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 $\gamma \left(L^{-1}{U}_{\mathrm{Lt}}\right)$, γ continuous, is studied under a Eulerian scaling $u_t^L=L^{-1}{U}_{\mathrm{Lt}}$, 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 ΔT, 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.