Mathematical Models and Optimal Control in Mosquito Transmitted Diseases

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A number of studies reported that ITN possession does not necessarily translate into

use. Human behavior change interventions, including information, education, communica-

tion (IEC) campaigns and post-distribution hang-up campaigns are strongly recommended,

especially where there is evidence of their effectiveness in improving ITN usage [1, 25, 55].

Here we consider the model from [2] for the effects of ITNs on the transmission dynamics

of malaria infection. Other articles considered the impact of intervention strategies using

ITN (see, e.g., [18, 47]). However, only in [2] the human behavior is incorporated into the

model. We follow [45], introducing in the model of [2] a supervision control, u, which

represents IEC campaigns for improving the ITN usage. In Section 7.2, we present the

mathematical model for malaria transmission with one control function u. In Section 7.3,

we propose an optimal control problem for the minimization of the number of infectious

humans while controlling the cost of control interventions. Finally, in Section 7.4, some

numerical results are analyzed and interpreted from the epidemiological point of view.

Future research aims to expand the developed model to include the effects of mosquito re-

pellency incorporated in paints and textiles to reduce mosquito transmission of infectious

diseases.

7.2

CONTROLLED MODEL

We consider a mathematical model presented in [2] for the effects of ITN on the trans-

mission of malaria infection and introduce a time-dependent supervision control u. The

model considers transmission of malaria infection of mosquito (also referred as vector)

and human (also referred as host) population. The host population is divided into two

compartments, susceptible (Sh) and infectious (Ih), with a total population (Nh) given by

Nh = Sh +Ih. Analogously, the vector population is divided into two compartments, sus-

ceptible (Sv) and infectious (Iv), with a total population (Nv) given by Nv = Sv +Iv. The

model is constructed under the following assumptions: all newborns individuals are as-

sumed to be susceptible and no infected individuals are assumed to come from outside the

community.

The human and mosquito recruitment rates are denoted by Λh and Λv, respectively.

The disease is fast progressing, thus the exposed stage is minimal and is not considered.

Infectious individuals can die from the disease or become susceptible after recovery while

the mosquito population does not recover from infection. ITNs contribute for the mortality

of mosquitos. The average number of bites per mosquito, per unit of time (mosquito-human

contact rate), is given by

β = βmax(1−b),

where βmax denotes the maximum transmission rate and b the proportion of ITN usage.

It is assumed that the minimum transmission rate is zero. The value of β is the same for

human and mosquito population, so the average number of bites per human per unit of

time is βNv/Nh (see [2] and the references cited therein). Thus, the force of infection for

susceptible humans (λh) and susceptible vectors (λv) are given by

λh = ^p1βIv

Nh

and

λv = ^p2βIh

Nh

,