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

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O(ε²)

QSSA

full model

S(t)

I(t)

260

240

220

1.5

1

0.5

0

equilibrium

O(ε)

QSSA

full model

S(t)

I(t)

270

240

210

180

150

3

2.5

2

1.5

1

0.5

0

Figure 6.1: Trajectory in (S,I)-plane of (6.4) and trajectories with two approximations of

V : the O(ε⁰) or QSSA approximation and for terms up to O(ε) (left) and up to O(ε²)

(right).

only when ε > 0.175482. When ε = 0.175482 the spurious equilibrium equals S = N and

I = 0. Consequently the first order approximation approach can fail for ε > 0.175482 when

the initial data of the trajectory is far from the expected endemic equilibrium close in the

full model.

The second-order approximation does not have this shortcoming. However, for the

initial condition used the calculated transient dynamics is also poor for the second-order

approximation with ε = 1. The technique is applicable only when the starting point is suffi-

ciently close to the equilibrium as in the case of the QSSA trajectory in Figure 6.1 (QSSA

green curve). In (6.4) a value for ε was quantified by its interpretation as the ratio of the

rates of changes of the state variables of the host and vector population near equilibrium.

It was argued that ε = 1/365 is suitable value for the SIR-UV model. Simulations (not

shown) for the first- and second-order approximations indicate that when starting close to

the M0-plane defined in the Appendix 6.A, there are no spurious equilibria but higher-

order approximations do not improve much the QSSA solutions when ε = 1/365.

6.3

TWO-STRAIN DENGUE MODELS

In this section we describe the two-strain dengue models, namely the Host-only and

the host-vector model. The analysis of these models relies heavily on numerical techniques

where the time dependency on parameter values is obtained by simulations and bifurcation

analysis. Besides equilibria and limit cycles, both models show also chaotic dynamics.

6.3.1

Host-only models

In host-only models [2, 1, 12, 26, 34], the vector dynamics is considered implicitly in

the parameters by assuming that the size of the infected vector population is proportional

to that of the infected host population. In that respect this modeling approach closely re-