By Janos Englander, Brian Rider

Markov approaches and their purposes to partial differential equations Kuznetsov's contributions.- Stochastic equations on projective structures of groups.- Modeling festival among influenza strains.- Asymptotic effects for close to serious Bienaym\'e-Galton-Watson and Catalyst-Reactant Branching Processes.- a few course huge deviation effects for a branching diffusion.- Longtime habit for collectively Catalytic Branching.- Super-Brownian movement: Lp-convergence of martingales in the course of the pathwise backbone decomposition

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12, World Scientific Publishing Co. , River Edge, NJ, 1993. MR 1249274 (94k:58043) Raj11. Chandiraraj Robinson Edward Raja, A stochastic difference equation with stationary noise on groups, 2011, Canad. J. , to appear. 4153/CJM-2011-094-6. Tak09. Yoichiro Takahashi, Time evolution with and without remote past, Advances in discrete dynamical systems, Adv. Stud. , vol. 53, Math. Soc. Japan, Tokyo, 2009, pp. 347– 361. MR 2582432 (2011b:60019) Yor92. Marc Yor, Tsirel son’s equation in discrete time, Probab.

The hypothesis would be that the swine strain must confer some immunity against the seasonal flu. But, clearly the seasonal strain does not confer any immunity against the swine strain: after all even young people (the group most severely affected by the swine strain) have usually been exposed to the seasonal strain and do not seem to be protected against the swine strain. Hence, for this argument to work the swine strain must confer some immunity against the seasonal strain, but the seasonal strain cannot confer any immunity against the swine strain.

We will also show that geometry and space may be critical for coexistence. Our models deal with competition over only one flu season. In the real world, because of mutations the fight between two strains may go on for several flu seasons before one strain outcompetes the other. This picture is consistent with the very skinny shape of the phylogenetic tree for influenza; see, for instance Koelle et al. [5] and van Nimwegen [8]. In this chapter, the two competing strains are assumed not to undergo mutations, and therefore the time scale we focus on is one flu season.

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