97 research outputs found
Structured populations with distributed recruitment: from PDE to delay formulation
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.Comment: 28 pages, to appear in Mathematical Methods in the Applied Science
Stability results for a hierarchical size-structured population model with distributed delay
In this paper we investigate a structured population model with distributed delay. Our model incorporates two different types of nonlinearities. Specifically we assume that individual growth and mortality are affected by scramble competition, while fertility is affected by contest competition. In particular, we assume that there is a hierarchical structure in the population, which affects mating success. The dynamical behavior of the model is analysed via linearisation by means of semigroup and spectral methods. In particular, we introduce a reproduction function and use it to derive linear stability criteria for our model. Further we present numerical simulations to underpin the stability results we obtained
Positive Steady States of Evolution Equations with Finite Dimensional Nonlinearities
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 xed point problem. In case of irreducible governing semigroups we consider evolution equations with non-monotone nonlinearities of dimension two, and we establish a new xed point theorem for set-valued maps. In case of reducible governing semigroups we establish results for monotone nonlinearities of any nite 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 dierential equations arising in structured population dynamics. In particular, we establish existence of positive steady states of a size-structured juvenile- adult and a structured consumer-resource population model, as well as for a selection-mutation model with distributed recruitment process
Asymptotic behaviour of a structured population model on a space of measures
In this paper we consider a physiologically structured population model with
distributed states at birth, formulated on the space of non-negative Radon
measures. Using a characterisation of the pre-dual space of bounded Lipschitz
functions, we show how to apply the theory of strongly continuous positive
semigroups to such a model. In particular, we establish the exponential
convergence of solutions to a one-dimensional global attractor
Modelling Evolution of Virulence in Populations with a Distributed Parasite Load
Modelling evolution of virulence in host-parasite systems is an actively developing area of research with ever-growing literature. However, most of the existing studies overlook the fact that individuals within an infected population may have a variable infection load, i.e. infected populations are naturally structured with respect to the parasite burden. Empirical data suggests that the mortality and infectiousness of individuals can strongly depend on their infection load; moreover, the shape of distribution of infection load may vary on ecological and evolutionary time scales. Here we show that distributed infection load may have important consequences for the eventual evolution of virulence as compared to a similar model without structuring. Mathematically, we consider an SI model, where the dynamics of the infected subpopulation is described by a von Förster-type model, in which the infection load plays the role of age. We implement the adaptive dynamics framework to predict evolutionary outcomes in this model. We demonstrate that for simple trade-off functions between virulence, disease transmission and parasite growth rates, multiple evolutionary attractors are possible. Interestingly, unlike in the case of unstructured models, achieving an evolutionary stable strategy becomes possible even for a variation of a single ecological parameter (the parasite growth rate) and keeping the other parameters constant. We conclude that evolution in disease-structured populations is strongly mediated by alterations in the overall shape of the parasite load distribution
Assessing the Impact of (Self)-Quarantine through a Basic Model of Infectious Disease Dynamics
We introduce a system of differential equations to assess the impact of (self-)quarantine of symptomatic infectious individuals on disease dynamics. To this end we depart from using the classic bilinear infection process, but remain within the framework of the mass-action assumption. From the mathematical point of view, the model we propose is interesting due to the lack of continuous differentiability at disease-free steady states, which implies that the basic reproductive number cannot be computed following established mathematical approaches for certain parameter values. However, we parametrise our mathematical model using published values from the COVID-19 literature, and analyse the model simulations. We also contrast model simulations against publicly available COVID-19 test data, focusing on the first wave of the pandemic during March–July 2020 in the UK. Our simulations indicate that actual peak case numbers might have been as much as 200 times higher than the reported positive test cases during the first wave in the UK. We find that very strong adherence to self-quarantine rules yields (only) a reduction of 22% of peak numbers and delays the onset of the peak by approximately 30–35 days. However, during the early phase of the outbreak, the impact of (self)-quarantine is much more significant. We also take into account the effect of a national lockdown in a simplistic way by reducing the effective susceptible population size. We find that, in case of a 90% reduction of the effective susceptible population size, strong adherence to self-quarantine still only yields a 25% reduction of peak infectious numbers when compared to low adherence. This is due to the significant number of asymptomatic infectious individuals in the population
On the net reproduction rate of continuous structured populations with distributed states at birth
We consider a nonlinear structured population model with a distributed
recruitment term. The question of the existence of non-trivial steady states
can be treated (at least!) in three different ways. One approach is to study
spectral properties of a parametrized family of unbounded operators. The
alternative approach, on which we focus here, is based on the reformulation of
the problem as an integral equation. In this context we introduce a density
dependent net reproduction rate and discuss its relationship to a biologically
meaningful quantity. Finally, we briefly discuss a third approach, which is
based on the finite rank approximation of the recruitment operator.Comment: To appear in Computers and Mathematics with Application
- …