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SSG6.3%20%20Differential%20Eq%20Linear

# SSG6.3%20%20Differential%20Eq%20Linear - 6.3 LINEAR MODELS...

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6.3. LINEAR MODELS IN BIOLOGY 767 6.3 Linear Models in Biology An important class of models are described by the linear differential equation dy dt = c 0 + c 1 y where the constants c 0 and c 1 are model parameters that have specific physical or biological interpretations. For example, in Section 6.1 we saw how models with c 0 = 0 and c 1 = r were used to describe exponential population growth ( c 1 = R ) and radioactive decay ( c 1 = λ ). In this section, we discuss further applications where the constant coefficient c 0 is non-zero. Mixing models Mixing models are formulated on the premise that the density of individuals or concentration of molecules, which are generically characterized in terms of a number of objects per unit area or volume, form a homogeneous pool such that the flow of objects into the pool is controlled by an external constant rate while the flow of objects out of the pool is in proportion to the density of objects in the pool. This latter assumption implies that the greater the density of objects in the pool, the faster the total flow of objects out of the pool. Mixing Model Let y ( t ) represent the density of objects in a pool at time t . If objects flow into this pool at a constant total rate a > 0 and out of this pool at a rate by ( t ) > 0, i.e at a constant per-capita rate b > 0, then the density of objects in the pool over time is governed by the equation dy dt = Rate In Rate Out = a b y. Example 1. Modeling HIV Human immunodeficiency virus-type 1 (HIV-1) has many puzzling quantitative features. For instance, most HIV patients undergo a 10 year period during which the concentration of virus in the plasma is very low. It is only after this quiescent period that a patient experiences the onset of AIDS. The reason for this quiescent period is unknown, and it was presumed that during this period the virus was relatively inactive. Using models, Perelson and colleagues © 2010 Schreiber, Smith & Getz

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768 6.3. LINEAR MODELS IN BIOLOGY quantified viral levels in the blood of infected individuals during this quiescent period. Specifically, Perelson and colleagues let the concentration of viral particles in the blood plasma be represented by the variable V ( t ). They assumed that HIV viral particles infused into the blood, from production sites in lymphatic tissue, at a constant rate P > 0 and were eliminated from the blood at a rate cV ( t ), where c > 0 is referred to as the elimination rate constant . From these assumptions they obtained the mixing model dV dt = P c V, where t is measured in days. Both P and c are unknown constants. a. Data showed that after being put on a potent antiviral drug, the viral concentration fell exponentially in the blood. Assuming that the drug killed the production of new virus completely in lymphatic tissue, Perelson et al. estimated the half-life of the viral particles to be 0 . 2 days. Use this information to estimate the elimination rate constant, c .
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SSG6.3%20%20Differential%20Eq%20Linear - 6.3 LINEAR MODELS...

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