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■Bio-mathematics, Statistics and Nano-Technologies: Mosquito Control Strategies
and the Jacobian matrix evaluated at this point reads
J(E0) =
−µ
0
−β(1−pv) N
M
0
0
0
−(γ +µ)
β(1−pv) N
M
0
0
0
ϑM
N
−ν
0
ϑM
N
0
0
−(1−αv)βpv
N
M
−µ
0
0
0
(1−αv)βpv
N
M
0
−(γ +µ)
.
This matrix has two explicit eigenvalues, −µ, that do not affect stability. In addition, the
determinant of the remaining minor factorizes as follows,
det(E0) = (γ +µ)[βϑ(1−αvpv)−(γ +µ)ν],
thereby indicating that another possible eigenvalue is −(γ +µ), a fact that is easily verified
by direct substitution into the cubic characteristic equation. The remaining quadratic, after
factoring out these known eigenvalues, is
λ2 +λ(γ +µ+ν)+ν(γ +µ)−βϑ(1−αvpv) = 0.
Note that the corresponding discriminant is
∆= (γ +µ+ν)2 −4[ν(γ +µ)−βϑ(1−αvpv)] = (γ +µ−ν)2 +4βϑ(1−αvpv) ≥0.
Its roots are therefore real, and one eigenvalue is
λ−= −1
2[(γ +µ+ν)+
√
∆] ≤0.
On the other hand, we find the fifth eigenvalue
λ+ = −1
2
(γ +µ+ν)2 −∆
(γ +µ+ν)+
√
∆
= 2βϑ(1−αvpv)−(γ +µ)ν
(γ +µ+ν)+
√
∆
.
Thus to ensure stability, the following condition must hold:
β < βvTC = ν
γ +µ
(1−αvpv)ϑ.
(6.13)
We recall that in the reference case without control measures we have
βTC = (γ +µ)ν
ϑ
(6.14)
When β crosses from below the critical threshold βvTC, a transcritical bifurcation occurs
through which the endemic equilibrium is found. Further, since all eigenvalues are real,
Hopf bifurcations will never occur.