Public Thesis defense - ELI

Fractional diffusion models for epidemiological and immunological applications by Afshin FARHADI

Pour l’obtention du grade de Docteur en sciences agronomiques et ingénierie biologique

Lévy flights are a specific model of random walks that appear to be ubiquitous across many different fields where the dispersal process is faster than dictated by Brownian motion. Lévy flights consist of a succession of random displacements whose step lengths have a heavy-tailed probability distribution. Unlike Brownian motion, whose step-length probability distribution decays exponentially, the distribution of Lévy flights decays algebraically. This leads to heavier tails and hence a larger probability of very long displacements that would be almost impossible with a Brownian motion. This thesis consists in deriving a space-fractional-order diffusion model that explicitly represents the effect of Lévy flights, deriving a numerical algorithm for solving the model equations, and then applying it to study the dispersion of living organisms in the field of life-science problems. In immunology, it has been observed that CD8+ T cells adopt a Lévy flight foraging pattern in response to Toxoplasma gondii infection in the brain. Here, we show that the Lévy search pattern enables T cells to spread over the whole brain tissue and hence they can rapidly destroy infected cells distributed throughout the brain tissue. However, with the Brownian motion assumption, T cells travel through the brain slowly, leading to a slower decline of the infected cells far away from the source of T cells. In nature, the existence of landscape and physiological limitations prevents the occurrence of arbitrary large displacements by the individuals following a Lévy flight. Instead, truncated Lévy flights are introduced, which lead to truncated space-fractional-order diffusion models with a truncation parameter. As an application of such equations, we propose a simple epidemic model with the assumption that infected individuals follow a truncated Lévy flight, and then we investigate the effect of different values of the truncation parameter on the epidemic speed. Finally, we explore the obtained results for a more realistic model, i.e., the West Nile virus epidemic, which happens among wild birds and mosquitoes. We suggest that truncated space-fractional-order diffusion models can provide appropriate estimations of the epidemic speed that is underestimated and overestimated by the models based on pure Brownian and Lévy movements, respectively. Our proposed model leads to accelerating epidemic waves that finally reach a constant speed representing the maximum speed of the epidemic.

Jury members :

• Prof. Emmanuel Hanert (UCLouvain), supervisor
• Prof. Jacques Mahillon (UCLouvain), chairperson
• Prof. Pierre-Antoine Absil (UCLouvain), secretary
• Prof. Patrick Bogaert (UCLouvain)
• Prof. G. Hossien Erjaee (University of California, USA)
• Dr. Dana Copot (University of Ghent)

Pay attention :

The public defense of Afshin Farhadi scheduled for Wednesday 06 July at 02:00 p.m. will also take place in the form of a video conference

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