A structured population model with diffusion in structure space

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≥ 0. 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→ 0 ⁺ . Infection/reinfection is then modeled 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.