Fluid limit for a genetic mutation model. Issue 3 (15th November 2019) - Bajo Caraballo, Carlos; Grigorescu, Ilie
Abstract: We trace the time evolution of the number Ut 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 - 1 U Lt ), γ continuous, is studied under a Eulerian scaling u t L = L - 1 U Lt, L → ∞. The fluid limit ut, 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.