MMChapter4_4p8

# Then lim t x t 1 a and lim t y t 1 b j d flores usd

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Then lim t →∞ x ( t ) = 1 a and lim t →∞ y ( t ) = 1 b . J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 34 / 41

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The half-life of the drug in particular compartments can be estimated. For example, in the GI tract, where dx/dt = - ax, x ( t ) = x 0 e - at , the half-life is the value t where x ( t ) = x 0 / 2. Thus, the half-life is t = ln (2) /a . Suppose time is measured in hours and half-life of a particular drug in the GI tract is 1 / 2 hours and in the blood it is 5 hours, then a = 2 ln (2) and b = ln (2) 5 . The solution for the pharmacokinetics model is graphed in Figure 7 J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 35 / 41
0 5 10 15 20 25 30 35 40 45 50 0 1 2 3 4 5 6 7 8 Time Drug Concentration Pharmacokinetics Model GI tract Blood Figure 7: Drug concentration in the GI tract and blood for the pharmacokinetics model J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 36 / 41

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Matlab code function Xp=gi(t,x); % % Pharmacokinetics model % Sec. 4.10 page 163 % Code for the figure % 4.6 page 165 % the parameters a=2*log(2); b=log(2)/5; % the system of ODE’s Xp=[-a*x(1)+1; a*x(1)-b*x(2)]; end J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 37 / 41
Matlab code function Xp=gi(t,x); % % Pharmacokinetics model % Sec. 4.10 page 163 % Code for the figure % 4.6 page 165 % the parameters a=2*log(2); b=log(2)/5; % the system of ODE’s Xp=[-a*x(1)+1; a*x(1)-b*x(2)]; end function MyRun() x0=[0 0]; tspan=[0,50]; [t,x]=ode45(@gi,tspan,x0); %graphing the solution. p=plot(t,x(:,1),’b’,t,x(:,2),’r-.’); xlabel(’Time’); ylabel(’Drug Concentration’); title(’Pharmacokinetics Model’); set(p,’LineWidth’,3); pLegnd=legend(’GI tract’,’Blood’); set(pLegnd,’Location’, ’NorthWest’) end J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 37 / 41

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Consider the case where drug injection is periodic. This is a reasonable situation because prescription drugs are often taken at specific intervals of time. Suppose a drug is prescribed as follows: d ( t ) = 2 , 0 t 1 / 2 , 0 , 1 / 2 < t < 6 , where d ( t + 6) = d ( t ) ([Yeargers, E. K., Shonkwiler, R.W. & Herod J.V., 1996]). The drug is taken orally every six hours and released into GI tract over a half-hour period. J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 38 / 41
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 39 / 41

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References I F. Brauer & J. A. Nohel Qualitative Theory of Ordinary Differential Equations. Reprint , Dover 1989. J. M. Cushing Differential Equations: An Applied Approach. Prentice Hall 2004. C. Moler & C. Van Loan Nineteen dubious ways to compute the exponential of a matrix. SIAM Review 20:801-836, 1978. C. Moler & C. Van Loan Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review 45:3-49, 2003. J. D. Flores ( USD ) Math-735: Math Modeling Spring 2013 40 / 41
References II B. Noble Applied Linear Algebra. Prentice Hall 1969.
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