Section: New Results

Invasion processes and modeling in epidemiology

Participants : Bedr'Eddine Ainseba, Michel Langlais, Arnaud Ducrot, Pascal Zongo, Mahieddine Kouche, Pierre Magal, Chahrazed Benosman.

The research program in mathematical population dynamics as presented in 3.1 is mostly dedicated to predator-prey sytems or host-parasite sytems in heterogeneous environments. The new results include :

• A quasi comprehensive description of the complex intertwined dynamics for a SI epidemic model with density dependent incidence and a host population exhibiting an Allee effect is derived in a joint work with H Malchow and F Hilker. Typically we showed that unforeseen tri-stability dynamics, oscillations, limit cycles and Bogdanov-Takens bifurcation can occur. A a consequence the host population can also go extinct for SI models with density dependent incidence. This may have profound implications for biological conservation as well as pest management.

• A comprehensive analysis of the spatially structured SI model with logistic dynamics and vertical transmission, including the existence of wave fronts, is currently developed with A. Ducrot and P. Magal.

  Related analysis for cross-diffusion systems is worked out with M. Bendahmane.

• New results with S Anita and W.E. Fitzgibbon concerning the stabilisation of predators in a spatially distributed predator-prey system show that reducing the prey density may drive predators to extinction. This includes the case of non coincident spatial domains; in that case locally reducing prey density is closely related to herd management to prevent prey from unwelcome wild predators.

• The underlying dynamics of solutions to this predator-prey system posed on non coincident spatial domains without controllability effects is analysed with A Ducrot. An interesting feature is the impact of the distributed numerical response to predation on existence of stationary solutions, their stability and the occurence of waves.

• New results concerning the spread of a fungal disease in a vineyard are derived at two different spatio-temporal scales. First at a local scale, the wine tree level, the building and analysis of a sharply detailed discrete model with A Calonnec et al. allows to better understand the local contamination out of a primary focus at the beginning of Spring. Next at a meso scale, the plot level, a Reaction-Diffusion model for short and long distances dispersal of spores coupled to a system of ODEs for local production of spores is derived and analysed with A Calonnec, J Burie and A Ducrot to understand the global dynamics over a year.

  This is also the core of ANR ARCHIDEMIO and of the ARC INRIA M2A3PC. A friendly user interface for the discrete model at the plant scale was designed by two M1 students supported by ANR and ARC.

Toxoplasma gondii

With E. Gilot-Fromont, M.L. Poulle and M. Lélu we look for invasion and persistence criteria in simple ODE models featuring cat and rodent populations spatially distributed over spatial domains. First results were presented at EPIDEMICS 2 in Athens, december 09.

Blue Tongue Virus

With P. Ezanno, M. Charron and H. Segeers we look for invasion and persistence criteria in a single cattle herd / midge population system depending on whether this herd is a fattening one or one with insemination. Vertical transmission and vaccination play important roles. First results were presented at EPIDEMICS 2 in Athens, december 09.

Travelling waves for epidemic models

With P. Magal we investigate the existence of travelling wave solution for an age structured Kermack and McKendricks model with diffusion where both infectivity and recovery can depend on the duration of infection . We prove in [34] that the basic reproduction number leads to the existence or non-existence of such particular solutions. Moreover, when such solutions exist, there is a continuum of admissible wave speed.

With P. Magal and S. Ruan, we extend a previous work to a more general framework of multigroup interactions [28] . This means that a disease may circulate within several species and can be transmitted from one species to an other one. Together with some irreductibility assumptions on the graph of transmission, we provide the existence a travelling wave solutions in this context. Here again, the global basic reproduction number leads to the existence and non-existence of such solutions.

Malaria epidemiological modeling

P. Zongo is a PhD student at the university of Ouagadougou, co supervised by A. Ducrot. He defended his PhD in May 2009. He has an AUF financial support. He works on a model of malaria development in endemic areas. New results obtained with A. Ducrot to understand the dynamics of the ODE model by using basic reproduction number theory as well as bifurcation theory have been published in Journal of Biological dynamics [29] .

With A. Ducrot and J. Arino, P. Zongo studies a new metapopulation model for malaria development. The main idea is to understand what is the effect of migration on malaria epidemics. Moreover we aim to understand the role of the migration from rural to urban area on the transmission of malaria. This work has been submitted to Journal of Mathematical biology.

Brucellosis

Brucellosis is a serious animal (ovine, bovine,...) disease that can be transmitted to humans. This disease causes important economical damages in north Africa and south America. In this work , we construct and analyse an ovine brucellosis mathematical model. In this model, the population is divided into susceptible and infected subclasses. Susceptible individuals can contract the disease in two ways: (i) direct mode - caused by contact with infected individuals; (ii) indirect mode - related to the presence of virulent organisms in the environment. We derive a net reproductive number and analyse the global asymptotic behaviour of the model. We also perform some numerical simulations, and investigate the effect of a slaughtering policy. (Journal of Biological Dynamics, Volume 4, Issue 1 2010 , pages 2 - 11 )

Modeling transfers of proteins in cancer cells

In a joint work with P. Hinow, F. Le Foll, G. F. Webb, P. Magal studies the problem of transfer in a population structured by a continuum variable corresponding to the quantity being transferred. The transfer of the quantity occurs between individuals according to specified rules. The model is of Boltzmann type with kernels corresponding to the transfer process. We prove that the transfer process preserves total mass of the transferred quantity and the solutions of the simple model converge weakly to Radon measures. We generalize the model by introducing proliferation of individuals and production and diffusion of the transferable quantity. It is shown that the generalized model admits a globally asymptotically stable steady state, provided that transfer is sufficiently small. We discuss an application of our model to cancer cell populations, in which individual cells exchange the surface protein P-glycoprotein, an important factor in acquired multi-drug resistance against cancer chemotherapy. The work has been published in 2009 SIAM J. Appl. Math.