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

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this paper for a detailed discussion. The Hopf bifurcations occur also for the host-vector

model Figure 6.2b but there is no torus bifurcation TR.

6.4.2

Results for seasonally-forced systems

In [1, 41] dealing with dengue fever modelling, we considered a seasonally changing

mosquito population due to climate factors such as temperature or rainfall. For the host-

vector model the number of mosquitos M is modeled as a seasonally-forced term and is

given explicitly by a cosine function. For the host-only model in [1] the infection rate β(t)

was changed periodically:

M(t) = M0(1+ηcosω(t+ϕ)) ,

β(t) = β0(1+ηcosω(t+ϕ)) .

(6.9)

Parameter M0 is the mean vector population size in the host-vector model and β0 the mean

value for the host-only model. In the host-vector case the number of mosquitos changes

over time but because we assume a changing area size, the mosquito density remains the

same. The results for the seasonally forced host-only model (parameter β(t) with η = 0.35),

and the host-vector model (parameter M(t) with η = 0.1) are compared in Fig. 6.3 in the

lower range of the ratio of likelihoods of transmission from hosts with secondary and hosts

with primary infection to vectors, φ ∈[0,1.2].

In conclusion, these results indicate that the dynamics predicted by both models is

qualitatively but not quantitatively the same.

a

b

φ

I

T

P ⁺

T R

P ⁻

H

1.2

1

0.8

0.6

0.4

0.2

0

3

2.4

1.8

1.2

0.6

0

φ

I

P

P

T R

1.2

1

0.8

0.6

0.4

0.2

0

3

2.4

1.8

1.2

0.6

0

Figure 6.3: Two-strain non-autonomous a host-only model (6.6) and b host-vector

model (6.7) with parameter α = 2. The bullets mark the global maximum and minimum

values for limit cycles for total infected I. Red indicates stable and blue unstable solutions.