Arnaud Ducrot

Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations

We consider one-dimensional reaction-diffusion equations for a large class of spatially periodic nonlinearities – including multistable ones – and study the asymptotic behavior of solutions with Heaviside type initial data. Our analysis reveals some new dynamics where the profile of the propagation is not characterized by a single front, but by a layer of several fronts which we call a terrace. Existence and convergence to such a terrace is proven by using an intersection number argument, without much relying on standard linear analysis. Hence, on top of the peculiar phenomenon of propagation that our work highlights, several corollaries will follow on the existence and convergence to pulsating traveling fronts even for highly degenerate nonlinearities that have not been treated before.

A metapopulation model for malaria with transmission-blocking partial immunity in hosts

A metapopulation malaria model is proposed using SI and SIRS models for the vectors and hosts, respectively. Recovered hosts are partially immune to the disease and while they cannot directly become infectious again, they can still transmit the parasite to vectors. The basic reproduction number

An Age-Structured Within-Host Model for Multistrain Malaria Infections

{\mathcal{R}_0}

A mathematical model for malaria involving differential susceptibility, exposedness and infectivity of human host.

is shown to govern the local stability of the disease free equilibrium but not the global behavior of the system because of the potential occurrence of a backward bifurcation. Using type reproduction numbers, we identify the reservoirs of infection and evaluate the effect of control measures. Applications to the spread to non-endemic areas and the interaction between rural and urban areas are given.

Propagation of Salmonella within an Industrial Hen House

The aim of this paper is to investigate the spatial invasion of some infectious disease. The contamination process is described by the age since infection. Compared with the classical Kermack and McKendrick’s model, the vital dynamic is not omitted, and we allow some constant input flux into the population. This problem is rather natural in the context of epidemic problems and it has not been studied. Here we prove an existence and non-existence result for travelling wave solutions. We also describe the minimal wave speed. We are able to construct a suitable Lyapunov like functional decreasing along the travelling wave allowing to derive some qualitative properties, namely their convergence towards equilibrium points at x =± ∞.

Self-ignition of Polymerization fronts with convection : the ``Rainstorm Effect''

In this paper we propose an age-structured malaria within-host model taking into account multistrains interaction. We provide a global analysis of the model depending upon some threshold

Convergence to a pulsating travelling wave for an epidemic reaction-diffusion system with non-diffusive susceptible population

\mathcal T_0

Steady state concentration for a phenotypic structured problem modeling the evolutionary epidemiology of spore producing pathogens

. When