# Maple worksheet with Lectures, Part 17 # # follows section 7.11 of Yeargers # with modifications # # define the differentials just for illustration

diff(x(t),t)=-a*x+D(t);

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diff(y(t),t)=a*x-b*y;

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# define the differentials instead using the # "standard" compartmental syntax (again, # just for illustration)

diff(x1(t),t)=-a21*x1(t)+D(t);

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diff(x2(t),t)=a21*x1(t)-a02*x2(t);

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# define and plot the dosing function # Note: the definition of D is # different from the text # due to changes in Maple since the # text was written # # we use a step function, signum, that returns # +1 if the argument is positive and -1 if # the sign is negative {signum(x)=x/abs(x)} # we use an "environment variable" to tell # the function what value to take when # its argument is zero (otherwise it's not # defined)

_Envsignum0:=0:

Dose1:=t->sum((signum(t-n*6)-signum(t-(n*6+1/2)))*2,n=0..10);

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plot(Dose1(t),t=0..20);

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# now that we have the dosing function, we can # numerically integrate and plot

with(plots): with(DEtools):

Warning, the name changecoords has been redefined

# define the rate constants: x->y has a # half-life of 0.5 hours, y->environment has # a half-life of 5 hours. rate constants and # half-lives are related by the equation # rate = ln(2)/halflife

a:=ln(2)/(1/2);b:=ln(2)/5;

NiM+SSJhRzYiLCQtSSNsbkc2JEkqcHJvdGVjdGVkR0YqSShfc3lzbGliR0YlNiMiIiNGLQ==

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# use a "handle" to store the result of plots # of x vs. t and y vs. t

J:=DEplot({diff(x(t),t)=Dose1(t)-a*x(t),diff(y(t),t)=a*x(t)-b*y(t)},{x(t),y(t)},t=0..50,{[0,0,0]},stepsize=0.5,scene=[t,x(t)],linecolor=RED):

K:=DEplot({diff(x(t),t)=Dose1(t)-a*x(t),diff(y(t),t)=a*x(t)-b*y(t)},{x(t),y(t)},t=0..50,{[0,0,0]},stepsize=0.5,scene=[t,y(t)],linecolor=BLACK):

display({J,K});

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

# do this again using a variable to hold the # set of differentials (more readable=better # style)

deq:={diff(x(t),t)=Dose1(t)-a*x(t),diff(y(t),t)=a*x(t)-b*y(t)}:

J:=DEplot(deq,{x(t),y(t)},t=0..50,{[0,0,0]},stepsize=0.5,scene=[t,x(t)],linecolor=RED):

K:=DEplot(deq,{x(t),y(t)},t=0..50,{[0,0,0]},stepsize=0.5,scene=[t,y(t)],linecolor=BLACK):

display({J,K});

# now generate the phase plane to illustrate # the limit cycle

phaseportrait(deq,[x(t),y(t)],t=0..50,{[0,0,0]},stepsize=0.5);

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