The basic reproduction number $R_0$ is a fundamental quantity in epidemiological modeling, reflecting the typical number of secondary infections that arise from a single infected individual. While $R_0$ is widely known to scientists, policymakers, and the general public, it has received comparatively little attention in the controls community. This note provides two novel characterizations of $R_0$: a stability characterization and a geometric program characterization. The geometric program characterization allows us to write $R_0$-constrained and budget-constrained optimal resource allocation problems as geometric programs, which are easily transformed into convex optimization problems. We apply these programs to allocating vaccines and antidotes in numerical examples, finding that targeting $R_0$ instead of the spectral abscissa of the Jacobian matrix (a common target in the controls literature) leads to qualitatively different solutions.
Motivated by the question of optimal vaccine allocation strategies in heterogeneous population for epidemic models, we study various properties of the \emph{effective reproduction number}. In the simplest case, given a fixed, non-negative matrix $K$, this corresponds mathematically to the study of the spectral radius $R_e(η)$ of the matrix product $\mathrm{Diag}(η)K$, as a function of $η\in\mathbb{R}_+^n$. The matrix $K$ and the vector $η$ can be interpreted as a next-generation operator and a vaccination strategy. This can be generalized in an infinite dimensional case where the matrix $K$ is…
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 si…
Nonsmooth convex optimization to estimate the Covid-19 reproduction number space-time evolution with robustness against low quality data
Univ. Lille, CNRS, Centrale Lille（里尔大学、法国国家科学研究中心、中央理工学院里尔）；CNRS, ENS de Lyon, Laboratoire de Physique（法国国家科学研究中心、里昂高等师范学院、物理实验室）；INRIA, ENS de Lyon, Laboratoire d’Informatique（法国国家信息与自动化研究所、里昂高等师范学院、计算机实验室）
Dengue is a vector-borne disease transmitted from an infected human to an Aedes mosquito, during a blood-meal. Dengue is still a major public health problem. A model for the disease transmission is presented, composed by human and mosquitoes compartments. The aim is to simulate the effects of seasonality, on the vectorial capacity and, consequently, on the disease development. Using entomological information about the mosquito behavior under different temperatures and rainfall, simulations are carried out and the repercussions analyzed. The basic reproduction number of the model is given, as w…
This paper deals with the problem of estimating variables in nonlinear models for the spread of disease and its application to the COVID-19 epidemic. First unconstrained methods are revisited and they are shown to correspond to the application of a linear filter followed by a nonlinear estimate of the effective reproduction number after a change-of-coordinates. Unconstrained methods often fail to keep the estimated variables within their physical range and can lead to unreliable estimates that require aggressively smoothing the raw data. In order to overcome these shortcomings a constrained es…
Discrete convex functions are used in many areas, including operations research, discrete-event systems, game theory, and economics. The objective of this paper is to investigate basic operations such as direct sum, splitting, and aggregation that are related to network induction of discrete convex functions as well as discrete convex sets. Various kinds of discrete convex functions in discrete convex analysis are considered such as integrally convex functions, L-convex functions, M-convex functions, multimodular functions, and discrete midpoint convex functions.
In this paper, we study the rate of convergence of the cyclic projection algorithm applied to finitely many basic semi-algebraic convex sets. We establish an explicit convergence rate estimate which relies on the maximum degree of the polynomials that generate the basic semi-algebraic convex sets and the dimension of the underlying space. We achieve our results by exploiting the algebraic structure of the basic semi-algebraic convex sets.
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 accordi…
A model for the transmission of dengue disease is presented. It consists of eight mutually-exclusive compartments representing the human and vector dynamics. It also includes a control parameter (adulticide spray) in order to combat the mosquito. The model presents three possible equilibria: two disease-free equilibria (DFE) --- where humans, with or without mosquitoes, live without the disease --- and another endemic equilibrium (EE). In the literature it has been proved that a DFE is locally asymptotically stable, whenever a certain epidemiological threshold, known as the basic reproduction …
In this paper a class of robust two-stage combinatorial optimization problems is discussed. It is assumed that the uncertain second stage costs are specified in the form of a convex uncertainty set, in particular polyhedral or ellipsoidal ones. It is shown that the robust two-stage versions of basic network and selection problems are NP-hard, even in a very restrictive cases. Some exact and approximation algorithms for the general problem are constructed. Polynomial and approximation algorithms for the robust two-stage versions of basic problems, such as the selection and shortest path problem…
Consider a finite system of non-strict real polynomial inequalities and suppose its solution set $S\subseteq\mathbb R^n$ is convex, has nonempty interior and is compact. Suppose that the system satisfies the Archimedean condition, which is slightly stronger than the compactness of $S$. Suppose that each defining polynomial satisfies a second order strict quasiconcavity condition where it vanishes on $S$ (which is very natural because of the convexity of $S$) or its Hessian has a certain matrix sums of squares certificate for negative-semidefiniteness on $S$ (fulfilled trivially by linear polyn…
Motivated by applications in robotics and computer vision, we study problems related to spatial reasoning of a 3D environment using sublevel sets of polynomials. These include: tightly containing a cloud of points (e.g., representing an obstacle) with convex or nearly-convex basic semialgebraic sets, computation of Euclidean distances between two such sets, separation of two convex basic semalgebraic sets that overlap, and tight containment of the union of several basic semialgebraic sets with a single convex one. We use algebraic techniques from sum of squares optimization that reduce all the…
We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence, a highly desirable feature as convex problems are in principle easier to solve. In addition, if a basic semi-algebraic set K is convex but its defining polynomials are not, we provide a certificate of convexity if a sufficient (and almost necessary) condition is satified. This condition can be checked …
University of Texas, Arlington（德克萨斯大学阿灵顿分校）；Cognitiv；University of California, Berkeley（加州大学伯克利分校）
For general quadratically-constrained quadratic programming (QCQP), we propose a parabolic relaxation described with convex quadratic constraints. An interesting property of the parabolic relaxation is that the original non-convex feasible set is contained on the boundary of the parabolic relaxation. Under certain assumptions, this property enables one to recover near-optimal feasible points via objective penalization. Moreover, through an appropriate change of coordinates that requires a one-time computation of an optimal basis, the easier-to-solve parabolic relaxation can be made as strong a…
