Chapter 3: Linear & Non-Linear Interaction Models

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Preston Lester
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1 Chapter 3: 155/226 Chapter develops the models above to examine models which involve interacting species or quantities. Models lead to simultaneous differential equations for coupled quantites due to the interaction. Firstly, deal with Linear models which differ from the chemostat in that they can be uncoupled to reduce the pair of first order O.D.E.s to a single second order one and hence be solved exactly. As in previous chapter discuss steady states for this solution for the example of Diabetes Detection. Move on to more realistic non-linear interaction models. These don t have an exact solution & demonstrate how stability concepts may be used with Phase-Plane models to show model behaviour under various conditions. 156/226

2 Models of the form dx = Ax +By +P dy = Cx +Dy +Q (4.1) for constant A,B,C,D are said to be linear (i.e. no product terms in x,y), first order, & if P,Q = 0, homogeneous. Such models are also known as compartmental models as model quantities can be thought of as being compartmentalised & are hence useful for modelling mixing problems. 157/226 : Diabetes Detection Example: Diabetes Detection Glucose, an end product of carbohydrate digestion, is converted into energy in the body s cells. Insulin, A hormone secreted by the pancreas, facilitates glucose absorption by cells other than those of the brain & nervous system. A delicate balance is maintained between glucose (G) & insulin (H) in the bloodstream: if H is too low, too little glucose is absorbed (& is normally excreted); if H is too high too much glucose is absorbed by organs other than the brain. end result in either case can be coma & even death. Diabetes Mellitus patients need regular injections of insulin supplements to compensate for lack of pancreatic insulin. 158/226

3 : Diabetes Detection cont d Present a simple model for G/ H interaction & use this to discuss a clinical test for detection of mild forms of diabetes. Model must account for 4 features in glucose-insulin regulation: 1 G liver absorbs more glucose, converting it & storing it as glycogen. G reverses this process, 2 H more glucose absorbed from bloodstream through cell membranes. 3 G stimulates the pancreas to produce insulin at a faster rate; G lowers insulin production rate. 4 Insulin is constantly being produced by the pancreas & degraded by the liver. 159/226 : Diabetes Detection cont d Model ignores biochemistry & treates whole process as a compartmental one: bloodstream is a single box in which G and H are instantaneously uniform. Provided no recent digestion, glucose & insulin concentrations should be in equilibrium, interested in how system responds to change in equilibrium. Let G(t) & H(t) denote excess glucose & insulin conc ns at time t. G = H = 0 at equilibrium & +ive & -ive values refer to deviations from equilibrium. If either G,H are given a non-zero value, body will try to restore the equilibrium. 160/226

4 : Diabetes Detection cont d Assuming rates of change of G,H depend only on values G, H, may write: dg = αg βh (4.2) dh = γg δh for some constants α,β,γ,δ. To see why these constants must be all +ive, examine second eqn in Eqn.(4.2) & condition 4 above. If, initially, G = 0 & H > 0, dh/ < 0; this corresponds to the liver in action as per condition 4. Similar reasoning may be applied for other constants. 161/226 : Diabetes Detection cont d Eqn.s(4.2) can be reorganised by expressing everything in terms of a single variable: G +(α+δ) G +(αδ +βγ)g = 0 (4.3) which has an exact solution of form G(t) = c 1 e λ 1t +c 2 e λ 2t for constants c 1,c 2. Eigenvalues λ 1,λ 2 can be found from substituting this solution into Eqn.(4.3), from this, get: λ 2 +(α+δ)λ+(αδ +βγ) = 0 (4.4) Insulin concentration H may be got from: H = 1 ( ) G +αg β (4.5) 162/226

5 : Diabetes Detection cont d To test for diabetes, a glucose tolerance test is administered. In this, after fasting overnight, an injection of glucose is given. Blood samples are then taken at subsequent times and the glucose concentration is measured to test the glucose-insulin regulatory system. This should return to equilibrium faster for healthy patients than diabetic ones. Mathematically, test G = g 0 & H = 0 at t = 0, assuming that the glucose concentration spikes while the insulin concentration is instantaneously zero. 163/226 : Diabetes Detection cont d Initial conditions for Eqn.(4.3) can be found from inputting the above into Eqn.(4.5): G = g 0 and G = αg 0 at t = 0 (4.6) Depending on the values of the constants α,β,γ,δ (discussed below) find a number of solution types for Eqn.(4.3) shown in Table 4.1. Note: since all α,β,γ,δ are +ive, all the exponentials in the solutions to Eqn.(4.3) will decay to zero with time (i.e. all are either -ive or have -ive real part). 164/226

6 : Diabetes Detection cont d Solns of Eqn.(4.4) Solns of Eqn.(4.3) (a+λ λ 1,λ 2 real G = g 2 )e λ 1 t (a+λ 1 )e λ 2 t 0 λ 2 λ 1 one real, λ λ = υ ±iω G = g 0 ( (1 (α+λ))e λt G = g 0 cosωt α ω sinωt) e υt Table 4.1 : Diabetes Model Solutions 165/226 : Diabetes Detection cont d Results are backed up by exp t; in 1961 Bolie extrapolated data from dogs to humans. Using above model with exp l data determined that λ 1 = 1.36 & λ 2 = 2.34 and the following equations held: with the solutions dg = 2.92G 4.34H dh = 0.208G 0.78H (4.7) G(t) = g 0 ( 0.56e λ 1 t +1.56e λ 2t ) H(t) = 0.202g 0 ( 0.56e λ 1 t e λ 2t ) (4.8) 166/226

7 : Diabetes Detection cont d Putting G = 0, see that G returns exponentially to zero in about 1 hour (if our time units from the experimental results are hours) for a normal individual. So, by measuring G, can see whether curve & time to return to normality mirror these results. Results for an initial Glucose concentration g 0 = 1[mol][litre] 1 are shown in Fig /226 : Diabetes Detection cont d Concentration Time,t [hrs] Figure 4.1 : Insulin & Glucose Concentrations for g 0 = 1 in Diabetes 168/226

8 : Antibiotic Pharmacokinetics Kinetics of drug absorption, distribution, & elimination is known as pharmacokinetics. A knowledge of pharmacokinetics is needed to administer drugs optimally. i.e. dose should be less than the lethal dose and greater than the effective dose for as long as is determined. This range is known as the therapeutic range of the drug and may (e.g. for chemotherapy drugs) be quite narrow Hence knowing pharmacokinetics of some (especially novel) drugs and formulations is critical. For this pharmacokinetic modelling use differential equations. 169/226 : Antibiotic Pharmacokinetics cont d Tetracyclines (so-called because of four hydrocarbon rings) form quite an old, generally well-tolerated & broad-spectrum group of antibiotics. Tetracyclines such as doxycycline used for prophylaxis against Bacillus anthracis (anthrax) & is effective against Yersinia pestis, the infectious agent of bubonic plague. Also widely used for malaria treatment & prophylaxis, due to the fact that it can be taken for long time periods without the risk of many side-effects (as for Mefloquine, for example). 170/226

9 : Antibiotic Pharmacokinetics cont d For Tetracycline taken orally, represent respective amounts of tetracycline in G.I. tract & plasma at time t by x 1 (t) and x 2 (t). Equations associated with this are: dx 1 = Ax 1 dx 2 = +Ax 1 Bx 2 (4.9) i.e. amount of antibiotic in G.I. tract at rate defined by A & amount in plasma being excreted at rate defined by B. 171/226 : Antibiotic Pharmacokinetics cont d Representing eqn.(4.9) as a matrix and: ( ) A 0 K(A,B) = A B [ ] x1 x = matrix representation of the tetracycline equations is: x 2 ẋ = Kx (4.10) Assuming that A B, x 1 (0) = D, & x 2 (0) = 0, it can be shown that: dx 1 dx 2 = De At = D A [ B A e At e Bt] (4.11) 172/226

10 : Antibiotic Pharmacokinetics cont d Plasma conc n (x 2 (t)) can be fitted as: [ x 2 (t) = 2.65 e 0.15(t 0.41) e 0.715(t 0.41)] (4.12) Plasma concentration can be seen in Fig Concentration Time, t [hrs] Figure 4.2 : Plasma Concentrations for Antibiotic 173/226

The Chemostat: Stability at Steady States. Chapter 5: Linear & Non-Linear Interaction Models. So, in dimensional form, α 1 > 1 corresponds to

Introduction & Simple Models Logistic Growth Models The Chemostat: Stability at Steady States 1 So, in dimensional form, α 1 > 1 corresponds to K max < V F. As K max is max bacterial repro rate with unlimited