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Building New Models: Rethinking and Revising ODE Model Assumptions

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An Introduction to Undergraduate Research in Computational and Mathematical Biology

Part of the book series: Foundations for Undergraduate Research in Mathematics ((FURM))

Abstract

Ordinary Differential Equation (ODE) models are ubiquitous throughout the sciences, and form the backbone of the branch of mathematics known as applied dynamical systems. However, despite their utility and their analytical and computational tractability, modelers must make certain simplifying assumptions when modeling a system using ODEs. Relaxing (or otherwise changing) these assumptions may lead to the derivation of new ODE or non-ODE models and sometimes these new models can yield results that differ meaningfully in the context of a given application. The goal of this chapter is to explore some approaches to relaxing these ODE model assumptions to derive models which can then be analyzed in ways that parallel or build upon an existing ODE model analysis. To accomplish this, the first part of this chapter (Sect. 2) reviews some common methods for the application and analysis of ODE models. The next section (Sect. 3) explores various ways of deriving new models by modifying the assumptions of existing ODE models. This allows investigators to explore the extent to which ODE model results are robust to changes in model assumptions, and to answer questions that are better addressed using non-ODE models. The last part of this chapter suggests a few specific project ideas (Sect. 4) and encourages undergraduate researchers to share their results through presentations and publications (Sect. 5).

Electronic Supplementary Material The online version of this chapter (https://doi.org/10.1007/978-3-030-33645-5_1) contains supplementary material, which is available to authorized users.

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Notes

  1. 1.

    This function is often referred to as the right hand side , abbreviated RHS, of the ODE.

  2. 2.

    It is worth mentioning here that multiple different stochastic models can yield the same mean field ODE model.

  3. 3.

    The deterministic equation derived in this first step is called a mean field model because, for a given point X(t) in state space, we take the distribution of possible steps the system might take on the next time step, and define a new model by replacing that random quantity with its mean—in this case, the mean of the binomial random variable U(n = X(t), p = rΔt). One could visualize this on a grid of state space values with vectors pointing from x(t) to x(t + Δt), forming a vector field.

  4. 4.

    For each point in state space, one can think of the derivative \(\frac {d\mathbf {x}}{dt}\) as a vector pointing in a direction that is tangent to the trajectory passing through that point.

  5. 5.

    See upper level dynamical systems texts and online resources for the Taylor expansion of multivariate functions.

  6. 6.

    This definition is used instead of the more common \(p(\lambda )=\det (\mathbf {J}-\lambda \mathbf {I})\) since it ensures that the roots are the eigenvalues of J and that the polynomial is monic, i.e., has a leading coefficient of 1 on the λ n term.

  7. 7.

    Here we give the correct n = 5 case, which is missing a few terms in [4].

  8. 8.

    These basic reproduction numbers are best derived from stability criteria, which often require some rearrangement using a similar interpretation to arrive at the proper form of \(\mathcal {R}_0\).

  9. 9.

    For readers unfamiliar with topological equivalence, this essentially means that the gross qualitative features of solutions are the same across equivalent parameter sets, e.g., the number of equilibrium points and their stability will be the same, even though their exact numerical values may change. See [160, pg. 156], or more advanced treatments of topological equivalence in the texts mentioned above.

  10. 10.

    See the definition of an invariant set, and different types of attractors, in texts like [73, 104].

  11. 11.

    Here, integration by parts will yield an equation for the solution curves x(t), but we proceed assuming no such curves are available as this is typically the case in practice.

  12. 12.

    Here we are glossing over the formal details from singular perturbation theory [103] for deriving equations that approximate the fast- and slow-timescale dynamics of this model.

  13. 13.

    To view the documentation, load the package deSolve with the command library( deSolve) then type ?ode into the R console. See the Appendix for additional resources to get started using R.

  14. 14.

    Here the notation log( x) is used for the natural log function, instead of ln( x) , following the convention used in most modern scientific programming languages. Likewise, exp( x) =e x.

  15. 15.

    For related resources in R, see the packages such as CollocInfer [87], deBInfer [19], or browse the relevant CRAN Task Views (https://cran.r-project.org/web/views/).

  16. 16.

    Here the subscript 0 is used to distinguish the true parameters values μ 0 and σ 0 from the candidate values μ and σ that one might plug in to the likelihood function.

  17. 17.

    Sufficient criteria for the existence of one or more MLEs are that the parameter space is compact and the log likelihood surface is continuous. In practice, non-unique MLEs are more commonly the problem, especially when working with nonlinear ODE models.

  18. 18.

    It is worth pointing out that this model does not have a unique best-fit parameterization, unless certain parameters are held constant (i.e., are already known), even when estimating parameters from ideal data (e.g., a large sample with little or no noise)! This problem and the ways of resolving the issue are detailed in the latter half of this section.

  19. 19.

    See dnorm in R.

  20. 20.

    The performance can be improved somewhat, e.g., by setting the flag kkt=FALSE to avoid unnecessary computations or using parscale when parameter values vary by multiple orders of magnitude. See the documentation for optim and optimx for details.

  21. 21.

    Convergence code 0 indicates no errors, and convergence code 10 indicates degeneracy of the Nelder–Mead simplex. The code 10 often occurs when the model is not identifiable and reaches a “flat spot” in the objective surface.

  22. 22.

    Since the time to the next event under multiple different Poisson processes—one for each event type—is the minimum of the corresponding exponentially distributed event times, it is itself exponentially distributed with a rate that is the sum of the individual rates.

  23. 23.

    Additional intuition for this algorithm can be found in the section in [92] regarding “competing clocks” in a Poisson process framework.

  24. 24.

    The dwell time, or passage time, is the duration of time an individual spends in a given state. As discussed in Sect. 2.1, ODEs with linear loss rate terms implicitly assume that—in the context of Sect. 3.3.1—the times individuals spend in that state are exponentially distributed.

  25. 25.

    For simplicity, we here assume the initial cohort enters state I at t = 0 and thus follows the same dwell time distribution G.

  26. 26.

    Erlang distributions are gamma distributions with integer-valued shape parameters. Compared to exponential distributions, the Erlang density function is more hump shaped and the variance can be made arbitrarily small, as is sometimes desired in applications.

  27. 27.

    Readers familiar with Poisson processes may recall that homogeneous Poisson processes have inter-event times that are exponentially distributed, and the time to the kth event under a homogeneous Poisson process with rate r is Erlang distributed with rate r and shape k.

  28. 28.

    The coefficient of variation is the standard deviation divided by the mean.

  29. 29.

    More generally, if there are multiple lag values τ i, i = 1, …, k, the right hand side is of the form f(t, x(t), x(t − τ 1), …, x(t − τ k), θ).

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Acknowledgements

My outlook on undergraduate research at the interface of biology, mathematics, and statistics has been shaped by many influential mentors and peers that I was lucky to have known as an undergraduate at the University of Southern Colorado (now Colorado State University-Pueblo), at the Mathematical and Theoretical Biology Summer Institute (MTBI), as a graduate student in the Center for Applied Mathematics at Cornell University, and as a postdoctoral researcher in the Aquatic Ecology Laboratory and the Mathematical Biosciences Institute at The Ohio State University. It has truly been a privilege to know and learn from such an outstanding collection of people. I thank the students in my courses for exposing me to a broad array of project topics that have further influenced my outlook on undergraduate research across the sciences. I thank my colleague and wife Dr. Deena Schmidt; my colleagues Michael Cortez, Marisa Eisenberg, Colin Grudzien, and Andrey Sarantsev; my students Amy Robards, Jace Gilbert, Adam Kirosingh, Narae Wadsworth, and Catalina Medina; and reviewers Andrew Brouwer and Kathryn Montovan for many helpful comments, criticisms, and suggestions that ultimately improved this chapter. Finally, I thank my children Alex (age 7), Ellie (age 3), and Nat (age 3) for their unyielding companionship and boundless energy, which were instrumental to my preparing this chapter over such an extended period of time.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Nevada, Reno (UNR), Reno, NV, USA

    Paul J. Hurtado

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Editors and Affiliations

  1. Department of Mathematics, University of Portland, Portland, OR, USA

    Hannah Callender Highlander

  2. Mathematics & Statistics, Valparaiso University, Valparaiso, IN, USA

    Alex Capaldi

  3. Bates College, Lewiston, ME, USA

    Carrie Diaz Eaton

Appendix

Appendix

1.1 6 Getting Started Writing in LATE X and Programming in R

Installing the free software LATE X and R should be straightforward, but here are some installation tips for Microsoft Windows and Mac OS X users (Linux users can find similar installation instructions using the resources mentioned below). Readers are encouraged to ask other students and faculty at their institution about additional resources.

There are two pieces of software each, for LATE X and R, that should be installed: the basic LATE X and R software, and an enhanced user interface that facilitates learning for new users and helps established users with their day-to-day workflow (e.g., helpful menus, code autocompletion and highlighting, custom keyboard shortcuts, advanced document preparation capabilities, etc.). Educators will appreciate that both TeXstudio and R Studio have a consistent user interface across operating systems, making them ideal for group or classroom environments where students may be running a mix of operating systems.

If installing both LATE X and R (recommended), install the base software first (in either order) before installing TeXstudio and/or R Studio (in either order). More detailed instructions and resources are provided below.

1.2 6 Installing and Using LATE X

There are different implementations of LATE X available: MiKTeX is a popular Microsoft Windows option (http://miktex.org/), and TeX Live a popular Mac OS X option. TeX Live comes as part of a full 2 gigabyte installation called MacTeX (www.tug.org/mactex/; which includes the popular editors TeXstudio and TeXShop) or can be installed through a smaller 110 megabyte bundle BasicTeX (www.tug.org/mactex/morepackages.html). Configure LATE X to install packages “on the fly” without prompting you for permission. This can be done during (preferred) or after installation. Also download and install Ghostscript (www.tug.org/mactex/morepackages.html).

Next, install the TeXstudio editor (www.texstudio.org/), preferably after R is installed. Various settings can be changed after installation, including color themes, and configuring TeXstudio to compile a type of LATE X document that includes blocks of R code known as a Sweave or knitr document (use knitr).

For additional LATE X resources, see the author’s website (www.pauljhurtado.com/R/), the LaTeX wikibook [166], the AMS Short Math Guide for LATE X [42], and references and resources listed therein.

1.3 6 Installing and Using R

Download R from www.r-project.org/ and use the default installation process. Once R is installed (and, preferably once LATE X is installed), install R Studio from www.rstudio.com. By installing R Studio after LATE X, you will be able to create multiple document types to generate PDFs, including R Markdown documents and knitr documents. Helpful online resources include the “cheat sheets” on the R Studio website, introductory courses by DataCamp (www.datacamp.com) and Software Carpentry (www.software-carpentry.org), [157] for a gentle introduction to R and some applications to population modeling, and R resources on the author’s website (www.pauljhurtado.com/R/).

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Hurtado, P.J. (2020). Building New Models: Rethinking and Revising ODE Model Assumptions. In: Callender Highlander, H., Capaldi, A., Diaz Eaton, C. (eds) An Introduction to Undergraduate Research in Computational and Mathematical Biology. Foundations for Undergraduate Research in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-33645-5_1

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