M etodos Matem aticos e de Computa c ao I

Transcription

1 Métoos Matemáticos e e Computação I

2 Complex Systems 01/22 General structure Microscopic scale Iniviual behavior Description of the constituents Moel Macroscopic scale Collective behavior Emergence of regularities with size

3 Lotka-Volterra s moel 02/22 D. A. Mac Lulich, University of Toronto Stuies Biological Series 43 (1937)

4 Lotka-Volterra s moel 03/22 Population of hare H: Population of hare P: Population of lynx t H = αh βhp αh: Population of hares proportional to its size (Malthus moel) βhp: Decreasing of hare s population & interaction term (meeting of the two population - proportional to H an P)

5 Lotka-Volterra s moel 04/22 Population of lynx H: Population of hare P: Population of lynx t P = δp+γhp δp: Decreasing of lynx s population proportional to its size (eath rate) +γhp: Increasing of the lynx s population & interation term (malthusian effect & availability of sources - hare)

6 Lotka-Volterra s moel 05/22 Lotka-Volterra s equation H = αh βhp t P = δp + γhp t Mean-fiel moel Absence of geographic structure Nonlinear ifferential equation

7 Lotka-Volterra s moel (stationary solution) H = αh βhp t 06/22 P = δp + γhp t Stationary solution: t H(t) = t P(t) = 0 Trivial solution: H = P = 0 Solution: H = γ δ an P = α β Coexistence of the two populations

8 Lotka-Volterra s moel (reuce variables) H = αh βhp t 07/22 P = δp + γhp t Reefinition of the populations: h := γ δ H an p := β α P Rescaling of time: τ := t αδ Hare s birth - lynx eath ratio: ρ := α β h = ρh (1 p) τ τ p = 1 p (1 h) ρ Single parameter (ρ)

9 Lotka-Volterra s moel (reuce variables) h = ρh (1 p) τ τ p = 1 p (1 h) ρ ρ := α δ 08/22 Trivial solution: (h, p ) = (0, 0) Non-trivial solution: (h, p ) = (1, 1) Stability of the solutions?

10 Lotka-Volterra s moel (reuce variables) h = f (h, p) τ p = g(h, p) τ f (h, p) := ρh (1 p) g(h, p) := 1 ρ p (1 h) ρ := α δ 09/22 Trivial solution: (h, p ) = (0, 0) non-trivial solution: (h, p ) = (1, 1) τ h = f (h, p ) + h f (h, p ) (h h ) + p f (h, p ) (p p ) + τ τ p = g(h, p ) + h g(h, p ) (h h ) + p g(h, p ) (p p ) + ( ) h = J(h, p ) p ( ) h h p p J(h, p ) := ( h f (h, p ) h g(h, p ) p f ) (h, p ) p g(h, p )

11 Lotka-Volterra s moel (reuce variables) h = f (h, p) τ p = g(h, p) τ f (h, p) := ρh (1 p) g(h, p) := 1 ρ p (1 h) ρ := α δ 10/22 Stability analysis: eigenvalues of J ( J(h, p h ) := f (h, p ) p f ) ( ) (h, p ) ρ (1 p ) ρh h g(h, p ) p g(h, p = p ) ρ 1 ρ (1 h ) Eigenvalues of J (h = p = 0): ρ an 1 ρ (sale point) ( ) ( ) ( ) ( ) h 1 h 0 Unstable manifol: = Stable manifol: = p 0 p 1 Eigenvalues of J (h = p = 1): i an i (neutral stability)

12 h N. Boccara. Moeling Complex Systems. Lotka-Volterra s moel (reuce variables) h = f (h, p) τ p = g(h, p) τ f (h, p) := ρh (1 p) Phase portrait - trajectory in (h, p) space g(h, p) := 1 ρ p (1 h) ρ := α δ 11/22 ρ = 0.8 p h = ρ2 h p ( ) 1 p 1 h p h = p = 1 ln h h + ρ 2 (ln p p) = cte.

13 Population N. Boccara. Moeling Complex Systems. D. A. Mac Lulich, University of Toronto Stuies Biological Series 43 (1937) Lotka-Volterra s moel (reuce variables) h = f (h, p) τ p = g(h, p) τ Numerical solution f (h, p) := ρh (1 p) g(h, p) := 1 ρ p (1 h) ρ := α δ 12/22 t

14 Epiemiology 13/22 Moels for isease propagation Moels for gossip propagation Moels for infections Network resilience Percolation et cætera

15 virus isease Moels of epiemiology 14/22 SIS moel SIR moel et cætera Col Influenza... Ebola Chickenpox...

16 SIR moel 15/22 S λ I γ R S: Susceptible I: Infecte R: Remove/Recovere Mean-fiel version (complete graph) t S = λsi I = λsi γi t t R = γi

17 SIR moel 16/22 t S = λsi I = λsi γi t t R = γi t S + t I + t R = 0 S(t) + I (t) + R(t) = constant = N Stationary solution: I = 0 Absence of active state Trivial stationary solution: I = S = 0 Non-trivial stationary solution: I = 0 an S = ρ ρ := γ λ

18 SIR moel 17/22 t S = λsi I = λsi γi t t R = γi Initial conition: S(0) = S 0 > 0, I (0) = I 0 > 0 an R(0) = 0 [ ] t I (0) = λi 0 S 0 ρ J. D. Murray. Mathematical Biology (2002) > 0, S 0 > ρ < 0, S 0 < ρ ( ρ := γ ) λ t S(t) 0, t 0 ] [S t I (t) = λi ρ 0 if S 0 ρ

19 J. D. Murray. Mathematical Biology (2002) SIR moel 18/22 t S = λsi I = λsi γi t t R = γi R(0) = 0 S I = 1 + ρ S I + S ρ ln S = constant = I 0 + S 0 ρ ln S 0 Maximum infecte population s size: ( ) ( ) ρ ρ I max = I 0 + S 0 ρ + ρ ln = N ρ + ρ ln S 0 S 0

20 SIS moel 19/22 S λ I δ S S: Susceptible I: Infecte Mean-fiel (complete graph) S = λsi + δi t I = λsi δi t

21 SIS moel 20/22 S = λsi + δi t I = λsi δi t t S + t I = 0 S(t) + I (t) = constant = N Stationary solutions: Trivial stationary solutions: I = 0 an S = N Non-trivial stationary solutions: r := δ λ { S = r I = N r Active state

22 SIS moel 21/22 S = λsi + δi t I = λsi δi t t S + t I = 0 S(t) + I (t) = constant = N I (t) Population N = 100 λ = 0.1 δ = 1.0 I (0) = 0.01 S(t) Time

23 SIS moel SIR moel 22/22 SIS moel SIR moel Possibility of an active state Absence of active state Epiemy: active state Epiemy: fraction of actives