Fluid limit for a genetic mutation model

We trace the time evolution of the number U t of nondeleterious mutations, present in a gene modeled by a word of length L and DNA fragments by characters labeled 0, 1,…, N . For simplification, deleterious mutations are codified as equal to 0. The discrete case studied in Grigorescu ( Stochastic Models , 29, 2013 p. 328), is a modified version of the Pólya urn, where the two types are exactly the zeros and nonzeros. A random continuous‐time binary mutation model, where the probability of creating a deleterious mutation is 1/ N , while the probability of recovery γ ( L - 1U Lt ) , γ continuous, is studied under a Eulerian scalingu t L = L - 1U Lt , L  → ∞. The fluid limit u t , emerging due to the high frequency scale of mutations, is the solution of a deterministic generalized logistic equation. The power law γ ( u ) =  cu a captures important features in both genetical and epidemiological interpretations, with c being the intensity of the intervention, a the strength/virulence of the disease, and 1/ N the decay rate/infectiousness. Among other applications, we obtain a quantitative study of Δ T , the maximal interval between tests. Several stochastic optimization problems, including a generalization of the Shepp urn ( The Annals of Mathematical Statistics , 40, 1969 p. 993), are proposed.