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

in a secondary infection may develop a severe form of dengue and require hospitalization,

thus reducing their contact with the vector in the daily routine. The surprising result from

the numerical bifurcation analysis is that the introduction of a vector population tends to

stabilise the chaotic dynamics, which has been noticed with model assumptions in sev-

eral contexts of host-only models. Only by introducing an ecologically-motivated seasonal

forcing of the vector population is the chaotic regime restored for a wide variety of param-

eters. A review of seasonality effects in epidemiological models is given in [13].

6.4

COMPARISON OF HOST-ONLY AND HOST-VECTOR MODEL

6.4.1

Results for autonomous systems

Here we discuss the model under the assumption of a constant vector population over

time. All results for the autonomous system (6.7) use reference parameter values given in

App. 6.B from [2, 34, 44]. The results are compared with those for earlier models in the

literature: [12, 2, 34, 26]. Important is to recall that in [1] a realistic description of recorded

disease incidents was produced for empirical data sets of dengue fever in Thailand.

In Figure 6.2 the two parameter diagrams are shown for the host-only model (6.6) from

[34] (panel a) and host-vector model (6.7) (panel b). The bifurcation pattern for the host-

only model in Figure 6.2a is dominated by two Hopf bifurcations: one supercritical H⁺

and the other H⁻. Furthermore, there is one Torus bifurcation TR also called a Neimark-

Sacker bifurcation. These results have been discussed in [34] and the reader is referred to

a

b

P

H⁺

H–T R

H–T R

T R

•

•

H⁻

φ

α

[∗]

[†]

52

12

9

6

3

0

250

200

150

100

52

0

H⁻

H⁻

P

H⁺

ZH

•

GH

•

φ

α

[∗]

[†]

52

12

9

6

3

0

250

200

150

100

52

0

Figure 6.2: Two-parameter (φ,α) bifurcation diagram for the range φ ∈[0,12] and α ∈

[0,250]. a the host-only model (6.6) from [2] with B(t) similar to (6.7) with η = 0.1 and

b the host-vector model with M(t) (6.7) with η = 0.1 in Eq. 6.9. For α →∞the Hopf

bifurcation converges to the φ parameter value [†] for the model of [12] and [∗] marks the

value φ = 2.5 in [26].