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■Bio-mathematics, Statistics and Nano-Technologies: Mosquito Control Strategies
In the next section we use the bifurcation plots where the measure parameters pv and
αv vary to see the effects on the equilibrium values of the state variables. This information
can be used as an analytic approach to predict the effects in disease-control campaigns.
6.5.1.2
Sensitivity analysis of the SIRvUV model
We start with a numerical bifurcation analysis where the infection rate β varies and
we fix the control parameters pv and αv. In Figure 6.5 the bifurcation diagram is shown
for the non-vaccinated where αv = 0 and pv = 0 and the vaccinated case where αv = 0.2
and pv = 0.5. When the parameter β increases past the transcritical bifurcation point TC,
the disease-free equilibrium becomes unstable and the endemic equilibrium stable. Below
the TC point the system is disease-free and above it the disease persists endemically. For
the reference case where we have, using (6.14): βTC = 26.02 in Figure 6.5a and in Figure
6.5b: βvTC = 28.90.
The results in Figures 6.4 and 6.5 show that the TC threshold value βvTC increases
when both control parameters increase. Hence, there is complete eradication of the disease
in a larger parameter range of infection rates β. For the reference value β = 104 the number
of infected individuals I +Iv is somewhat lower when αv = 0.2 and pv = 0.5 and further-
more the number of infected mosquitos V is much lower, see Figure 6.5b with respect to
Figure 6.5a. For the sensitivity analysis with respect to the control parameters pv and αv
we perform a numerical bifurcation analysis with the parameter values given in Table 6.B.1
after [42] and we fix the infection rate at β = 104. In Fig. 6.6a,b the effect of vaccination
a
b
0.0
4.0
8.0
0
0.5
1.0
0.0
500
1000
0
20
40
60
80
100
TC
V
I
S
β
0.0
4.0
8.0
0
0.5
1.0
0.0
500
1000
0
20
40
60
80
100
TC
V
I, Iv
S, Sv
β
Figure 6.5: One-parameter diagram for the bifurcation parameter β for SIRvUV model.
Stable equilibria for the state variables S,I,V are shown in red and unstable in blue. In
panel a without vaccination (αv = pv = 0) where βTC = 26.02. In panel b with vacci-
nation. Stable equilibria for the state variables Sv,Iv are shown in green and unstable in
magenta). The vaccination parameters are αv = 0.2 and pv = 0.5 and βvTC = 28.90. All
other parameter values are given in Table 6.B.1.