Problem 2
This problem considers a cardiovascular model due to Ottesen
[16] involving the arterial pressure, P_{a}(t) = y_{1}(t), the
venous pressure, P_{v}(t) = y_{2}(t), and the heart rate, H(t) = y_{3}(t). Ottesen studies conditions under which the delay causes
qualitative differences in the solution and in particular,
oscillations in P_{a}(t). Delays t = 1.0, 1.4, 3.9, 5.0, 7.5,10 are considered in [16]. There are a number of
parameters in the model, so you might wish simply to declare them
as global values rather than pass them through dde23 to
the function for evaluating the equations, or even to hard code
them. Plot y_{1}(t) = P_{a}(t) for several values of t. You
should find that the solutions obtained for different values of t differ dramatically. Solve on [0,350] the equations


 
1 c_{a} R

y_{1}(t) + 
1 c_{a} R

y_{2}(t) + 
1 c_{a}

V_{str} y_{3}(t) 
 


1 c_{v} R

y_{1}(t)  
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1 c_{v} R

+ 
1 c_{v} r

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ø

y_{2}(t) 
 



where



1 1 + ( y_{1}(tt)/a_{s} )^{bs}


 


1 1 + ( a_{p}/y_{1}(t) )^{bp}


 


a_{H} T_{s} 1 + g_{H} T_{p}

 b_{H} T_{p} . 


For t £ 0, the solution has the constant value


 

 


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è


1 R V_{str}

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ø


æ ç
è


1 1 + r/R

ö ÷
ø

P_{0} 


As in [16], use c_{a} = 1.55, c_{v} = 519, R = 1.05,r = 0.068, V_{str} = 67.9, a_{0} = a_{s} = a_{p} = 93,a_{H} = 0.84, b_{0} = b_{s} = b_{p} = 7, b_{H} = 1.17,g_{H} = 0, P_{0} = 93 . The following figures for t = 1
and t = 7.5 show qualitatively different solutions.
One of the figures of [16] shows the solution components
when the peripheral pressure R is reduced exponentially from its
value of 1.05 to 0.84 beginning at t = 600. For this computation
the delay was 4 and the interval [0,1000]. You can easily
modify the previous program to solve this problem. All you have to
do is inform the solver of the loworder discontinuity at a known
time by setting the value of the 'Jumps' option to 600,
modify the function for evaluating the DDEs to include
if t <= 600
R = 1.05;
else
R = 0.21 * exp(600t) + 0.84;
end
and use the specified delay and interval. All the solution
components are of interest. The figure shows the sharp change in
the heart rate due to the change in R at t = 600.
References
 [16]

J.T. Ottesen,
Modelling of the BaroflexFeedback Mechanism With TimeDelay,
J. Math. Biol., 36 (1997), 4163.