Unformatted text preview: ODE of vectors Example: Projectile motion with air drag Systems of coupled ODEs Example: Spread of an epidemic Midterm project due date extended until next Tuesday (3/30) No HW this week – finish your projects! Reading for Differential Equations in Appendix B. Often in science the quantities we are interested in are vectors which require a magnitude and direction. Typically we express a vector by it’s individual components: = (vx , vy , vz ) v
Computationally, our 1D problems now become 3 1-D problems. We just perform the calculations on each of the components individually. dvx 1 = Fdx (vx ) dt m dvy 1 = Fdy (vy ) − g dt m 1 Fd = − Cd ρA| | vv 2 Restrict to 2-D, so only need x and y components of position (r) and velocity (v). Force of gravity is only in y direction, drag force is always opposite to velocity (x and y components constantly change). drx = vx dt dry = vy dt In many ‘systems’ we encounter in science, the evolution of a variable depends on the values of other variables, … which in turn depend on the value of the original variable. We say ODEs which describe the evolution of these variables are coupled. Examples: Predator – prey models (Lotka-Volterra) Spread of infectious diseases Chemical reactions (Belousov-Zhabotinski) Climate models (Lorentz) Simple model of how an infectious disease (like the flu) might spread in a ‘closed’ population. Two groups of people: number of susceptibles (S) who can catch the disease and infectives (I) which have the disease. dS = −rSI dt
dI = rSI − aI dt The parameters r and a are characteristic of the disease (i.e. how infectious the disease is and rate of recovery respectively) Think about the logic behind these equations! Also need initial (t=0) conditions: S (0) = S0 I (0) = I0 Need to discretize using forward Euler method. For time = 0 to T Sk+1 − Sk in n time steps: = −rSk Ik T ∆t = n For a time step k: ∆t Ik+1 − Ik = rSk Ik − aIk ∆t So to compute the S and I values for the next time step (k+1): Ik+1 = Ik + ∆t(rSk Ik − aIk ) Sk+1 = Sk − ∆trSk Ik Data was collected during a flu outbreak in a British boarding school (closed population) in 1978. This simple ODE system was used to successfully model the data by adjusting r and a values (non-linear fitting using a model rather than an analytic function).
Data from Mathematical Biology by J.D. Murray – excellent book! An important question to ask is “Are there values for the variables for which the system does NOT evolve?” These values define steady-state solutions and define “fixed points” for the equations. Remember, variables are not changing if their derivatives are 0. There is one fixed point for the epidemic model: I = 0 since I is in every term on the RHS of all equations. Let’s take a look at a model which has a more interesting fixed point. System of ODEs describing the change in population of predators (y) and prey (x). dx = x(a − cy − bx) dt
dy = −y (d − ex) dt Interpretation of parameters: a - prey birth rate, b – prey death rate (other than being eaten), c – rate of consumption, d – predator death rate, e – predator birth rate Find fixed points by setting both derivatives to 0 and solving for x and y: a bd d There are others, y0 = − x0 = see them? c ce e ...
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