SCIENTIFIC PROGRAMS AND ACTIVITIES

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES Short Thematic Program on Delay Differential Equations May 2015 Organizing Committee Odo Diekmann (Utrecht) Sue Ann Campbell (Waterloo) Stephen Gourley (Surrey) Yuliya Kyrychko (Sussex) Eckehard Schöll (TU Berlin) Michael Mackey (McGill) Hans-Otto Walther (Giessen) Glenn Webb (Vanderbilt) Jianhong Wu (York)

 

Abstracts

Structured populations: three talks for an overview on numerical approximation

AUTHORS and TALKS' TITLES
TALK 1 - Julia Sánchez Sanz (BCAM Spain): Structured populations: computing eigenvalues of linear models
TALK 2 - Rossana Vermiglio (University of Udine, Italy): Structured populations: back to ODEs for nonlinear models
TALK 3 - Dimitri Breda (University of Udine, Italy): Structured populations: how challenging is Daphnia?

We are interested in structured populations representing consumer-resource interaction. Recently, they are modeled via a Delay (or renewal or Volterra functional) Equation for the consumer birth rate coupled to a Delay Differential Equation for the available resource (briefly, DE/DDE).

With the aim at investigating the dynamical behavior under parameter variation, we start by focusing on the stability of equilibria. The first talk, indeed, presents a pseudospectral approach to compute the eigenvalues of the infinitesimal generator of general linear systems of coupled DEs/DDEs. When the latter are obtained by linearizing nonlinear problems around the equilibrium of interest, these eigenvalues can determine the local asymptotic stability and their variation may lead to detect Hopf bifuractions.

Solved the question of equilibria, in the second talk we make a considerable step forward by extending the pseudospectral approach directly to nonlinear problems. This way, we reduce the original delay system to a finite number of nonlinear Ordinary Differential Equations (ODEs). We start from DDEs to arrive to coupled DEs/DDEs. The final objective is to apply existing, efficient and complete bifurcation tools for ODEs in order to prolong the investigation of the dynamics beyond equilibria, e.g., periodic orbits and their bifurcations.

Both approaches, linear and nonlinear, are applied to analyze a model of "Daphnia consuming algae". As this model features several additional difficulties like distributed delays, juveniles-adults discontinuities and external ODEs, it is commonly considered as a challenging prototype in the literature. The analysis of its dynamics is thus rather prohibitive from an analytical point of view. This reflects as well on the numerical side and the third talk, in fact, serves to summarize such difficulties and to discuss possible solutions to succeed in obtaining reliable and efficient approximations.

Beyond the speakers, many people are involved in the several aspects of this broad work, a (maybe non-exhaustive) list of their names follow in alphabetical order: Odo Diekmann (Utrecht), Philipp Getto (Dresden), Mats Gyllenberg (Helsinki), Davide Liessi (Udine), Stefano Maset (Trieste), Andre de Roos (Amsterdam), Francesca Scarabel (Helsinki).

 

Åke Brännström, Umeå University
On the convergence of the Escalator Boxcar Train
(Joint work with Linus Carlsson and Daniel Simpson)

The Escalator Boxcar Train (EBT) is a numerical method used in theoretical biology to investigate the dynamics of physiologically structured population models. It works by discretizing the population into a finite number of 'cohorts' whose dynamics are described by ordinary differential equations. The method was developed more than two decades ago, but had long resisted attempts to give a formal proof of convergence. Using a modern framework of measure-valued solutions, we investigate the EBT method and show that the sequence of approximating solution measures generated by the EBT method converges weakly to the true solution measure under weak conditions on the growth rate, birth rate, and mortality rate.

 

 

Hermann Brunner, University of Newfoundland and Hong Kong Baptist University
Numerical analysis of Volterra functional integral equations with state-dependent delays

In order to simulate the behavior of solutions of Volterra functional integral and integro-differential equations with state-dependent delays one is interested in designing high-order (and locally superconvergent) numerical schemes based on, for example, piecewise polynomial collocation. In this talk we describe the key ideas underlying the so-called time transformations that allow the transformation of a state-dependent Volterra functional equation into a problem with (prescribed) non-state-dependent delay. In this classical framework (which also provides the breaking points induced by the original state-dependent delay) the question of the attainable order of collocation solutions be readily answered. While much of the analysis is now understood, many problems regarding the computational implementation of such collocation methods remain to be addressed.

The talk describes ongoing joint work with Stefano Maset (University of Trieste).

Wayne Enright, University of Toronto
Exploiting special problem stucture when Reliably Investigating the solution of systems of DDEs with multilple delays

In recent years we have developed a class of reliable order p methods for the approximate solution of general DDEs. In the theoretical analysis of these methods we have identified several trade-offs that do arise and have to be addressed when applying these methods to DDEs that exhibit special structure. Similar trade-offs also arise when one is concerned with investigating other important properties of the solutions of DDEs. I will give examples of these trade-offs and how they arise when investgating the sensitivities of the solution of DDES; the computation of very accurate approximations of DDEs; and the task of determining the parameters of a DDE which "best fit" some prescribed observed data.

 

Teresa Faria, University of Stirling
Permanence of nonautonomous cooperative population models with delays

For a large family of nonautonomous scalar delay differential equations
(DDEs)
used in population dynamics, some criteria for permanence are given. The method described here is based on comparative results with auxiliary monotone systems. In particular, it applies to a nonautonomous scalar model proposed as an alternative to the usual delayed logistic equation. The method is extended to cooperative $n$-dimensional systems of DDEs, for which sufficient conditions for permanence are given.

Jozsef Farkas, University of Stirling
Modelling structured populations: from partial differential equation to delay formulation

We consider a structured model with distributed states at birth. The model can be formulated as a PDE and also as a delayed integral equation.
We investigate the connection between solutions of the PDE and the delayed integral equation.

Joint work with Angel Calsina (Universitat Autonoma de Barcelona).

 

Philipp Getto, Basque Center for Applied Mathematics
A differential equation with state-dependent delay from stem cell population dynamics

In many stem cell models the maturation of stem cells is regulated by fully mature cells [4]. This may lead to formulation of continuously maturity structured population models as differential equations with state dependent delay [1,4,5]. The existing qualitative theory for such equations [7] requires smoothness of the delay functional as a function of the state, which is a history. I here focus on a submodel for the delay and discuss its smoothness [2,3,5,6].

References

[1] T. Alarcon, Ph. Getto, A. Marciniak-Czochra, M.dM. Vivanco, A model for stem cell population dynamics with regulated maturation delay, Discr. Cont. Dyn. Sys. b. Supplement 2011 32-43.

[2] Henri Cartan, Differential Calculus, Hermann, 1971

[3] O. Diekmann, S. van Gils, S.M. Verduyn Lunel, H.-O. Walther, Delay Equations, Functional-, Complex-, and Nonlinear Analysis. Springer Verlag, New York, 1995.

[4] Ph. Getto and A. Marciniak-Czochra, Mathematical modelling as a tool to understand cell self-renewal and differentiation, book chapter in M dM. Vivanco (ed.) ``Mammary stem cells - Methods in Molecular Biology'' Humana press, in press.

[5] Ph. Getto and M. Waurick, A differential equation with state-dependent delay from cell population biology
arXiv:1411.3097
[6] F. Hartung, On differentiability of solutions with respect to parameters in neutral differential equations with state-dependent delays, J. Math. Anal. Appl., 324:1 (2006) 504-524.

[7] F. Hartung, T. Krisztin, H.-O. Walther, J. Wu, Functional Differential Equations with state dependent delays: Theory and Applications, Chapter V in Handbook of Differential Equations: Ordinary Differential Equations, Volume

4, Elsevier.

Stephen Gourley, University of Surrey - Guildford
Delay equation models for populations that experience competition at immature life stages

Models of intra- and inter-specific competition at immature life stages will be presented. These will include a simple delay model for a single species that experiences larval competition. Its solutions are bounded for any birth function. In some situations the technique of reducing an age-structured model to a system of delay equations applies. In the case of immature competition the delay equations cannot always be written down explicitly because their right hand sides depend on the solutions of the nonlinear ordinary differential equations that arise when one solves the nonlinear age-structured equations that determine the maturation rates in terms of the birth rates. This situation arises in the case of competition between two strains or species. However, in a simple two-strain competition model, vital properties of those right hand sides can be indirectly inferred using monotone systems theory. I also discuss extensions to the case when individual larvae experience competition from other larvae at various stages of development. This is joint work with Rongsong Liu and Gergely Rost.

Pierre Magal, Université de Bordeaux
Some age-structured model describing hospital acquired infection

The first part of presentation will be devoted to some results about a class of epidemic models with criss-cross transmission and age of infection. Some global stability result will be presented.

The second part of the presentation will be devoted to some singular perturbation results for such a class of systems. Namely we will talk about slow-fast dynamics and the question addressed here is : does it make sense to simplify the systems by using such an idea.

Under some simplifying assumptions, age-structured models can be rewritten as a system of delay differential equations. We will present some recent convergence results and discuss the convergence globally in time.

We will conclude this presentation by mentioning a Tikhonov like result for a class of abstract non-densely defined Cauchy problems. We will show how this kind of results can help to understand the convergence for singularly perturbed age-structured models and also delay differential equations.

John Mallet-Paret, Brown University
Tensor Products, Positive Operators, and Delay-Differential Equations

We study linear nonautonomous delay-differential equations, such as
$$
\dot x(t) = -a(t)x(t)-b(t)x(t-1).
\eqno(*)
$$
Such equations can occur as the linearization of a nonlinear delay equation
$$
\dot x(t) = -f(x(t),x(t-1))
$$
around a given solution (often around a periodic solution), and they are crucial in understanding the stability of such solutions.
%Such nonlinear equations occur in a variety of scientific models, and despite their simple appearance,
%can lead to a rather difficult mathematical analysis.

We develop an associated linear theory to equation ($*$) by taking the $m$-fold wedge product
(in the infinite-dimensional sense of tensor products) of the dynamical system generated by ($*$).
Remarkably, in the case of a ``signed feedback,'' namely where $(-1)^m b(t) > 0$ for some integer $m$,
the associated linear system is given by an operator which is positive with respect to
a certain cone in a Banach space. This leads to detailed information about stability properties of ($*$),
and in particular, information about its charateristic multipliers.

 

Stefano Maset, Universita' di Trieste
An abstract framework in the numerical solution of boundary value problems for functional differential equations

In this talk, we consider the numerical solution of boundary value problems for general neutral functional differential equations. The problems are restated in an abstract form and, then, a general discretization of the abstract form is introduced and a convergence analysis of this discretization is developed. Moreover, concrete results on concrete problems are also discussed.

Connell McCluskey, Wilfrid Laurier University
Lyapunov Functionals for Structured Delay Systems

I will discuss recent work on using Lyapunov functionals for delay differential equations and age-structured PDEs that arise in mathematical epidemiology. Typically, a Lyapunov functional for these systems is found by modifying a Lyapunov function that works for the corresponding ODE. This raises the question: When does a Lyapunov function for an ODE imply that a Lyapunov functional for corresponding DDEs or PDEs exist?

Yukihiko Nakata,University of Tokyo
Dynamics of a reinfection epidemic model and an application to a childhood infectious disease

Dynamical aspects of epidemic models with reinfection are not fully understood. We introduce an extension of the SIRS epidemic model structuring populations in terms of infection-age and recovery-age. Tracking individuals' history after infection, we derive a coupled system of a delay differential equation and a renewal equation for two dynamical variables: susceptible population and force of infection. For some cases local and global stability analysis will be presented. It will be shown a possibility of double Hopf bifurcation indicated in a parameter plane, as an elaboration of an analysis in (Diekmann and Montijn, 1982). Stability analysis is used for a possible explanation of periodic outbreak of mycoplasma pneumoniae observed in Japan.

Israel Ncube, Alabama A&M University
Analysis of a transcendental equation arising in modelling applications

Employing direct analytic approaches inspired by ideas from classical complex variable theory, we investigate the problem of stability of a transcendental quasi-polynomial often arising in modelling applications using delay differential equations.
The aim of the talk is to present some preliminary results on what is essentially on-going research work.

Shigui Ruan, University of Miami
Hopf Bifurcation and Normal Forms in a structured evolutionary epidemiological model of influenza A drift

In this paper, we first introduce the Hopf bifurcation theorem and the normal form theory for semilinear Cauchy problems in which the linear operator is not densely defined and is not a Hille-Yosida operator and present procedures to compute the Taylor's expansion and normal form of the reduced system restricted on the center manifold. We then apply the main results and computation procedures to determine the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions in a structured evolutionary epidemiological model of influenza A drift and an age structured population model. (Based on joint papers with Zhihua Liu and Pierre Magal).

Horst R. Thieme, Arizona State University,
Spreading speeds for a fox rabies model with infection-age dependent diffusion

An epidemic outbreak is considered for rabies in a spatially distributed fox population where the susceptible foxes do not move but infected foxes diffuse with diffusion coefficients that depend on their infection-age (time since infection). This takes into account that foxes in an early phase of the latency period would hardly diffuse while foxes with full-blown rabies may diffuse considerably. Since an outbreak situation is considered, the population turnover of the fox population is ignored. This allows to transform a system consisting of an ODE for the susceptible foxes and an age-dependent diffusion equation for the infected foxed (similar equations can be found in [4, 8, 11, 13]) to a single space-time integral equation of renewal type for the cumulative number of infected foxes [1, 6, 7, 9] and to find an implicit formula for the spreading speed of the rabies epidemic [2, 3, 5, 10, 12]. As implicit as it is, it still allows to study the dependence of the spreading speed on the diffusion coefficients, the length of the latency period, the per capita infection and disease death rates and other demographic or epidemiologic parameters [5].

References
[1] O. Diekmann, Limiting behaviour in an epidemic model, Nonlinear Analysis, TMA, 1 (1977), 459-470

[2] O. Diekmann, Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol. 6 (1978), 109-130

[3] O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Differential Equations 33 (1979), 5873

[4] Gourley, S., Y. Kuang, A delay reaction-diffusion model of the spread of bacteriophage infection, SIAM J. Appl. Math. 65 (2005), 550-566

[5] Liu, Hao, Spatial Spread of Rabies in Wildlife, Dissertation, Arizona State University, December 2013

[6] Jones, D.A., G. Rost, H.L. Smith, H.R. Thieme, Spread of phage infection of bacteria in a petri dish, SIAM J. Applied Math. 72 (2012), 670-688

[7] Jones, D.A., H.L. Smith, H.R. Thieme, Spread of viral infection of immobilized bacteria, Networks and Heterogeneous Media 8 (2013), 327-342

[8] So, J.W.-H., J. Wu, X. Zou, A reaction-diffusion model for a single species with age structure, I. Travelling wavefronts on unbounded domains, Proc. Roy. Soc. London Ser. A 457 (2001), 1841-1853

[9] Thieme, H.R., A model for the spatial spread of an epidemic, J. Math. Biology 4 (1977), 337-351

[10] Thieme, H.R., Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations. J. Reine Angew. Math. 306 (1979), 94-121

[11] Thieme, H.R., X.-Q. Zhao, A nonlocal and delayed predator-prey reaction-diffusion model. Nonlinear Analysis RWA 2 (2001), 145-160

[12] Thieme, H.R., X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models. JDE 195 (2003), 430-470

[13] Webb, G.F., Population models structured by age, size, and spatial position, Structured Population Models in Biology and Epidemiology (Pierre Magal, Shigui Ruan, eds.), Springer, Berlin Heidelberg, 2008

 

Xingfu Zou, University of Western Ontario
Some DDE models with spatial non-locality caused by mobility and delay

In this talk, I will review some mathematical models for population dynamics and infectious disease dynamics. These models contain spatial non-locality caused, in the case of population dynamics, by the maturation delay and mobility of the immature individuals, and in the case of infectious disease dynamics, by the mobility of latent individuals. We will see how the situation of the spatial domain and the boundary conditions will affect such non-locality, and how the mobility of immature/latent individuals will have an impact on the model dynamics.