A contact process with stronger mutations on trees

We consider a spatial stochastic model for a pathogen population growing inside a host that attempts to eliminate the pathogens through its immune system. The pathogen population is divided into different types. A pathogen can either reproduce by generating a pathogen of its own type or produce a pathogen of a new type that does not yet exist in the population. Pathogens with living ancestral types are protected against the host’s immune system as long as their progenitors are still alive. Each pathogen type without living ancestral types is eliminated by the immune system after a random period, independently of the other types. When a pathogen type is eliminated from the system, all pathogens of this type die simultaneously. In this paper, we determine the conditions on the set of model parameters that dictate the survival or extinction of the pathogen population when the dynamics unfold on graphs with an infinite tree structure.

💡 Research Summary

The paper introduces and rigorously analyzes a spatial stochastic model for a pathogen population that evolves under the combined forces of reproduction, beneficial mutation, and host immune response. Each pathogen occupies a vertex of a graph and can give birth to a new pathogen on a neighboring empty vertex at rate λ. With probability 1–r the offspring inherits the parent’s type; with probability r it mutates to a completely new type that is assumed to be “stronger” than its ancestor. Crucially, a type is not removed by the immune system until its ancestral type has disappeared; only then does an independent exponential clock of rate 1 start ticking for that type, and when it rings all individuals of that type are eliminated simultaneously. This delayed killing mechanism makes the process non‑Markovian, distinguishing it from the earlier Schinazi‑Schweinsberg model (S2) where killing clocks start immediately.

The authors focus on two infinite tree graphs. The undirected homogeneous tree T_d has degree d + 1 at every vertex, while the directed tree T⁺_d has each vertex (except the root) with one parent and d children. In T⁺_d a vertex can be occupied at most once, which yields a monotone dependence of the survival probability on both λ and r. By exploiting this monotonicity the authors derive an explicit critical birth rate

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