Population Dynamics and Pattern Formation in Antigen Antibody Co-evolution

The adaptive immune system provides specific protection against foreign threats by removing recognized antigens. Nevertheless, rapidly mutating viruses such as HIV can cause chronic infections by evading antibody recognition, leading to an evolutionary arms race. Here we introduce a minimal predator-prey model to study co-evolution of antibody and antigen. We show that intermediate replication rate and mutation rate of the antigen can lead to chronic infection. The effect of cross-reactivity, represented by a nonlocal interaction, is investigated. Interestingly, asymmetric cross-reactivity may lead to pattern formation in the shape space, indicating localization of antigens. On the one hand, instability resulting from the asymmetry could enable late clearance of antigens. On the other hand, an escape phase could emerge under homeostatic constraints.