Toward an integrated workforce planning framework using structured equations

Strategic Workforce Planning is a company process providing best in class, economically sound, workforce management policies and goals. Despite the abundance of literature on the subject, this is a notorious challenge in terms of implementation. Reasons span from the youth of the field itself to broader data integration concerns that arise from gathering information from financial, human resource and business excellence systems. This paper aims at setting the first stones to a simple yet robust quantitative framework for Strategic Workforce Planning exercises. First a method based on structured equations is detailed. It is then used to answer two main workforce related questions: how to optimally hire to keep labor costs flat? How to build an experience constrained workforce at a minimal cost

An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age structured equations

For monotone linear differential systems with periodic coefficients, the (first) Floquet eigenvalue measures the growth rate of the system. We define an appropriate arithmetico-geometric time average of the coefficients for which we can prove that the Perron eigenvalue is smaller than the Floquet eigenvalue. We apply this method to Partial Differential Equations, and we use it for an age-structured systems of equations for the cell cycle. This opposition between Floquet and Perron eigenvalues models the loss of circadian rhythms by cancer cells.Comment: 7 pages, in English, with an abridged French versio

A non-expanding transport distance for some structured equations

Structured equations are a standard modeling tool in mathematical biology. They areintegro-differential equations where the unknown depends on one or several variables, representing the state or phenotype of individuals. A large literature has been devoted to many aspects of these equations and in particular to the study of measure solutions.Here we introduce a transport distance closely related to the Monge-Kantorovich distance,which appears to be non-expanding for several (mainly linear) examples of structured equations

Structured Equations for Complex Living Systems - Modeling, Asymptotics and Numerics

Complex living systems differ from those systems whose evolution is well described by the laws of Classical Physics. In fact, they are endowed with self-organizing abilities that result from the interactions among their constituent individuals, which behave according to specific functions, strategies or traits. These functions/strategies/traits can evolve over time, as a result of adaptation to the surrounding environment, and are usually heterogeneously distributed over the individuals, so that the global features expressed by the system as a whole cannot be reduced to the superposition of the single functions/strategies/traits. Quoting Aristotle, we can say that, within these systems, "the whole is more than the sum of its parts". As a result, when we study the dynamics of complex living systems, there are new concepts that come into play, such as adaptation, herding and learning, which do not belong to the traditional vocabulary of physical sciences and make the dynamics of these systems hardly to be forecast. Moving from the above considerations, the subject of my PhD was the development and the study of structured equations for population dynamics (partial differential equations and integro-differential equations) applied to modelling the evolution of complex living systems. In particular, I designed models for multicellular systems, living species and socio-economic systems with the aim of inspecting mechanisms underlying the emergence of collective behaviors and self-organization. In the framework of structured equations, individuals belonging to a given system are divided into different populations and heterogeneously distributed characteristics are modelled by suitable independent variables, the so-called structuring variables. For each population, a function describing the distribution of the individuals over the structuring variables is introduced, which evolves through a partial differential equation, or an integro-differential equation, whose parameter functions are defined according to the phenomena under study. I decided to use such mathematical framework since it makes possible to effectively model the afore mentioned complexity aspects of living systems and provides an efficient way to reduce complexity in view of the mathematical formalization. With particular reference to multicellular systems, I focused on the design and the study of mathematical models describing the evolutionary dynamics of cancer cell populations under the selective pressures exerted by therapeutic agents and the immune system. Proliferation, mutation and competition phenomena are included in these models, which rely on the idea that the process leading to the emergence of resistance to anti-cancer therapies and immune action can be considered, at least in principles, as a Darwinian micro-evolution. It is worth noting that most of these models stem from direct collaborations with biologists and clinicians. Besides local and global existence results for the mathematical problems linked to the models, my PhD thesis presents results related to concentration phenomena arising in phenotype-structured equations and opinion-structured equations (i.e., the weak convergence of the solutions to sums of Dirac masses), and with the derivation of macroscopic models from space-velocity structured equations. From the applicative standpoint such concentration phenomena provide a possible mathematical formalization of the selection principle in evolutionary biology and the emergence of opinions; macroscopic models, instead, offer an overall view of the systems at hand. Numerical simulations are performed with the aim of illustrating, and extending, analytical results and verifying the consistency of the model with empirical dat

How does variability in cells aging and growth rates influence the malthus parameter?

The aim of this study is to compare the growth speed of different cell populations measured by their Malthus parameter. We focus on both the age-structured and size-structured equations. A first population (of reference) is composed of cells all aging or growing at the same rate $\bar v$. A second population (with variability) is composed of cells each aging or growing at a rate $v$ drawn according to a non-degenerated distribution $\rho$ with mean $\bar v$. In a first part, analytical answers -- based on the study of an eigenproblem -- are provided for the age-structured model. In a second part, numerical answers -- based on stochastic simulations -- are derived for the size-structured model. It appears numerically that the population with variability proliferates more slowly than the population of reference (for experimentally plausible division rates). The decrease in the Malthus parameter we measure, around 2% for distributions $\rho$ with realistic coefficients of variations around 15-20\%, is determinant since it controls the {\it exponential} growth of the whole population

Tracking the evolution of cancer cell populations through the mathematical lens of phenotype-structured equations

This work was supported in part by the French National Research Agency through the “ANR blanche” project Kibord [ANR-13-BS01-0004].Background: A thorough understanding of the ecological and evolutionary mechanisms that drive the phenotypic evolution of neoplastic cells is a timely and key challenge for the cancer research community. In this respect, mathematical modelling can complement experimental cancer research by offering alternative means of understanding the results of in vitro and in vivo experiments, and by allowing for a quick and easy exploration of a variety of biological scenarios through in silico studies. Results: To elucidate the roles of phenotypic plasticity and selection pressures in tumour relapse, we present here a phenotype-structured model of evolutionary dynamics in a cancer cell population which is exposed to the action of a cytotoxic drug. The analytical tractability of our model allows us to investigate how the phenotype distribution, the level of phenotypic heterogeneity, and the size of the cell population are shaped by the strength of natural selection, the rate of random epimutations, the intensity of the competition for limited resources between cells, and the drug dose in use. Conclusions: Our analytical results clarify the conditions for the successful adaptation of cancer cells faced with environmental changes. Furthermore, the results of our analyses demonstrate that the same cell population exposed to different concentrations of the same cytotoxic drug can take different evolutionary trajectories, which culminate in the selection of phenotypic variants characterised by different levels of drug tolerance. This suggests that the response of cancer cells to cytotoxic agents is more complex than a simple binary outcome, i.e., extinction of sensitive cells and selection of highly resistant cells. Also, our mathematical results formalise the idea that the use of cytotoxic agents at high doses can act as a double-edged sword by promoting the outgrowth of drug resistant cellular clones. Overall, our theoretical work offers a formal basis for the development of anti-cancer therapeutic protocols that go beyond the ‘maximum-tolerated-dose paradigm’, as they may be more effective than traditional protocols at keeping the size of cancer cell populations under control while avoiding the expansion of drug tolerant clones.Publisher PDFPeer reviewe

Dynamics of Structured Equations of Infectious Diseases

Από το μοντέλο ευλογιάς του Daniel Bernoulli το 1760 έως την πρόσφατη πανδημία COVID-19, τα Μαθηματικά έχουν χρησιμοποιηθεί στην Πληθυσμιακή Βιολογία για να εξηγήσουν και να προβλέψουν τα ξεσπάσματα μολυσματικών ασθενειών. Με τις μολυσματικές ασθένειες να αποτελούν την κύρια αιτία θανάτου παγκοσμίως, ιδιαίτερα στα μικρά παιδιά σε χώρες χαμηλού εισοδήματος, η μελέτη μοντέλων πληθυσμού και ιδιαίτερα μοντέλων «με διάκριση πληθυσμού κατά ένα χαρακτηριστικό» στην Επιδημιολογία παραμένει επιτακτική ανάγκη. Οι εξισώσεις των τελευταίων μοντέλων διακρίνουν τα άτομα το ένα από το άλλο σύμφωνα με χαρακτηριστικά όπως η ηλικία, ο τόπος κατοικίας, η κοινωνικο-οικονομική κατάσταση και οι μετακινήσεις. Η παρούσα μεταπτυχιακή διπλωματική εργασία αποτελεί μια εισαγωγή στα Μαθηματικά Μοντέλα Επιδημιολογίας. Κεντρικό θέμα είναι η μελέτη διαφόρων επιδημιολογικών μοντέλων, βάσει των οποίων μπορεί να αναπτυχθεί και να εξαπλωθεί μια λοιμώδης νόσος σε έναν κλειστό πληθυσμό. Έτσι, θα παρουσιάσουμε και θα αναλύσουμε τέτοια μοντέλα συνήθων διαφορικών εξισώσεων και επιδημιολογικά μοντέλα μερικών διαφορικών εξισώσεων για να εξηγήσουμε πώς αυτά τα χαρακτηριστικά επηρεάζουν τη δυναμική των μοντέλων και κατά συνέπεια τις επιδημιολογικές διαδικασίες.From the smallpox model of Daniel Bernoulli in 1760 to recent COVID-19 pandemic, Mathematics have been used in population biology to explain and predict the infectious diseases outbreaks. With infectious diseases being a leading cause of death worldwide, particularly in low income countries, especially in young children, the study of population models and especially, structured population models in Epidemiology remains an urgent need. Structured equations distinguish individuals from one another according to characteristics such as age, location, status, and movement. This M.Sc. Thesis is an introduction to the Mathematical Models of Epidemiology. The central theme is the study of various epidemiological models, based on which an infectious disease can develop and spread in a closed population. So, we will present and analyze such models of ordinary differential equations and structured epidemiological models of partial differential equations to explain how these characteristics affect the dynamics of the models and consequently the epidemiological processes

Tracking the evolution of cancer cell populations through the mathematical lens of phenotype-structured equations

Background: A thorough understanding of the ecological and evolutionary mechanisms that drive the phenotypic evolution of neoplastic cells is a timely and key challenge for the cancer research community. In this respect, mathematical modelling can complement experimental cancer research by offering alternative means of understanding the results of in vitro and in vivo experiments, and by allowing for a quick and easy exploration of a variety of biological scenarios through in silico studies. Results: To elucidate the roles of phenotypic plasticity and selection pressures in tumour relapse, we present here a phenotype-structured model of evolutionary dynamics in a cancer cell population which is exposed to the action of a cytotoxic drug. The analytical tractability of our model allows us to investigate how the phenotype distribution, the level of phenotypic heterogeneity, and the size of the cell population are shaped by the strength of natural selection, the rate of random epimutations, the intensity of the competition for limited resources between cells, and the drug dose in use. Conclusions: Our analytical results clarify the conditions for the successful adaptation of cancer cells faced with environmental changes. Furthermore, the results of our analyses demonstrate that the same cell population exposed to different concentrations of the same cytotoxic drug can take different evolutionary trajectories, which culminate in the selection of phenotypic variants characterised by different levels of drug tolerance. This suggests that the response of cancer cells to cytotoxic agents is more complex than a simple binary outcome, i.e., extinction of sensitive cells and selection of highly resistant cells. Also, our mathematical results formalise the idea that the use of cytotoxic agents at high doses can act as a double-edged sword by promoting the outgrowth of drug resistant cellular clones. Overall, our theoretical work offers a formal basis for the development of anti-cancer therapeutic protocols that go beyond the 'maximum-tolerated-dose paradigm', as they may be more effective than traditional protocols at keeping the size of cancer cell populations under control while avoiding the expansion of drug tolerant clones. Reviewers: This article was reviewed by Angela Pisco, Sébastien Benzekry and Heiko Enderling

Global Bifurcation of Positive Equilibria in Nonlinear Population Models

Existence of nontrivial nonnegative equilibrium solutions for age structured population models with nonlinear diffusion is investigated. Introducing a parameter measuring the intensity of the fertility, global bifurcation is shown of a branch of positive equilibrium solutions emanating from the trivial equilibrium. Moreover, for the parameter-independent model we establish existence of positive equilibria by means of a fixed point theorem for conical shells.Comment: 17 page