Construct ODE (Ordinary Differential Equation) Models .
Simple epidemic models Construct ODE (Ordinary DifferentialEquation) models Relationship between the diagram and theequations Alter models to include other factors.
SimpleepidemicsSIS modelSIR modelDiagramModelDiagramModelSolve quationsTime isimplicitSolutionover timeEquilibria(ODEs 0)Stability ofequilibriaSIR with vaccinationDiagramModelSI with treatmentSIR with mutationLabLong term behaviourwith and withouttreatmentExploring parameters:Less infectious versionDiagramModel
Ordinary Differential Equations(ODEs) ODEs deal with populations, notindividuals We assume the population is well-mixed We keep track of the inflow and theoutflow.ODEs Ordinary Differential Equations
SIS epidemic Susceptibleà Infectedà Susceptible You get sick, then recover, but withoutimmunity E.g. the common cold.
Diagram Susceptibles become infected at rate a Infecteds recover at rate b.
SIS equations Becoming infected depends on contactbetween Susceptibles and Infecteds (aSI) Recovery is at a constant rate,proportional to number of Infecteds (b).a infection rateb recovery rate
Total population is constant Add equationstogether N S I (totalpopulation) dN/dt 0 à N is aconstant.S' bI-aSII' aSI-bIS SusceptibleI Infected
Solving directly Since N S I, this means S N-II' aSI-bI Let A aN-b be a constantS SusceptibleI Infecteda infection rateb recovery rate
Separate the variables Put the I’s on one side and the t’s on theother(including dI and dt)I Infecteda infection rateA aN-b (constant)N total pop.b recovery rate
Time series solution Rearrange using partial fractions Integrate Use initial condition I(0) I0S SusceptibleI Infecteda infection rateN total pop.b recovery rate(SeeEpidemicNotes)
b 0.1b recovery ratea infection rateN populationb 0.7(a 0.2 and N 3x106)
Phase portraits Since N S I,I -S N This is a straightline in I and S Time is implicit.S SusceptibleI InfectedN total pop.
Equilibrium points Equilibria occur when derivatives arezero:S SusceptibleI Infecteda infection rateb recovery rateS' bI-aSII' aSI-bI
Two equilibria Thus our equilibrium points are The latter always exists, the former is onlybiologically reasonable if p N.S SusceptibleN total pop.b recovery rateI Infectedp b/aa infection rate
N total pop.b recovery ratep b/aa infection rate
StabilityS' a(p-S)II' a(S-p)I S p à S' 0, I' 0 S p à S' 0, I' 0S SusceptibleN total pop.b recovery rateI Infectedp b/aa infection rate
N total pop.b recovery ratep b/aa infection rate
Stability implications When p b/a N, the recovery rate is high,so infecteds recover quickly and thepopulation moves to a population ofsusceptibles When p b/a N, the infection rate is highand the infection stabilises at an endemicequilibrium.N total pop.b recovery ratea infection rate
SimpleepidemicsSIS modelSIR modelDiagramModelDiagramModelSolve quationsTime isimplicitSolutionover timeEquilibria(ODEs 0)Stability ofequilibriaSIR with vaccinationDiagramModelSI with treatmentSIR with mutationLabLong term behaviourwith and withouttreatmentExploring parameters:Less infectious versionDiagramModel
SIR epidemics Susceptibleà Infectedà Removed Removed can be recovered, immune, ordead.a infection rateb recovery rate
SIR equations Becoming infected depends on contactbetween Susceptibles and Infecteds (aSI) Recovery is at a constant rate,proportional to number of Infecteds (b).a infection rateb recovery rate
SIR with vaccination A vaccine sends some Susceptiblesdirectly to the Recovered (immune) state N S I R.S SusceptibleR Recoveredb recovery rateI Infecteda infection ratec vaccination rate
Vaccination equations Vaccination is assumed to be a fixednumber of shots per time period (c).S SusceptibleI InfectedR Recovereda infection rateb recovery ratec vaccination rate
SIR with mutation If the virus mutates, Recovereds lose theirimmunity.S SusceptibleR Recoveredb recovery rateI Infecteda infection ratee mutation rate
Mutation equations A time-delay T allows a ‘grace period’before people are susceptible again They become susceptible at a rate (e)depending on their status at time t-T.S SusceptibleI InfectedR Recovereda infection rateb recovery rate
Delay Differential Equations These are called delay-differentialequations They are harder to analyse than ordinarydifferential equations, but are often morerealistic.
Using math to solve real ith aticalconclusion
Summary From simple assumptions, we can makemodels that might be simple, or might becomplicated Mathematical modelling is like mapmaking We need to decide which factors areimportant and which we can safely ignore“All models are wrong. but some are useful”- George Box.
SimpleepidemicsSIS modelSIR modelDiagramModelDiagramModelSolve quationsTime isimplicitSolutionover timeEquilibria(ODEs 0)Stability ofequilibriaSIR with vaccinationDiagramModelSI with treatmentSIR with mutationLabLong term behaviourwith and withouttreatmentExploring parameters:Less infectious versionDiagramModel
Ordinary Differential Equations(ODEs) ODEs deal with populations, not individuals We assume the population is well-mixed We keep track of the inflow and the outflow. ODEs Ordinary Differential Equations. SIS epidemic Susceptible!Infected!Susceptible You get sick, then recover, but without
The linear ODE is called homogeneous if g(x) 0, nonhomogeneous, otherwise. If an ODE is not of the above form, we call it a non-linear ODE. 1.1 First-order linear ODE The general form of a rst-order linear ODE is y0 p(x)y g(x): The basic principle to solve a rst-order linear ODE is to make left hand side a derivative of an
equations to second-order linear ordinary differential equations. Moreover, Lie discovered that every second-order ordinary differential equation can be reduced to second-order linear ordinary differential equation with-out any conditions via contact transformation. Having mentioned some methods above, there are yet still other methods to solve .
Introduction to Advanced Numerical Differential Equation Solving in Mathematica Overview The Mathematica function NDSolve is a general numerical differential equation solver. It can handle a wide range of ordinary differential equations (ODEs) as well as some partial differential equations (PDEs). In a system of ordinary differential equations there can be any number of
I Definition:A differential equation is an equation that contains a function and one or more of its derivatives. If the function has only one independent variable, then it is an ordinary differential equation. Otherwise, it is a partial differential equation. I The following are examples of differential equations: (a) @2u @x2 @2u @y2 0 (b .
DIFFERENTIAL EQUATIONS FIRST ORDER DIFFERENTIAL EQUATIONS 1 DEFINITION A differential equation is an equation involving a differential coefficient i.e. In this syllabus, we will only learn the first order To solve differential equation , we integrate and find the equation y which
3 Ordinary Differential Equations K. Webb MAE 4020/5020 Differential equations can be categorized as either ordinary or partialdifferential equations Ordinarydifferential equations (ODE's) - functions of a single independent variable Partial differential equations (PDE's) - functions of two or more independent variables
Keywords Second-order ordinary linear differential equations, iterative solutions, Green functions, computer algebra systems 1. Introduction There is a group of second-order linear ordinary differential equations (ODE) that play a prominent role throughout the realm of Mathematical Physics [1], [4]. Hermite's equation y 2xy y 0 (1)
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