Angel Calsina | Universitat Autònoma de Barcelona (original) (raw)

Papers by Angel Calsina

To describe the dynamics of a size-structured population and its unstructured resource, we formul... more To describe the dynamics of a size-structured population and its unstructured resource, we formulate bookkeeping equations in two different ways. The first, called the PDE formulation, is rather standard. It employs a first order partial differential equation, with a non-local boundary condition, for the size-density of the consumer, coupled to an ordinary differential equation for the resource concentration. The second is called the DELAY formulation and employs a renewal equation for the population level birth rate of the consumer, coupled to a delay differential equation for the (history of the) resource concentration. With each of the two formulations we associate a constructively defined semigroup of nonlinear solution operators. The two semigroups are intertwined by a non-invertible operator. In this paper we delineate in what sense the two semigroups are equivalent. In particular, we i) identify conditions on both the model ingredients and the choice of state space that guara...

We study the basic reproduction number (R0) in an epidemic model where infected individuals are i... more We study the basic reproduction number (R0) in an epidemic model where infected individuals are initially asymptomatic and structured by the time since infection. At the beginning of an epidemic outbreak the computation of R0 relies on limited data based mostly on symptomatic cases, since asymptomatic infected individuals are not detected by the surveillance system. R0 has been widely used as an indicator to assess the dissemination of infectious diseases. Asymptomatic individuals are assumed to either become symptomatic after a fixed period of time or they are removed (recovery or disease-related death). We determine R0 understood as the expected secondary symptomatic cases produced by a symptomatic primary case through a chain of asymptomatic infections. R0 is computed directly by interpreting the model ingredients and also using a more systematic approach based on the next-generation operator. Reported Covid-19 cases data during the first wave of the pandemic in Spain are used to...

Mathematical Methods in the Applied Sciences, 2020

In the framework of population dynamics, the basic reproduction number R0 is, by definition, the ... more In the framework of population dynamics, the basic reproduction number R0 is, by definition, the expected number of offspring that an individual has during its lifetime. In constant and time periodic environments it is calculated as the spectral radius of the so-called next-generation operator ([12, 18]). In continuously structured populations defined in a Banach lattice X with concentrated states at birth one cannot define the next-generation operator in X. In the present paper we present an approach to compute the basic reproduction number of such models as the limit of the basic reproduction number of a sequence of models for which R0 can be computed as the spectral radius of the next-generation operator. We apply these results to some examples: the (classical) size-dependent model, a size structured cell population model, a size structured model with diffusion in structure space (under some particular assumptions) and a (physiological) age-structured model with diffusion in structure space.

Discrete & Continuous Dynamical Systems - B, 2017

In this work we establish conditions which guarantee the existence of (strictly) positive steady ... more In this work we establish conditions which guarantee the existence of (strictly) positive steady states of a nonlinear structured population model. In our framework the steady state formulation amounts to recasting the nonlinear problem as a family of eigenvalue problems combined with a fixed point problem. Amongst other things our formulation requires us to control the growth behaviour of the spectral bound of a family of linear operators along positive rays. For the specific class of model we consider here, this presents a considerable challenge. We are going to show that the spectral bound of the family of operators, arising from the steady state formulation, can be controlled by perturbations in the domain of the generators (only). These new boundary perturbation results are particularly important for models exhibiting fertility controlled dynamics. As an important by-product of the application of the boundary perturbation results we employ here we recover (using a recent theorem by H. R. Thieme) the familiar net reproduction number (or function) for models with single state at birth, which include for example the classic McKendrick (linear) and Gurtin-McCamy (non-linear) age-structured models.

Bulletin of mathematical biology, Nov 3, 2017

A spatially structured linear model of the growth of intestinal bacteria is analysed from two gen... more A spatially structured linear model of the growth of intestinal bacteria is analysed from two generational viewpoints. Firstly, the basic reproduction number associated with the bacterial population, i.e. the expected number of daughter cells per bacterium, is given explicitly in terms of biological parameters. Secondly, an alternative quantity is introduced based on the number of bacteria produced within the intestine by one bacterium originally in the external media. The latter depends on the parameters in a simpler way and provides more biological insight than the standard reproduction number, allowing the design of experimental procedures. Both quantities coincide and are equal to one at the extinction threshold, below which the bacterial population becomes extinct. Optimal values of both reproduction numbers are derived assuming parameter trade-offs.

Discrete and Continuous Dynamical Systems - Series B, 2017

An autonomous semi-linear model for the proliferation of bacteria within a heterogeneous populati... more An autonomous semi-linear model for the proliferation of bacteria within a heterogeneous population of animals is developed. It is assumed that bacteria grow inside the intestines and that they can be either attached to the epithelial wall or as free particles in the lumen. A condition involving ecological parameters is given, which can be used to decide the existence of endemic equilibria as well as local stability properties of the non-endemic one. Some implications on phage therapy are addressed.

Journal of Mathematical Analysis and Applications, 2016

In this paper, we study the asymptotic (large time) behavior of a selection-mutation-competition ... more In this paper, we study the asymptotic (large time) behavior of a selection-mutation-competition model for a population structured with respect to a phenotypic trait, when the rate of mutation is very small. We assume that the reproduction is asexual, and that the mutations can be described by a linear integral operator. We are interested in the interplay between the time variable t and the rate ε of mutations. We show that depending on α > 0, the limit ε → 0 with t = ε −α can lead to population number densities which are either Gaussian-like (when α is small) or Cauchy-like (when α is large).

Nonlinear Analysis: Real World Applications, 2016

We introduce and investigate an SIS-type model for the spread of an infectious disease, where the... more We introduce and investigate an SIS-type model for the spread of an infectious disease, where the infected population is structured with respect to the different strain of the virus/bacteria they are carrying. Our aim is to capture the interesting scenario when individuals infected with different strains cause secondary (new) infections at different rates. Therefore, we consider a nonlinear infection process, which generalises the bilinear process arising from the classic mass-action assumption. Our main motivation is to study competition between different strains of a virus/bacteria. From the mathematical point of view, we are interested whether the nonlinear infection process leads to a well-posed model. We use a semilinear formulation to show global existence and positivity of solutions up to a critical value of the exponent in the nonlinearity. Furthermore, we establish the existence of the endemic steady state for particular classes of nonlinearities.

Mathematical Methods in the Applied Sciences, 2016

In this work first we consider a physiologically structured population model with a distributed r... more In this work first we consider a physiologically structured population model with a distributed recruitment process. That is, our model allows newly recruited individuals to enter the population at all possible individual states, in principle. The model can be naturally formulated as a first order partial integro-differential equation, and it has been studied extensively. In particular, it is well-posed on the biologically relevant state space of Lebesgue integrable functions. We also formulate a delayed integral equation (renewal equation) for the distributed birth rate of the population. We aim to illustrate the connection between the partial integro-differential and the delayed integral equation formulation of the model utilising a recent spectral theoretic result. In particular, we consider the equivalence of the steady state problems in the two different formulations, which then leads us to characterise irreducibility of the semigroup governing the linear partial integro-differential equation. Furthermore, using the method of characteristics, we investigate the connection between the time dependent problems. In particular, we prove that any (non-negative) solution of the delayed integral equation determines a (non-negative) solution of the partial differential equation and vice versa. The results obtained for the particular distributed states at birth model then lead us to present some very general results, which establish the equivalence between a general class of partial differential and delay equation, modelling physiologically structured populations.

After an analysis of some actual models for bacteria-phage interaction we present a new different... more After an analysis of some actual models for bacteria-phage interaction we present a new differential equations system where the susceptible cell population is physiologically structured by the number of viral receptors on which an attach-detach mechanism is regarded. The interaction takes place in a limited resources environment, modeled via the logistic equation, and in company of a phage resistant bacterial strain. In a first formulation we consider finite attach and detach rates over bacterial phage receptors. A second and simpler model is obtained when neglecting viral detachments. Both formulations are first presented in a discrete manner and converted to their corresponding continuous fashions.

Publicacions Matemàtiques, 1994

In this paper we examine a prey-predator system with a characteristic of the predator subject to ... more In this paper we examine a prey-predator system with a characteristic of the predator subject to mutation. The ultimate equilibrium of the system is found by Maynard-Smith et al. by th e so called ESS (Evolutionary Stable Strategy). Using a system of reaction-diffusion equations with non local terms, we conclud e the ESS result for the diffusion coefficient tending to zero, without resorting to any optimization criterion .

SIAM Journal on Mathematical Analysis, 2014

We study the question of existence of positive steady states of nonlinear evolution equations. We... more We study the question of existence of positive steady states of nonlinear evolution equations. We recast the steady state equation in the form of eigenvalue problems for a parametrised family of unbounded linear operators, which are generators of strongly continuous semigroups; and a fixed point problem. In case of irreducible governing semigroups we consider evolution equations with non-monotone nonlinearities of dimension two, and we establish a new fixed point theorem for set-valued maps. In case of reducible governing semigroups we establish results for monotone nonlinearities of any finite dimension n. In addition, we establish a non-quasinilpotency result for a class of strictly positive operators, which are neither irreducible nor compact, in general. We illustrate our theoretical results with examples of partial differential equations arising in structured population dynamics. In particular, we establish existence of positive steady states of a size-structured juvenileadult and a structured consumer-resource population model, as well as for a selection-mutation model with distributed recruitment process.

SIAM Journal on Applied Mathematics, 1999

The first part of this paper is devoted to a complete description of the dynamics of a continuous... more The first part of this paper is devoted to a complete description of the dynamics of a continuously structured population model coupled with a dynamical resource. In the model, it is assumed that the energy each individual obtains from the resource is channeled between growth and reproduction in a proportion that depends on the individual's size. In the second part, an optimal allocation of this energy is obtained that turns out to be a convergence-stable ESS and is described by what is called a "bang-bang" strategy.

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2013

We consider a selection–mutation equation for the density of individuals with respect to a contin... more We consider a selection–mutation equation for the density of individuals with respect to a continuous phenotypic evolutionary trait. We assume that the competition term for an individual with a given trait depends on the traits of all the other individuals, therefore giving an infinite-dimensional nonlinearity. Mutations are modelled by means of an integral operator. We prove existence of steady states and show that, when the mutation rate goes to zero, the asymptotic profile of the population is a Cauchy distribution.

Nonlinear Analysis: Real World Applications, 2011

We consider a nonlinear cyclin content structured model of a cell population divided into prolife... more We consider a nonlinear cyclin content structured model of a cell population divided into proliferative and quiescent cells. We show, for particular values of the parameters, existence of solutions that do not depend on the cyclin content. We make numerical simulations for the general case obtaining, for some values of the parameters convergence to the steady state but also oscillations of the population for others.

Nonlinear Analysis: Real World Applications, 2014

We introduce a size structured pde model for a susceptible and a resistant bacteria populations t... more We introduce a size structured pde model for a susceptible and a resistant bacteria populations that grow as they feed from a resource that is added at a constant rate. Bacteria populations evolve in the presence of a phage population that is adsorbed by the susceptible bacteria but also by a bulk population of phage free receptors on cell debris and on infected cells. Assuming that the individual cell growth is non negative, we compute an age function that allow us to change variables and to obtain an equivalent system with structure by the cell age and where the cell volume becomes a state variable. We characterize the steady states of these models.

Mathematical Models and Methods in Applied Sciences, 2006

In this paper we present a proof of existence and uniqueness of solution for a class of PDE model... more In this paper we present a proof of existence and uniqueness of solution for a class of PDE models of size structured populations with distributed state-at-birth and having nonlinearities defined by an infinite-dimensional environment. The latter means that each member of the population experiences an environment according to a sort of average of the population size depending on the individual size, rank or any other variable structuring the population. The proof of the local existence and uniqueness of solution as well as the continuous dependence on initial continuous is based on the general theory of quasi-linear evolution equations in nonreflexive Banach spaces, while the global existence proof is based on the integration of the local solution along characteristic curves.