Bifurcation in epidemic models

(7)
(8)
Bifurcations in an SIRS Model (cont.)
where p= α(d + ν) , k d +γ m= , d +ν A = N0 q= k , d +ν
γ . d +ν
Saddle-Node Bifurcation
(0, 0) is an equilibrium. This is a disease free equilibrium. It is always stable. According to the formula from van den Driessche and Watmough (Math. Biosci,2002), we can calculate the reproduction number of this model and obtain R0 = 0. we should know further details of dynamical behaviors: When is the disease persistent? When does the disease die out?
SN Continue (cont.)
I ’ = I2 (A − I − R)/(1 + p I2) − m I R’=qI−R A = 11.5 p = 0.2 q = 4.2 m=5
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4 R 3 2 1 0 0 0.5 1 1.5 I 2 2.5 3
Hopf Bifurcation
E1 is a saddle, impossible for a Hopf bifurcation, E2 is a node or a focus, it is possible to have a Hopf bifurcation, a necessary condition is that the trace of the Jacobian matrix at E2 is zero. The sign of the trace is determined by
Saddle-Node Bifurcation (cont.)
(i) there is no positive equilibrium if A2 < 4m(mp + q + 1) := A0 ; (ii) there is one positive equilibrium if A2 = 4m(mp + q + 1); (iii) there are two positive equilibria if A2 > 4m(mp + q + 1). Two positive equilibria E1 = (I1 , R1 ) and E2 = (I2 , R2 ), are: √ A− A2 −4m(mp+q+1) I1 = , R1 = qI1 ; √ 2(mp+q+1) A+ A2 −4m(mp+q+1) I2 = , R2 = qI2 . 2(mp+q+1)
Bogdanov-Takens bifurcation is a codimension 2 bifurcation. Its advantage is we have a very clear picture to the phase portrait around the equilibrium. Phase portrait is given in the following figure.
(mq+2m−1−q+2m2 p)2 . (m−1)(mp+p+1)
(9)
Stability
Theorem E2 is stable if one of the following inequalities holds (m − 1)(mp + p + 1)A2 > (mq + 2m − 1 − q + 2m2 p)2 , m < 1, q<
(1) Conditions: Assume: f is smooth; x = 0 is an equilibrium when α = 0. Let J be the Jacobian matrix of f at x = 0. Suppose 0 is an eigenvalue of J with multiplicity 2 but
Let a 2D ODE: ˙ x = f (x, y , c), ˙ y = g(x, y , c). It has a non-hyperbolic equilibrium point at (0, 0) when c = 0 such that eigenvalues of the Jacobian matrix λ = α(c) + iβ(c) satisfies: α(0) = 0, Then if place.
Basic Types
Saddle Node bifurcations Hopf bifurcations Codimension 2 bifurcations. Backward bifurcations
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Hopf Bifurcation
Bogdanov-Takens Bifurcation (cont.)
A = 0. When α = 0, if system (2) has the following normal form at x = 0 dx1 = x2 + higher order dt dx2 2 = x1 ± x1 x2 + higher order dt (3) (4)
Bogdanov-Takens Bifurcation (cont.)
bifurcation curves (saddle-node, Hopf, homoclinic):
2 SN := {(β1 , β2 ) : 4β1 − β2 = 0} H := {(β1 , β2 ) : β1 = 0, β2 < 0} 6 2 HL := {(β1 , β2 ) : β1 = − β2 + o(), β2 < 0} 25
Saddle-Node Bifurcation (cont.)
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SN Continue
E1 is a saddle and E2 is a node or focus. A2 is a saddle-node bifurcation parameter. Generally speaking, the policy for disease control is to drive A2 below A0 . But we have shown: Even if A2 above A0 , it is possible that the disease dies out. So, it is unnecessary to drive A2 below A0 so that disease dies out.
Bogdanov-Takens Bifurcation (cont.)
β2 SD
(4)
(1)
β1
HL
(3)
(2)
SD
H
Backward Bifurcations
In classical epidemic models, we have forward bifurcations:
Backward Bifurcations (cont.)
σ=
3F 3 ∂3 1 ∂ 3 F1 1 [ ∂u3 + ∂ ∂u∂v 2 + ∂u2F2 + ∂ F32 ] 3 16 ∂v ∂v 2 2 ∂ 2 F1 1 + √ [ ∂u∂v ( ∂ F21 + ∂ F21 ) ∂u ∂v 16 k1 ∂ 2 F2 ∂ 2 F2 ∂ 2 F2 − ∂u∂v ( ∂u2 + ∂v 2 )− 2 2 ∂ 2 F1 ∂ 2 F2 + ∂ F21 ∂ F22 ]. ∂u 2 ∂u 2 ∂v ∂v
(1)
Hopf Bifurcation (cont.)
Bifurcation direction is determined by the sign of σ. If σ < 0, the limit cycle is stable. If σ > 0, the limit cycle is unstable.
Bogdanov-Takens Bifurcation
Bogdanov-Takens Bifurcation is a codimension 2 bifurcation. Review of the basic theory: Consider dx = f (x, α), dt x ∈ R2 , α ∈ R2 . (2)
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Backward Bifurcation
Backward bifurcations: It is not enough to eradicate a disease if R0 < 1.
Backward Bifurcation (cont.)
∂α ∂c (0)
β(0) = 0.
= 0, the Hopf bifurcation takes
Hopf Bifurcation (cont.)
Bifurcation directions: when we transform the original system into standard form, we obtain du = −ωv + F (u, v ), 1 dt dv = ωu + F (u, v ), 2 dt Set
Stability and Bifurcation of Epidemic Models
Wendi Wang
Department of Mathematics Southwest University Chongqing, 400715 PR. China
May 16, 2006
Outline
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Bogdanov-Takens Bifurcation (cont.)
identity so that we have the following normal form dx1 = x2 dt dx2 2 = β1 + β2 x1 + x1 − x1 x2 . dt Here, we neglect higher order terms. By this normal formal, we can obtain three (5) (6)
4 3 f = (q p − p2 )I2 − A pI2 2 −(2 + q + 2 p)I2 + AI2 − 1
Hopf Bifurcation (cont.)
I2 contains a square root. Using Maple, we find Hopf bifurcation points m > 1, q > 2mp+1 , m−1 A2 =
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R0
Bifurcations in an SI; νR, dt 1 + αI 2 dI = kI 2 S − (d + γ)I, dt 1 + αI 2 dR = γI − (d + ν)R, dt It can be reduced to: dI = I 2 (A − I − R) − mI, dt 1 + pI 2 dR = qI − R, dt
Then system (2) admits Bogdanov-Takens bifurcation at x = 0. (2) Bifurcation Curves: Suppose B-T bifurcation exists. When α = 0, we use several topological transformations near to