Announcements. Topics: Work On: - sec0ons 1.2 and 1.3 * Read these sec0ons and study solved examples in your textbook!

Transcription

1 Announcements Topics: - sec0ons 1.2 and 1.3 * Read these sec0ons and study solved examples in your textbook! Work On: - Prac0ce problems from the textbook and assignments from the coursepack as assigned on the course web page (under the link SCHEDULE + HOMEWORK )

2 Models A mathema&cal model is a descrip0on of a biological pavern, observa0on, or rule using mathema0cal concepts and language (such as func0ons and equa0ons). When we have a model, we can apply tools of calculus to study how a living system changes.

3 Ebola virus (EBOV) outbreak in West Africa Research needed to understand the spread of infec0on, so that it can be controlled effec0vely new research since previous models are not adequate

4 Research goal: study infec0ons in three countries Guinea, Sierra Leone and Liberia Start with data (WHO=World Health Organiza0on) hvp:// epidemic-a-pandemic-alert-and-response/outbreaknews.html Data taken from 22 March to 20 August 2014

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6 How to make sense of these numbers? Research ques0on: based on the data available un0l 20 August 2014, predict the number of infected individuals and the number of deaths for September 2014, to determine if present control measures work, or need to be improved

7 Math model Divide the popula0on into four groups: S=suscep0ble E=exposed I=infected R=recovered and build mathema0cal rela0onships between these groups

8 We will learn math which will help us understand all this!

9 Look at the first equa0on ds dt = β(t) S I N deriva0ve=rate of change, so this equa0on describes how the number of suscep0ble people changes over 0me due to the infec0on

10 Look at the first equa0on ds dt = β(t) S I N the deriva0ve is nega0ve, so the number of suscep0ble people decreases (due to exposure and infec0on)

11 Look at the first equa0on ds dt = β(t) S I N the rate of change is propor0onal to the number of suscep0ble people S and to the propor0on I/N of the total popula0on N who are infected the constant of propor0onality changes with 0me, so is a func0on of t

12 Look at the first equa0on ds dt = β(t) S I N this is called a differen0al equa0on to solve for S we have to un-apply the deriva0ve, i.e., we have to use integra0on

13 Represent data visually (red=number of infected; black=number of deaths) Which curves best describe the WHO data? We will study logis0c (leg and middle), exponen0al growth (right), and many other curves

14 When the differen0al equa0ons are solved, researchers are able to compute the effec0ve reproduc0on number Re (= number of secondary infec0ons generated by an infected individual ager control measures are put into place) The predic0ons are that by September 2014, Guinea and Sierra Leone Re<1 (control measures work, infec0ons will be declining) Liberia Re about 1.6 (need much bever control measures in order to stop the outbreak)

15 20 August 2015: Epidemic is over!

16 end of the epidemic, June-August 2015

17 20 August 2015, a year later: how good was the model? Data as reported to World Health Organiza0on:

18 important fact about logis0c growth (we will discuss details later) the value where there is a change in the pavern of increase is the inflec0on point height at inflec0on = one-half of the horizontal asymptote, i.e., one-half of maximum

19 20 August 2015, a year later: how good was the model? inflec0on bit below 400 es0mated total 800 actual: 2524 inflec0on around 400 es0mated total 800 actual: 3951 no inflec0on detected ager 1000 deaths total deaths > 2000 actual: 4806

20 all predic0ons made in 2014 were underes0mates many possible reasons, including: the model described in this paper is not adequate the model assumed that the control measures that existed in September 2014 would not change (in reality, due to the lack of resources, the measures weakened at 0mes) data that was available in August 2014 (based on which the model was run) was inaccurate; for instance, deaths in remote areas were not reported there were secondary infec0ons coming from outside the countries studied, which were not predicted by the model

21 Important message: We need to learn math in order to understand a vastly increasing number of publica0ons in biology and health sciences which use mathema0cs and sta0s0cs Even if you do not plan to become a researcher, you will need to read and understand all kinds of documents, manuals, and reports which use quan0ta0ve informa0on

22 Dynamical Systems Discrete-&me dynamical systems describe a sequence of measurements made at equally spaced intervals Con&nuous-&me dynamical systems, usually known as differen0al equa0ons, describe measurements that change con0nuously

23 Example of a discrete model! Let! I t!denote!the!number!of!people!infected!with!flu,!where!t!is!time!in!days!(so! that! t + 1!represents!tomorrow!if! t represents!today).!it!has!been!determined!that! 60I I t +1 = t 0.5I t + 1.5! and!that!initially!there!are!two!cases!of!flu,!i.e.,! I 0 = 2!!! Then! etc.!!! 60I I 1 = 0 0.5I = 60(2) 0.5(2) = 48! 60I I 2 = 1 0.5I = 60(48) = ! 0.5(48) I I 3 = 2 0.5I = 60(113) = ! 0.5(113) + 1.5

24 Conversions To be studied independently however we can take up ques0ons in tutorials look at tables in sec0on 1.2, do conversion, etc.

25 Rela0ons and Func0ons A rela&on between two variables is the set of all pairs of values that occur. A func0on is a special type of rela0on.

26 Func0ons A func&on f is a rule that assigns to each real number x in some set D (called the domain) a unique real number f(x) in a set R (called the range).

27 Most (almost all) data collected in life sciences cons0tutes a RELATION and not a FUNCTION Example: the graph on the next slide shows the cranial capacity (i.e., the brain volume) calculated from the skulls of early humans and modern humans, between 3 million years in the past and today

28 four skulls, roughly the same age but with different cranial capacity cranial capacity in millilitres 2,100 1,700 1, M 2.5 M 2 M 1.5 M 1 M 0.5 M NOW years ago (M=million) This diagram shows a rela0on, and not a func0on

29 Using sta0s0cal methods such as regression (these methods are covered in sta0s0cs courses in levels 2 and above), we can iden0fy a func0on which approximates the data cranial capacity in millilitres 3 M 2.5 M 2 M 1.5 M 1 M 0.5 M NOW years ago (M=million) 2,100 1,700 1,

30 And then we work with the func0on we obtained. Why? Because we have no choice. It is not possible to work with rela0ons and obtain quan0ta0ve results desired in our research in the life sciences. cranial capacity in millilitres 3 M 2.5 M 2 M 1.5 M 1 M 0.5 M NOW years ago (M=million) 2,100 1,700 1,

31 cranial capacity in millilitres cranial capacity in millilitres 3 M 2.5 M 2 M 1.5 M 1 M 0.5 M NOW years ago (M=million) 2,100 1,700 1, M 2.5 M 2 M 1.5 M 1 M 0.5 M NOW years ago (M=million) 2,100 1,700 1, Of course, we can say something for example, the data on the right suggests some kind of exponen0al growth. But in order to quan0fy that growth, and further work with it, we need to have a func0on

32 Domain The domain of a func0on f is the largest set of real numbers (possible x-values) for which the func0on is defined (as a real number). Example: Find the domain of the following func0ons.

33 Graphs The graph of a func0on f is a curve that consists of all points (x,y) where x is in the domain of f and y=f(x). Example: Sketch the graph and find the domain and range of f (x) = x 2 +8x 17. y x

34 Piecewise Func0ons A piecewise func0on f(x) is a func0on whose defini0on changes depending on the value of x. Example: Absolute Value Func8on The absolute value of a number x, denoted by x, is the distance between x and 0 on the real number line. $ f (x) = x = % x if x 0 & x if x < 0

35 Piecewise Func0ons Example: Sketch the graph of f(x). f (x) = # 1 % x, % $ % 2, % & x 1, x <1 x =1 x >1 y x

36 Variables and Parameters A variable represents a measurement that can change during the course of an experiment. A parameter represents a measurement that remains constant during an experiment but can change between different experiments.

37 Variables and Parameters Example: Body Mass Index (BMI) BMI = m h 2 where and h m is a person s mass in kilograms is their height in metres. BMI is the dependent variable; and h are the two independent variables. m

38 Variables and Parameters We can study how a func0on depends on one of its variables at a 0me by holding all other variables constant. For example, to study how BMI depends on mass, we fix height to be constant (i.e., collect data from all people of the same height).

39 Body Mass Index Height as a Parameter BMI = 0.416m m