Methods of Mathematical Modelling 1

what is the model commonly used to model epidemics and the explanation

SEIR model
Susceptible, Exposed, Infected, Recovered

what are the two approaches of modelling

• phenomenological (aka statistical, empirical, data-driven, black-box)

• mechanistic (aka white-box)

what is phenomenological modelling

fitting curves to existing data in order to make predictions

what is mechanistic modelling + pros and cons

• constructed using knowledge of the actual phenomena and the equations and variables have more meaning

• require more time, experience and effort but are often more accurate

how is the “state of the system” characterised

by knowing the values of some/all the variables

what is a deterministic model

ignores fluctuations and works as a sort of “average behaviour” model

what is a stochastic model

a model that accounts for fluctuations using probabilistic techniques

what are the three types of questions

forward, inverse and control

what is a forward question

given all information about the current state of a system and its variables, can you use this information to predict other properties/ how it will function

what is an inverse question

you’re missing information about a variable, can you use the information available to figure it out

what is a control question

how best to make a system perform a certain way/ optimise its performance

how to express the function u applied k times to y₀

u^[k](y₀)

definition of a stationary point y* for a difference equation

y*=u(y*)
when u is the function of the difference equation

what does a stationary point being stable/unstable mean in a difference equation

if values of y_k approach/diverge from y* when starting nearby

how to tell if a stationary point is stable/unstable in a difference equation

stable if |u’(y*)|<1
unstable if >1

define an n-periodic point

if y_n=p and y₀=p and y_k≠p for 0<k<n

what is the order of a difference equation

how many previous states of y_k appear in the equation

what is a solution to a difference equation

when we can express y_k in terms of k only, not previous states of y

what guess should you try to solve a second order difference equation

y_k=λ^k

general form of a differential equation

F(t, x(t), x’(t),…x^(k))=0

what are the independent and dependent variables in a differential equation

t= independent variable
x(t)= dependent variable

what is the order of a differential equation

the order of its highest derivative

what does autonomous mean for a differential eq

F doesn’t dependent explicitly on the independent variable ie t doesn’t appear by itself in the equation

when is a DE linear

none of the x⁽ⁱ⁾(t) are squared or anything

when is a DE homogenous

if F( all the derivatives)=0
ie the “constant term”=0
“constant term” refers to some function of t

what is an explicit solution

when the dependent variable is given as a function of the independent variable
eg x(t)=18e^3t

what is an implicit solution

if the solution satisfies an equation that relates to the independent and dependent variable, and contains no derivatives

what is a well-posed solution

it exists, is unique and is stable

general form of a first-order, autonomous equation

x’(t)=f( x(t) )

when is x* a stationary solution

when f(x*)=0, x(t)=x* is a stationary solution

what does it mean for a stationary solution to be stable

if nearby solutions remain close as t grows the solution is stable

when is a stationary solution stable

if f’(x*)<0 is it stable

how to account for an initial condition x₀=x(t₀) when solving a separable DE

when integrating, integrate with limits t₀ to t and x(t₀) to x(t)

what is the integrating factor

e^-R(t) where R(t)= integral of t₀ to t of the “function coefficient” of x(t)

what does it mean for solutions to be linearly independent

one is not a multiple of the other

if a DE has two linearly independent solutions x₁ and x₂, how can you derive more solutions

all solutions are of the form x = l₁x₁+l₂x₂

what are the three principles of dimensional analysis

• all variables have unique, known dimensions

• dimensional homogeneity

• dimensional completeness

what is dimensional homogeneity

all terms summed in a mathematical statement have the same dimensions

what is dimensional completeness

you can combine independent variables to produce a variable of the same dimensions as the dependent variable

how to nondimensionalise the equation
d=u(i₁,i₂,…i_m)

• [d] = [i₁]^a[i₂]^b….

• write d, i₁, i_2… in terms of the fundamental dimensions

• solve for a,b,…(normally have to write in terms of at least one letter as have more variables than equations)

• pull out some dimensions to match [d] then the rest are nondimensional

• put them in a function u

what are the two principles of nondimensionalising differential equations

• select scales s.t as many parameters as possible are normalised to 1

• select scales s.t all non-dimensional parameters in the non-dimensional problem are “small” and don’t become infinity when certain dimensional parameters become big or small

what are the dimensions of the integrating factor

dimensionless