Mathematical Models in Biology and Medicine
Instructor: Eberhard O. Voit, Ph.D.
The course introduces the student to a representative set of models that elucidate the
nature of biological and medical phenomena. Upon completion of the course, the
student will understand the rationale behind the models, explore their potential and
limits, and execute standard analyses and simulations. Examples that are discussed
include classical models, such as biochemical system models and parasite-host
models, as well as modern models taken from the current literature.
Text:
Leah Edelstein-Keshet: Mathematical Models in Biology,
Birkhäuser Series (McGraw Hill), NY 1987
Additional Reading: E. O. Voit: Computational Analysis of Biochemical
Systems, Cambridge University Press, 2000
Grading:
Three exams @ 25% each
Homework @ 25% (Emphasis on effort rather than correctness)
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Detailed Syllabus (subject to change)
Introduction
What is a biomedical system?
What is systems analysis?
What is a model?
Correlative versus explanatory models
Discrete versus continuous systems
Deterministic versus stochastic systems
Stochastic phenomena with deterministic features (gas laws)
Deterministic phenomena with stochastic features (chaos)
Paradigm shift from reductionism to integrated systems approach
Simple phenomena that cannot be analyzed intuitively
Recursive Models of Growth
Ch. 1
N(t+1) = a N(t)
example: bacterial population growth
general behavior
comparison with exponential growth
effect of a (a<-1, a=-1, -1<a<0, a=0, 0<a<1, a=1, a>1)
questions of non-integer N(t)
inclusion of death
Recursion with memory
Fibonacci sequence as a model for population growth;
golden section
definition of growth rate
questions of steady states and convergence
propagation of annual plants
Ch. 1
conversion into two-variable recursion without memory
matrix representation and analysis
stability and extinction
Nonlinear recursions
logistic map
general features and analyses
graphical evaluation
transition from simple steady state to periodicity to chaos
Ch. 2
Populations with age structure
matrix representation
Leslie models
effect of delayed pregnancy on population dynamics
Hahn model of cell population growth
Voit and Dick model for cell populations with distributed
cycle durations; semi-stochastic; phase-specific cancer treatment
Markov Models
Motivation, relationship to conditional probabilities
Definitions
Two-state models
Random walk models
Matrix notation for transition probabilities
n-step probabilities
Chapman-Kolmogorov equations
Limiting probabilities
solution of systems of linear equations (if necessary)
substitution, matrix manpulations, determinant methods
Conversion of 2nd-order Markov chains into simple chains
>>> more in courses like Stochastic Processes and Time Series Analysis
General Nature of Continuous Biological Systems
Simple models of complex systems (e.g., growth)
Open systems, closed systems; system versus environment
General mathematical system description
types of variables, parameters, constants
processes and interactions
introduction of differential equations, rationale
formulation of general compartment models
General Methods of Analysis
Ch. 4
Computer simulation
Linearization
Nonlinear approximation
Simplifications
temporal
time scales
constancy at non-dominant time scales
spatial
ordinary versus partial differential equations
functional
validation of approximation
acceptance of inaccuracies
extrapolation
Types of approximation
Ch. 2 Appendix
Taylor’s theorem
linearization
one and more dimensions
polynomial approximation
value and problems of high-order polynomials
power-law approximation
one and more dimensions
approximation versus regression
concept of operating point
piecewise approximation
Linear system description
general discussion
superposition and reduction to submodels
Canonical nonlinear system representation
justification and validation of power-law models
generalized mass action systems
S-systems
Lotka-Volterra systems
model design from subject area information
flow diagrams
identification of variable types
translation into equations
simulation
steady states
analytical solution, if possible
numerical solution
local stability, based on linearization (if necessary)
concept of structural stability
>>> more in Biochemical Systems Analysis (Fall)
Kinetic Models
Elemental chemical reactions
formulation with methods of previous section
rate constants and kinetic orders
exponential decay, half-time
Enzyme catalyzed reactions
conceptual model
mathematical formulation with methods of previous section
quasi steady-state assumption
derivation of Michaelis-Menten rate law
interpretation of parameters
parameter estimation
nonlinear versus linearization-based
modulated reactions
types of inhibition
competitive inhibition
conceptual model
mathematical formulation
Ch. 4
Ch. 4
Ch. 4
Ch. 7
derivation of simplified rate laws
parameter estimation
Hill rate law
>>> more in Biochemical Systems Analysis (Fall)
Growth Processes Revisited
Ch. 6
Growth as a feature of very complex systems
temporally dominant processes
approximation
exponential growth with time- and size-dependent growth rate
logistic growth
Weibull, Gompertz, and Bertalanffy functions
classification of growth functions
growth functions resulting from controlled cell cycle models
Population Dynamics
Chs. 5,6
Review of relevant methods discussed so far
Standard approaches
growth models + competition + predation +
immigration, emigration, hunting, fishing
populations composed of subpopulations (age, health status, ...)
Types of questions that can be addressed
overall dynamics
survival, extinction, coexistence
stability
oscillations
responses to changes
temporary (perturbations)
persistent (environmental conditions, mutations)
control of populations
endangered species
pests
Methods
Ch. 5
qualitative analysis
algebraic analysis
numerical analysis
simulation
Limitations
Simple competition models
two species with mutual inhibition
qualitative analysis
special cases (parameter values = 0, symmetry)
phase plane
distinction of cases
analysis of nullclines, steady states, stability
interpretation
Lotka-Volterra model of predation
limitation to centers
extensions, allowing damped oscillations
Predator-prey systems with nonlinear nullclines
Whale-krill system with fishing
Chemostat models
Ch. 4
Models of infectious diseases
Ch. 6
SIR model (susceptibles, infected, removed)
concepts, definitions
standard analyses
dependence on initial values
identification of invariants and decoupling of equations
SIR model with recovery
SIR models with asymptomatics
eradication of epidemics
Comparison of SIR models and corresponding Markov models
Diffusion and Dispersion
Partial differential equations
Random walk revisited
Diffusion equation
Dispersion
Plume models
Ch. 9