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

6.5

MODELING AND ANALYSIS OF CONTROL MEASURES FOR DENGUE

FEVER

The aim of vaccination is to protect the entire host population against dengue. Math-

ematical models for vaccination are reviewed in [32] with an overview of potential ap-

proaches. An important measure for the vaccination effectiveness, called efficacy, is the

portion p of the population that acquires immunity. The threshold that assures eradication

of dengue in the population is called the critical vaccination threshold pc. In Fig. 2 of [32]

the relationship between the basic reproduction number R0 > 1 and pc is shown.

The basic reproduction number R0 (Section 6.2.1) for the vector-borne case is studied

in [53] and in [16] for the analysis of a model very similar to the SIRvUV model (6.10).

In [39] a simple example of a single-strain SIR-UV model is described and analyzed. In

addition, the relationship between R0 and the critical vaccination proportion pc is derived

as pc > (1 −1/R0)/p, 0 ≤p ≤1 is the efficacy of the vaccination. That expression is

only a rough estimate for the fraction of the population that must receive protection im-

munity especially due to the fact the dengue is a multi-serotype virus with cross-immunity

effects. Exposure to a single serotype is generally assumed to give lifelong immunity to

that serotype. However, the effect of immunity to one serotype on infections of the other

serotype can lead to a more severe disease [32], ranging from clinically unapparent to the

fatal hemorrhagic form and the associated dengue shock syndrome.

We model two control measures: vaccination and vector control. The model is the

single-strain dengue model [44, 42] shown in (6.4). We use the parameter values given in

Table 6.B.1 after [41]. The aim of vector control is again to protect the entire host pop-

ulation against dengue. Implementing a measure focusing on diminishing the number of

mosquitos one should not model the effectiveness of the change of their total number M

like done in modelling seasonality (6.9) but a change of mosquitos’ death rate ν.

We analyze an extension of the single-strain host-vector model (6.4) by including con-

trol measures and therefore the focus is on changing parameter values that quantify the

mechanisms enforcing the control, such as vaccination or mosquito control via insecti-

cides, repellents, and other means.

6.5.1

Description of a model with vaccination

We use the simple model (6.4) and introduce new parameters to model vaccination,

see also [16], namely: 0 ≤pv ≤1 the fraction of newborns vaccinated and 0 ≤αv ≤1 the

vaccine efficacy. Two new compartments for the vaccinated individuals are introduced:

susceptible Sv and infected Iv. The model with vaccination, which we label SIRvUV,