A structured population model with diffusion in structure space

12 Jul 2018, 10:30
New Law School/--107 (University of Sydney)

New Law School/--107

University of Sydney



Fabio A. Milner (School of Mathematical and Statistical Sciences, Arizona State University)


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.

Primary author

Fabio A. Milner (School of Mathematical and Statistical Sciences, Arizona State University)


Andrea Pugliese (Department of Mathematics, University of Trento, Italy)

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