Numerical Methods in Cancer Models
|
|
- Rafe Lynch
- 5 years ago
- Views:
Transcription
1 Numerical Methods in Cancer Models Doron Levy Department of Mathematics and Center for Scientific Computation and Mathematical Modeling (CSCAMM) University of Maryland, College Park
2 Plan Motivation 1 What is cancer? 2 Delayed di erential equations 3 Agent-based models 4 PDEs
3 Delayed Di erential Equations in Cancer Models Analysis & Numerical Methods Doron Levy Department of Mathematics and Center for Scientific Computation and Mathematical Modeling (CSCAMM) University of Maryland, College Park
4 Outline Motivation 1 Motivation Cancer Immunology Stem Cell Transplantation 2 Zero Crossings Time Scales 3 Numerical Methods 4
5 Outline Motivation The immune response to leukemia Stem cell transplantation 1 Motivation Cancer Immunology Stem Cell Transplantation 2 Zero Crossings Time Scales 3 Numerical Methods 4
6 Motivation The immune response to leukemia Stem cell transplantation What is leukemia? Normal cells: stem cells turn into mature cells Leukemia: A malignant transformation of a stem cell or a progenitor cell Myeloid or Lymphocytic Acute or Chronic Doron Levy Montre al, May 2013
7 CML Motivation The immune response to leukemia Stem cell transplantation 3phases Chronic: uncontrolled proliferation Accelerated Acute: Uncontrolled proliferations. Cells do not mature Philadelphia chromosome Translocation (9;22) Oncogenic BCR-ABL gene fusion The ABL gene expresses a tyrosine kinase. Growth mechanisms Easy to diagnose Drug targeting this genetic defect (a tyrosine kinase inhibitor)
8 Treating leukemia Motivation The immune response to leukemia Stem cell transplantation Chemotherapy Bone Marrow or Stem Cell transplant Chemo + radiotherapy + transplantation Imatinib (Gleevec) Molecular targeted therapy - suppresses the corrupted control system $32K-$98K/year
9 Problems with existing therapies The immune response to leukemia Stem cell transplantation Remission vs. Cure: Can CML be cured? Yes! but only with a bone marrow (or stem cell) transplant Requires a (matching) donor Ariskyprocedure(+unpredictablesidee ects) Imatinib? Does not cure the disease: stopping it causes a relapse New Medical Data: There is an anti-leukemia immune response (Lee lab) The strength and dynamics of the specific anti-leukemia immune response can be measured Number of cells Activity (count signaling molecules)
10 The observed immune response The immune response to leukemia Stem cell transplantation A di erent immune response for each patient. However: Question: At the beginning of the treatment: no immune response Peak: around 6-12 months (after starting the drug treatment) Later: waning immune response What is the relation between the dynamics of the cancer, the drug, and the immune response?
11 Amathematicalmodel The immune response to leukemia Stem cell transplantation r y divide x2 n nτ Leukemic stem cell (y 0 ) a y (a y ) Progenitor b y (b y ) Differentiated c y (c y ) cell (y 1 ) cell (y 2 ) Terminal cell (y 3 ) s T T cells (T) p 0 e -cnc kc d 0 + q C p(c,t) d 1 + q C p(c,t) d 2 + q C p(c,t) d 3 + q C p(c,t) d T Ingredients: Leukemia cells: stem cells,, fully functional cells Mutations, Drug (Imatinib), Anti leukemia immune response Kim, Lee, Levy: PLoS Computational Biology, 08 Michor et al. (Nature 05). Cronkite and Vincent (69), Rubinow (69), Rubinow & Lebowitz (75), Fokas, Keller, and Clarkson (91), Mackey et al (99,...), Neiman (00), Moore & Li (04), Michor et al (05), Komarova & Woodarz (05).
12 Michor s model + immune response The immune response to leukemia Stem cell transplantation Cells without mutations: Anti-cancer T cells: ẏ 0 = [r y (1 u) d 0 ]y 0 q c p(c, T )y 0, ẏ 1 = a y y 0 d 1 y 1 q c p(c, T )y 1, ẏ 2 = b y y 1 d 2 y 2 q c p(c, T )y 2, ẏ 3 = c y y 2 d 3 y 3 q c p(c, T )y 3. T = s t d t T p(c, T )C +2 n q T p(c n, T n )C n, p(c, T ) = p 0 e cnc kt, C = X (y i + z i ), C n = C(t n ).
13 The immune response to leukemia Stem cell transplantation Accounting for the immune response Cell Concentration (k/µl) Leukemia No immune response 0.01 T cells Time (months)
14 Stopping imatinib (simulation) The immune response to leukemia Stem cell transplantation Cell Concentration (k/µl) x T cell concentration Imatinib removed at month 12 Leukemia Time (months)
15 The immune response to leukemia Stem cell transplantation Cancer vaccines: a mathematical design Cell Concentration (k/µl) Leukemia cells Inactivated leukemia cells (vaccines) Anti-leukemia T cells Time (months) log 10 [concentration] log 10 [concentration] SC DC Time (months) PC TC Time (months) Inactivated leukemia cells Anti-cancer T cells V = d V V q c p(c, T )V + s v (t) T = s t d t T p(c, T )(C + V )+2 n p(c n T n )(q T C n + V n )
16 Model populations Motivation The immune response to leukemia Stem cell transplantation Host Cells Cancer Anti-donor T cells General blood cells Donor cells Anti-cancer T cells (cancer-specific) Anti-host T cells General blood cells
17 Everything takes time Motivation The immune response to leukemia Stem cell transplantation
18 Anti-Cancer T Cells Motivation The immune response to leukemia Stem cell transplantation σ σ TC/C Interaction Ignore React Proliferate nτ q 1 TC/C TC/C p 2 TC/C p 1 ρ Reload q 2 TC/C q 3 TC/C kct C x2 n υ T C d TC Death Die or Become anergic kt D T C p 2 TD/TC p 1 TD/TC Survive Perish TD/TC Interaction
19 Anti-Host T Cells Motivation The immune response to leukemia Stem cell transplantation TH /C Interaction Ignore React q 3 TH/C Die or become anergic TH/C p 2 pth/c 1 TH /H Interaction σ kct H Proliferate qth/c 1 nτ ρ Reload x2 n υ q 2 TH/C σ kht H nτ Ignore React pth/h 2 pth/h 1 ρ q 1 TH/H x2 n υ q 2 TH/H T H d TH Death kt D T H σ Proliferate nτ x2 n Reload p 2 TD/TH p 1 TD/TH υ Survive Perish q 2 TH/TD q 1 TH/TD p 2 TH/TD p 1 TH/TD ρ No flow qth/td 3 =0 Killed by T D Ignore React T H /T D Interaction
20 Anti-Donor T Cells Motivation The immune response to leukemia Stem cell transplantation TD/TC Interaction Ignore React No flow qtd/tc 3 = 0 T D /D Interaction ptd/tc 2 ptd/tc 1 q 2 TD/TC σ kt C T D Proliferate qtd/tc 1 nτ ρ Reload x2 n υ σ kdt D nτ Ignore React ptd/d 2 ptd/d 1 ρ q 1 TD/D x2 n υ q 2 TD/D T D d TD Death p 2 TH/TD σ Survive kt H T Perish D pth/td 1 Proliferate nτ x2 n υ Reload q 2 TD/TH ptd/th 2 ptd/th 1 q 1 TD/TH ρ No flow qtd/th 3 =0 Killed by T H Ignore React T D /T H Interaction
21 General Donor and Host Blood Cells The immune response to leukemia Stem cell transplantation Stem Cells Stem Cells S D S H D H d D p TD/D 1 kdt D Perish ρ TD/D Interaction d H p 1 TH/H kht H Perish ρ TH/H Interaction Death Death
22 Cancer Cells Motivation The immune response to leukemia Stem cell transplantation C/TH Interaction Perish pc/th 1 kt H C logistic growth r C C p 1 C/TC kt C C Perish C/TC Interaction Death rate is included in net logistic growth term ρ Death ρ
23 Equations... Motivation The immune response to leukemia Stem cell transplantation dt H dt = d TH T H kct H kt D T H kht H + p T H /C 2 kc(t )T H (t )+p T D /T H 2 p T H /T D 2 kt D (t )T H (t ) + p T H /H 2 kh(t )T H (t ) +2 n p T H /C 1 q T H /C 1 kc(t n )T H (t n ) +2 n p T H /H 1 q T H /H 1 kh(t n )T H (t n ) +2 n p T D /T H 2 p T H /T D 1 q T H /T D 1 kt D (t n )T H (t n ) + p T H /C 1 q T H /C 2 kc(t )T H (t ) + p T H /H 1 q T H /H 2 kh(t )T H (t ) + p T D /T H 2 p T H /T D 1 q T H /T D 2 kt D (t )T H (t ).
24 Time Delays Motivation The immune response to leukemia Stem cell transplantation Time for reactive T cell-antigen interaction = 5min Time for unreactive interactions = 1min Time for cell division = day T cell recovery time after killing another cell = 1 day
25 Relapse Motivation The immune response to leukemia Stem cell transplantation Cell Concentration in 10 3 cells/µl General host cells H Anti host T cells T H Cell Concentration in 10 3 cells/µl 6 x Cancer cells C eventually overwhelms T H Time in Days never goes to Time in Days
26 Remission Motivation The immune response to leukemia Stem cell transplantation Cell Concentration in 10 3 cells/µl General host cells H Anti host cells T H Cell Concentration in 10 3 cells/µl 6 x Cancer cells C C = 0 at time Time in Days Time in Days
27 Oscillations Motivation The immune response to leukemia Stem cell transplantation Cell Concentration in 10 3 cells/µl A: Stable oscillation Anti host T cells Cancer cells General host cells Cell Concentration in 10 3 cells/µl B: Unstable Oscillation Time in Days Time in Days
28 Extinction instead of stability The immune response to leukemia Stem cell transplantation Cell Concentration in 10 3 cells/µl Without state constraint Anti host T cells T H Cancer cells C The value of C crosses 0 at time Time in Days Cell Concentration in 10 3 cells/µl With state constraint Cancer cells C Anti host T cells T H The value of C crosses 0 at time , and does not recover Time in Days
29 The immune response to leukemia Stem cell transplantation Initial anti-host cells vs. initial host cells Initial anti host T cell concentration T H,0 (cells/µl) Results up to time 2000 Successful Unsuccessful Unresolved Higher initial host blood cell concentrations improve the chances of a successful cure Greater initial anti-host T cell concentrations slightly favor the chances of cure Initial host cell concentration H 0 (10 3 cells/µl)
30 The immune response to leukemia Stem cell transplantation Number of cell divisions vs. cancer growth rate Average # of T cell divisions n Results up to time 1000 Successful Unsuccessful Unresolved A higher average number of T cell divisions favor complete remission Higher cancer growth rate make complete remission slightly more likely Cancer growth rate r C (1/day)
31 Outline Motivation Zero Crossing Time Scales 1 Motivation Cancer Immunology Stem Cell Transplantation 2 Zero Crossings Time Scales 3 Numerical Methods 4
32 Zero crossing: Example Zero Crossing Time Scales AsimpleDDE: dx dt = rx(t 1), x(t) =1, t < 0. Solve: t 2 [0, 1). Then dx dt = rx(t 1) = r x = rt + c =1 rt. If r > 0thenx(t) =0forT 2 [0, 1)
33 Zero crossing: example Zero Crossing Time Scales AsimpleDDE: dx dt = rx(t 1), x(t) =1, t < 0. Proceed: t 2 [1, 2) Then dx dt = rx(t 1) = r + r 2 t r 2 x =1 rt + r 2 2 (t 1)2.
34 Zero crossing: example Zero Crossing Time Scales AsimpleDDE: dx dt = rx(t 1), x(t) =1, t < 0. The general solution: t 2 [n, n + 1) Xn+1 k (t k + 1)k x(t) = ( r) k! k=0 Question: For what r does that exist a T 2 [n, n + 1) such that Xn+1 k (T k + 1)k ( r) k! k=0 = 0?
35 Zero crossing: example Zero Crossing Time Scales AsimpleDDE: dx dt = rx(t 1), x(t) =1, t < 0. The general solution: t 2 [n, n + 1) Xn+1 k (t k + 1)k x(t) = ( r) k! k=0 Question: For what r does that exist a T 2 [n, n + 1) such that Xn+1 k (T k + 1)k ( r) k! k=0 = 0?
36 Time Scales Motivation Zero Crossing Time Scales A toy problem x n+1 = (1 y n )x n y n+1 = x 2 n + k + y n The map iterated twice x n+2 = (1+x 2 n k y n )(1 y n )x n y n+2 = ((1 y n ) 2 + 1)x 2 n +2k + y n
37 Time Scales Motivation Zero Crossing Time Scales A toy problem x n+1 = (1 y n )x n y n+1 = x 2 n + k + y n The map iterated twice x n+2 = (1+x 2 n k y n )(1 y n )x n y n+2 = ((1 y n ) 2 + 1)x 2 n +2k + y n
38 This corresponds to... Motivation Zero Crossing Time Scales
39 Outline Motivation 1 Motivation Cancer Immunology Stem Cell Transplantation 2 Zero Crossings Time Scales 3 Numerical Methods 4
40 Approach #1 Motivation A mesh in time that is based on the delay. Numerical methods for ODEs that use the mesh points only (multistep methods) Example: y 0 (t) =f (t, y(t), y(t (t))), t 0 apple t apple t f, y(t) = (t), t apple t 0. Asetofmeshpoints: ={t 0, t 1...,t N = t f }, such that t n (t n) 2. Forward Euler: y n+1 = y n + h n+1f (t n, y n, y q), q < n. Same idea with Adams-like methods, Heun, etc.
41 Approach #2: Feldstein Free the mesh selection from the delay Use extranodal points for the approximation of the delayed term y(t (t)). Example: (t 0 apple (t) apple t) y 0 (t) =f (t, y(t), y( (t))), t 0 apple t apple t f, Assume h =(t f y(t 0 )=y 0, t 0)/m. Foranyt n = t 0 + nh, define (tn) t 0 q(n) =floor. h Anumericalmethod: yn+1 = y n + hf (t n, y n, z n), y 0 = y(t 0), where z n = y q(n), i.e., a piecewise-constant approximation of y( (t)). Alternatively, a piecewise-linear approximation: z n =(1 r(n))y q(n) + r(n)y q(n)+1.
42 Approach #3: Bellman s method of steps Assume a constant delay. In the first interval [t 0, t 0 + ] theddehasthe form: y 0 (t) =f (y(t), (t )), y(t 0 )= (t 0 ). In the second interval [t 0 +,t 0 +2 ], define y 1 = y(t y 2 (t) =y(t). Then: 8 y1 >< 0(t) =f (t,y 1(t), (t 2 )), y2 0(t) =f (t, y 2(t), y 1 (t)), y 1 (t 0 + ) = (t 0 ), >: y 2 (t 0 + ) =y(t 0 + ) ) and And so on... For every timestep, a larger system. However, this system can be solved using standard methods for ODEs.
43 Approach #4: Methods based on continuous extensions Continuous Extension: very low cost method to get an accurate approximation of the solution at every point in the interval (t n + h n+1 ) = n,1( )y n n,in+1( )y n in 0 +h n+1 (y n,...,y n in ;, g, n), 0 apple apple 1. where g (f, y) =f (t, y, (t (t, y))). This is how every Matlab routine provides solutions at the sampled points
44 Approach #4: Methods based on continuous extensions Consider the DDE y 0 (t) =f (t, y(t), y(t (t, y(t)))), t 0 apple t apple t f, y(t) = (t), t apple t 0. Using continuous extensions, solving the DDE amounts to solving the ODE: w 0 n+1 (t) =f (t, w n+1 (t), x(t (t, w n+1 (t)))), t n apple t apple t n+1 where w n+1 (t n )=y n, 8 < (s), s apple t 0, x(s) = (s), t 0 apple s apple t n, : w n+1 (s), t n apple s apple t n+1, and is the continuous extension interpolant.
45 Additional Reading Bellen and Zennaro, Numerical Methods for Delay Di erential Equations, Oxford Shampine and Thompson, Numerical Solution of Delay Di erential Equations
46 Outline Motivation 1 Motivation Cancer Immunology Stem Cell Transplantation 2 Zero Crossings Time Scales 3 Numerical Methods 4
47 SQRT = ^ ; = D X DT DX T T DT DX DT DX T T DT X T! T T # d 2x dx dx(t 1 ) dx(t 2 )! x = 0 2 dt" dt dt dt #ROSSING SET 7 ;SQRT SQRT SQRT DX T T DT DX DT X DX T T DT # 5NMARKED AREAS IN THE FIGURE ABOVE REGIONS STABLE REPRESENT! DENOTE THE NUMBER OF 4HE NUMBERS T 5NMARKED AREAS IN THE FIGURE ABOVE REPRESENT STABLE REGIONS 4HE NUMBERS DENOTE THE NUMBER OF EIGENVALUES IN THE RIGHT HALF PLANE.OTICE THAT # REPRESENTS AN ISOLATED STABLE REGION " " # # X CORRESPONDING "LUE LINESDENOTE BRANCHES TO INDICES OF TYPE U V 'REEN LINES DENOTETBRANCHES CORRESPONDING T TO INDICES OF TYPE U V "!! T T T T T T T T 5NMARKED IN THE FIGURE ABOVE 5NMARKED IN THE FIGURE ABOVE " AREASAREAS REPRESENT STABLE REGIONS REPRESENT STABLE REGIONS 4HE4HE NUMBERS DENOTE THE THE NUMBER OF OF NUMBERS DENOTE NUMBER EIGENVALUES IN THE RIGHTRIGHT HALFHALF PLANE EIGENVALUES IN THE PLANE X ISOLATED.OTICE THAT # REPRESENTS AN STABLE REGION EIGENVALUES IN THE RIGHT HALF PLANE!! SQRT = ^ ; = #ROSSING SET 7 ;SQRT SQRT SQRT SQRT = ^ = ; = #ROSSING SET 7 SQRT SQRT SQRT ^ ; = T ;SQRT T T D X DT DX T T DT DX T T X X D X DTDX DT DX DT DX T T DT DX T T DT DT T D X DT T Motivation #ROSSING SET 7 ;SQRT SQRT SQRT numerical methods X X T TT T T T " "#.OTICE THATTHAT # REPRESENTS AN ISOLATED.OTICE # REPRESENTS AN ISOLATED STABLE REGION STABLE REGION "LUE LINES DENOTE BRANCHES CORRESPONDING TO INDICES OF TYPE U V 'REEN LINES DENOTE BRANCHES CORRESPONDING TO INDICES OF TYPE U V "LUE LINES BRANCHES CORRESPONDING "LUEDENOTE LINES DENOTE BRANCHES CORRESPONDING TO INDICES OF TYPE TO INDICES OF U TYPEV U V 'REEN'REEN LINES DENOTE BRANCHES CORRESPONDING LINES DENOTE BRANCHES CORRESPONDING TO INDICES OF TYPE V U V TO INDICES OF U TYPE T T # Doron Levy T T T T Montre al, May 2013
48 Stability crossing curves Ref: Gu, Niculescu, Chen, On stability crossing curves for general systems with two delays, J. Mathematical Analysis & Applications, 311 (2005), pp Stability crossing curves: The set of delays for which the characteristic equation has at least one imaginary zero (or pair of imaginary zeros). Associated with change of stability
49 The two delays case A DDE with two constant delays x(t)+c 1 x(t 1 )+c2x(t 2 )+c 3 x 0 (t)+c 4 x 0 (t 1 )+c5x 0 (t 2 )=0. The characteristic equation: h(s) =h 0 (s)+h 1 (s)e 1s + h 2 (s)e 2s. Let a k (s) =h k (s)/h 0 (s). Then a(s, 1, 2 )=1+a 1 (s)e 1s + a 2 (s)e 2s =0. Stability: a question of the number of the roots of the characteristic equation with a real part on the right hand side of the plane.
50 The two delays case For an imaginary s = i! to satisfy a(s, 1, 2 ) = 0, the vector corresponding to the three terms must form a triangle: a(s, 1, 2 )=1+a 1 (s)e 1s + a 2 (s)e 2s =0. Hence, their magnitudes must satisfy the triangle inequalities: a 1 (i!) + a 2 (i!) 1, 1 apple a 1 (i!) a 2 (i!) apple1.
51 The two delays case The triangle inequalities determine which i! may be zeros of a(s) The set of all such! are the crossing set. Any given! defines a collection of pairs ( 1, 2 ). 1 = \a 1(i!)+(2u 1) ± 1! 2 = \a 2(i!)+(2v 1) 2! where from the law of cosine: 0, u = u ± 0, u± 0 +1,... 0, u = v ± 0, v ± 0 +1,... 1,2 = cos 1 1+ a1,2 (i!) 2 a 2,1 (i!) 2 2 a 1,2 (i!), and u ± 0, v ± 0 are the smallest possible integers such that the corresponding 1,2 are nonnegative.
52 The two delays case The crossing set always consists of a finite number of intervals of finite length Any interval of! s defines a collection of curves in R 2 The general case is a union of the following sets:
53 Example Motivation DDE: dx dt = 2x(t 1)+x(t 2 ).
STABILITY CROSSING BOUNDARIES OF DELAY SYSTEMS MODELING IMMUNE DYNAMICS IN LEUKEMIA. Silviu-Iulian Niculescu. Peter S. Kim. Keqin Gu. Peter P.
DISCRETE AND CONTINUOUS doi:.3934/dcdsb.2.3.29 DYNAMICAL SYSTEMS SERIES B Volume 3, Number, January 2 pp. 29 56 STABILITY CROSSING BOUNDARIES OF DELAY SYSTEMS MODELING IMMUNE DYNAMICS IN LEUKEMIA Silviu-Iulian
More informationA LYAPUNOV-KRASOVSKII FUNCTIONAL FOR A COMPLEX SYSTEM OF DELAY-DIFFERENTIAL EQUATIONS
U.P.B. Sci. Bull., Series A, Vol. 77, Iss., 015 ISSN 13-707 A LYAPUNOV-KRASOVSKII FUNCTIONAL FOR A COMPLEX SYSTEM OF DELAY-DIFFERENTIAL EQUATIONS Irina Badralexi 1, Andrei Halanay, Ileana Rodica Rădulescu
More informationGRÜNWALD-LETNIKOV SCHEME FOR SYSTEM OF CHRONIC MYELOGENOUS LEUKEMIA FRACTIONAL DIFFERENTIAL EQUATIONS AND ITS OPTIMAL CONTROL OF DRUG TREATMENT
GRÜNWALD-LETNIKOV SHEME FOR SYSTEM OF HRONI MYELOGENOUS LEUKEMIA FRATIONAL DIFFERENTIAL EQUATIONS AND ITS OPTIMAL ONTROL OF DRUG TREATMENT ESMAIL HESAMEDDINI AND MAHIN AZIZI DEPARTMENT OF MATHEMATIAL SIENES,
More informationPeriodic Solutions in a Mathematical Model for the Treatment of Chronic Myelogenous Leukemia
Math. Model. Nat. Phenom. Vol. 7, No.,, pp. 35-44 DOI:.5/mmnp/7 Periodic Solutions in a Mathematical Model for the Treatment of Chronic Myelogenous Leukemia A. Halanay Department of Mathematics I, Politehnica
More informationNew Computational Tools for Modeling Chronic Myelogenous Leukemia
Math. Model. Nat. Phenom. Vol. 4, No. 2, 2009, pp. 119-139 DOI: 10.1051/mmnp/20094206 New Computational Tools for Modeling Chronic Myelogenous Leukemia M. M. Peet 1, P. S. Kim 2, S.-I. Niculescu 3 and
More informationA Mathematical Model of Chronic Myelogenous Leukemia
A Mathematical Model of Chronic Myelogenous Leukemia Brent Neiman University College Oxford University Candidate for Master of Science in Mathematical Modelling and Scientific Computing MSc Dissertation
More informationDynamic Prediction of Disease Progression Using Longitudinal Biomarker Data
Dynamic Prediction of Disease Progression Using Longitudinal Biomarker Data Xuelin Huang Department of Biostatistics M. D. Anderson Cancer Center The University of Texas Joint Work with Jing Ning, Sangbum
More informationLecture 1 From Continuous-Time to Discrete-Time
Lecture From Continuous-Time to Discrete-Time Outline. Continuous and Discrete-Time Signals and Systems................. What is a signal?................................2 What is a system?.............................
More informationarxiv: v1 [q-bio.to] 22 Mar 2019
A QUANTITATIVE STUDY ON THE ROLE OF TKI COMBINED WITH WNT/β-CATENIN SIGNALING AND IFN-α IN THE TREATMENT OF CML THROUGH DETERMINISTIC AND STOCHASTIC APPROACHES SONJOY PAN SOUMYENDU RAHA SIDDHARTHA P. CHAKRABARTY
More informationSolving Delay Differential Equations (DDEs) using Nakashima s 2 Stages 4 th Order Pseudo-Runge-Kutta Method
World Applied Sciences Journal (Special Issue of Applied Math): 8-86, 3 ISSN 88-495; IDOSI Publications, 3 DOI:.589/idosi.wasj.3..am.43 Solving Delay Differential Equations (DDEs) using Naashima s Stages
More informationMulti-state Models: An Overview
Multi-state Models: An Overview Andrew Titman Lancaster University 14 April 2016 Overview Introduction to multi-state modelling Examples of applications Continuously observed processes Intermittently observed
More informationStability Analysis of a Simplified Yet Complete Model for Chronic Myelogenous Leukemia
Stability Analysis of a Simplified Yet Complete Model for Chronic Myelogenous Leukemia Marie Doumic-Jauffret,2 Peter S. Kim 3 Benoît Perthame,2 March 4, 2 To appear in Bulletin of Mathematical Biology
More informationSection 1.1 Algorithms. Key terms: Algorithm definition. Example from Trapezoidal Rule Outline of corresponding algorithm Absolute Error
Section 1.1 Algorithms Key terms: Algorithm definition Example from Trapezoidal Rule Outline of corresponding algorithm Absolute Error Approximating square roots Iterative method Diagram of a general iterative
More informationModeling with differential equations
Mathematical Modeling Lia Vas Modeling with differential equations When trying to predict the future value, one follows the following basic idea. Future value = present value + change. From this idea,
More informationHepatitis C Mathematical Model
Hepatitis C Mathematical Model Syed Ali Raza May 18, 2012 1 Introduction Hepatitis C is an infectious disease that really harms the liver. It is caused by the hepatitis C virus. The infection leads to
More informationSTABILITY ANALYSIS OF DELAY DIFFERENTIAL EQUATIONS WITH TWO DISCRETE DELAYS
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 20, Number 4, Winter 2012 STABILITY ANALYSIS OF DELAY DIFFERENTIAL EQUATIONS WITH TWO DISCRETE DELAYS XIHUI LIN AND HAO WANG ABSTRACT. We use an algebraic
More informationFuzzy modeling and control of HIV infection
Fuy modeling and control of HIV infection Petrehuş Paul, Zsófia Lendek Department of Automation Technical University of Cluj Napoca Memorandumului 28, 4114, Cluj Napoca, Romania Emails: {paul.petrehus,
More informationChapter 7 Fall Chapter 7 Hypothesis testing Hypotheses of interest: (A) 1-sample
Bios 323: Applied Survival Analysis Qingxia (Cindy) Chen Chapter 7 Fall 2012 Chapter 7 Hypothesis testing Hypotheses of interest: (A) 1-sample H 0 : S(t) = S 0 (t), where S 0 ( ) is known survival function,
More informationEigenvalues and eigenvectors.
Eigenvalues and eigenvectors. Example. A population of a certain animal can be broken into three classes: eggs, juveniles, and adults. Eggs take one year to become juveniles, and juveniles take one year
More informationChapter 4 Fall Notations: t 1 < t 2 < < t D, D unique death times. d j = # deaths at t j = n. Y j = # at risk /alive at t j = n
Bios 323: Applied Survival Analysis Qingxia (Cindy) Chen Chapter 4 Fall 2012 4.2 Estimators of the survival and cumulative hazard functions for RC data Suppose X is a continuous random failure time with
More informationMODELLING TUMOUR-IMMUNITY INTERACTIONS WITH DIFFERENT STIMULATION FUNCTIONS
Int. J. Appl. Math. Comput. Sci., 2003, Vol. 13, No. 3, 307 315 MODELLING TUMOUR-IMMUNITY INTERACTIONS WITH DIFFERENT STIMULATION FUNCTIONS PETAR ZHIVKOV, JACEK WANIEWSKI Institute of Applied Mathematics
More informationMathematics, Box F, Brown University, Providence RI 02912, Web site:
April 24, 2012 Jerome L, Stein 1 In their article Dynamics of cancer recurrence, J. Foo and K. Leder (F-L, 2012), were concerned with the timing of cancer recurrence. The cancer cell population consists
More informationTransport Equations in Biology; Adaptive Dynamics
Transport Equations in Biology; Adaptive Dynamics Matt Becker University of Maryland, College Park Applied Mathematics, Statistics, and Scientific Computation March 26, 2014 Becker Cancer RIT University
More informationAsynchronous oscillations due to antigenic variation in Malaria Pf
Asynchronous oscillations due to antigenic variation in Malaria Pf Jonathan L. Mitchell and Thomas W. Carr Department of Mathematics Southern Methodist University Dallas, TX SIAM LS, Pittsburgh, 21 Outline
More information6.003 Homework #1. Problems. Due at the beginning of recitation on September 14, 2011.
6.003 Homework #1 Due at the beginning of recitation on September 14, 2011. Problems 1. Solving differential equations Solve the following differential equation y(t) + 3 dy(t) + 2 d2 y(t) = 1 dt dt 2 for
More informationA Population-level Hybrid Model of Tumour-Immune System Interplay: model construction and analysis.
A Population-level Hybrid Model of Tumour-Immune System Interplay: model construction and analysis. Giulio Caravagna Dipartimento di Informatica, Sistemistica e Comunicazione, Università degli Studi Milano-Bicocca.
More informationModified Milne Simpson Method for Solving Differential Equations
Modified Milne Simpson Method for Solving Differential Equations R. K. Devkate 1, R. M. Dhaigude 2 Research Student, Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad,
More informationMultistate models in survival and event history analysis
Multistate models in survival and event history analysis Dorota M. Dabrowska UCLA November 8, 2011 Research supported by the grant R01 AI067943 from NIAID. The content is solely the responsibility of the
More informationMath 345 Intro to Math Biology Lecture 19: Models of Molecular Events and Biochemistry
Math 345 Intro to Math Biology Lecture 19: Models of Molecular Events and Biochemistry Junping Shi College of William and Mary, USA Molecular biology and Biochemical kinetics Molecular biology is one of
More informationCS520: numerical ODEs (Ch.2)
.. CS520: numerical ODEs (Ch.2) Uri Ascher Department of Computer Science University of British Columbia ascher@cs.ubc.ca people.cs.ubc.ca/ ascher/520.html Uri Ascher (UBC) CPSC 520: ODEs (Ch. 2) Fall
More informationLecture 9. Statistics Survival Analysis. Presented February 23, Dan Gillen Department of Statistics University of California, Irvine
Statistics 255 - Survival Analysis Presented February 23, 2016 Dan Gillen Department of Statistics University of California, Irvine 9.1 Survival analysis involves subjects moving through time Hazard may
More informationOrdinary differential equations - Initial value problems
Education has produced a vast population able to read but unable to distinguish what is worth reading. G.M. TREVELYAN Chapter 6 Ordinary differential equations - Initial value problems In this chapter
More informationLecture 4: Numerical solution of ordinary differential equations
Lecture 4: Numerical solution of ordinary differential equations Department of Mathematics, ETH Zürich General explicit one-step method: Consistency; Stability; Convergence. High-order methods: Taylor
More informationIntroduction to stochastic multiscale modelling of tumour growth
Introduction to stochastic multiscale modelling of tumour growth Tomás Alarcón ICREA & Centre de Recerca Matemàtica T. Alarcón (ICREA & CRM, Barcelona, Spain) Lecture 1 CIMPA, Santiago de Cuba, June 2016
More informationAntibody Identification I
Antibody Identification I Amanda Smith, MLS(ASCP) CM Immunohematologist Laboratory of Immunohematology and Genomics April 18, 2017 New York Blood Center 1 Introduction Red cell alloantibodies other than
More informationChronic Granulomatous Disease Medical Management
Chronic Granulomatous Disease Medical Management N I C H O L A S H A R T O G, M D D i r e c t o r o f P e d i a t r i c / A d u l t P r i m a r y I m m u n o d e f i c i e n c y C l i n i c A s s i s t
More informationIntermediate Differential Equations. John A. Burns
Intermediate Differential Equations Delay Differential Equations John A. Burns jaburns@vt.edu Interdisciplinary Center for Applied Mathematics Virginia Polytechnic Institute and State University Blacksburg,
More informationLinear Multistep Methods I: Adams and BDF Methods
Linear Multistep Methods I: Adams and BDF Methods Varun Shankar January 1, 016 1 Introduction In our review of 5610 material, we have discussed polynomial interpolation and its application to generating
More informationA model for transfer phenomena in structured populations
A model for transfer phenomena in structured populations Peter Hinow 1, Pierre Magal 2, Glenn F. Webb 3 1 Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455,
More informationx(t) = x(t h), x(t) 2 R ), where is the time delay, the transfer function for such a e s Figure 1: Simple Time Delay Block Diagram e i! =1 \e i!t =!
1 Time-Delay Systems 1.1 Introduction Recitation Notes: Time Delays and Nyquist Plots Review In control systems a challenging area is operating in the presence of delays. Delays can be attributed to acquiring
More informationChapter 2: Time-Domain Representations of Linear Time-Invariant Systems. Chih-Wei Liu
Chapter : Time-Domain Representations of Linear Time-Invariant Systems Chih-Wei Liu Outline Characteristics of Systems Described by Differential and Difference Equations Block Diagram Representations State-Variable
More informationESE 531: Digital Signal Processing
ESE 531: Digital Signal Processing Lec 9: February 13th, 2018 Downsampling/Upsampling and Practical Interpolation Lecture Outline! CT processing of DT signals! Downsampling! Upsampling 2 Continuous-Time
More informationMathematical models of discrete and continuous cell differentiation
Mathematical models of discrete and continuous cell differentiation Anna Marciniak-Czochra Institute of Applied Mathematics, Interdisciplinary Center for Scientific Computing (IWR) and BIOQUANT University
More informationIntroduction to some topics in Mathematical Oncology
Introduction to some topics in Mathematical Oncology Franco Flandoli, University of Pisa y, Finance and Physics, Berlin 2014 The field received considerable attentions in the past 10 years One of the plenary
More informationarxiv: v1 [q-bio.to] 21 Apr 2018
arxiv:1805.01009v1 [q-bio.to] 21 Apr 2018 Approximate Analytical Solution of a Cancer Immunotherapy Model by the Application of Differential Transform and Adomian Decomposition Methods Abstract Alireza
More informationODE Homework 1. Due Wed. 19 August 2009; At the beginning of the class
ODE Homework Due Wed. 9 August 2009; At the beginning of the class. (a) Solve Lẏ + Ry = E sin(ωt) with y(0) = k () L, R, E, ω are positive constants. (b) What is the limit of the solution as ω 0? (c) Is
More informationResearch Article On the Stability Property of the Infection-Free Equilibrium of a Viral Infection Model
Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume, Article ID 644, 9 pages doi:.55//644 Research Article On the Stability Property of the Infection-Free Equilibrium of a Viral
More informationA Variable-Step Double-Integration Multi-Step Integrator
A Variable-Step Double-Integration Multi-Step Integrator Matt Berry Virginia Tech Liam Healy Naval Research Laboratory 1 Overview Background Motivation Derivation Preliminary Results Future Work 2 Background
More informationDiscrete versus continuous-time models of malaria infections
Discrete versus continuous-time models of malaria infections Level 2 module in Modelling course in population and evolutionary biology (701-1418-00) Module author: Lucy Crooks Course director: Sebastian
More informationMATH 100 Introduction to the Profession
MATH 100 Introduction to the Profession Modeling and the Exponential Function in MATLAB Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2011 fasshauer@iit.edu MATH
More informationSummer School on Delay Differential Equations and Applications
Summer School on Delay Differential Equations and Applications Dobbiaco (BZ), Italy, June 26 30, 2006 The numerical solution of delay differential equations Page 1 of 211 M. Zennaro Dipartimento di Matematica
More informationPackage threg. August 10, 2015
Package threg August 10, 2015 Title Threshold Regression Version 1.0.3 Date 2015-08-10 Author Tao Xiao Maintainer Tao Xiao Depends R (>= 2.10), survival, Formula Fit a threshold regression
More informationDynamical models of HIV-AIDS e ect on population growth
Dynamical models of HV-ADS e ect on population growth David Gurarie May 11, 2005 Abstract We review some known dynamical models of epidemics, given by coupled systems of di erential equations, and propose
More informationNumerical Algorithms for ODEs/DAEs (Transient Analysis)
Numerical Algorithms for ODEs/DAEs (Transient Analysis) Slide 1 Solving Differential Equation Systems d q ( x(t)) + f (x(t)) + b(t) = 0 dt DAEs: many types of solutions useful DC steady state: state no
More informationSome Results Concerning Uniqueness of Triangle Sequences
Some Results Concerning Uniqueness of Triangle Sequences T. Cheslack-Postava A. Diesl M. Lepinski A. Schuyler August 12 1999 Abstract In this paper we will begin by reviewing the triangle iteration. We
More informationArmitage and Doll (1954) Probability models for cancer development and progression. Incidence of Retinoblastoma. Why a power law?
Probability models for cancer development and progression Armitage and Doll (95) log-log plots of incidence versus age Rick Durrett *** John Mayberry (Cornell) Stephen Moseley (Cornell) Deena Schmidt (MBI)
More informationThe second-order 1D wave equation
C The second-order D wave equation C. Homogeneous wave equation with constant speed The simplest form of the second-order wave equation is given by: x 2 = Like the first-order wave equation, it responds
More informationNumerical Methods - Initial Value Problems for ODEs
Numerical Methods - Initial Value Problems for ODEs Y. K. Goh Universiti Tunku Abdul Rahman 2013 Y. K. Goh (UTAR) Numerical Methods - Initial Value Problems for ODEs 2013 1 / 43 Outline 1 Initial Value
More informationLAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC
LAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC R. G. DOLGOARSHINNYKH Abstract. We establish law of large numbers for SIRS stochastic epidemic processes: as the population size increases the paths of SIRS epidemic
More informationAn Optimal Control Approach to Cancer Treatment under Immunological Activity
An Optimal Control Approach to Cancer Treatment under Immunological Activity Urszula Ledzewicz and Mohammad Naghnaeian Dept. of Mathematics and Statistics, Southern Illinois University at Edwardsville,
More information5. Parametric Regression Model
5. Parametric Regression Model The Accelerated Failure Time (AFT) Model Denote by S (t) and S 2 (t) the survival functions of two populations. The AFT model says that there is a constant c > 0 such that
More informationMathematical Modeling of Immune Responses to Hepatitis C Virus Infection
East Tennessee State University Digital Commons @ East Tennessee State University Electronic Theses and Dissertations 12-2014 Mathematical Modeling of Immune Responses to Hepatitis C Virus Infection Ivan
More informationQ2 (4.6) Put the following in order from biggest to smallest: Gene DNA Cell Chromosome Nucleus. Q8 (Biology) (4.6)
Q1 (4.6) What is variation? Q2 (4.6) Put the following in order from biggest to smallest: Gene DNA Cell Chromosome Nucleus Q3 (4.6) What are genes? Q4 (4.6) What sort of reproduction produces genetically
More informationNuclear Spectroscopy: Radioactivity and Half Life
Particle and Spectroscopy: and Half Life 02/08/2018 My Office Hours: Thursday 1:00-3:00 PM 212 Keen Building Outline 1 2 3 4 5 Some nuclei are unstable and decay spontaneously into two or more particles.
More informationMultistep Methods for IVPs. t 0 < t < T
Multistep Methods for IVPs We are still considering the IVP dy dt = f(t,y) t 0 < t < T y(t 0 ) = y 0 So far we have looked at Euler s method, which was a first order method and Runge Kutta (RK) methods
More information1.2. Introduction to Modeling. P (t) = r P (t) (b) When r > 0 this is the exponential growth equation.
G. NAGY ODE January 9, 2018 1 1.2. Introduction to Modeling Section Objective(s): Review of Exponential Growth. The Logistic Population Model. Competing Species Model. Overview of Mathematical Models.
More informationMath 308 Exam I Practice Problems
Math 308 Exam I Practice Problems This review should not be used as your sole source of preparation for the exam. You should also re-work all examples given in lecture and all suggested homework problems..
More informationDepartmental Curriculum Planning
Department: Btec Subject: Biology Key Stage: 4 Year Group: 10 Learning aim A: Investigate the relationships that different organisms have with each other and with their environment Learning aim B: Demonstrate
More informationAdaptive Prediction of Event Times in Clinical Trials
Adaptive Prediction of Event Times in Clinical Trials Yu Lan Southern Methodist University Advisor: Daniel F. Heitjan May 8, 2017 Yu Lan (SMU) May 8, 2017 1 / 19 Clinical Trial Prediction Event-based trials:
More informationModeling the Immune System W9. Ordinary Differential Equations as Macroscopic Modeling Tool
Modeling the Immune System W9 Ordinary Differential Equations as Macroscopic Modeling Tool 1 Lecture Notes for ODE Models We use the lecture notes Theoretical Fysiology 2006 by Rob de Boer, U. Utrecht
More information12/1/17 OUTLINE KEY POINTS ELEMENTS WITH UNSTABLE NUCLEI Radioisotopes and Nuclear Reactions 16.2 Biological Effects of Nuclear Radiation
OUTLINE 16.1 Radioisotopes and Nuclear Reactions 16.2 Biological Effects of Nuclear Radiation PET scan X-ray technology CT scan 2009 W.H. Freeman KEY POINTS Radioactivity is the consequence of an unstable
More informationComputational Genetics Winter 2013 Lecture 10. Eleazar Eskin University of California, Los Angeles
Computational Genetics Winter 2013 Lecture 10 Eleazar Eskin University of California, Los ngeles Pair End Sequencing Lecture 10. February 20th, 2013 (Slides from Ben Raphael) Chromosome Painting: Normal
More informationIntersection of Objects with Linear and Angular Velocities using Oriented Bounding Boxes
Intersection of Objects with Linear and Angular Velocities using Oriented Bounding Boxes David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the
More informationNonlinear Observers. Jaime A. Moreno. Eléctrica y Computación Instituto de Ingeniería Universidad Nacional Autónoma de México
Nonlinear Observers Jaime A. Moreno JMorenoP@ii.unam.mx Eléctrica y Computación Instituto de Ingeniería Universidad Nacional Autónoma de México XVI Congreso Latinoamericano de Control Automático October
More informationIn Silico Design of New Drugs for Myeloid Leukemia Treatment
In Silico Design of New Drugs for Myeloid Leukemia Treatment Washington Pereira and Ihosvany Camps Computational Modeling Laboratory LaModel Exact Science Institute ICEx Federal University of Alfenas -
More informationHomework 3 posted, due Tuesday, November 29.
Classification of Birth-Death Chains Tuesday, November 08, 2011 2:02 PM Homework 3 posted, due Tuesday, November 29. Continuing with our classification of birth-death chains on nonnegative integers. Last
More informationSolving Ordinary Differential Equations
Solving Ordinary Differential Equations Sanzheng Qiao Department of Computing and Software McMaster University March, 2014 Outline 1 Initial Value Problem Euler s Method Runge-Kutta Methods Multistep Methods
More informationENGI 9420 Lecture Notes 8 - PDEs Page 8.01
ENGI 940 ecture Notes 8 - PDEs Page 8.0 8. Partial Differential Equations Partial differential equations (PDEs) are equations involving functions of more than one variable and their partial derivatives
More informationDelay Differential Equations Part II: Time and State dependent Lags
Delay Differential Equations Part II: Time and State dependent Lags L.F. Shampine Department of Mathematics Southern Methodist University Dallas, Texas 75275 shampine@smu.edu www.faculty.smu.edu/shampine
More informationCHAPTER 3 HEART AND LUNG TRANSPLANTATION. Editors: Mr Mohamed Ezani Md. Taib Dato Dr David Chew Soon Ping Dr Ashari Yunus
CHAPTER 3 HEART AND LUNG TRANSPLANTATION Editors: Mr Mohamed Ezani Md. Taib Dato Dr David Chew Soon Ping Dr Ashari Yunus Expert Panel: Mr Mohamed Ezani Md. Taib (Chairperson) Dr Abdul Rais Sanusi Datuk
More informationAPPM 2360 Lab 3: Zombies! Due April 27, 2017 by 11:59 pm
APPM 2360 Lab 3: Zombies! Due April 27, 2017 by 11:59 pm 1 Introduction As you already know, in the past month, zombies have overrun much of North America, including all major cities on both the East and
More informationBiology Eighth Edition Neil Campbell and Jane Reece
BIG IDEA I The process of evolution drives the diversity and unity of life. Enduring Understanding 1.C Life continues to evolve within a changing environment. Essential Knowledge 1.C.1 Speciation and extinction
More informationIntroduction to numerical simulations for Stochastic ODEs
Introduction to numerical simulations for Stochastic ODEs Xingye Kan Illinois Institute of Technology Department of Applied Mathematics Chicago, IL 60616 August 9, 2010 Outline 1 Preliminaries 2 Numerical
More informationMathematical Modeling for the (Mathematically) Faint of Heart
Mathematical Modeling for the (Mathematically) Faint of Heart Jennifer Galovich St. John s University (MN) and Virginia Bioinformatics Institute MD-VA-DC MAA Section Meeting April 13, 2013 How can we develop
More informationCompeting paths over fitness valleys in growing populations. Michael D. Nicholson 1 and Tibor Antal 2
Competing paths over fitness valleys in growing populations Michael D. Nicholson 1 and Tibor Antal 2 1 School of Physics and Astronomy, University of Edinburgh 2 School of Mathematics, University of Edinburgh
More informationModeling using conservation laws. Let u(x, t) = density (heat, momentum, probability,...) so that. u dx = amount in region R Ω. R
Modeling using conservation laws Let u(x, t) = density (heat, momentum, probability,...) so that u dx = amount in region R Ω. R Modeling using conservation laws Let u(x, t) = density (heat, momentum, probability,...)
More informationObservability. It was the property in Lyapunov stability which allowed us to resolve that
Observability We have seen observability twice already It was the property which permitted us to retrieve the initial state from the initial data {u(0),y(0),u(1),y(1),...,u(n 1),y(n 1)} It was the property
More informationDynamics of Erythroid Progenitors and Erythroleukemia
Math. Model. Nat. Phenom. Vol. 4, No. 3, 2009, pp. 210-232 DOI: 10.1051/mmnp/20094309 Dynamics of Erythroid Progenitors and Erythroleukemia N. Bessonov a,b, F. Crauste b1, I. Demin b, V. Volpert b a Institute
More informationCHAPTER 3 HEART AND LUNG TRANSPLANTATION. Editors: Mr Mohamed Ezani Md. Taib Dato Dr David Chew Soon Ping Dr Ashari Yunus
CHAPTER 3 Editors: Mr Mohamed Ezani Md. Taib Dato Dr David Chew Soon Ping Dr Ashari Yunus Expert Panel: Mr Mohamed Ezani Md. Taib (Chairperson) Dr Abdul Rais Sanusi Datuk Dr Aizai Azan Abdul Rahim Dr Ashari
More informationPlant and animal cells (eukaryotic cells) have a cell membrane, cytoplasm and genetic material enclosed in a nucleus.
4.1 Cell biology Cells are the basic unit of all forms of life. In this section we explore how structural differences between types of cells enables them to perform specific functions within the organism.
More informationControlling Systemic Inflammation Using NMPC. Using Nonlinear Model Predictive Control with State Estimation
Controlling Systemic Inflammation Using Nonlinear Model Predictive Control with State Estimation Gregory Zitelli, Judy Day July 2013 With generous support from the NSF, Award 1122462 Motivation We re going
More informationAP Biology Notes Outline Enduring Understanding 1.C. Big Idea 1: The process of evolution drives the diversity and unity of life.
AP Biology Notes Outline Enduring Understanding 1.C Big Idea 1: The process of evolution drives the diversity and unity of life. Enduring Understanding 1.C: Life continues to evolve within a changing environment.
More informationu n 2 4 u n 36 u n 1, n 1.
Exercise 1 Let (u n ) be the sequence defined by Set v n = u n 1 x+ u n and f (x) = 4 x. 1. Solve the equations f (x) = 1 and f (x) =. u 0 = 0, n Z +, u n+1 = u n + 4 u n.. Prove that if u n < 1, then
More informationBasic Procedures for Common Problems
Basic Procedures for Common Problems ECHE 550, Fall 2002 Steady State Multivariable Modeling and Control 1 Determine what variables are available to manipulate (inputs, u) and what variables are available
More informationNumerical solution of ODEs
Numerical solution of ODEs Arne Morten Kvarving Department of Mathematical Sciences Norwegian University of Science and Technology November 5 2007 Problem and solution strategy We want to find an approximation
More informationScientific Computing with Case Studies SIAM Press, Lecture Notes for Unit V Solution of
Scientific Computing with Case Studies SIAM Press, 2009 http://www.cs.umd.edu/users/oleary/sccswebpage Lecture Notes for Unit V Solution of Differential Equations Part 1 Dianne P. O Leary c 2008 1 The
More information14 Branching processes
4 BRANCHING PROCESSES 6 4 Branching processes In this chapter we will consider a rom model for population growth in the absence of spatial or any other resource constraints. So, consider a population of
More informationENGI 9420 Lecture Notes 8 - PDEs Page 8.01
ENGI 940 Lecture Notes 8 - PDEs Page 8.01 8. Partial Differential Equations Partial differential equations (PDEs) are equations involving functions of more than one variable and their partial derivatives
More informationSpatial Moran model. Rick Durrett... A report on joint work with Jasmine Foo, Kevin Leder (Minnesota), and Marc Ryser (postdoc at Duke)
Spatial Moran model Rick Durrett... A report on joint work with Jasmine Foo, Kevin Leder (Minnesota), and Marc Ryser (postdoc at Duke) Rick Durrett (Duke) ASU 1 / 27 Stan Ulam once said: I have sunk so
More informationTopic 3: Fourier Series (FS)
ELEC264: Signals And Systems Topic 3: Fourier Series (FS) o o o o Introduction to frequency analysis of signals CT FS Fourier series of CT periodic signals Signal Symmetry and CT Fourier Series Properties
More information