Supplementary Materials for 'Mathematical Modeling for Epidemiology and Ecology'

My new book, Mathematical Modeling for Epidemiology and Ecology, was published by Springer-Verlag in April 2023. Supplementary materials include programs and data files.

Other BioMath Educational Resources

My BioMath educational resources include books, book chapters, pedagogical papers, and teaching modules.

Resources in Mathematics Education for Biology

• Teaching Modules: Educational Modules for Mathematical Epidemiology and Educational Modules for COVID-19
• Ledder (2021), Qualitative Analysis of a Resource Management Model and Its Application to the Past and Future of Endangered Whale Populations, CODEE 14, DOI: 10.5642/codee.202114.01.03. This paper grew out of a talk I gave at the 2020 Joint Mathematics Meetings in Denver.
• C. Diaz-Eaton et al (2019), A "Rule-of-Five" Framework for Models and Modeling to Unify Mathematicians and Biologists and Improve Student Learning, PRIMUS 29, 799-829, DOI: 10.1080/10511970.2018.1489318. This paper grew out of a NimBios workshop on mathematics education in biology. It provides a pedagogical framework for mathematical modeling.
• G. Ledder and M. Homp (2021), Using a COVID-19 Model in Various Classroom Settings to Assess Effects of Interventions, PRIMUS, DOI: 10.1080/10511970.2020.1861143. Michelle Homp and I present a primer on mathematical modeling and mathematical epidemiology and discuss the teaching modules listed above.
• G. Ledder (2017), Empirical Modeling: Choosing Models and Fitting Them to Data, College Math. J. 47, 109-119, DOI: 10.4169/college.math.j.47.2.109. In this paper, I discuss linear least squares, the semilinear least squares method I introduced in my 2013 book (see below), and the Akaike Information Criterion for model selection.
• G.Ledder (2017), Scaling for Dynamical Systems in Biology, Bull Math Biol 79, 2747-2772. Asymptotic methods are ubiquitous in models for physical science, but not often used in biological science. This is unfortunate, as many biological models have features that lend themselves to asymptotic methods. The first step in asymptotic analysis is scaling, which is not easy in biology. In this paper, I present some of the basic principles I use in scaling, using my onchocerciasis model as an example.
• G.Ledder (2013), Mathematics for the Life Sciences: Calculus, Modeling, Probability, and Dynamical Systems, Springer. Available from Springer. Second edition will be finished soon.
• G.Ledder, J.Carpenter, T. Comar, ed., Undergraduate Mathematics for the Life Sciences: Models, Processes, & Directions. This book-length manuscript has been accepted for the MAA Notes series and will be published in the first half of 2013. Preface, annotated Table of Contents, and editor's introduction.
• G.Ledder (2008), An experimental approach to mathematical modeling in biology, PRIMUS 18, 119-138. I had the good fortune to be able to serve as guest editor for the January 2008 issue of the undergraduate mathematics pedagogy journal PRIMUS. A number of excellent papers can be found in that issue, as well as this paper of mine, which was slipped in unobtrusively by the guest editor.

Course Notes (for standard math courses)

These include stand-alone notes for topics in calculus, partial differential equations, asymptotics, and nonlinear optimization, and a full set of notes for queueing theory.

Multivariable Calculus

These notes were used to accompany the Hughes-Hallett calculus text, but they are written so as to be largely independent of the text, except of course for differences in notation.

• Quadric Surfaces describes the various quadric surfaces aligned along a coordinate axis.
• Derivatives summarizes the different types of derivatives for scalar functions of more than one variable.
• Differential explains the proper use of the notation of the differential and why it should not be used when doing linear approximation.
• Extrema summarizes the concepts and methods of local and global extrema for functions of two variables.
• Definite Integral Graphic summarizes the components of definite integrals in one, two, or three dimensions.
• Definite Integrals describes the various types of definite integrals, including accumulation over time and aggregation over various spatial domains, including lines, curves, plane regions, surfaces, and volumes.
• Iterated Integrals presents the method for evaluating iterated integrals and general instructions for setting them up.
• Triple Integral Variable Order briefly describes how to choose the most convenient order for setting up triple integrals, based on the characteristics of the region.
• Coordinate Systems is a set of sketches that show how to think of the three common coordinate systems. In particular, a sketch of the rz-plane with radial coordinate ρ and angle of declination φ; shows how to use polar coordinate transformations to connect spherical and cylindrical coordinates.
• Path Integrals presents the concept and evaluation for integrals of scalar functions on curves.
• Line Integrals presents the concept and evaluation for integrals of vector functions along curves.
• Flux Integrals presents the concept and evaluation for integrals of vector functions across surfaces.

Differential Equations

• Euler methods presents the Euler method and Improved Euler methods. This material is taken from my DE book.
• First Order Linear presents the variation of parameters method for solving first order linear equations. This method has some advantages over the usual integrating factor method: it is easy to remember and paves the way for variation of parameters for second-order equations.
• Linear DEs is a comprehensive guide to solving linear differential equations of all orders.
• Undetermined Coefficients presents the method for undetermined coefficients in terms of generalized exponential functions.
• Laplace Transform Overview describes the basic structure of the Laplace transform method.
• Partial Fraction Decomposition presents a complete method for partial fraction decomposition.
• First Translation Theorem explains in detail how to use the theorem that gives the Laplace transform of the product of a transformable function with an exponential function.
• Summary - First Translation Theorem briefly summarizes the previous document.
• Piecewise Continuous Functions introduces the Heaviside function and uses it to obtain single formulas for piecewise-defined functions.
• Second Translation Theorem presents the use of the theorem in two different forms for calculating Laplace transforms of functions with switches and inverting the associated transforms.

Partial Differential Equations

• Green's Function Primer introduces Green's functions using a boundary value problem for an ODE.
• Solving Linear PDEs on Finite Domains generalizes the method of separation of variables so as to include eigenfunction expansion for problems with nonhomogeneous differential equations.

Asymptotic Analysis and Modeling

• Asymptotics Intro introduces asymptotics with examples (8 pages).
• Dominant Balance Example Problem presents the method for using dominant balance arguments to find asymptotic approximations to solutions of nonlinear problems (2 pages).
• Singular Perturbation presents Van Dyke's method for obtaining matched asymptotic expansions for first-order problems, second-order initial value problems, and second-order boundary value problems (9 pages).
• Laplace's Method is a careful presentation of Laplace's method for obtaining asymptotic approximations to definite integrals. Stirling's formula is here, along with relatively simple problems (5 pages).
• Scaling is a study of scaling for a progressively complex set of physical problems, including a continuously stirred tank reactor, a projectile problem, and a model of a damped nonlinear spring (7 pages).
• Steady Flow Across a Flat Plate presents the derivation from first principles of the classic problem for steady flow across a flat plate, sometimes called the Blasius problem (5 pages).

Operations Research

• Queueing Theory Overview provides an overview of the problems and uses of queueing theory.
• Queueing Theory Notes is a self-contained chapter-length treatment of queueing theory, suitable for use as a stand-alone module in an operations research course or as an introduction to a queueing theory course.

Projects (for standard math courses)

These include projects for various applied math courses.

Projects

• Art Forgery: An investigation of the celebrated van Meegeran art forgery case. (requires differential equations)
• Continuously Stirred Tank Reactor: An investigation of a chemical mixing problem with an exothermic chemical reaction having a temperature-dependent rate constant. This is a classic problem in chemical engineering that I used as an Honors project for a very capable differential equations student. The project is divided into three phases of work.

Talks

I have given a large number of expository and pedagogical talks. I've also posted some recent research talks in mathematical biology.

Expository Talks

• Mathematical Modeling in Biology. (Given as a guest lecture for the CURE Workshop, June, 2021) This talk focused on general principles of modeling, using the SEIR epidemic model as an illustration.
• Introduction to Mathematical Models in Epidemiology. (Given as a guest lecture for Math in the City, January, 2021)
• The Past, Present, and Future of Endangered Whale Populations: Qualitative Analysis and Modeling. (Originally given at the Joint Math Meetings, January, 2020) (Revised in December 2020 to match CODEE paper)
• Mathematical Modeling in Population Dynamics. (Given at Sweet Briar College and University of Richmond, November, 2008)
• Population Growth in a Structured Population. (Given at Benedictine College, College of DuPage, and Marymount University, November, 2008)
• A Mathematical Model for a Self-Limiting Population. (Given at the Joint Math Meetings, January, 2005)
• A Simple Introduction to Integral Equations. (Given at Mathematics on the Northern Plains, Dordt College, April, 2004)
• A Math Problem for a Mission to Mars. (Given at Mathematics on the Northern Plains, Dordt College, April, 2004)

Pedagogical Talks

• A COVID-19 Model and its Matlab Implementation. (Given as an interactive presentation at the SIMIODE Workshop, February, 2021) This talk focused on the P3 module that can be found at UNL COVID-19 Teaching Module.
• Teaching Mathematical Epidemiology at a Variety of Levels Using Multiple Representation Theory. (Given at the SMB Annual Meeting, August, 2020)
• Mathematical Modeling Via Multiple Representations, with Carrie Diaz-Eaton. (Given at the SMB Annual Meeting, August, 2019)
• Dynamical Systems for Students with Minimal Calculus Background. (Given at the SMB Annual Meeting, July, 2015)
• Using Virtual Laboratories to Teach Mathematical Modeling. (Given at the SMB Annual Meeting, July, 2012) (The talk also used demonstrations of my BUGBOX virtual laboratory software.)
• Designing Math Courses: Pedagogical Issues. (Given at the Joint Math Meetings, January, 2009)
• A Terminal Post-Calculus-I Mathematics Course for Biology Students. (Given at the Society for Mathematical Biology annual meeting, Toronto, August, 2008)
• Teaching Biology in Mathematics Classes. (Given at the Nebraska-South Dakota MAA Sectional meeting, Wayne State College, April, 2008)
• An "Experimental" Course in Mathematical Ecology. RSTE Talk 1, RSTE Talk 2, Bugbox Maplet (Given at the Joint Math Meetings, January, 2007)
• Mathematical Modeling in Biology. (Given at the Math Biology workshop, U.S. Military Academy, April, 2006)
• Brief notes on undergraduate research. (Given as a UNL presentation, January, 2006)
• Assessment Using Online Management Systems. (Given at the Joint Math Meetings, January, 2006)

Research Talks

• A Model for COVID-19 with Limited Vaccination. (Given at the SMB Annual Meeting, June, 2021)
• The Local Control Theory of Plant Resource Allocation. (Given at the University of Ottawa Applied Mathematics Colloquium, October, 2020)
• A Consumer-Resource Model with Synchronized Reproduction. (Given at the ICMA Meeting by Richard Rebarber, September, 2019)
• A Mathematical Model for Onchocerciasis with Intermittent Treatment. (Given at the ICMA Meeting, October, 2017)
• A Carbon Economy Model for Tree Growth (Given at the Joint Mathematics Meetings, January, 2015)
• Using Scaling and Asymptotics to Simplify Dynamical Systems. (Given at the SMB Annual Meeting, June, 2013)
• An Optimization Model that Links Masting to Seed Herbivory. (Given at the SMB Annual Meeting, July, 2012)
• Predator-Prey Interaction in Structured Models. (Given at the Joint Math Meetings, January, 2008)

Research Papers

My research work has been in mathematical ecology, population dynamics, epidemiology, plant physiology, and plant life history theory.

Research Papers

• G.Ledder, R. Rebarber, T. Pendleton, A.N. Laubmeier, J. Weisbrod (2021), A discrete/continuous time resource competition model and its implications, J. Biol. Dyn. 15:sup1, S168-S189, DOI: 10.1080/17513758.2020.1862927. This paper examines a model in which two consumers compete for resources, with each of the consumers reproducing at discrete times while the resource reproduces continuously.
• G.Ledder, S.E. Russo, E.B. Muller, A. Peace, R.M. Nisbet (2020), Local control of resource allocation is sufficient to model optimal dynamics in syntrophic systems: a model for root:shoot allocation in plants, Theor. Ecol. 13, 481–501. DOI: 10.1007/s12080-020-00464-9. This paper develops a model of plant resource allocation between roots and shoots that is based on local control of resources, similar to what happens in obligate syntrophy. Local control produces results that are optimal in several senses.
• V. Couvreur, G.Ledder, S. Manzoni, D.A. Way, E.B. Muller, S.E. Russo (2018), Water transport through tall trees: A vertically explicit, analytical model of xylem hydraulic conductance in stems, Plant, Cell, Environ. 41, 1821–1839. DOI: 10.1007/s12080-020-00464-9. We offer an innovative model for stem hydraulics that allows many of the properties of stems to be functions of path length from the base of the tree.
• G.Ledder, D. Sylvester, R.R. Bouchat, J.A. Thiel (2018), Continuous and pulsed epidemiological models for onchocerciasis with implications for eradication strategy, Math. Biosci. Eng. 15, 841-862. DOI: 10.3934/mbe.2018038. We offer an innovative model for stem hydraulics that allows many of the properties of stems to be functions of path length from the base of the tree.
• G.Ledder (2017), Scaling for Dynamical Systems in Biology, Bull. Math. Biol. 79, 2747-2772. Asymptotic methods are ubiquitous in models for physical science, but not often used in biological science. This is unfortunate, as many biological models have features that lend themselves to asymptotic methods. The first step in asymptotic analysis is scaling, which is not easy in biology. In this paper, I present some of the basic principles I use in scaling, using my onchocerciasis model as an example.
• G. Ledder (2014), The basic dynamic energy budget model and some implications, Letters Biomath., 1, 221-233, DOI: 10.1080/23737867.2014.11414482. This paper presents a simple introduction to DEB models, using standard mathematical notation rather than the specialized system favored by most DEB practitioners, but somewhat unintelligible to the uninitiated.
• J.D. Logan, G. Ledder, W. Wolesensky (2009), Type II functional response for continuous, physiologically structured models, J. Theo. Biol., 259, 373-381, DOI: 10.1007/s00285-003-0263-1. This paper generalizes the Holling type II functional response model to more complicated settings.
• G. Ledder (2007), Forest defoliation scenarios, Math. Biosci. Eng., 4, 15-28. This paper represents the "final word" on the spruce budworm model created by Ludwig, Jones, and Holling and previously analyzed by Brauer and Castillo-Chavez. Using asymptotic analysis, I identify various types of long-term behavior and link them to regions of the parameter space.
• G. Ledder, J.D. Logan, A. Joern (2004), Dynamic energy budget models with size-dependent hazard rates, J. Math. Biol., 48, 605-622, DOI: 10.1007/s00285-003-0263-1. It is often assumed in DEB models that the death rate of organisms is simply a Poisson process; that is, that longevity is exponentially distributed. In this paper, we see that a consequence of this fact is that the optimal time for transitioning from growth to reproduction occurs when the population is still large; that is, it is optimal for a significant fraction of individuals to mature, but at a small size. This is not typical in nature, where most species have life histories in which a vanishingly small fraction of offspring survive to adulthood, where they have long careers in reproduction. The way to resolve this anomaly is to posit a hazard rate that is a sharply decreasing function of size rather than the constant implied by the exponential distribution.

BUGBOX Mathematical Modeling Simulations

BUGBOX software is designed for inquiry-based learning of mathematical modeling for population dynamics. The software creates a virtual laboratory space inhabited by virtual insects. Students conduct experiments and devise mathematical models to predict the experiment results.

BUGBOX Simulations

In BUGBOX-predator, students do experiments to find a relationship between prey density and consumption by one predator. The program is a virtual adaptation of the human simulation described in a classic 1959 paper by C. S. Holling. [Netlogo version posted on 2022/07/16.]

In BUGBOX-population, students observe the changes in the insect population to discover aspects of the insects' life cycle. They can construct a simple linear difference equation model, using their observations to estimate the parameter values. Four different scenarios give a sequence of increasing complexity. [Netlogo version posted on 2022/07/16.]

Everyone is welcome to use BUGBOX for their courses. I would just appreciate an email to let me know.

BUGBOX-predator was originally written in python 2, with extensive use of the pygame and pgu packages, and converted into an executable using py2exe. BUGBOX-population was originally written in python 3, with the built-in tkinter GUI package, and converted into a windows executable using cx_Freeze.

Netlogo has become the standard software for age_nt-based modeling such as is required for the BUGBOX software. There is extensive documentation on-line. People interested in writing agent-based models in Netlogo should also consult Agent-Based and Individual-Based Modeling, 2nd edition, by Steven F. Railsback and Volker Grimm.

Large Print macros for LaTeX

It is somewhat tricky to produce a large print version of a LaTeX document for visually impaired students. At this link, I have worked out a general scheme that allows me to quickly convert documents to large print.

I am now a self-funded postdoc (i.e., an emeritus professor) and have moved to Denver Colorado. I check email regularly and am still active in scholarship.

Feel free to use any of my materials for courses at other schools. I would appreciate being informed about which of my materials have been used where.