Symmetry breaking in a model of antigenic variation with immune delay

TL;DR

This paper analyzes how immune delay influences symmetric dynamics in a malaria antigenic variation model, identifying bifurcations and periodic solutions, with results applicable to various infectious disease models.

Contribution

It introduces a novel analytical approach using isotypic decomposition to study immune delay effects and identifies bifurcation conditions in a malaria model.

Findings

Periodic solutions emerge at Hopf bifurcations due to immune delay.

Explicit boundary conditions for bifurcations are derived for small delays.

Numerical simulations illustrate diverse dynamical behaviors.

Abstract

Effects of immune delay on symmetric dynamics are investigated within a model of antigenic variation in malaria. Using isotypic decomposition of the phase space, stability problem is reduced to the analysis of a cubic transcendental equation for the eigenvalues. This allows one to identify periodic solutions with different symmetries arising at a Hopf bifurcation. In the case of small immune delay, the boundary of the Hopf bifurcation is found in a closed form in terms of system parameters. For arbitrary values of the time delay, general expressions for the critical time delay are found, which indicate bifurcation to an odd or even periodic solution. Numerical simulations of the full system are performed to illustrate different types of dynamical behaviour. The results of this analysis are quite generic and can be used to study within-host dynamics of many infectious diseases.