疟疾抗原变异模型的稳定性与分支
Stability and bifurcations in a model of antigenic variation in malaria
摘要：
我们研究了最近提出的疟疾抗原变异模型的特性，该模型包含多个表位以及持久和短暂的免疫反应。我们表明，在一个消失的衰减率为持久的免疫反应的情况下，该系统表现出所谓的“无参数分叉”，由于在相空间中的平衡点的超曲面的存在。当持久免疫反应的衰减率不为零时，平衡点的超曲面退化，并产生许多其他稳态，其中许多与系统的排列对称性有关。通过数值计算系统的横向李雅普诺夫指数，研究了系统在完全对称状态下的鲁棒性。这个练习的结果表明，对于持久免疫应答的消失衰减，完全对称状态在参数空间的大部分中并不鲁棒，相反，所有变体都发展出它们自己的时间动态，从而有助于整体时间演化。同时，如果持久免疫应答的衰减速率增加，则完全对称状态可以变得稳健，只要持久免疫应答的增长速率迅速。
英文摘要：
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.
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