1. Introduction to Models

1.1 Mechanistic Models and Descriptive Models

They are mostly used to study physical and biological processes: because we want to know how they work: understanding, explanation; because we want to know how they will go on in the future; prediction; because we want to influence them : control.The use of mathematical models has produced many important results that could not easily be obtained by other means. It has become an indispensable tool in many areas.In order to use this tool, one must first formulate a mathematical description (a mathematical model) of the real-world (biological, industrial, economic, etc.) process. Then, within the domain of mathematics, one can analyze the behavior of the model by mathematical analysis, computer simulation and other techniques. This will produce conclusions and predictions, which, finally, must be translated back into real-world conclusions.This means that using models for studying processes involves mathematics, but also the translations of real-world phenomena into mathematics, and back. Therefore, this course will not only be about mathematics, and we shall give much attention to the translation and interpretation process.And remember: mathematical models are a tool. A powerful tool, but still a tool. If the tool will not help you in understanding the real-world process you are studying, there is no reason to use it.

The idea behind mathematical models is the same as in science in general: if you want to understand (or predict, or control) a process, you must look inside the process and find out how it works. If you want to predict the weather, it is not enough to just observe rainy days and sunny days, the temperature etc.; you must find out more about underlying mechanism like air currents, evaporation, condensation points and so forth.With mathematical models, it is the same. For a good model of a process, you must go into the relevant sub-processes and find out how they interact to produce the overall process. In other words: a good model should be a mechanistic model.

Sometimes, the term model is also used in a wider sense, covering every use of mathematical techniques on empirical data; also if no attempt is made to go into the underlying mechanism. In such cases, the term descriptive model or empirical model is used.Most of this course will be about mechanistic models. But first we shall discuss two examples of descriptive models, which usually are much simpler.

Example 1 In food technology, one studies a process by which chips of a roughly cylindrical shape are produced from coarsely ground maize by a process called extrusion. The chips should preferably be as crispy as possible, and therefore the diameter of the cylinder produced from a given amount of maize should be as large as possible.Experiments show than the cylinder diameter depends on the water content of the maize.The figure shows the results of these experiments.The water content can be controlled in the production process, and the question now is: how large should be one choose the water content in order to get the largest possible diameter? The question can be answered by constructing an empirical model. This model states that the relationship between water content (x) and cylinder diameter (y) has the form of parabola:y = ax2 + bx + cThe parameters a, b, c are then chosen in such a way that the parabola has the best possible fit to the observations (later in this course, we shall show how to do this). Once the parabola has been found, it is easy to determine its maximum value. x x x x x x x x x x x

x x x x x x x x x x x

optimumExerciseThe optimal water content (i.e. the water content which will product the largest diameter) will depend on the parabola that was found from the data. The parabola is completely determined by the 3 parameters a, b, c.Express the optimal diameter in terms of the parameters a, b, and c.

Classroom discussionWhy not solve the problem in the easy way by just taking the maximal observed point?Once you have answered that question, you run into a second problem: how do you know the curve is a parabola?

Example 2Government regulations for toxic substances set standards for maximum allowable concentrations. Therefore, one needs to know concentration levels that give no effect, or only very weak effects. Of course these are difficult to observe experimentally. Instead, one does animal experiments at levels that are much higher (because the effect can then be much better observed). So one obtains the higher end of the dose-response curve. The lower end (which is the one that is of interest for regulation purpose) is then obtained by an empirical model. In the graph, you see how the procedure works; the empirical model is often chosen to be a straight line.

No-effect dose x x x x x

Classroom discussionDiscuss the procedure of example 2, and compare with example 1.

1.2. Common FunctionsIn the construction and analysis of models, we shall meet all kinds of mathematical functions. Now of course the number of possible functions is limitless, but in chemical, biological etc. applications, some functions occur much more frequently than others. Here we shall discuss some of them. We shall first present them in their most basic form, and then some commonly occurring variations obtained by shifting, stretching etc. We shall also briefly discuss Frankenstein functions: functions that are built up from fragments of different basic functions and which are often encountered in applications.

1.2.1. Straight Line

General form: F(x) = ax + b

The value of a reflects the slope of the line. If a > 0, the line increases (from left to right); if a < 0, it decreases.

1.2.2. Parabola

Basic form : f(x) = x2.General form: f(x) = ax2 + bx + cIf a > 0, the parabolas top is down; if a < 0, the top is up.Shifting the parabola (up/down, left/right) results in different values for b and c, respectively.

1.2.3. Hyperbola

Basic form:

Hyperbolas always consist of two unconnected branches. However, in many applications one of the branches (the dotted one in the graph) has no physical meaning, and is ignored. Hyperbola always process two asymptotes.

There exists a rather complicated general formula that covers all hyperbolas, but we will not need it. Hyperbolas occurring in applications are nearly always of amore restricted subtype, the so-called rectangular hyperbolas (meaning that the two asymptotes make an angle of 90), with the additional property that one asymptote is horizontal (and, consequently, the other is vertical).

The general form for such an hyperbola is

In fact, one does not really need four parameters; three is sufficient:

In the most common application (enzyme kinetics, microbiology, animal ecology), the hyperbola passes through the origin. This has the consequence that b = 0. So most hyperbolas you will see have the simple form (with only two parameters).

1.2.4. Exponential

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1- 0 1

Basic form : f(x) = ex.General form: f(x) = aebx. If b > 0, the curve has the same shape as the basic exponential; if b < 0, one gets the familiar curve of exponential decay (radio-activity and other applications).Another common variant is the functionf(x) = a(1 e-bx)Which, for large values of x, goes towards the value a.In this course, we shall also encounter sums and differences of exponentials, like the illustrated function f(x) = e-2x e-3x.

1.2.5. Logarithm

Basic form: f(x) = ln xThe logarithm is the inverse function of the exponential: if y = ex then x = ln y. You can also see this in the graph: if you take the graph of the exponential and -then switch the x and y axes, you obtain the graph of the logarithmic function.The function has a vertical asymptote at x = 0. For x < 0, the function is not defined (negative numbers possess no logarithm).As x become larger and larger, f(x) will increase slower and slower, as in the hyperbole. But, unlike the hyperbole, there is no upper limit; ii you go on long enough, die function will reach any value.

1.2.6. "Frankenstein" Functionsf1(x)

f2(x)

xIn practical situations, one often meets relationships that are composed of fragments of different functions, like the famous Frankenstein monster that was put together from different human bodies. In the illustrated example, the function if consists of fragments taken from a parabola, f1(x) = 0.8 x2, and a straight line, f2(x) =1.3.The customary way to describe such a function:f(x) =f1(x)for x 1,27f(x) =f2(x)for x 1,27where the value 1,27 first must be calculated by intersecting f1 and f2.There is another, and simpler, way to describe this sort of function, as follow:f(x) = min (f1(x), f2(x))In this case:f(x) = min (0,8x2, 1,3)The `min' (for 'minimum') defines the function as follows: it simply takes, for each value of x, the lowest value of f1(x) and f2(x).The functions `min' and 'max' (which is defined in a similar way) are useful in the construction of models. As you will see later in this they can in the simulation program.

1.3. Problems

Consider the hyperbola . Find the values for p, q, r, such that describes the same hyperbola.

1-2

Consider the hyperbolic, function Prove that it passes through the origin. If x becomes very large (x ), what will be the (asymptotic) upper lima of y? Determine the value of x for which y assumes half its upper limit value. Let a = 3 and d = 2. Sketch the hyperbola over the range -5 x 5. Also sketch its asymptotes.

1-3.Consider the function y = f(t) = e-2t - e-3t we showed earlier as an example of a commonly occurring function.Calculate ymax

1.4. Plot the function f(x) = min(-2x + 4,2) Plot the function f(x) = max(min(-2x 4,2), 0).

Problems 1- 5 and 1- 6 will he you first exercises in the construction and analysis of mathematical models you will get a mechanism, and you must analyze its consequences. Mathematically, these models are still simple and you will only need very hide theory for them. But you will see, they may still be unfamiliar and difficult. Just try them, and see how far you can get.

1-5.Someone is taking a 50 mg paludrin pill as a protection against malaria. The paludrin, however, gradually disappears from the body (by excretion ands or biotransformation), and after 24 hours only 50% remains.a. How much remains after days? After 3, 4 days?b. How long will it take until just l mg is left in the body?Now instead of just one pill, the person will take a pill each day in order to have permanent protection. Let us say the pill is taken at 07:59 daily.c. What will be the paludrin body load at 08:00 on days 1, 2, 3, 4? At day n?d. Plot the values obtained in c. You will notice that the day-by-day body load (as taken at 08:00), gradually increases. Will it increase indefinitely, or will it move towards a limit value? If so, what will be this limit value?e. If the body load for a good protection against malaria is 80 mg, what should be the daily dose?f. Up to now, we have been looking at the body load in day-by-day time steps. Of course, it changes continuously: Try to express the body load during the first day as a continuous function of time.g. Also do this for the second day.h. Suppose that the so called halflife of the pill (the time it takes to decrease to half its original value) is not 24 hours but 48 hours. For this case, answer questions c to e.i. The same as h, but now for a half-life of 33 hours, for a half-life of T hours.

1-6.The People's Republic of China has strict rules for berth control. The rules are quite simple:1. Every couple is allowed to have one child.2. If the first child is a girl, the couple is allowed to have a second child. That will be all. (You will probably know that, in China, there is generally a very strong preference for boys.)One may ask what the consequences of such a system will be. Two questions in particular:a. What will this mean to the growth rate of the Chinese population? In this form, the question will probably too difficult. Instead, simplify the question in such a way that you can answer it, e.g. by something like comparing the sizes of "the present generation" and "the next generation".b. What will happen to the sex ratio in China? Will there be more men than women, or less, or equal?The present rules are very strict, and one can think of more moderate variants that still take account of the general preference for boys.c. Variant: if the second child is again a girl, a couple is allowed to have a third child. Answer questions a and b.d. Variant: couples are allowed to go on having children until they have a boy. Answer questions a and b.