Well‐posedness,positivity, and time asymptotics properties for a reaction–diffusion model of plankton communities,involving a rational nonlinearity with singularity

In this work, we consider a reaction–diffusion system, modeling the interaction between nutrients, phytoplankton, and zooplankton. Using a semigroup approach in , we prove global existence, uniqueness, and positivity of the solutions. The nonlinearity is handled by providing estimates in , allowing to deal with most of the functional responses that describe predator/prey interactions (Holling I, II, III, Ivlev) in ecology. The paper finally exhibits some time asymptotic properties of the solutions.     

Threshold behaviour of a SI epidemiological model with two structuring variables

In this article, we study a SI epidemic model describing the spread of a disease in a perfectly mixed managed population, representing an animal herd in a fattening farm. The epidemic process is characterized by a non-neglectable and variable incubation period, during which individuals are infectious but cannot be easily detected. The susceptible and infected populations are structured according to age and, for infected, to time remaining before the end of the incubation, where they show detectable clinical signs. We study the well posedness and the asymptotic behaviour of the problem and show that in some cases, even if the farm is fed with healthy animals, disease persistence can occur. We give an explicit formula for the basic reproduction number \({\mathcal{R}_0}\) and the biological interpretation of this threshold on a specific example. We finally illustrate the asymptotic behaviour of the model by numerical simulations.     

Identifiability analysis of an epidemiological model in a structured population

We investigate the parameter identifiability problem for a system of nonlinear integro-partial differential equations of transport type, representing the spread of a disease with a long infectious but undetectable period in an individual population. After obtaining the expression of the model input-output relationships, we give sufficient conditions on the initial and boundary conditions of the system that guarantee the parameter identifiability on a finite time horizon. We finally illustrate our findings with numerical simulations.