Kasetsart University Workshop. Mathematical modeling using calculus & differential equations concepts

Transcription

1 Kasetsart University Workshop Mathematical modeling using calculus & differential equations concepts Dr. Anand Pardhanani Mathematics Department Earlham College Richmond, Indiana USA A copy of these slides is available at pardhan/kaset/workshop/

2 Warmup Consider the following models for the rate of change of some population P (t) (a) (b) (c) dp = 0.4P dp = 0.4P dp = 0.4P ( 1 P 200 ( 1 P 200 ) ) ( 1 50 P ) For each model, sketch a rough graph of P vs t for the following 3 initial values: P (0) = 20. P (0) = 100. P (0) = 250. Hint: It is not necessary to solve the problem by integration!

3 Warmup Recall a simple and powerful calculus concept: The sign of the derivative tells us a lot. If f (x) < 0, then f > 0, then f = 0, then f

4 Warmup Solution (a) P = 0.4P Sign of P (as function of P ) P= P The sign graph shows P if P = 0 P if P > 0 P if P < 0 Thus the solutions for P (0) = 20, P (0) = 100, and P (0) = 250 look like: P(t) t

5 (b) P = 0.4P ( 1 P ) 200 Sign of P P= P= The sign graph shows P if P = 0 or 200 P if 0 < P < 200 P if P < 0 or P > 200 P 400 P(t) 200 t

6 (c) P = 0.4P ( 1 P ) ( 1 50 ) 200 P Sign of P P= P= P= The sign graph shows 400 P(t) P if P = 50 or 200 P if 50 < P < 200 P if P < 0, 0 < P < 50 or P > 200 P 200 t

7 Illustration: Predator-prey modeling R(t) = population of rabbits (prey) F (t) = population of fox (predator) Exercise: Fill in reasonable forms for the missing terms (in the boxes) dr = rabbit births rabbit deaths df = fox births fox deaths What assumptions did you make? How did they affect your model?

8 Predator-prey modeling Lotka-Volterra model: Rabbit birth rate = constant Rabbit death rate fox population Fox death rate = constant Fox birth rate rabbit population This leads to models of the form: dr = a R b RF df = c F + d RF where a, b, c, d are positive constant parameters. One way to interpret the parameters: a = per-capita birth rate of rabbits b F = per-capita death rate of rabbits due to F (t) foxes c = per-capita death rate of foxes d R = per-capita birth rate of foxes due to R(t) rabbits Exercise: Find the equilibrium solutions (or, fixed points). Interpret their meaning.

9 Numerical example: dr df Predator-prey modeling = 0.08 R RF = 0.02 F RF Simulate using Maple: Effect of initial conditions (i) R(0) = 500, F (0) = 50 (ii) R(0) = 500, F (0) = 100 Effect of parameter values Carefully vary each parameter and explore the effect. What can you tell about the solution behavior?

10 Brief recap via a short quiz! 1. Suppose the population of some species is given by the model dp = (8 P )2 (12 P ) Find the equilibrium solutions, determine their stability, and sketch a few typical solution curves on a graph of P vs. t. 2. Shown below are different models of two interacting populations. Determine what type of interaction is occurring in each case: (i) predator-prey, or (ii) competing species, or (iii) cooperating species (a) dm = 0.012M MN dn = 0.003N MN (b) dm = 0.003M MN dn = 0.012N MN M, N denote species populations, and t denotes time.

11 Objective: Infectious disease modeling Predict how an infectious disease spreads through a population. Some examples of such diseases include: Swine flu, measles, TB, malaria, HIV, etc. Compartmental modeling strategy: OR

12 Model 1: SIS Assumptions: Infectious disease modeling Total population is constant. No births/deaths. At any time, a person is either infected or susceptible, but not both! Infected persons become susceptible again after they recover. Model: Exercise: Fill in reasonable forms for the missing terms (in the boxes) ds = rate of infection + rate of recovery di = + rate of infection rate of recovery Hint: Rate of infection depends on S(t) and I(t). Rate of recovery depends only on I(t). [why?]

13 Infectious disease modeling A simple form of the terms: Rate of infection = βsi Rate of recovery = γi } β and γ are constants Thus the model becomes ds = βsi + γi di = βsi γi Exercise: What are the units of β and γ? Does the model assume S + I is constant? Let S +I = N. Use this and reduce the model to a single differential equation in I. Does this model look familiar?! What is the long-term behavior of the solution? What do β and γ represent? β how contagious the disease is, and how much contact between S and I γ average time of recovery from disease

14 Infectious disease modeling Basic reproduction number: Let R 0 = βn γ This s a key parameter used in disease modeling. The value R 0 = 1 represents a bifurcation point in this model. The solution behavior changes at this point. Details of how/why: Using S + I = N we get: di = β(n I)I γi = βi(n I γ β ) The two equilibrium solutions are: I = 0 and I = N γ β Exercise: What does each equilibrium solution represent in real life? Explore the effect of R 0 using Maple simulations. Case1: Let N = 50, 000, β = , γ = 1/14 Case2: change β to You can use any initial conditions s.t. S(0) + I(0) = N

15 Model 2: SIR Assumptions: Infectious disease modeling Total population is constant. No births/deaths. At any time, a person is either infected, susceptible or recovered. Recovered persons cannot get the infection again. Model: Exercise: A simple SIR model can be made by rearranging terms in the SIS model. Here it is again for reference ds = βsi + γi di = βsi γi Rearrange terms and construct the SIR model ds = di = dr =

16 Infectious disease modeling Here is one way to do it: ds = βsi di = βsi γi dr = γi Compare with the SIS model: What do the parameters mean? How can we estimate their values? What are the equilibrium solutions? What is the basic reproduction ratio?

17 Infectious disease modeling Simulate an example using Maple: Let N = 50, 000, β = , γ = 1/14, S(0) = 45, 400, I(0) = 2, 100, R(0) = 2, 500 Thus, the equations become S (t) = S I I (t) = S I I 14 Exercise: R (t) = I 14 (1) Why is S + I + R = 0? What does that mean? (2) The sign of which derivative tells you whether the infection is spreading or diminishing? (3) There is an important critical value of S. When S is above this value the infection spreads, and when it is below, the infection decreases. Can you find this value? (4) If a vaccine exists, it is possible (according to this model) to vaccinate enough people to stop the spread of the disease. What is the minimum number of susceptible people that must be vaccinated to stop the spread?

18 A modeling project In 2003 the SARS epidemic broke out in Hong Kong. Data about the number of infected people per week is available for a 10-week period (see Science magazine, vol. 300, 2003) Determine which of the 2 models (SIS or SIR) can be applied to model this situation. Explain why. Construct the model. Simulate it using Maple and check whether it works. [E.g., let S(0) = 1, 000, 000, I(0) = 5, R(0) = 0]

19 A modeling project Some hints: * The given SARS data is a graph of I(t). * Try to estimate the value of γ first. * A rough estimate of I during the initial period can be obtained from the graph. * Can you estimate β if you know γ and I? If yes, how? If not, what else do you think is needed? * Keep γ fixed and try to vary β over some range. Use Maple simulations and compare your graph of I(t) with the given SARS data.

20 Extensions of previous models Predator-prey extensions: Competing species or cooperating species E.g. dx dy = 0.08 X XY = 0.02 Y XY Improved predator-prey models E.g. dx = 0.08 X dy ( 1 X = 0.02 Y XY 3-species hybrid models E.g. dx = 0.08 X dy = 0.05 Y dz ( 1 X ) XY ) XY 0.03 XZ ) XY 0.05 Y Z ( 1 Y = 0.02 Z XZ Y Z Q: Can you explain what interactions are happening here?

21 Extensions of previous models Infectious disease extensions: More complex population compartments E.g., SEIR (Susceptible, Exposed, Infected, Recovered); SEIRS; SIQRS; etc. Modeling the spread of ideas or opinions E.g. Public opinion of government policies S = no opinion I = for the policy Z= against the policy ds = β i(s + Z)I β z (S + I)Z + γ i I + γ z Z di = β i(s + Z)I γ i I dz = β z(s + I)Z γ z Z where β i, β z, γ i, γ z are constant parameters.

22 80 70 Iraq war opinion: no opinion war was wrong decision war was right decision 60 y (percent) t (months)

23 Review what we understand All models are wrong. But some are useful! A famous quote 1. What is mathematical modeling? 2. How do we construct mathematical models? 3. Suppose I want to model a predator-prey system in which the population is a function of position and time. E.g., R(x, y, t) = rabbit population density F (x, y, t) = fox population density How would that change our old model: dr = a R b RF df = c F + d RF

24 The Mathematical Contest in Modeling Annual event for undergraduate students worldwide. It is held every year around the 1st weekend of Feb. The contest consists of open-ended, real-life problems for which there is no known correct solution. Contest participants work in teams. Each team consists of 3 students, plus a faculty advisor. All work is submitted electronically. I strongly encourage Kasetsart students to participate next year! See their website Each year s prize winning entries are published in the UMAP Journal. Examples of prize winning entries from previous years can also be found by doing a Google search with keywords such as mathematical contest in modeling solutions. We will look at examples of some topics and prize winning entries. But, keep in mind, there is no single correct solution to these open-ended problems.

25 Modeling natural phenomena What are natural phenomena? Heating/cooling processes The motion of fluids, electrons, objects, etc. Chemical reactions Climate, weather and geological processes Conservation is a key concept in modeling nature: E.g. Conservation of mass Conservation of momentum Conservation of energy Conservation of (electric) charge When we write the conservation principle in mathematical form, we get a model.

26 Modeling natural phenomena An illustration: Fluid flow Consider an infinitesimal volume V, with surface S, in the flow field (a 2D example is shown in the sketch). Rate of change of mass inside V = d ρ dv V Rate of mass flux through the surface = ρu ds S d By conservation of mass: ρ dv = ρu ds V S ρ t dv = (ρu) dv V V ρ t = (ρu) OR ρ t + (ρu) = 0 This is the famous continuity equation of fluid flow.

27 Modeling natural phenomena Similarly, using conservation of momentum, together with constitutive relations, we get a very complete and accurate mathematical model for fluid flow ρ t + (ρu) = 0 ( ) u ρ t + u u = p + τ + f ρ = density, u = velocity vector, p = pressure, τ = stress tensor, f = body force vector. These are the famous Navier-Stokes equations. In 2D Cartesian coordinates, for incompressible fluids (e.g., water), with no body forces they simplify to u x + v y = 0 ( ) u ρ t + u u x + v u = P ( 2 ) y x + µ u x + 2 u 2 y 2 ρ ( v t + u v x + v v y ) = P y + µ ( 2 v x v y 2 )

28 A simple tsunami model What causes a tsunami? Vertical displacement of large mass of water in deep ocean. Typical characteristics Very long wave-length, small amplitude waves. Wave-length >> ocean depth e.g., wave-length = km, ocean depth = 2-3 km. Simple modeling approach Wave energy and motion is predominantly in 1 direction. So we can use a 1D, depth-averaged approximation of the Navier-Stokes equations. This is called the shallow water model.

29 A simple tsunami model Shallow water wave equations In 1D, they have the form h t + (hu) x = 0 (hu) + (hu gh 2 ) = body forces t x The unknowns are h(x, t) = water level, u(x, t) = velocity. Using these we can show that speed of tsunami wave is v(x) g b(x) where b(x) = depth of ocean at position x. Exercise: Compute the speed of tsunami waves for each case below (1) b(x) = 4300m (average depth of Pacific ocean) (2) b(x) = 3400m (average depth of Indian ocean) (3) b(x) = 11000m (deepest point in the Pacific) Conclusion?

30 A simple tsunami model Tsunamis in deep water travel very fast, but they have very small amplitude. This is captured very well by the shallow water model. Q: As the tsunami nears the shore, what happens according to the above model? This leads to what consequence? The amplitude A of tsunami waves can also be modeled similarly, which leads to the relation Exercise: A 1 b 1/4 Suppose a tsunami in deep ocean has amplitude 1 m when the water depth is 4000 m. Using the above relationship, find the wave amplitude when it is near the shore at water depths of about 2 m. For comparison, the 2011 earthquake-induced tsunami in Japan was about 7 m high when it hit the coast. Let s view a matlab simulation of the shallow water equations for a tsunami runup!

31 Pattern formation in nature Nature is full of fascinating patterns. Some examples include: Fractal patterns in plants, landscapes and crystals Fibonacci patterns in fruit and flowers Animal coat patterns Migration patterns in butterflies, whales and birds Sand dunes, fossils and geological patterns. Mathematics is also full of fascinating patterns! Is there some connection with nature? Let us explore some PDE models and the patterns in their solutions.

32 Pattern formation in nature Consider the following image Suppose the pattern represents the concentration of some chemical. For example, low concentrations correspond to lighter colors, and high concentrations to darker colors. Let u demote the local concentration of this chemical. Exercise: (1) What does the graph of u look like on the square patch? As a first step torwards this, consider a narrow horizontal strip of the square. Make a graph of the pattern (u vs. x) you might see on this horizontal strip. (2) Conjecture a formula for a function whose garph might have similar features. (3) Now return to the 2D case with the square patch. Sketch a qualitatively reasonable graph of u(x, y). (4) Conjecture a formula for a function u(x, y) whose garph might have some similar features.

33 Reaction-diffusion models We consider 2 chemicals in a 2D domain. Let u(t, x, y) = concentration of one chemical v(t, x, y) = concentration of other chemical Example of a typical R-D model [ u u 2 fv ] (u q) (u + q) u t = D 1 2 u + 1 ɛ v t = D 2 2 v + u v Here everything except u, v are constant parameters. This is called the Oregonator model. See typical solution patterns on the website pardhan/kaset/workshop/

34 PDE models Models and research results from various other application areas.