Steady states in a structured epidemic model with Wentzell boundary condition

Calsina, Àngel; Farkas, Jozsef Zoltan

doi:10.1007/s00028-012-0142-6

Cited by 26 publications

(45 citation statements)

References 36 publications

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“…[33,34]. To overcome some of the difficulties, we have developed a very general framework to treat steady state problems of some classes of nonlinear partial differential or partial integro-differential equations, see [10,11,12]. The power of the method we developed becomes apparent for models with so-called infinite dimensional nonlinearities, when in fact the existence of positive steady states cannot be completely characterised by an appropriately defined net reproduction function.…”

Section: Prologuementioning

confidence: 99%

Net reproduction functions for nonlinear structured population models

Farkas

2018

Math. Model. Nat. Phenom.

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The goal of this note is to present a general approach to define the net reproduction function for a large class of nonlinear physiologically structured population models. In particular, we are going to show that this can be achieved in a natural way by reformulating a nonlinear problem as a family of linear ones; each of the linear problems describing the evolution of the population in a different, but constant environment. The reformulation of a nonlinear population model as a family of linear ones is a new approach, and provides an elegant way to study qualitative questions, for example the existence of positive steady states. To define the net reproduction number for any fixed (constant) environment, i.e. for the linear models, we use a fairly recent spectral theoretic result, which characterizes the connection between the spectral bound of an unbounded operator and the spectral radius of a corresponding bounded operator. For nonlinear models, varying the environment naturally leads to a net reproduction function. ProloguePhysiologically structured population models have been developed and studied extensively in the past decades, see for example the monographs [18,35,38,45] for an in-depth introduction and review of the topic. Our general research agenda, to relate the stability of steady states of physiologically structured population models to biologically meaningful quantities, allowed us to obtain a number of interesting results, see e.g. [27,29,30,31,32]. In particular, as we have seen for example in [1,27,29,30,31,32], the existence and stability of steady states of nonlinear structured population models can often be characterised using an appropriately defined net reproduction function. In fact we note that stability questions of nonlinear matrix population models were already investigated in the same spirit, see for example the paper [46]. However, we would like to emphasize that previously net reproduction functions were defined for concrete nonlinear models on an 'ad hoc' basis, but typically via analysing the corresponding steady state equations, see e.g. [1,27,28,29,30,31,32] for more details. It is our goal in this paper to present a general framework, which is applicable to different classes of nonlinear models.The existence of positive steady states is an important and interesting question, when studying nonlinear population models; and to establish the existence of non-trivial steady states is often challenging. This is especially the case for models formulated as infinite dimensional dynamical systems, for example delay equations, partial differential equations or integro-differential equations, see e.g. [33,34]. To overcome some of the difficulties, we have developed a very general framework to treat steady state problems of some classes of nonlinear partial differential or partial integro-differential equations, see [10,11,12]. The power of the method we developed becomes apparent for models with so-called infinite dimensional nonlinearities, when in fact the existence 1991 Mathematics Subj...

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Section: Prologuementioning

confidence: 99%

Net reproduction functions for nonlinear structured population models

Farkas

2018

Math. Model. Nat. Phenom.

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“…These properties can be established by using perturbation results from [2] and [32], respectively; see also [9,11] for similar developments. Also note that if r satisfies condition (3.14) then O 0 is shown to be compact exactly in the same way as the operator Φ was shown to be compact earlier.…”

Section: Asymptotic Behaviourmentioning

confidence: 99%

Mathematical analysis of a clonal evolution model of tumour cell proliferation

Farkas¹,

Webb²

2016

J. Evol. Equ.

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Abstract. We investigate a partial differential equation model of a cancer cell population, which is structured with respect to age and telomere length of cells. We assume a continuous telomere length structure, which is applicable to the clonal evolution model of cancer cell growth. This model has a non-standard non-local boundary condition. We establish global existence of solutions and study their qualitative behaviour. We study the effect of telomere restoration on cancer cell dynamics. Our results indicate that without telomere restoration, the cell population extinguishes. With telomere restoration, exponential growth occurs in the linear model. We further characterise the specific growth behaviour of the cell population for special cases. We also study the effects of crowding induced mortality on the qualitative behaviour, and the existence and stability of steady states of a nonlinear model incorporating crowding effect. We present examples and extensive numerical simulations, which illustrate the rich dynamic behaviour of the linear and nonlinear models.

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“…Related models featuring structured population balances have been widely used to model biological population dynamics, for instance, stability of structured populations was investigated in [4,5], existence of solutions of continuous diffusive coagulation fragmentation models was studied in [6] e.g., similar models without diffusion were discussed in [7,8,9], existence of solutions using a fixed-point approach was investigated in [10,11,12], and finally size distribution and long-time asymptotics were studied in [13,14].…”

Section: Introductionmentioning

confidence: 99%

Mathematical Study of an Inflammatory Model for Atherosclerosis: A Nonlinear Renewal Equation

Meunier

Müller

2018

Acta Appl Math

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We present here a population structured model to describe the dynamics of macrophage cells. The model involves the interactions between modified LDL, monocytes/macrophages, cytokines and foam cells. The key assumption is that the individual macrophage dynamics depends on the amount of lipoproteins it has internalized. The obtained renewal equation is coupled with an ODE describing the lipoprotein dynamics. We first prove global existence and uniqueness for the nonlinear and nonlocal system. We then study long time asymptotics in a particular case describing silent plaques which undergo periodic rupture and repair. Finally we study long time asymptotics for the nonlinear renewal equation obtained when considering the steady state of the ODE. and we prove that .... +∞ 0 M (t, a)da is used to control the first momentum +∞ 0 aM (t, a)da.

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Steady states in a structured epidemic model with Wentzell boundary condition

Cited by 26 publications

References 36 publications

Net reproduction functions for nonlinear structured population models

Net reproduction functions for nonlinear structured population models

Mathematical analysis of a clonal evolution model of tumour cell proliferation

Mathematical Study of an Inflammatory Model for Atherosclerosis: A Nonlinear Renewal Equation

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