1. Announcements Topics: -sections 1.2, 1.3, and 2.1 * Read these sections and study solved examples in your textbook! Work On: -Practice problems from the textbook and assignments from the coursepack as assigned on the course web page (under the link “SCHEDULE + HOMEWORK”)

2. Models A mathematical model is a description of a biological pattern, observation, or rule using mathematical concepts and language (such as functions and equations). 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 infection, so that it can be controlled effectively new research since previous models are not adequate

4. Research goal: study infections in three countries Guinea, Sierra Leone and Liberia Start with data (WHO=World Health Organization) http://www.afro.who.int/en/clusters-a- programmes/dpc/epidemic-a-pandemic-alert-and- response/outbreak-news.html Data taken from 22 March to 20 August 2014

5. … …

6. How to make sense of these numbers? Research question: based on the data available until 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 population into four groups: S=susceptible E=exposed I=infected R=recovered and build mathematical relationships between these groups

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

9. Look at the first equation derivative=rate of change, so this equation describes how the number of susceptible people changes over time due to the infection

10. Look at the first equation the derivative is negative, so the number of susceptible people decreases (due to exposure and infection)

11. Look at the first equation the rate of change is proportional to the number of susceptible people S and to the proportion I/N of the total population N who are infected the constant of proportionality changes with time, so is a function of t

12. Look at the first equation this is called a differential equation to solve for S we have to un-apply the derivative, i.e., we have to use integration

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

14. When the differential equations are solved, researchers are able to compute the effective reproduction number Re (= number of secondary infections generated by an infected individual after control measures are put into place) The predictions are that by September 2014, Guinea and Sierra Leone … Re<1 (control measures work, infections will be declining) Liberia … Re about 1.6 (need much better control measures in order to stop the outbreak)

15. 20 August 2015, a year later: how good was the model? Data as reported to World Health Organization:

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

17. 20 August 2015, a year later: how good was the model? inflection bit below 400 estimated total 800 actual: 2524 inflection around 400 estimated total 800 actual: 3951 no inflection detected after 1000 deaths total deaths > 2000 actual: 4806

18. all predictions made in 2014 were underestimates 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 times) 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 infections coming from outside the countries studied, which were not predicted by the model

19. Important message: We need to learn math in order to understand a vastly increasing number of publications in biology and health sciences which use mathematics and statistics 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 quantitative information

20. Dynamical Systems Discrete-time dynamical systems describe a sequence of measurements made at equally spaced intervals Continuous-time dynamical systems, usually known as differential equations, describe measurements that change continuously

22. Conversions To be studied independently…

23. Relations and Functions A relation between two variables is the set of all pairs of values that occur. A function is a special type of relation.

24. Functions A function 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).

25. Most (almost all) data collected in life sciences constitutes 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

26. This diagram shows a relation, and not a function

27. Using statistical methods such as regression (these methods are covered in statistics courses in levels 2 and above), we can identify a function which approximates the data

28. And then we work with the function we obtained. Why? Because we have no choice. It is not possible to work with relations and obtain quantitative results desired in our research in the life sciences.

29. Of course, we can say something … for example, the data on the right suggests some kind of exponential growth. But in order to quantify that growth, and further work with it, we need to have a function

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

31. Graphs The graph of a function 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

32. Piecewise Functions A piecewise function f(x) is a function whose definition changes depending on the value of x. Example: Absolute Value Function The absolute value of a number x, denoted by |x|, is the distance between x and 0 on the real number line.

33. Piecewise Functions Example: Sketch the graph of f(x).

34. 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.

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

36. Variables and Parameters We can study how a function depends on one of its variables at a time 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).

37. Body Mass Index Height as a Parameter

38. Proportional and Inversely Proportional Relationships Example: Body Mass Index (BMI) Note: BMI is proportional to mass. If a person’s mass changes (and their height remains the same), then their BMI will change by the same amount.

39. Proportional and Inversely Proportional Relationships If Then So a 10% increase in body mass results in a 10% increase in BMI.

40. Proportional and Inversely Proportional Relationships Example: Body Mass Index (BMI) Note: BMI is inversely proportional to height squared. So an increase in height (with mass held constant), will result in a decrease in BMI.

41. Proportional and Inversely Proportional Relationships If Then So a 10% increase in height results in a 17% decrease in BMI.

42. Relationship Between Mass and Volume The fundamental relation between the mass and the volume of an object, or living organism, states that where is the density. Note: If density is constant, then mass is proportional to volume.

43. Linear Functions For a linear function, the change in output ( ) is proportional to the change in input ( ) If the change in input is scaled by some factor, then the change in output is scaled by the same factor. We call this constant the slope of the line

44. Linear Functions slope: point-slope equation: slope-y-intercept equation:

45. Linear Model for the Population of Canada Data: YearTime, t Population, P(t) (in thousands) 1996028 847 2001530 007 20061031 613

46. Linear Model for the Population of Canada Create a linear model for the population of Canada as a function of time using the first two data points.

47. Linear Model for the Population of Canada Use this model to predict Canada’s population in 2006: Actual observed population in 2006:

48. Power Functions A power function is a function of the form where is a constant. Note: Although can be any real number, we usually omit the case when

49. Power Functions Some special cases: a=2: a=3:

50. Power Functions Some special cases: a=1/2: a=1/3: square root function cube root function

51. Power Functions Some special cases: a=-1: a=-2: rational function

52. Models Involving Power Functions Example: Blood Circulation Time in Mammals Blood circulation time is the average time needed for the blood to reach a site in the body and come back to the heart. It has been determined that, for mammals, the blood circulation time is proportional to the fourth root of the body mass.

53. Example: Blood Circulation Time in Mammals Model: where is the blood circulation, in seconds, is the body mass, in kilograms, and is some proportionality constant. Models Involving Power Functions

54. Example: Blood Circulation Time in Mammals Graph:

55. Question If the body mass increases 10-fold, how does the blood circulation time change?

56. Question This means that the blood circulation time of an elephant weighing 5400 kg is about 1.78 times longer than the blood circulation time of a cow that weights 540 kg.