Narrow escape to small windows on a small ball modeling the viral entry into the cell nucleus

A certain class of viruses replicates inside a cell if they can enter the nucleus through one of many small target pores, before being permanently trapped or degraded. We adopt for viral motion a switching stochastic process model and we estimate here the probability and the conditional mean first passage time for a viral particle to attain alive the nucleus. The cell nucleus is covered with thousands of small absorbing nuclear pores and the minimum distance between them defines the smallest spatial scale that limits the efficiency of stochastic simulations. Using the Neuman-Green's function method to solve the steady-state Fokker-Planck equation, we derive asymptotic formula for the probability and mean arrival time to a small window for various pores' distributions, that agree with stochastic simulations. These formulas reveal how key geometrical parameters defines the cytoplasmic stage of viral infection.