  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.
