Asymptotic Control for a Class of Piecewise Deterministic Markov Processes Associated to Temperate Viruses

We aim at characterizing the asymptotic behavior of value functions in the control of piece-wise deterministic Markov processes (PDMP) of switch type under nonexpansive assumptions. For a particular class of processes inspired by temperate viruses, we show that uniform limits of discounted problems as the discount decreases to zero and time-averaged problems as the time horizon increases to infinity exist and coincide. The arguments allow the limit value to depend on initial configuration of the system and do not require dissipative properties on the dynamics. The approach strongly relies on viscosity techniques, linear programming arguments and coupling via random measures associated to PDMP. As an intermediate step in our approach, we present the approximation of discounted value functions when using piecewise constant (in time) open-loop policies.Comment: In this revised version, statements of the main results are gathered in Section 3. Proofs of the main results (Theorem 4 and Theorem 7) make the object of separate sections (Section 5, resp. Section 6). The biological example makes the object of Section 4. Notations are gathered in Subsection 2.1. This is the final version to be published in SICO

Controllability Metrics on Networks with Linear Decision Process-type Interactions and Multiplicative Noise

This paper aims at the study of controllability properties and induced controllability metrics on complex networks governed by a class of (discrete time) linear decision processes with mul-tiplicative noise. The dynamics are given by a couple consisting of a Markov trend and a linear decision process for which both the "deterministic" and the noise components rely on trend-dependent matrices. We discuss approximate, approximate null and exact null-controllability. Several examples are given to illustrate the links between these concepts and to compare our results with their continuous-time counterpart (given in [16]). We introduce a class of backward stochastic Riccati difference schemes (BSRDS) and study their solvability for particular frameworks. These BSRDS allow one to introduce Gramian-like controllability metrics. As application of these metrics, we propose a minimal intervention-targeted reduction in the study of gene networks

Border Avoidance: Necessary Regularity for Coefficients and Viscosity Approach

Motivated by the result of invariance of regular-boundary open sets in \cite{CannarsaDaPratoFrankowska2009} and multi-stability issues in gene networks, our paper focuses on three closely related aims. First, we give a necessary local Lipschitz-like condition in order to expect invariance of open sets (for deterministic systems). Comments on optimality are provided via examples. Second, we provide a border avoidance (near-viability) counterpart of \cite{CannarsaDaPratoFrankowska2009} for controlled Brownian diffusions and piecewise deterministic switched Markov processes (PDsMP). We equally discuss to which extent Lipschitz-continuity of the driving coefficients is needed. Finally, by applying the theoretical result on PDsMP to Hasty's model of bacteriophage (\cite{hasty\_pradines\_dolnik\_collins\_00}, \cite{crudu\_debussche\_radulescu\_09}), we show the necessity of explicit modeling for the environmental cue triggering lysis

Mathematical Methods, Modelling and Applications

This volume deals with novel high-quality research results of a wide class of mathematical models with applications in engineering, nature, and social sciences. Analytical and numeric, deterministic and uncertain dimensions are treated. Complex and multidisciplinary models are treated, including novel techniques of obtaining observation data and pattern recognition. Among the examples of treated problems, we encounter problems in engineering, social sciences, physics, biology, and health sciences. The novelty arises with respect to the mathematical treatment of the problem. Mathematical models are built, some of them under a deterministic approach, and other ones taking into account the uncertainty of the data, deriving random models. Several resulting mathematical representations of the models are shown as equations and systems of equations of different types: difference equations, ordinary differential equations, partial differential equations, integral equations, and algebraic equations. Across the chapters of the book, a wide class of approaches can be found to solve the displayed mathematical models, from analytical to numeric techniques, such as finite difference schemes, finite volume methods, iteration schemes, and numerical integration methods

Spatio-temporal modelling of climate-sensitive disease risk: towards an early warning system for dengue in Brazil

The transmission of many infectious diseases is affected by climate variations, particularly for diseases spread by arthropod vectors such as malaria and dengue. Previous epidemiological studies have demonstrated statistically significant associations between infectious disease incidence and climate variations. Such research has highlighted the potential for developing climate-based epidemic early warning systems. To establish how much variation in disease risk can be attributed to climatic conditions, non-climatic confounding factors should also be considered in the model parameterisation to avoid reporting misleading climate-disease associations. This issue is sometimes overlooked in climate related disease studies. Due to the lack of spatial resolution and/or the capability to predict future disease risk (e.g. several months ahead), some previous models are of limited value for public health decision making. This thesis proposes a framework to model spatio-temporal variation in disease risk using both climate and non-climate information. The framework is developed in the context of dengue fever in Brazil. Dengue is currently one of the most important emerging tropical diseases and dengue epidemics impact heavily on Brazilian public health services. A negative binomial generalised linear mixed model (GLMM) is adopted which makes allowances for unobserved confounding factors by including spatially structured and unstructured random effects. The model successfully accounts for the large amount of overdispersion found in disease counts. The parameters in this spatio-temporal Bayesian hierarchical model are estimated using Markov Chain Monte Carlo (MCMC). This allows posterior predictive distributions for disease risk to be derived for each spatial location and time period (month/season). Given decision and epidemic thresholds, probabilistic forecasts can be issued, which are useful for developing epidemic early warning systems. The potential to provide useful early warnings of future increased and geographically specific dengue risk is investigated. The predictive validity of the model is evaluated by fitting the GLMM to data from 2001-2007 and comparing probabilistic predictions to the most recent out-of-sample data in 2008-2009. For a probability decision threshold of 30% and the pre-defined epidemic threshold of 300 cases per 100,000 inhabitants, successful epidemic alerts would have been issued for 94% of the 54 microregions that experienced high dengue incidence rates in South East Brazil, during February - April 2008.Leverhulme Trus