Intermediate Differential Equations Delay Differential Equations John A. Burns jaburns@vt.edu Interdisciplinary Center for Applied Mathematics Virginia Polytechnic Institute and State University Blacksburg, Virginia 2461-531 MATH 4245 - Fall 212
Population Dynamics Use growth of protozoa as example A population could be Bacteria, Viruses Cells (Cancer, T-cells ) People, Fish, Cows, Things that live and die t time p( t) b( t) d( t) sec, hrs, days, years. Number of cells at time t Probability that a cell divides in unit time at time t Probability that a cell dies in unit time at time t ASSUME a closed population
Population Dynamics Number of new cells on t, t t Number of cell deaths on Change in cell population t, t t b( t) p( t) t d( t) p( t) t p( t t) p( t) b( t) p( t) t d( t) p( t) t p( t t) t p( t) b( t) d( t) p( t) r( t) p( t) TAKE LIMIT AS t Fundamental LAW d of population growth p( t) r( t) p( t) dt Thomas R. Malthus (1766-1834)
Population Dynamics b( t) d( t) r( t) BIRTH RATE DEATH RATE GROWTH RATE d dt DO AN EXPERIMENT p ( ) p 1 r 1 p(1) 1e 25 Malthus ASSUMED CONSTANT RATES p( t) r p( t) r p( t) e p t b( t) d( t) r( t) r e r b r d 25 1 r ln e b 2.5 r.9163 d ln(2.5)
Population After 5 Days 1 9 8 p( t) 1 e.9163t 7 6 5 4 3 2 1.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Population After 7.5 Days 1 9 8 p( t) 1 e.9163t 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8
Population After 1 days 1 x 14 9 8 p( t) 1 e.9163t 7 6 NOT WHAT REALLY HAPPENS 5 4 3 2 1 1 2 3 4 5 6 7 8 9 1
Incubation Period Fundamental LAW d p( t) r( t) p( t) of population growth dt More accurate? -- d p ( t ) r ( t ) p ( t r ) dt r > is an incubation period r() t r b d
Example d p ( t ) p ( t ) dt d p t dt ( ) p ( t /2) NOT a good model of a population growth
Improved Model COMPETITION FOR FOOD AND SPACE Malthus ASSUMED PLENTY OF SPACE AND FOOD Pierre-Fancois Verhulst (184-1849) ) ( ) (, ) ( 1 1 t p d d t d p(t) b b t b ) ( 1 1 ) ( 1 ) ( 1 1 t p K r t p r d b r t r ) ( ) ( ) ( ) ( 1 1 t p d b d b t d t b t r
Logistics Growth Rate K b 1 r d 1 p( t) r( t) p( t) 1 r( t) r 1 p ( t ) K LE 1 p( t) r 1 p( t) p( t) K K b 1 r d CARRYING CAPACITY 1 NATURAL REPRODUCTIVE RATE IS THIS A BETTER (MORE ACCURATE) MODEL??
A Comparison: First 5 Days p( t) ( ) r p t 1 p( t) r 1 p( t) p( t) K 1 9 8 p( t) 1 e.9163t 1 9 8 p(t) 7 7 6 6 5 5 4 4 3 3 2 2 1 1.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Malthusian LAW of population growth.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Logistic LAW of population growth
A Comparison: First 7.5 Days p( t) ( ) r p t 1 p( t) r 1 p( t) p( t) K 1 9 5 1, 5, 45 8 4 7 35 6 3 5 25 4 2 3 15 2 1 1 5 1 2 3 4 5 6 7 8 Malthusian LAW of population growth 1 2 3 4 5 6 7 8 Logistic LAW of population growth
A Comparison: First 1 Days p( t) ( ) r p t 1 p( t) r 1 p( t) p( t) K 1 x 14 1, 9 9, 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 1 2 3 4 5 6 7 8 9 1 Malthusian LAW of population growth 1 2 3 4 5 6 7 8 9 1 Logistic LAW of population growth
Logistic Equation: 15 Days HOWEVER WE OFTEN OBSERVE 1 p( t) r 1 p( t) p( t) K 1 9 8 7 p(t) K 6 5 4 3 2 1 5 1 15
Periodic Populations: Blowflys DATA Hutchinson s Equation
Delayed Logistics Equation r b( t) b b p(t r), d( t) d 1 r b d r( t) b( t) d( t) b d b p( t r) K 1 r( t) r 1 p( t r) K 1 r b 1 HE 1 ( ) 1 ( ) ( ) p t r p t r p t K
1 p( t) 1 p( t) p( t) 1 Delayed Logistic 1 p t p t p t 1 ( ) 1 ( /2) ( ) p() 1 p( s) 1, s s
Different Initial Functions 1 p t p t p t 1 ( ) 1 ( /2) ( ) p( s) 1, s s p s e s s 5s ( ),
Different Initial Functions 1 p t p t p t 1 ( ) 1 ( /2) ( ) p( s) (1 s) / r, s p( s) (1 s) / r, s s s
Predator - Prey Models PREDATOR macrophage PREY bacteria
Predator - Prey Models PREDATOR Macrophage PREY Ecoli
Interacting Species Predator - Prey Models Vito Volterra Model (1925) Alfred Lotka Model (1926) x( t) y( t) a, b, c, d THINK OF SHARKS AND SHARK FOOD Numbers of prey Numbers of predators Parameters d x ( t ) x ( t ) a by ( t ) dt d y ( t ) y ( t ) c dx ( t ) dt
Delayed Predator - Prey Models IT TAKES SOME FINITE TIME FOR THE PREDATOR TO NOTE THAT THE FOOD (PREY) HAS INCREASED HENCE THE RATE OF INCREASE IN THE PREDATOR POPULATION IS DELAYED x( t) Numbers of prey y( t) Numbers of predators r delayed response a, b, c, d Parameters d x ( t ) x ( t ) a by ( t ) dt d y ( t ) y ( t ) c dx ( t r ) dt Delayed Logistic Equation
Immune System & HIV HIV VIRUS CD4+T
Immune System & HIV T(t) concentration of uninfected targeted helper T cells, T * (t) concentration of infected T cells producing virus, V(t) concentration of virus. MORE ACCURATE Perelson, Banks.
Cancer Models
Background In the U.S. 4% chance for the average person to develop cancer Breast cancer is the 2nd most common cancer among American women Risk factors: incidence in family, oral contraceptives, obesity Normal cells have many checkpoints During checkpoints reproduction is stopped if abnormality is detected Cancer cells don t have these checkpoint. Unmanageable proliferation leads to loss of genetic information Recent cancer cells more mutated than older cancer cells.
The Cancer Cell Cycle 4 stages to cell cycle: G1 (presynthetic) S (synthetic) G2 (postsynthetic) mitosis G (quiescent) Immune cells cytotoxic T-cells flow increases to area of tumor cells INTERFACE STAGE Paclitaxel is a common drug used for Breast, Ovarian, Head and Neck Cancer - attack tumor cells during a cell cycle
Cell Population Dynamics T () t I -- Population of cells in Interface stage T () M t -- Population of cells in Mitosis stage T () Q t -- Population of cells in Quiescent stage It () ct () -- Population of Immune cells (cytotoxic T-cells) -- Concentration of the drug Paclitaxel G. Newbury, A Numerical Study of a Delay Differential Equation Model for Breast Cancer, MS Thesis, Department of Mathematics, Virginia Tech, Blacksburg, VA, August, 27. M. Villasana and G. Ochoa, Heuristic Design of Cancer Chemotherapies, IEEE Transactions of Evolutionary Computation, 8 (24), 513-521. R. Yafia, Dynamics Analysis and Limit Cycle in a Delayed Model for Tumor Growth with Quiescence, Nonlinear Analysis, Modeling and Control, 11 (26), 95 11.
Delay Equation Model (1) (2) (3) (4) dtq () t a5ti ( t ) a6tq ( t) d4tq ( t) c5tq ( t) I( t) u1( t) TQ ( t) dt dti () t 2 a4tm ( t) a5ti ( t ) a6tq ( t) c1t I ( t) I( t) d2ti ( t) a1t I ( t ) dt dtm () t dt a T ( t ) d T ( t) a T ( t) c T ( t) I( t) u ( t) T ( t) 1 I 3 M 4 M 3 M 2 M n di() t I( t)[ TQ ( t) TI ( t) TM ( t)] k c n dt [ T ( t) T ( t) T ( t)] Q I M 4 M 6 Q 1 3 2 I( t) T ( t) c I( t) T ( t) c I( t) T ( t) d I( t) u () t I( t) I u () t g ( w( t), c( t) ) i i
Delay Equation Model (5) (6) dw1 () t dt dw2 () t dt w ( t) c( t), w () 1 2 1 1 w ( t) c( t), w () 2 2 w( t) r w ( t) r w ( t) 1 1 2 2 To compare with existing models. u t 1 k6wt () () k (1 e ) k2wt () k4wt () u () t k (1e ) u () t k (1e ) 5 2 1 We also investigated the ODE model: = 3 3
Typical Parameters and Inputs concentration c(t) T T [ T (), T (), T (), I()] [.8, 1.3, 1.2, ].9 Q I M T T [ T (), T (), T (), I()] [.7,.8,.6, ].12 Q I M lim It ( ).12 t NORMAL LEVEL 1.9.8.7.6.5.4.3.2.1 PARAMETERS 5 1 15 2 25 3 35 4 time t c(t) -- Pulsed concentration of drug
Typical Short Time Simulation T T [ T (), T (), T (), I()] [.7,.8,.6, ].9 Q I M? IS THE DRUG WORKING?
Longer Time Simulation T T [ T (), T (), T (), I()] [.7,.8,.6, ].9 Q I M!! NO!!! SOLUTION GOES TO + AS t +? HOW DO WE KNOW THIS WILL HAPPEN? 3 years
{ (IVP) Delay Differential Equations (Σ) (IC ) (IC 1 ) x( t) f x( t), x( t r), q x( t ) x R x( s) ( s), r s n n n m n f ( x, z, q) : D R R R R EXAMPLES
Example 1 x( t) cos( t) x( t) x( t / 2), x() 1 x( t) sin( t) cos( t / 2) x( t / 2) x( t) cos( t) sin( t) x( t) sin( t) cos( t) cos( t / 2) sin( t / 2) x( t / 2)
Example 1 1.5 1.5 -.5-1 -1.5 5 1 15
Example 2 x( t) x( t 1), x() 1 3.5 3 2.5 x t 2 ( ) ( t 1) / 2 3/ 2 2 1.5 x() t t 1.5 xt ( ) 1.5 1 1.5 2 2.5 3
Example 2 x( t) x( t 1), x() 1 8 7 6 5 x t t t t 2 3 ( ) 1 ( 1) / 2 ( 2) /3! 4 3 2 x t t t 2 ( ) 1 ( 1) / 2 1 x( t) t 1.5 1 1.5 2 2.5 3? WHAT IS A REASONABLE INITIAL VALUE PROBLEM?
Example 2 x( t) x( t 1), x() 1 3.5 3 2.5 2 1.5 1.5 1.5 1 1.5 2 2.5 3
Example 2 x( t) x( t 1), x() 1 8 7 6 5 4 3 2 1 1.5 1 1.5 2 2.5 3
IVP for Delay Equations x( t) f ( x( t), x( t r)), x() x( s) ( s), r s () s r t r t
x( t) f ( x( t), x( t r)) x() Solution x( s) ( s), r s PAST HISTORY t x( t) f ( x( s), x( s r) ds x( t) f ( x( t), x( t r)) n n n f ( x, z) : D R R R
Existence n n n f ( x, z) : R R R Theorem D1. Assume f: R n R n ---> R n is a continuous function on R n R n and r >. If R n and () is continuous on the interval [-r,), then there exists at least one solution to the initial value problem (IVP) x( t) f ( x( t), x( t r)) x() x( s) ( s), r s UNIQUENESS IS MORE COMPLEX
Some Notation x ( ) :[ r,) R t n xt ( s) x( t s), r s x () t x () s t xt ( s) xt () r s t r t s t
xt Retarded Equations ( s) x( t s), r s x ( r) x( t r) t x ( s) x( s) ( s), r s x( t) f ( x( t), x( t r)) x() x( s) ( s), r s x( t) f ( x( t), x ( r)) x() x ( s) ( s), r s t n F( x, ( )) : R C[ r,] R F( x, ( )) f ( x, ( r)) n
Retarded Equations x( t) f ( x( t), x( t r)) x() x( t) f ( x( t), x ( r)) x() t x( s) ( s), r s x ( s) ( s), r s F( x, ( )) f ( x, ( r)) x( t) f ( x( t), x( t r)) F( x( t), x t ( )) ( x(), x ( )) (, ( )) n R C[ r,] STATE SPACE OF INITIAL DATA
More Retarded Equations x( t) x( t) P( s) x( t s) ds x() r x( s) ( s), r s F( x, ( )) x P( s) ( s) ds F( x( t), x ( )) x( t) P( s) x ( s) ds x( t) P( s) x( t s) ds t r t r r x( t) x( t) P( s) x( t s) ds F( x( t), xt ( )) r
Definition n F( x, ( )) : R C[ r,] R n The function F: R n C[-r,] ---> R n is Lipschitzian on the open set R n C[-r,] if there constant > such that F( x, ( )) F( y, ( ) ( x y sup ( s) ( s) ) rs for all ( x, ( ), ( y, ( )) RC[ r,] FUNDAMENTAL UNIQUENESS THEOREM
Uniqueness Theorem Theorem D2. Assume F: R n C[-r,] ---> R n is continuous and Lipschitzian on every compact subset R n C[-r,]. If (, ()), then there exists a unique solution to the initial value problem (IVP) ( x(), x ( )) (, ( )) n F( x, ( )) : R C[ r,] R x( t) F( x( t), x t ( )) R C[ r,] n n Jack Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.
Applications in Life Sciences POPULATIONS WITH DELAYED RESPONSE TO A RESOURCE d N ( t ) r ( t ) N ( t ) dt 1 r( t) r 1 N( t) K LE d 1 N ( t ) r 1 N ( t ) N ( t ) dt K 1 r( t) r 1 N( t r) K HE d 1 N ( t ) r 1 N ( t r ) N ( t ) dt K
Hutchinson s Equation DATA Hutchinson s Equation
Delay Predator-Prey Models
Delay Predator-Prey Models
HIV Models T(t) concentration of uninfected targeted helper T cells, T * (t) concentration of infected T cells producing virus, V(t) concentration of virus. MORE ACCURATE WITH DELAY Nelson, Murray and Perelson
HIV Models MORE COMPLEX WITH CONTINUOUS DELAY
Delay Glucose-Insulation System
Delay Glucose-Insulation System
dt () t Delay Cancer Models a T ( t ) a T ( t) d T ( t) c T ( t) I( t) Q (1) 5 I 6 Q 4 Q 5 Q dt (2) dti () t dt 2 a T ( t) a T ( t ) a T ( t) c T ( t) I( t) d T ( t) a T ( t ) 4 M 5 I 6 Q 1 I 2 I 1 I (3) dtm () t dt a T ( t ) d T ( t) a T ( t) c T ( t) I( t) 1 I 3 M 4 M 3 M (4) n di() t I( t)[ TQ ( t) TI ( t) TM ( t)] k n dt [ T ( t) T ( t) T ( t)] Q I M c I( t) T ( t) c I( t) T ( t) d I( t) 4 M 6 Q 1 c 2 I( t) T ( t) I G. Newbury, A Numerical Study of a Delay Differential Equation Model for Breast Cancer, MS Thesis, Department of Mathematics, Virginia Tech, Blacksburg, VA, August, 27.
Delay Cancer Models
Delay Cancer Models
Delay Epidemic & Biochemical Networks
Delay Immune Response