International Journal of Differential Equations

Volume 2016, Article ID 5915768, 12 pages

http://dx.doi.org/10.1155/2016/5915768

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

College of Engineering, Technology, and Physical Sciences, Department of Mathematics, Alabama A & M University, 4900 Meridian Street North, Huntsville, AL 35762, USA

Received 4 July 2015; Revised 26 November 2015; Accepted 1 December 2015

Academic Editor: Salim Messaoudi

Copyright © 2016 Israel Ncube. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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]).