A structured population model is described and analyzed, in which individual dynamics is stochastic. The model consists of a PDE of advection-diffusion type in the structure variable. The population may represent, for example, the density of infected individuals structured by pathogen density $x$, $x\ge0$. The individuals with density $x=0$ are not infected, but rather susceptible or recovered. Their dynamics is described by an ODE with a source term that is the exact flux from the diffusion and advection as $x\to0^+$. Infection/reinfection is then modelled moving a fraction of these individuals into the infected class by distributing them in the structure variable through a probability density function. Existence of a global-in-time solution is proven, as well as a classical bifurcation result about equilibrium solutions: a net reproduction number $R_0$ is defined that separates the case of only the trivial equilibrium existing when $R_0<1$ from the existence of another - nontrivial - equilibrium when $R_0>1$. Numerical simulation results are provided to show the stabilization towards the positive equilibrium when $R_0>1$ and towards the trivial one when $R_0<1$, result that is not proven analytically. Simulations are also provided to show the Allee effect that helps boost population sizes at low densities.