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

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αv

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Figure 6.4: Bifurcation diagram for the transcritical bifurcation TC fixed by βvTC ((6.13))

in the three-parameter (αv,pv,β)-space. The dot • marks the parameter values αv = 0.2,

pv = 0.5 used for Figure 6.5b.

Note that βvTC (6.13) increases without bounds whenever αv and pv approach 1. This

implies that when either the vaccine efficacy and the fraction of vaccinated newborns ap-

proach 100%, the endemic equilibrium is hard to attain, and the disease is essentially erad-

icated. But these percentages are difficult to attain in practical situations. In summary,

knowing the infection rate β, vaccination is successful when the two control parameters

are chosen below the TC-surface in Figure 6.4 where the values for all epidemiological

and demographic parameter are from Table 6.B.1 in Appendix 6.B.

In order to evaluate the vaccination control measure, we compute the expression for

the threshold parameter values. We give the expressions for these TC thresholds for the

two control parameters pv and αv separately. These expressions can be used in a sensitivity

analysis in order the judge the effectiveness of changing these parameters separately. For

the proportion pv of vaccinated immediately after birth it reads:

pvTC = ^{βϑ−(γ +µ)ν}

αvβϑ

,

and for vaccine efficacy αv

αvTC = ^{βϑ+(γ +µ)ν}

pvβϑ

.

Note that only the expressions at the occurrence of the TC, the infection rate parame-

ters β and ϑ, the birth and death rates µ, ν respectively of host and vector, appear in these

expressions, but not N and M.