all the roots of (32) have negative real part if, and only if,Similarly, for the equation with , the Routh-Hurwitz criterion implies that all the roots of this equation have negative real part if, and only if,

Proposition 1. For , the equilibrium of (2) is asymptotically stable if, and only if, the inequality in (33) is fulfilled.

We arrive at the first of our main results.

Theorem 2. Suppose that conditions (34) and (31) are satisfied and that (20) has no positive real roots for , where . Since , it follows that is asymptotically stable when and unstable when . When , a simple root Hopf bifurcation occurs.