Mathematical Models and Optimal Control in Mosquito Transmitted Diseases

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7.3

OPTIMAL CONTROL PROBLEM

We formulate an optimal control problem that describes the goal and restrictions of

the epidemic. In [2] it is found that the ITN usage must attain 75% (b = 0.75) of the host

population in order to extinct malaria. Therefore, educational campaigns must continue

encouraging the population to use ITNs. Moreover, it is very important to assure that ITNs

are in good conditions and each individual knows how to use them properly. Having this

in mind, we introduce a supervision control function, u, where the coefficient 1 −u rep-

resents the effort to reduce the number of susceptible humans that become infected by

infectious mosquitos bites, assuring that ITNs are correctly used by the fraction b of the

host population.

We consider the state system (7.1) of ordinary differential equations in R⁴ with the set

of admissible control functions given by

Ω=

u(·) ∈L∞(0,tf)|0 ⩽u(t) ⩽1, ∀t ∈[0,tf]

.

The objective functional is given by

J1(u) =

Z tf

0

A1Ih(t)+ ^C

2 ^u2(t)dt,

(7.2)

where the weight coefficient, C, is a measure of the relative cost of the interventions as-

sociated to the control u and A1 is the weight coefficient for the class Ih. The aim is to

minimize the infectious humans while keeping the cost low. More precisely, we propose

the optimal control problem of determining (S^∗

h^,I∗

h^,S∗

v^,I∗

v^{) associated to an admissible}

control u^∗(·) ∈Ωon the time interval [0,tf], satisfying (7.1), the initial conditions Sh(0),

Ih(0), Sv(0) and Iv(0) (see Table 7.1) and minimizing the cost function (7.2), i.e.,

J1(u^∗(·)) = min

Ω^J1(u(·)).

(7.3)

The existence of an optimal control u^∗(·) comes from the convexity of the Lagrangian of

(7.2) with respect to the control and the regularity of the system (7.1) (see, e.g., [11, 13] for

existence results of optimal solutions). Applying the Pontryagin maximum principle [34],

we derive the optimal solution (u^∗,S^∗

h^,I∗

h^,S∗

v^,I∗

v^{) of the proposed optimal control problem}

(see Appendix 7.A).

More generally, one could take the following cost function:

J2(u) =

Z tf

0

A1Ih(t)+A2Iv(t)+ ^C

2 ^u2(t)dt,

where A2 is the weight constant on infectious mosquitos (for numerical simulations we

considered A2 = 25). It turns out that when we include in the objective function the num-

ber of infectious mosquitos, the distribution of the total host population Nh and vector

population Nv by the categories Sh, Ih and Sv, Iv, respectively, is the same for both cost

functions J1 and J2 (see Figures 7.1 and 7.2). On the other hand, the effort on the control

is higher for the cost function J2 (see Figure 7.3). Therefore, we choose to use the cost

function J1 in our numerical simulations (Section 7.4).