An Infinite-Dimensional SIS Model

Delmas, Jean‐François; Dronnier, Dylan; Zitt, Pierre-André

doi:10.48550/arxiv.2006.08241

Cited by 4 publications

(30 citation statements)

References 51 publications

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“…The goal of this paper is to prove a number of properties of R e , that shed a light on how to vaccinate in the best possible way. In previous works [7,10], we introduced a general infinite-dimensional kernel framework in which the matrix formulation appears as a special finite-dimensional case. We state our results in this general framework, but for ease of the presentation, we shall stick to the matrix formulation in this introduction.…”

Section: Introductionmentioning

confidence: 99%

Effective reproduction number: Convexity, invariance and cordons sanitaires

Delmas¹,

Dronnier²,

Zitt³

2021

Preprint

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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.

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Section: Introductionmentioning

confidence: 99%

Effective reproduction number: Convexity, invariance and cordons sanitaires

Delmas¹,

Dronnier²,

Zitt³

2021

Preprint

Self Cite

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show abstract

“…The recent advances in graph limits theory [4,39] justify describing the transmission of the disease by a kernel defined on a probability space. We already used this type of model in [10,11,13], in particular for an SIS dynamics, see also [11,Section 2] for other epidemic models.…”

Section: T H E K E R N E L M O D E Lmentioning

confidence: 99%

“…By analogy with the discrete setting and also based on [10,11], we define the basic reproduction number in this context thanks to the following formula:…”

Section: T H E K E R N E L M O D E Lmentioning

confidence: 99%

“…In heterogeneous generalizations of classical compartmental models, also called metapopulation models, see [5,10,37], the population is stratified into homogeneous groups sharing the same characteristics (time to recover from the disease, interaction with the other groups, . .…”

Section: Introductionmentioning

confidence: 99%

“…The reproduction number can then be identified as the spectral radius of the so-called next generation matrix, see [48]. With this definition, it is still true that the outbreak dies out if R 0 is smaller than 1 and invades the population otherwise; see [10,28,46,47,48] for instance.…”

Section: Introductionmentioning

confidence: 99%

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Optimal vaccination: Various (counter) intuitive examples

Delmas,

Dronnier,

Zitt

2021

Preprint

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In previous articles, we formalized the problem of optimal allocation strategies for a (perfect) vaccine in an infinite-dimensional metapopulation model. The aim of the current paper is to illustrate this theoretical framework with multiple examples where one can derive the analytic expression of the optimal strategies. We discuss in particular the following points: whether or not it is possible to vaccinate optimally when the vaccine doses are given one at a time (greedy vaccination strategies); the effect of assortativity (that is, the tendency to have more contacts with similar individuals) on the shape of optimal vaccination strategies; the particular case where everybody has the same number of neighbors.

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A note on Fokker-Planck equations and graphons

Coppini

2021

Preprint

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Fokker-Planck equations represent a suitable description of the finite-time behavior for a large class of particle systems as the size of the population tends to infinity. Recently, the theory of graph limits have been introduced in the mean-field framework to account for heterogeneous interactions among particles. In many instances, such network heterogeneity is preserved in the limit which turns from being a single Fokker-Planck equation (also known as McKean-Vlasov) to an infinite system of non-linear partial differential equations (PDE) coupled by means of a graphon. While appealing from an applied viewpoint, few rigorous results exist on the graphon particle system. This note addresses such limit systems focusing on the relation between initial conditions and interaction network: if the system initial datum and the graphon degrees satisfy a suitable condition, a significantly simpler representation of the solution is available. This in turn implies that very different graphons can lead to exactly the same particle behavior, shedding some light on the network influence on the dynamics.Examples of such representation are provided. In particular, step kernels represent a class of graphons to which our result applies: this in turn opens the way to approximate the graphon particle system with a finite system of Fokker-Planck equations. As a byproduct, we show that when the initial condition is uniform, every graphon with constant degree leads to a behavior indistinguishable from the well-known mean-field limit.

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An Infinite-Dimensional SIS Model

Cited by 4 publications

References 51 publications

Effective reproduction number: Convexity, invariance and cordons sanitaires

Effective reproduction number: Convexity, invariance and cordons sanitaires

Optimal vaccination: Various (counter) intuitive examples

A note on Fokker-Planck equations and graphons

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