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

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.


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][3][4][5][6][7][8][9][10][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][4][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 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 timedelayed modification of Recker et al. [1] is expressible in the form [7,8] where the index  = 1, . . .,  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 crossreactive 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  ∈ R + is a discrete time delay.The index  = 1, . . .,  <  separates the parasitised red blood cell population, denoted by   , into  <  variants.The variants  =  + 1, . . .,  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][4][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 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   are identical to unity if the variants  and  have some minor epitopes in common; otherwise   = 0 [4].The matrix T 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 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  minor epitopes, with  antigenic variants per epitope.By recourse to (5), we may express (2) in vectorial form as where y = ( 1 ,  2 , . . .,   ) t , x = ( 1 ,  2 , . . .,   ) t , w = ( 1 ,  2 , . . .,   ) t , and 1  = (  times ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ 1, 1, . . ., 1) t .With appropriate initial conditions, it may be easily shown that system (6) is well posed [4]; that is, its solutions are nonnegative ∀ ≥ 0. Symmetry properties of ( 6) are encoded in the associated  adjacency matrix T. 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 S  represents the symmetric group of all permutations in a network of  nodes with an all-to-all coupling and Z  is the cyclic group of order , corresponding to rotations by 2/ [4,12,13].In particular, system ( 6) is equivariant under the action of the dihedral group D   , which is a 2 dimensional symmetry group of an   -gon (see [3] for a pertinent brief outline of equivariance bifurcation theory).
For ) , 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  0 is given by and yields the characteristic equation [7]  (, ) fl ( 1 ()   (, )) where and where all the parameters are nonnegative.Equation ( 12) consists of the factors Δ 1 (, ) fl  1 ()  (, ) and Δ 2 (, ) fl   (, ), with multiplicities  − 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  = 0 is given by The equilibrium  0 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.
Let us shift our attention momentarily to the factor Δ 2 (, ) of (12).First of all, we note that ( 22) has at least one positive real root if  0 < 0 and  8 > 0. The proof of this fact is indeed elementary.Assume that (22) has eight simple real roots, denoted by   ,  = 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  0 > 0,  8 > 0. To establish that one of these roots is positive, we proceed in the following manner.Assume that  8 ̸ = 0, and define the function where where  = 0, 1, 2, . . .and  = 1, 2, . . ., 8. When  =  Assume that  = () in ( 12), and differentiate with respect to .For the simple root case, we recall from (12) that Δ 2 (, ) = 0.As a consequence of this, we obtain the following: International Journal of Differential Equations By continuity, it follows that Re[()] becomes positive when  >  (0)   and the equilibrium  0 of (2) becomes unstable.As a result, a simple root Hopf bifurcation occurs when  passes through the critical time delay  (0)  .Consider the equation Δ 1 (, ) = 0 when  = 0; 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 Δ 2 (, ) = 0 with  = 0, the Routh-Hurwitz criterion implies that all the roots of this equation have negative real part if, and only if, Proposition 1.For  = 0, the equilibrium  0 of ( 2) is asymptotically stable if, and only if, the inequality in (33) is fulfilled.
We arrive at the first of our main results.
The characteristic equation ( 12) can be written in the form By inspection, we can see that (35) has purely imaginary roots  = ± of multiplicity  − 1 for parameters such that Δ 1 (±, ) = 0.That is, when The set of equations of ( 16) has four parameters, namely, (, , , ).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   .We consider  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   : Squaring and adding the expressions of (37) yield an equation for the corresponding critical value of  for Δ 1 : Solving (39) for the parameter  gives where where | ⋅ | is the Euclidean norm on R 3 .Let x() be a solution of (2), and define x  () = x( + ), − ≤  ≤ 0. If the solution x() is continuous, then x  () ∈ C. We may now express (2) as the functional differential equation where ) , with  fl Analogously, the linearisation of (2) about  0 is expressible as where the linear operator () : C → R 3 is defined as where O  and I  are the  ×  zero and identity matrices, respectively.The  ×  matrices P  and Q  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 () given by [17,18]  ()  =   ,  ∈ Dom ( ()) , where the eigenvalues of () correspond to the roots of the characteristic equation ( 12).
Definition 3. Let F : C → R 3 and Γ be a compact group.
The system x  () = F(x  ) is said to be Γ-equivariant if F(x  ) = F(x  ) for all  ∈ Γ.

Equivariant Hopf Bifurcation
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] From the above consideration, we get that the characteristic matrix of the linearisation of (2) about  0 is given by the 3 × 3 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  0 is given by the 3 × 3 block matrix of the form ) . (55) 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 det  3 () = 0, (56) 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][19][20] V  fl (1,   ,  2 , . . .,  (3−1) ) T ,  = 0, 1, . . ., 3 − 1,  =  2/3 . (57) We note the following identities: (58)

2 International
Journal of Differential Equations 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.

Figure 1 :
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 4 is connected.For clarity and to avoid cluttering the diagram, we have only shown the full network connections of variant 4.
[3] general dihedral group D   of order 2  , 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]).