We formalize and study the problem of optimal allocation strategies for a (perfect) vaccine in the infinite-dimensional SIS model. The question may be viewed as a bi-objective minimization problem, where one tries to minimize simultaneously the cost of the vaccination, and a loss that may be either the effective reproduction number, or the proportion of the infected population in the endemic state. We prove the existence of Pareto optimal strategies, describe the corresponding Pareto frontier in both cases, and study its convexity and stability properties. We also show that vaccinating according to the profile of the endemic state is a critical allocation, in the sense that, if the initial reproduction number is larger than 1, then this vaccination strategy yields an effective reproduction number equal to 1.

In this article, we introduce an infinite-dimensional deterministic SIS model which takes into account the heterogeneity of the infections and the social network among a large population. We study the long-time behavior of the dynamic. We identify the basic reproduction number R0 which determines whether there exists a stable endemic steady state (super-critical case: R0 > 1) or if the only equilibrium is disease-free (critical and sub-critical case: R0 ≤ 1). As an application of this general study, we prove that the so-called "leaky" and "all-or-nothing" vaccination mechanism have the same effect on R0. This framework is also very natural and intuitive to model lockdown policies and study their impact.

We consider the problem of optimal allocation strategies for a (perfect) vaccine in an infinite-metapopulation model (including SIS, SIR, SEIR, . . . ), when the loss function is given by the effective reproduction number Re, which is defined as the spectral radius of the effective next generation matrix (in finite dimension) or more generally of the effective next generation operator (in infinite dimension). We give sufficient conditions for Re to be a convex or a concave function of the vaccination strategy. Then, following a previous work, we consider the bi-objective problem of minimizing simultaneously the cost and the loss of the vaccination strategies. In particular, we prove that a cordon sanitaire might not be optimal, but it is still better than the "worst" vaccination strategies. Inspired by the graph theory, we compute the minimal cost which ensures that no infection occurs using independent sets. Using Frobenius decomposition of the whole population into irreducible sub-populations, we give some explicit formulae for optimal ("best" and "worst") vaccinations strategies. Eventually, we provide equivalence properties on models which ensure that the function Re is unchanged.

In this note, we study the long time behavior of Lotka-Volterra systems whose coefficients vary randomly. Benaïm and Lobry (2015) recently established that randomly switching between two environments that are both favorable to the same species may lead to different regimes: extinction of one species or the other, or persistence of both species. Our purpose here is to describe more accurately the range of parameters leading to these regimes, and the support of the invariant probability measure in case of persistence.

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