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1/957 Space8 Space IIInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

Space2/95Mathematicsand StatisticsZZdω ωM MMathematics 4MB3/6MB3Mathematical BiologyInstructor: David EarnLecture 7SpaceMonday 28 October 2019Instructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

Space3/95AnnouncmentsMidterm test:Date: Monday 4 November 2019Time: 11:30am–1:30pmLocation: in class, ETB-237Assignment 4 is due the day of the midterm.Due Monday 4 November 2019 before class.Make sure you personally can do the question on calculatingR0 on this assignment before the midterm test.Instructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

Space4/95Spatial Epidemic DynamicsInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

Space5/95Something to think aboutAll of our analysis has been of temporal patterns of epidemicsWhat about spatial patterns?What problems are suggested by observed spatial epidemicpatterns?Can spatial epidemic data suggest improved strategies forcontrol?Can we reduce the eradication threshold below pcrit 1 R10 ?Instructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

Space6/95Measles and Whooping Cough in 60 UK citiesMeaslesWhoopingCoughRohani, Earn & Grenfell (1999) Science 286, 968–971Instructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

Space7/95Better Control? Eradication?The term-time forced SEIR model successfully predicts pastpatterns of epidemics of childhood diseasesCan we manipulate epidemics predictably so as to increaseprobability of eradication?Can we eradicate measles?Instructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

Space8/95Idea for eradicating measlesTry to re-synchronize measles epidemics in the UK and,moreover, synchronize measles epidemics worldwide:synchrony is goodDevise new vaccination strategy that tends to synchronize. . .Avoid spatially structured epidemics. . .Time to think about the mathematics of synchrony. . .But analytical theory of synchrony in a periodically forcedsystem of differential equations is mathematicallydemanding. . .So let’s consider a much simpler biological model. . .Instructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceThe Logistic Map9/95TheLogistic MapInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceThe Logistic Map10/95Logistic MapSimplest non-trivial discrete time population model for asingle species (with non-overlapping generations) in a singlehabitat patch.Time: t 0, 1, 2, 3, . . .State: x [0, 1](population density)Population density at time t is x t . Solutions are sequences:x 0, x 1, x 2, . . .x t 1 F (x t ) for some reproduction function F (x ).For logistic map: F (x ) rx (1 x ), so x t 1 rx t (1 x t ).x t 1 [r (1 x t )]x t r is maximum fecundity (which isachieved in limit of very small population density).What kinds of dynamics are possible for the Logistic Map?Instructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceLogistic Map Time Series,11/95r 0.5x t 1 rx t (1 x t ),1.0The Logistic Mapr 0.5,x0 0.636620.80.6xt0.40.20.001020304050Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceLogistic Map Time Series,12/95r 0.9x t 1 rx t (1 x t ),1.0The Logistic Mapr 0.9,x0 0.636620.80.6xt0.40.20.001020304050Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceLogistic Map Time Series,13/95r 1x t 1 rx t (1 x t ),1.0The Logistic Mapr 1,x0 0.636620.80.6xt0.40.20.001020304050Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceLogistic Map Time Series,14/95r 1.1x t 1 rx t (1 x t ),1.0The Logistic Mapr 1.1,x0 0.636620.80.6xt0.40.20.001020304050Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceLogistic Map Time Series,15/95r 1.5x t 1 rx t (1 x t ),1.0The Logistic Mapr 1.5,x0 0.636620.80.6xt0.40.20.001020304050Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceLogistic Map Time Series,16/95r 2x t 1 rx t (1 x t ),1.0The Logistic Mapr 2,x0 0.318310.80.6xt0.40.20.001020304050Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceLogistic Map Time Series,17/95r 2.5x t 1 rx t (1 x t ),1.0The Logistic Mapr 2.5,x0 0.318310.80.6xt0.40.20.001020304050Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceLogistic Map Time Series,18/95r 3x t 1 rx t (1 x t ),1.0The Logistic Mapr 3,x0 0.318310.80.6xt0.40.20.001020304050Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceLogistic Map Time Series,19/95r 3.2x t 1 rx t (1 x t ),1.0The Logistic Mapr 3.2,x0 0.318310.80.6xt0.40.20.001020304050Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceLogistic Map Time Series,20/95r 3.5x t 1 rx t (1 x t ),1.0The Logistic Mapr 3.5,x0 0.318310.80.6xt0.40.20.001020304050Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceLogistic Map Time Series,21/95r 3.75x t 1 rx t (1 x t ),1.0The Logistic Mapr 3.75,x0 0.318310.80.6xt0.40.20.001020304050Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceLogistic Map Time Series,22/95r 3.83x t 1 rx t (1 x t ),1.0The Logistic Mapr 3.83,x0 0.318310.80.6xt0.40.20.001020304050Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceLogistic Map Time Series,23/95r 4x t 1 rx t (1 x t ),1.0The Logistic Mapr 4,x0 0.318310.80.6xt0.40.20.001020304050Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceThe Logistic Map24/95Logistic Map SummaryTime series show:r 1 Extinction.1 r 3 Persistence at equilibrium.r 3 period doubling cascade to chaos, then appearanceof cycles of all possible lengths, and more chaos, . . .How can we summarize this in a diagram?Bifurcation diagram (wrt r ).Ignore transient behaviour: just show attractor.Instructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceThe Logistic Map25/95Logistic Map, F (x ) rx (1 x ), 1 r 40.8x0.60.40.21.01.52.02.53.03.54.0rInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceThe Logistic Map26/95Logistic Map, F (x ) rx (1 x ), 2.9 r 40.8x0.60.40.23.03.23.43.63.84.0rInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceThe Logistic Map27/95Logistic Map, F (x ) rx (1 x ), 3.4 r 40.8x0.60.40.23.53.63.73.83.94.0rInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceThe Logistic Map28/95Logistic Map as a Tool to Investigate SynchronyVery simple single-patch model: only one state variable.Displays all kinds of dynamics from GAS equilibrium, toperiodic orbits, to chaos.This was extremely surprising to population biologists andmathematicians in the 1970s.May RM (1976) “Simple mathematical models with very complicated dynamics” Nature 261, 459–467Easier to work with logistic map as single patch dynamicsthan SIR or SEIR model.Can still understand how synchrony works conceptually.Now we are ready for the . . . . . Mathematics of Synchrony . . .Instructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony29/95Mathematics of SynchronySystem comprised of isolated patchese.g., cities, labelled i 1, . . . , nState of system in patch i specified by xie.g., xi (Si , Ei , Ii , Ri )Connectivity of patches specified by a dispersal matrixM (mij )System is coherent (perfectly synchronous) if the state is thesame in all patchesi.e., x1 x2 · · · xnInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony30/95Illustrative example: logistic metapopulationSingle patch model: x t 1 F (x t )Reproduction function: F (x ) rx (1 x )Multi-patch model:xit 1 t 1 xi.e.,nXj 1mij F (xj t )m11 · · · m1nF (x1t ) . . . . . . .tt 1mn1 · · · mnnF (xn )xn1 where M (mij ) is dispersal matrix.Colour coding of indices:row indices are redcolumn indices are cyanInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony31/95Basic properties of dispersal matrices M (mij )Discrete-time metapopulation model:xit 1 nXj 1mij F (xjt ),i 1, 2, . . . , n.mij proportion of population in patch jthat disperses to patch i. 0 mij 1 for all i and j(each mij is non-negative and at most 1)Total proportion that leaves or stays in patch j:(sum of column j) nXi 1mij 1nXi 1mij(every column sums to at most 1)Could be 1 if some individuals are lost (die) while dispersing.Instructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony32/95Basic properties of dispersal matrices M (mij )Discrete-time metapopulation model:xit 1 nXj 1mij F (xjt ),i 1, 2, . . . , n.Definition (No loss dispersal matrix)An n n matrix M (mij ) is said to be a no loss dispersalmatrix if all its entries are non-negative (mij 0 for all i and j)and its column sums are all 1, i.e.,nXi 1mij 1 ,for each j 1, . . . , n.The dispersal process is “conservative” in this case.A no loss dispersal matrix is also said to be “column stochastic”.Instructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony33/95Notation for coherent statesDiscrete-time metapopulation model:xit 1 nXj 1mij F (xjt ),i 1, 2, . . . , n.State at time t is xt (x1t , . . . , xnt ) Rn .If state x is coherent, then for some x R we havex (x1 , x2 , . . . , xn ) (x , x , . . . , x ) x (1, 1, . . . , 1)For convenience, definee (1, 1, . . . , 1) Rnso any coherent state can be written x e, for some x R.Instructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony34/95Constraint on row sums of dispersal matrix MLemma (Row sums are the same)If all initially coherent states remain coherent then the row sums ofthe dispersal matrix are all the same.Proof.Suppose initially coherent states remain coherent, i.e.,xt ae xt 1 be for some b R.Choose a such that F (a) 6 0. Thenxit 1 b nXj 1 mij F (xjt ) nXj 1mij Instructor: David EarnnXj 1bF (a)mij F (a) F (a)nXj 1mij(independent of i)Mathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony35/95Constraint on row sums of dispersal matrix MLemma (Row sums are all 1)If every solution {x t } of the single patch map F (x ) yields acoherent solution {x t e} of the full map then the row sums of thedispersal matrix are all 1.Proof.Suppose xt ae xt 1 F (a)e and F (a) 6 0. Thenxit 1 F (a) nXj 1 mij F (xjt ) nXj 1mij 1Instructor: David EarnnXj 1mij F (a) F (a)nXj 1mij(independent of i)Mathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony36/95ProjectYou should be thinking about your Project. . .Remember your group must give an oral presentation of yourproject as well (in the last class).Classes after the midterm are NOT optional. Your group isexpected to meet in class and take advantage of theinstructor’s presence to solve issues with your project.Project Notebook template is posted on project page.Movie night?Instructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony37/95Midterm TestStudent Name:Student Number:MATHEMATICS 4MB3/6MB3Midterm Test, Monday 11 March 2019Special Instructions and Notes:(i) This test has 12 pages. Verify that your copy is complete. Note that the final twopages are blank to provide additional space if needed.(ii) Clearly write your name and student number at the top of each page.(iii) Answer all questions in the space provided.(iv) It is possible to obtain a total of 50 marks. There are 10 multiple choicequestions (2 marks each) and 10 short answer questions (total of 30 marks).(v) For multiple choice questions, circle only one answer.(vi) No calculators, notes, or aids of any kind are permitted.(vii) PHAC refers to the Public Health Agency of Canada.GOOD LUCKInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony38/95Midterm TestThe test will cover everything from lectures andassignments/solutions up to and including today.Material connected with time series analysis andsynchrony/coherence will occur only in multiple choicequestions.You are assumed to be comfortable with:Elementary algebra, including finding the eigenvalues of 2 2matrices.Stability analyses of differential equations.Finding R0 by biological and mathematical [ρ(FV 1 )]methods.Converting flow charts or verbal descriptions intocompartmental ODE models.You will be presented with scenarios including graphs, andasked to write explanations that would be understandable bypeople at PHAC.Instructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony39/95Let’s review what we’ve done so far on spatial models. . .Logistic metapopulation modelNotion of coherenceNo-loss dispersal matrix M: column sums are all 1To retain homogeneous solutions: row sums are all 1Instructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony40/95Simple examples of no loss dispersal matricesEqual coupling: a proportion m from each patch dispersesuniformly among the other n 1 patches:mij 1 mi jm/(n 1) i 6 j(Nearest-neighbour coupling: a proportion m go to the twonearest patches:mij 1 mm/2 0i ji j 1 or j 1 (mod n)otherwiseReal dispersal patterns generally between these two extremesInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony41/95Key QuestionCan we find conditions on the dispersal matrix M, and/or thesingle patch reproduction function F , that guarantee (orpreclude) coherence asymptotically (as t )?If so, then this sort of analysis should help to identifysynchronizing vaccination strategies.Instructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony42/95Logistic Metapopulation Simulation (r 1, m 0.2)n 10,r 1,m 0.2,λ 5432109876543210987654321098765432150Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony43/95Logistic Metapopulation Simulation (r 2, m 0.2)n 10,r 2,m 0.2,λ 2109876543210987654321111101020304050Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony44/95Logistic Metapopulation Simulation (r 2, m 0.2)n 10,r 2,m 0.2,λ 0.7781.0kx 9876543210987654321111101020304050Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony45/95Logistic Metapopulation Simulation (r 2, m 0.02)n 10,r 2,m 0.02,λ 11102102120.0987654302021111020304050Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony46/95Logistic Metapopulation Simulation (r 2, m 0.02)n 10,r 2,m 0.02,λ 0.9781.0kx 02102120.0987654302021111020304050Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony47/95Logistic Metapopulation Simulation (r 2, m 0)n 10,r 2,m 0,λ 1111111102020220.0101111020304050Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony48/95Logistic Metapopulation Simulation (r 2, m 0)n 10,r 2,m 0,λ 11.0kx 11111111102020220.0101111020304050Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony49/95Logistic Metapopulation Simulation (r 3.2, m 0.2)n 10,r 3.2,m 0.2,λ 10987654321150240.2320.01101020304050Time tInstructor: David EarnMathematics 4MB3/6MB3Mathematical Biology

SpaceSynchrony50/95Logistic Metapopulation Simulation (r 3.2, m 0.2)n 10,r 3.2,m 0.2,λ 0.7781.0kx 3456

Space 2/95 Mathematics and Statistics Z M dω Z M ω Mathematics 4MB3/6MB3 Mathematical Biology Inst

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