arXiv subjects

Alan D. Rendall

Publications and source records attributed to Alan D. Rendall.

At least 19 recordsLinked to original sources

arXiv2025

Backward bifurcations and multistationarity

The theory of backward bifurcations provides a criterion for the existence of positive steady states in epidemiological models with parameters where the basic reproductive ratio is less than one. It is often seen in simulations that this phenomenon is accompanied by multistationarity, i.e. the existence of more than one positive steady state, but the latter circumstance is not implied by the general theory. The central result of this paper is a theorem which gives a criterion for the existence of one stable and one unstable positive steady state for parameters where the basic reproductive ratio is less than one. It also gives a criterion for the existence of one stable and one unstable positive steady state in the case that the basic reproductive ratio is greater than one. These steady states arise in a bifurcation. It is shown that in one case, a model for the in-host dynamics of hepatitis C, this result can be used as a basis for showing the existence of parameters for which there are two positive steady states, one of which is stable. Thus, in particular, multistationarity is proved in that case. It is also shown to what extent this theorem can be applied to some other models which have been studied in the literature and what new results can be obtained. In that context the new approach is compared with those previously known.

math.DS↗

arXiv2023

Dynamics of a mathematical model of virus spreading incorporating the effect of a vaccine

The COVID-19 pandemic led to widespread interest in epidemiological models. In this context the role of vaccination in influencing the spreading of the disease is of particular interest. There has also been a lot of debate on the role of non-pharmaceutical interventions such as the disinfection of surfaces. We investigate a mathematical model for the spread of a disease which includes both imperfect vaccination and infection due to virus in the environment. The latter is studied with the help of two phenomenological models for the force of infection. In one of these models we find that backward bifurcations take place so that for some parameter values an endemic steady state exists although the basic reproduction ratio $R_0$ is less than one. We also prove that in that case there can exist more than one endemic steady state. In the other model all generic transcritical bifurcations are forward bifurcations so that these effects cannot occur. Thus we see that the occurrence of backward bifurcations, which can be important for disease control strategies, is dependent on the details of the function describing the force of infection. By means of simulations the predictions of this model are compared with data for COVID-19 from Turkey. A sensitivity analysis is also carried out.

math.DS↗

arXiv2022

Phenomenology of an in-host model of hepatitis C

This paper carries out an analysis of the global properties of solutions of an in-host model of hepatitis C for general values of its parameters. A previously unknown stable steady state on the boundary of the positive orthant is exhibited. It is proved that the model exhibits Hopf bifurcations and hence periodic solutions. A general parametrization of positive steady states is given and it is determined when the number of steady states is odd or even, according to the value of a certain basic reproductive ratio. This implies, in particular, that when this reproductive ratio is greater than one there always exists at least one positive steady state. A positive steady state which bifurcates from an infection-free state when the reproductive ratio passes through one is always stable, i.e. no backward bifurcation occurs in this model.

q-bio.PE↗

arXiv2021

Analysis of a model of the Calvin cycle with diffusion of ATP

The dynamics of a mathematical model of the Calvin cycle, which is part of photosynthesis, is analysed. Since diffusion of ATP is included in the model a system of reaction-diffusion equations is obtained. It is proved that for a suitable choice of parameters there exist spatially inhomogeneous positive steady states, in fact infinitely many of them. It is also shown that all positive steady states, homogeneous and inhomogeneous, are nonlinearly unstable. The only smooth steady state which could be stable is a trivial one, where all concentrations except that of ATP are zero. It is found that in the spatially homogeneous case there are steady states with the property that the linearization about that state has eigenvalues which are not real, indicating the presence of oscillations. Numerical simulations exhibit solutions for which the concentrations are not monotone functions of time.

q-bio.MN↗

arXiv2021

Autophosphorylation and the dynamics of the activation of Lck

Lck (lymphocyte-specific protein tyrosine kinase) is an enzyme which plays a number of important roles in the function of immune cells. It belongs to the Src family of kinases which are known to undergo autophosphorylation. It turns out that this leads to a remarkable variety of dynamical behaviour which can occur during their activation. We prove that in the presence of autophosphorylation one phenomenon, bistability, already occurs in a mathematical model for a protein with a single phosphorylation site. We further show that a certain model of Lck exhibits oscillations. Finally we discuss the relations of these results to models in the literature which involve Lck and describe specific biological processes, such as the early stages of T cell activation and the stimulation of T cell responses resulting from the suppression of PD-1 signalling which is important in immune checkpoint therapy for cancer.

q-bio.MN↗

arXiv2020

Analysis of an initial value problem for an extracellular and intracellular model of hepatitis C virus infection

In this paper, a mathematical analysis of the global dynamics of a viral infection model in vivo is carried out. We study the dynamics of a hepatitis C virus (HCV) model, under therapy, that considers both extracellular and intracellular levels of infection. At present most mathematical modeling of viral kinetics after treatment only addresses the process of infection of a cell by the virus and the release of virions by the cell, while the processes taking place inside the cell are not included. We prove that the solutions of the new model with positive initial values are positive, exist globally in time and are bounded. The model has two virus-free steady states. They are distinguished by the fact that viral RNA is absent inside the cells in the first state and present inside the cells in the second. There are basic reproduction numbers associated to each of these steady states. If the basic reproduction number of the first steady state is less than one then that state is asymptotically stable. If the basic reproduction number of the first steady state is greater than one and that of the second less than one then the second steady state is asymptotically stable. If both basic reproduction numbers are greater than one then we obtain various conclusions which depend on different restrictions on the parameters of the model. Under increasingly strong assumptions we prove that there is at least one positive steady state (infected equilibrium), that there is a unique positive steady state and that the positive steady state is stable. We also give a condition under which every positive solution converges to a positive steady state. This is proved by methods of Li and Muldowney. Finally, we illustrate the theoretical results by numerical simulations.

q-bio.PE↗

arXiv2020

Unbounded solutions of models for glycolysis

The Selkov oscillator, a simple description of glycolysis, is a system of two ordinary differential equations with mass action kinetics. In previous work the authors established several properties of the solutions of this system. In the present paper we extend this to prove that this system has solutions which diverge to infinity in an oscillatory manner at late times. This system was originally derived from another system with Michaelis-Menten kinetics. It is shown that the Michaelis-Menten system, like that with mass action, has solutions which diverge to infinity in a monotone manner. It is also shown to admit subcritical Hopf bifurcations and thus unstable periodic solutions. We discuss to what extent the unbounded solutions cast doubt on the biological relevance of the Selkov oscillator and compare it with other models in the literature.

q-bio.MN↗

arXiv2019

A proof of unlimited multistability for phosphorylation cycles

The multiple futile cycle is a phosphorylation system in which a molecular substrate might be phosphorylated sequentially n times by means of an enzymatic mechanism. The system has been studied mathematically using reaction network theory and ordinary differential equations. It is known that the system might have at least as many as 2[n/2]+1 steady states (where [x] is the integer part of x) for particular choices of parameters. Furthermore, for the simple and dual futile cycles (n=1,2) the stability of the steady states has been determined in the sense that the only steady state of the simple futile cycle is globally stable, while there exist parameter values for which the dual futile cycle admits two asymptotically stable and one unstable steady state. For general n, evidence that the possible number of asymptotically stable steady states increases with $n$ has been given, which has led to the conjecture that parameter values can be chosen such that [n/2]+1 out of 2[n/2]+1 steady states are asymptotically stable and the remaining steady states are unstable. We prove this conjecture here by first reducing the system to a smaller one, for which we find a choice of parameter values that give rise to a unique steady state with multiplicity 2[n/2]+1. Using arguments from geometric singular perturbation theory, and a detailed analysis of the centre manifold of this steady state, we achieve the desired result.

q-bio.MN↗

arXiv2018

The minimal model of Hahn for the Calvin cycle

There are many models of the Calvin cycle of photosynthesis in the literature. When investigating the dynamics of these models one strategy is to look at the simplest possible models in order to get the most detailed insights. We investigate a minimal model of the Calvin cycle introduced by Hahn while he was pursuing this strategy. In a variant of the model not including photorespiration it is shown that there exists exactly one positive steady state and that this steady state is unstable. For generic initial data either all concentrations tend to infinity at lates times or all concentrations tend to zero at late times. In a variant including photorespiration it is shown that for suitable values of the parameters of the model there exist two positive steady states, one stable and one unstable. For generic initial data either the solution tends to the stable steady state at late times or all concentrations tend to zero at late times. Thus we obtain rigorous proofs of mathematical statements which together confirm the intuitive idea proposed by Hahn that photorespiration can stabilize the operation of the Calvin cycle. In the case that the concentrations tend to infinity we derive formulae for the leading order asymptotics using the Poincaré compactification.

q-bio.MN↗

arXiv2018

Dynamics of the Selkov oscillator

A classical example of a mathematical model for oscillations in a biological system is the Selkov oscillator, which is a simple description of glycolysis. It is a system of two ordinary differential equations which, when expressed in dimensionless variables, depends on two parameters. Surprisingly it appears that no complete rigorous analysis of the dynamics of this model has ever been given. In this paper several properties of the dynamics of solutions of the model are established. With a view to studying unbounded solutions a thorough analysis of the Poincaré compactification of the system is given. It is proved that for any values of the parameters there are solutions which tend to infinity at late times and are eventually monotone. It is shown that when the unique steady state is stable any bounded solution converges to the steady state at late times. When the steady state is unstable it is shown that for given values of the parameters either there is a unique periodic solution to which all bounded solutions other than the steady state converge at late times or there is no periodic solution and all solutions other than the steady state are unbounded. In the latter case each unbounded solution which tends to infinity is eventually monotone and each unbounded solution which does not tend to infinity has the property that each variable takes on arbitrarily large and small values at arbitrarily late times.

q-bio.SC↗

arXiv2017

Analytic description of singularities in Gowdy spacetimes

We use Fuchsian Reduction to construct singular solutions of Einstein's equations which belong to the class of Gowdy spacetimes. The solutions have the maximum number of arbitrary functions. Special cases correspond to polarized, or other known solutions. The method provides precise asymptotics at the singularity, which is Kasner-like. All of these solutions are asymptotically velocity-dominated. The results account for the fact that solutions with velocity parameter uniformly greater than one are not observed numerically. They also provide a justification of formal expansions proposed by Grubišić and Moncrief.

gr-qc↗

arXiv2017

Multiple steady states and the form of response functions to antigen in a model for the initiation of T cell activation

The aim of this paper is to study the qualitative behaviour predicted by a mathematical model for the initial stage of T cell activation. The state variables in the model are the concentrations of phosphorylation states of the T cell receptor complex and the phosphatase SHP-1 in the cell. It is shown that these quantities cannot approach zero and that the model possesses more than one positive steady state for certain values of the parameters. It can also exhibit damped oscillations. It is proved that the chemical concentration which represents the degree of activation of the cell, that of the maximally phosphorylated form of the T cell receptor complex, is in general a non-monotone function of the activating signal. In particular there are cases where there is a value of the dissociation constant of the ligand from the receptor which produces an optimal activation of the T cell. In this way the results of certain simulations in the literature have been confirmed rigorously and some important features which had not previously been seen have been discovered.

q-bio.MN↗

arXiv2016

A Calvin bestiary

This paper compares a number of mathematical models for the Calvin cycle of photosynthesis and presents theorems on the existence and stability of steady states of these models. Results on five-variable models in the literature are surveyed. Next a number of larger models related to one introduced by Pettersson and Ryde-Pettersson are discussed. The mathematical nature of this model is clarified, showing that it is naturally defined as a system of differential-algebraic equations. It is proved that there are choices of parameters for which this model admits more than one positive steady state. This is done by analysing the limit where the storage of sugars from the cycle as starch is shut down. There is also a discussion of the minimal models for the cycle due to Hahn.

q-bio.MN↗

arXiv2016

Stability of stationary solutions in models of the Calvin cycle

In this paper results are obtained concerning the number of positive stationary solutions in simple models of the Calvin cycle of photosynthesis and the stability of these solutions. It is proved that there are open sets of parameters in a model of Zhu et. al. for which there exist two positive stationary solutions. There are never more than two isolated positive stationary solutions but under certain explicit special conditions on the parameters there is a whole continuum of positive stationary solutions. It is also shown that in the set of parameter values for which two isolated positive stationary solutions exist there is an open subset where one of the solutions is asymptotically stable and the other is unstable. In related models , for which it was known that more that one positive stationary solution exists, it is proved that there are parameter values for which one of these solutions is asymptotically stable and the other unstable. A key technical aspect of the proofs is to exploit the fact that there is a bifurcation where the centre manifold is one-dimensional.

q-bio.SC↗

arXiv2016

Veiled singularities for the spherically symmetric massless Einstein-Vlasov system

This paper continues the investigation of the formation of naked singularities in the collapse of collisionless matter initiated in [RV]. There the existence of certain classes of non-smooth solutions of the Einstein-Vlasov system was proved. Those solutions are self-similar and hence not asymptotically flat. To obtain solutions which are more physically relevant it makes sense to attempt to cut off these solutions in a suitable way so as to make them asymptotically flat. This task, which turns out to be technically challenging, will be carried out in this paper. [RV] A. D. Rendall and J. J. L. Velázquez, A class of dust-like self-similar solutions of the massless Einstein-Vlasov system. Annales Henri Poincare 12, 919-964, (2011).

gr-qc↗

arXiv2015

Sustained oscillations in the MAP kinase cascade

The MAP kinase cascade is a network of enzymatic reactions arranged in layers. In each layer occurs a multiple futile cycle of phosphorylations. The fully phosphorylated substrate then serves as an enzyme for the layer below. This papers focusses on the existence of parameters for which Hopf bifurcations occur and generate periodic orbits. Furthermore it is explained how geometric singular perturbation theory allows to generalize results from simple models to more complex ones.

q-bio.MN↗

arXiv2015

Overload breakdown in models for photosynthesis

In many models of the Calvin cycle of photosynthesis it is observed that there are solutions where concentrations of key substances belonging to the cycle tend to zero at late times, a phenomenon known as overload breakdown. In this paper we prove theorems about the existence and non-existence of solutions of this type and obtain information on which concentrations tend to zero when overload breakdown occurs. As a starting point we take a model of Pettersson and Ryde-Pettersson which seems to be prone to overload breakdown and a modification of it due to Poolman which was intended to avoid this effect.

q-bio.MN↗

arXiv2015

Dynamical features of the MAPK cascade

The MAP kinase cascade is an important signal transduction system in molecular biology for which a lot of mathematical modelling has been done. This paper surveys what has been proved mathematically about the qualitative properties of solutions of the ordinary differential equations arising as models for this biological system. It focusses, in particular, on the issues of multistability and the existence of sustained oscillations. It also gives a concise introduction to the mathematical techniques used in this context, bifurcation theory and geometric singular perturbation theory, as they relate to these specific examples. In addition further directions are presented in which the applications of these techniques could be extended in the future.

q-bio.MN↗