### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 2079-2102 |

Number of pages | 24 |

Journal | Journal Of Mathematical Biology |

Volume | 77 |

Issue number | 6-7 |

DOIs | |

State | Published - Dec 1 2018 |

### Keywords

- Diffusion
- Stochastic model
- Structured population model

### ASJC Scopus subject areas

- Modeling and Simulation
- Agricultural and Biological Sciences (miscellaneous)
- Applied Mathematics

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## Cite this

*Journal Of Mathematical Biology*,

*77*(6-7), 2079-2102. https://doi.org/10.1007/s00285-018-1246-6