Stability and bifurcations in a model of antigenic variation in malaria

TL;DR

This paper analyzes a malaria antigenic variation model, revealing complex bifurcation behaviors and stability properties depending on immune response decay rates, with implications for understanding immune dynamics.

Contribution

It introduces a detailed bifurcation analysis of a malaria model incorporating multiple immune responses and symmetry considerations, highlighting conditions for stability and symmetry breaking.

Findings

Bifurcations without parameters occur when decay rate is zero.

The hypersurface of equilibria degenerates with non-zero decay rate.

Symmetry stability depends on immune response decay and growth rates.

Abstract

We examine the properties of a recently proposed model for antigenic variation in malaria which incorporates multiple epitopes and both long-lasting and transient immune responses. We show that in the case of a vanishing decay rate for the long-lasting immune response, the system exhibits the so-called "bifurcations without parameters" due to the existence of a hypersurface of equilibria in the phase space. When the decay rate of the long-lasting immune response is different from zero, the hypersurface of equilibria degenerates, and a multitude of other steady states are born, many of which are related by a permutation symmetry of the system. The robustness of the fully symmetric state of the system was investigated by means of numerical computation of transverse Lyapunov exponents. The results of this exercise indicate that for a vanishing decay of long-lasting immune response, the fully symmetric state is not robust in the substantial part of the parameter space, and instead all variants develop their own temporal dynamics contributing to the overall time evolution. At the same time, if the decay rate of the long-lasting immune response is increased, the fully symmetric state can become robust provided the growth rate of the long-lasting immune response is rapid.