Mathematical Models and Optimal Control in Mosquito Transmitted Diseases

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0

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100

0

0.2

0.4

0.6

0.8

1

time t (in days)

control u

b=0.75 and J1

b=0.75 and J2

Figure 7.3: Optimal control u for different cost functions J1 and J2 (parameter/constant

values from Table 7.1 and b = 0.75).

Then, system (7.A.5) is solved by a backward fourth-order Runge–Kutta scheme using

the current iteration solution of (7.1). The controls are updated by using a convex combina-

tion of the previous controls and the values from (7.A.6) (see Appendix 7.A). The iterative

method ends when the values of the approximations at the previous iteration are close to

the ones at the present iteration. For details see [10, 17, 46].

First of all we consider b = 0.75 and show that when we apply the supervision control

u, better results are obtained, that is, the number of infected humans vanishes faster when

compared to the case where no controls are used. If the control intervention u is applied,

then the number of infectious individuals vanishes after approximately 30 days. If no con-

trol is considered, then it takes approximately 70 days to assure that there are no infectious

humans (see Figure 7.4 for the fraction of susceptible and infectious humans and Figure 7.5

for the optimal control).

0

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80

100

700

750

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1000

1050

Time (in days)

Sh

b=0.75

without controls

with controls

(a) Susceptible humans

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50

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300

Time (in days)

Ih

b=0.75

without controls

with controls

(b) Infectious humans

Figure 7.4: Susceptible and infectious humans for b = 0.75 with and without control.