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■Bio-mathematics, Statistics and Nano-Technologies: Mosquito Control Strategies

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Figure 6.8: Bifurcation diagram of the SIRqVM model (6.15) for the transcritical bifurca-

tion TC fixed by βνTC (6.17) in the three-parameter (qb,pν,β)-space. The dot • marks the

parameter values qb = 1.0 and qν = 1.0 used in Figure 6.9.

chances of disease eradication. The endemic equilibrium E^∗reads:

S^∗= ^{N(µqbϑ+2νγ +2νµ)}

qbϑ(β +µ)

,

I^∗= ^{Nµ(βqbϑ−2νγ −2νµ)}

qbϑ(β +µ)(2γ +2µ)

,

(6.18a)

V ^∗= ^{µM∞(βqbϑ−2νγ −2νµ)}

qvβ(µqbϑ+2νγ +2νµ) ^,

M^∗= ^M∞

qν

.

(6.18b)

6.5.2.2

Sensitivity analysis of the SIRqVM model

Similar to the analysis for vaccination controls, we give bifurcation diagrams for the

two control parameters qb and qν where (6.17) implies

qbTC = ^{2(γ +µ)ν}

βϑ

.

We start with an increase of the mortality of vector population modelled in (6.15) with qb =

1 and qν = 1 and whereby 0 ≤β ≤104 is varied. The bifurcation diagram for parameter β is

shown in Figure 6.9. The transcritical bifurcation occurs at βνTC = 52.02. This means that

due to the control measures the system becomes endemic at a much higher rate of infection

β than in the original case without any control shown in Figure 6.5 where βTC = 26.02.

The numerical bifurcation results for the control parameter qb in Figure 6.10 show that

there is a critical vector control threshold value qb equal to the threshold parameter value

at the transcritical bifurcation TC.