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■Bio-mathematics, Statistics and Nano-Technologies: Mosquito Control Strategies
a
b
0.0
4.0
8.0
0
0.5
1.0
0.0
500
1000
0
0.2
0.4
0.6
0.8
1
V
I, Iv
S, Sv
αv
0.0
4.0
8.0
0
0.5
1.0
0.0
500
1000
0
0.2
0.4
0.6
0.8
1
TC
V
I, Iv
S, Sv
αv
Figure 6.7: One-parameter diagram for the bifurcation parameter for SIRvUV model for
αv and β = 104. The curves show the stable equilibrium values. Red curves are for the host
and vector compartments S,I,V and green for the vaccinated host compartments Sv,Iv. In
panel a: αv = 1 and in panel b: αv = 1. In the latter case the transcritical bifurcation TC
occurs for αvTC = 0.75.
the same as in (6.10a). The equations for the vector population (6.10c) are replaced by the
following for V and M where two control parameters qb and qν are introduced
˙S = −β
M SV +µ(N −S) ,
˙I = β
M SV −(γ +µ)I ,
dR
dt = γI −µR ,
(6.15a)
˙V = qb
ϑ
N (M −V )I −ν(1+qν
M
M∞
)V ,
˙M = ν(1−qν
M
M∞
)M .
(6.15b)
Note that the total number of mosquitos M(t) = U(t)+V (t) does not need to be constant
in time. We consider two scenarios for disease control campaigns: the parameter qb is a
factor that reduces the force of infection rate of the vector, while qν is proportional to the
demographic loss rate for the vector population where the number of mosquitos changes
as a logistic decay with a lower bound parameter 0 < M∞≤M. This represents a practi-
cal threshold value for the possibility of the vector population’s extermination. The vector
population’s growth is again exponential with linear birth rate proportional to M, and the
death rate is quadratic: proportional to M2.
We remark that the density dependent force of infection formulation seems to give
problems when the vector population is extinct: M = 0. However, for the host population
it is possible to use the densities U/M and V/M instead of the numbers U and V where M
is not used explicitly for the host population modelling. This also shows that the epidemi-
ological and demographic mechanisms can be combined as well as strictly separated.