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

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a

b

0.0

4.0

8.0

0

0.5

1.0

0.0

500

1000

0

0.2

0.4

0.6

0.8

1

V

I, Iv

S, Sv

pv

0.0

4.0

8.0

0

0.5

1.0

0.0

500

1000

0

0.2

0.4

0.6

0.8

1

TC

V

I, Iv

S, Sv

pv

Figure 6.6: One-parameter diagram for the bifurcation parameter for SIRvUV model for pv

and β = 104. The curves are the stable equilibrium values. Red curves are for the host and

vector compartments S,I,V and green curves for the vaccinated host compartments Sv,Iv.

In panel a: αv = 0 and in panel b: αv = 1. In the latter case the transcritical bifurcation TC

occurs for pvTC = 0.75.

where αv = 0 and αv = 1 respectively, where pv is increased from 0 to 1. For αv = 0 the

vector population is constant while for the non-vaccinated population the susceptible S

and infected I population deceases to zero while those Sv and Iv for the vaccinated popu-

lation increase such that the sums are independent. Only for pv = 1 the whole population

is disease-free.

For αv = 1 and all pv values there are no infected vaccinated individuals Iv while the

susceptible vaccinated individuals Sv increase linearly. Below the transcritical bifurcation

point TC the susceptible host population S remains the same and the number of infected

individuals I decreases. With αv = 1 there is a TC at pvTC = 0.75. Hence an disease-free

population exists now for an interval above this TC. The vaccinated host population Sv

increases linearly with pv and equals M at pv = 1. and the vaccinated infected host popu-

lation Iv is extinct. Above the TC point the non-vaccinated susceptible host S population

decreases and is extinct for pv = 1. The infected vector population V is also extinct above

the TC. This shows that there is a positive effect for the hosts I = 0 only above this TC

threshold value.

6.5.2

Model with vector control

We study now vector control measures for the model (6.4) by introducing vector con-

trol population parameters that represent the effect of insecticides and personal protection

targeting the mosquito (SIRqVM model). The equation for the host population remains