Quasispecies dynamics on a network of interacting genotypes and idiotypes: Formulation of the model

A quasispecies is the stationary state of a set of interrelated genotypes that evolve according to the usual principles of selection and mutation. Quasispecies studies have invariably concentrated on the possibility of errors during genotype replication and their role in promoting either the survival or the demise of the quasispecies. In a previous work [V. C. Barbosa, R. Donangelo, and S. R. Souza, J. Theor. Biol. 312, 114 (2012)], we introduced a network model of quasispecies dynamics, based on a single probability parameter ($p$) and capable of addressing several plausibility issues of previous models. Here we extend that model by pairing its network with another one aimed at modeling the dynamics of the immune system when confronted with the quasispecies. The new network is based on the idiotypic-network model of immunity and, together with the previous one, constitutes a network model of interacting genotypes and idiotypes. The resulting model requires further parameters and as a consequence leads to a vast phase space. We have focused on a particular niche in which it is possible to observe the trade-offs involved in the quasispecies' survival or destruction. Within this niche, we give simulation results that highlight some key preconditions for quasispecies survival. These include a minimum initial abundance of genotypes relative to that of the idiotypes and a minimum value of $p$. The latter, in particular, is to be contrasted with the stand-alone quasispecies network of our previous work, in which arbitrarily low values of $p$ constitute a guarantee of quasispecies survival.