Appendices

■141

6.A

TIME SCALE SEPARATION, EXAMPLE: SIR-UV MODEL

The singular perturbation technique has been applied to the SIR-UV model [42, 41].

The slow (S,I)-flow in Eqn. (6.4) occurs near a manifold that is a smooth surface in R³.

Letting ε = 0 in (6.2) determines the two-parameter nullspace for V in (6.4) consisting of

the critical manifold in (6.5).

M0 =



(S,I,V ) | V =

ϑMI

ϑI +νN ^{, 0 ≤S ≤N, 0 ≤I ≤N}



,

and one can demonstrate M0 is normally hyperbolic. Using M0 ’s local invariance, V can

be approximated by a power series

V = q0(S,I)+q1(S,I)ε+q2(S,I)ε² +...,

0 < ε ≪1 .

q0(I) =

ϑMI

ϑI +νN

(QSSA approximation)

q1(S,I) = −^νϑMN2

(ϑI +νN)³

 β

M ^{Sq0(I)−(γ +µ)I}



q2(S,I) = ^νϑ2MN3βI

(ϑI +νN)⁵



µ(N −S)−^β

M ^Sq0(I)



−

νϑMN²

(ϑI +νN)^{3 q1(S,I)}

−νϑMN³

(γ +µ)(νN −2ϑI)

(ϑI +νN)⁵

−^{ϑβS(νN −3ϑI)}

(ϑI +νN)⁶

 β

M ^{Sq0(I)−(γ +µ)I}



The coefficients qi(S,I), i = 0,1,2 are computed from substitution of this equation invari-

ance criterion. With i = 0 we get the QSSA expression Eqn. (6.5). For more details the

reader is refered to [42, 41].