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■Bio-mathematics, Statistics and Nano-Technologies: Mosquito Control Strategies

Table 7.1: Parameter values.

Symbol

Description

Value

Λh

Recruitment rate in humans

10³/(70×365)

Λv

Recruitment rate in mosquitos

10⁴/21

µh

Natural mortality rate in humans

1/(70×365)

δh

Disease induced mortality rate in humans

10⁻³

b

Proportion of treated net usage

0.25; 0.3; 0.4; 0.5;

0.6; 0.7; 0.75

γh

Recovery rate of infectious humans to be susceptible

1/4

µv1

Natural mortality rate of mosquitos

1/21

µmaxb

Mortality rate of mosquitos due to treated net

1/21

βmax

Maximum mosquito-human contact rate

0.1

p1

Probability of disease transmission from mosquito

1

p2

Probability of disease transmission from human to mosquito

1

A1

Weight constant on infectious humans

25

C

Weight constant on control

50

Sh(0)

Susceptible individuals initial value

800

Ih(0)

Infectious individuals initial value

200

Sv(0)

Susceptible vectors initial value

4,000

Iv(0)

Infectious vectors initial value

900

where p1 and p2 are the transmission probability per bite from infectious mosquitos to

humans, and from infectious humans to mosquitos, respectively. The death rate of the

mosquitos is modeled by µvb = µv1+µmaxb, where µv1 is the natural death rate and µmaxb

is the death rate due to pesticide on ITNs.

The coefficient 1−u represents the effort of susceptible humans that become infected

by infectious mosquitos bites, such as educational programs/campaigns for the correct use

of ITNs, supervision teams that visit every house in a certain region and assure that every

person has access to an ITN, know how to use it correctly, and recognize its importance

on the reduction of malaria disease transmission. The values of the parameters Λh, Λv, µh,

δh, γh, µv1, µmaxb, βmax, p1 and p2 are taken from [2] (see Table 7.1). The state system of

the controlled malaria model is given by



































˙Sh(t) = Λh −(1−u(t))λhSh(t)+γhIh(t)−µhSh(t),

˙Ih(t) = (1−u(t))λhSh(t)−(µh +γh +δh)Ih(t),

˙Sv(t) = Λv −λvSv(t)−µvbSv(t),

˙Iv(t) = p2λvSv(t)−µvbIv(t).

(7.1)

The rate of change of the total human and mosquito populations is given by

˙Nh(t) = Λh −µhNh(t)−δhIh(t),

˙Nv(t) = Λv −µvbNv(t).