While kinetic models are widely used to describe viral infection at various levels, most of them are focused on temporal aspects and understanding of corresponding spatio-temporal aspects remains limited. In this work, our attention is focused on the initial stage of infection of immobile cells by virus particles (“virions”) under flow conditions with diffusion. A practical example of this scenario occurs when humans or animals consume food from virion-containing sources. Mathematically, such situations can be described by using a model constructed in analogy with those employed in chemical engineering for analysis of the function of a plug-flow reactor with dispersion. As in the temporal case, the corresponding spatio-temporal model predicts either the transition to a steady state or exponential growth of the populations of virions and infected cells. The spatial distributions of these species are similar in both of these regimes. In particular, the maximums of the populations are shifted to the upper boundary of the infected region. The results illustrating these conclusions were obtained analytically and by employing numerical calculations without and with the dependence of the kinetic parameters on the coordinate. The model proposed has also been used in order to illustrate the effect of antiviral feed additives on feedborne infection towards curbing disease transmission. © 2020 Elsevier B.V.