The values of radii r i are given by r 1 4 r 2 5 2

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The values of radii r i are given by r 1 = 4 , r 2 = 5 / 2 and r 3 = 5. From Corollary 3, it follows that if a < - 4 , b < - 5 / 2, and c < - 5, then the eigenvalues are negatives or have negative real parts. J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 26 / 41
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This example illustrate that the conditions in Corollary 3 are sufficient, but not necessary. Example 4.16 Let the matrix A = - 2 1 0 1 - 2 1 0 1 - 2 . For this matrix r 1 = 1 = r 3 and r 2 = 2 but a 22 = - r 2 . However, the eigenvalues of A are negatives, λ 1 , 2 , 3 = - 2 , - 2 ± 2. J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 27 / 41
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Outline 1. Phase Plane Analysis 2. Eigenvalues. Real Eigenvalues Complex Eigenvalues Examples 3. Gershgorin’s Theorem 4. An Example: Pharmacokinetics Model 5. References J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 28 / 41
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An Example: Pharmacokinetics Model A model for the ingestion of a drug into the body is constructed ([Yeargers, E. K., Shonkwiler, R.W. & Herod J.V., 1996]). The drug is taken orally and is delivery to the gastrointestinal ( GI ) tract Figure 5. The drug, then, moves into the blood stream, without delay, at a rate proportional to its concentration in the GI tract and independent of its concentration in the blood. The drug is metabolized and cleared from the blood at a rate proportional to its concentration there. The model is based on the two compartments, GI tract and blood. Let x ( t ) denote the concentration of the drug in the GI tract and y ( t ) the concentration in the blood. In addition, let d ( t ) denote the dosage. Figure 6 represents the two-compartmental model. J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 29 / 41
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Figure 5: Upper and Lower human gastrointestinal tract. J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 30 / 41
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d(t) GI tract x a b blood y Figure 6: Compartmental diagram of drug concentration in the GI tract and blood. The two compartments can be modeled as a system of linear, nonhomogeneous differential equations: dx dt = - ax + d ( t ) dy dt = ax - by, a, b > 0 , a 6 = b. J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 31 / 41
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Because this system is linear, we know that the general solution to this model is X ( t ) = a At X 0 + e At Z t 0 e - As G ( s ) ds, where G ( s ) = ( d ( s ) , 0) T and A = - a 0 a - b . Since A has two negative eigenvalues, - a and - b , lim t →∞ e At X 0 = 0 . The homogeneous solution represents a transient solution. J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 32 / 41
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Therefore, lim t →∞ X ( t ) = lim t →∞ e At Z t 0 e - As G ( s ) ds, where e At = e - at 0 a e - bt - e - at a - b e - bt . Another method to solve this system is to solve for x first, a first-order nonhomogeneous equation, then use x to solve for y . J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 33 / 41
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Suppose there is continuous release of a drug into the GI tract, that is, suppose d ( t ) is constant. Let d ( t ) = 1 and initially, x (0) = 0 = y (0). The solution to the system can be shown to satisfy x ( t ) = 1 a ( 1 - e - at ) y ( t ) = 1 b + e - at a - b - ae - bt b ( a - b ) .
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