Multi-Strain Host-Vector Dengue Modeling: Dynamics and Control

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The bite efficiency ϑ is decreased by multiplication with a factor 0 ≤qb ≤1. This is

obtained by replacing ϑ by qbϑ in model (6.10). Scenario qb < 1 models the use of personal

household protection such as repellents, window screens or wearing long-sleeved clothes.

The mosquito population size M is in this case constant. The model accounts also for an

additional loss rate of the mosquitos qν due to, for instance, the use of insecticides in water

storage outdoor containers, or insecticide treated materials, coils and vaporizers.

6.5.2.1

Analysis of the SIRqVM model

There are two equilibria for system (6.15): the trivial, disease-free equilibrium E0 and

the interior, endemic equilibrium E^∗. The trivial equilibrium equals

E0 = (S0 = N, I0 = 0, V0 = 0, M0 = ^M∞

qν

) ,

(6.16)

and the Jacobian matrix evaluated at this point reads

J(E0) =





















−µ

0

−qνβ ^N

M ∞

0

0

−(γ +µ)

qνβ ^N

M ∞

0

0

qbϑ^M∞

qνN

−2ν

0

0

0

0

−ν





















.

This matrix has two explicit real negative eigenvalues, −µ and −ν that do not affect sta-

bility.

The stability is fixed by the 2 × 2 internal block matrix of J(E0). Indeed the trace

of this matrix is −(γ + µ + 2ν) and therefore there cannot be a Hopf bifurcation for the

disease-free equilibrium. The determinant of this matrix is zero at

βνTC = ^{2(γ +µ)ν}

qbϑ

.

(6.17)

and this determines the position of the transcritical bifurcation TC when β is the bifurca-

tion parameter. Note that this expression is independent of M∞as well as qν and therefore

both parameters have no effect on the position of the extinction threshold. Therefore, we

will not analyze their effects in detail. Figure 6.8 gives the surface in the three-parameter

space for this function (6.17). When β crosses from below the threshold βνTC, a tran-

scritical bifurcation occurs. Note that βνTC (6.17) increases without bounds whenever

qb approaches 0. This situation shows that for all qν values the system converges to the

disease-free equilibrium (6.16). However, this is hard to attain in practice.

From (6.17) and Figure 6.8 we see that reducing the biting rate qb increases the critical

threshold βνTC. The same effect is obtained independently by increasing the mosquito

mortality rate. The combined measures have of course a much stronger reinforcement of

the increase in the threshold for attaining the endemic equilibrium, thereby increase the