# Open Quasispecies Models: Stability, Optimization, and Distributed Extension

## TL;DR

This paper introduces open quasispecies models that incorporate extinction, analyzing their stability and dynamics both analytically and numerically, and extending them to distributed models for complex biological applications.

## Contribution

It proposes a novel open quasispecies framework with stability analysis, inverse fitness optimization, and continuous distributed models for large genotype spaces.

## Key findings

- Quasispecies distribution and error threshold are observed in the models.
- Extinction phenomena are incorporated alongside growth dynamics.
- Distributed models enable analysis of high-dimensional genotype spaces.

## Abstract

We suggest a natural approach that leads to a modification of classical quasispecies models and incorporates the possibility of population extinction in addition to growth. The resulting modified models are called open. Their essential properties, regarding in particular equilibrium behavior, are investigated both analytically and numerically. The hallmarks of the quasispecies dynamics, viz. the heterogeneous quasispecies distribution itself and the error threshold phenomenon, can be observed in our models, along with extinction. In order to demonstrate the flexibility of the introduced framework, we study the inverse problem of fitness allocation under the biologically motivated criterion of steady-state fitness maximization. Having in mind the complexity of numerical investigation of high-dimensional quasispecies problems and the fact that the actual number of genotypes or alleles involved in a studied process can be extremely large, we also build continuous-time distributed open quasispecies models. The obtained results may serve as an initial step to developing mathematical models that involve directed therapy against various pathogens.