Malaria is one of the leading causes of morbidity and mortality in Kenya and it kills an estimated 34,000 children under five every year. Economically, it is estimated that 170 million working days in Kenya are lost each year because of malaria illness.

http://kenya.usaid.gov/programs/health/72.

After 1990, pyrethroids were promoted as insecticides of choice especially for Insecticides Treated Nets (ITNs) and Indoor Residual Spray (IRS) [1] [2] , due to high efficacy, rapid rate of knockdown, strong mosquito excito-repellence and low mammalian toxicity [3] .

In Kenya, ITNs have mainly been distributed to pregnant women and children under 5 years old by the Kenya Ministry of Health and non-governmental organizations [4] [5] . Currently, ITN coverage for children under 5 years old has increased rapidly from 7% in 2004 to 67% in 2006; this increase has been associated with a 44% reduction in malaria deaths [6] . However there is an increasing case resistance of mosquitoes to pyrethroid. The likely zoonotic nature of P. falciparum and the behavioral changes of mosquitoes are many new features which indicate that malaria control is not yet achieved [7] - [9] . The gains made from ITNs and IRS therefore are threatened by the development of physiological or behavioral resistance in the malaria vectors, which is widely documented [10] . Anopheline mosquitoes exhibit two major mechanisms of pyrethroid resistance, they are:

1) Increased level of metabolic detoxification of the insecticide,

2) Reduced sensitivity in the target sites of the insecticide. The target site of the pyrethroids is the voltage- gated sodium channel.

The second type of resistance is caused when a point mutation in the region II of the para-type sodium channel genes causes a change in affinity between the insecticide and its binding site or the sodium channel, and it induces a phenotype termed knock-down resistance (KDR) in a range of insecticides [11] - [14] . Insensitivity at the sodium channel target site also leads to cross-resistance between different classes of insecticides [15] .

Reports from literature confirm that use of ITNs and IRS has led to a substantial reduction in mosquitoes, reduced malaria transmission and a 44% reduction in malaria deaths [16] - [18] . However, although there was a global reduction in overall malaria transmission, 57% of the population continued to live in areas where transmission remained moderate to intense in Africa [19] . The ITNs and IRS intervention can reduce malaria transmission by targeting mosquitoes when they feed upon sleeping humans and/or rest inside houses, livestock shelters or other man-made structures. Despite high coverage, malaria spreading mosquitoes can maintain robust transmission because they develop resistance hence limiting the achievable impact [20] [21] . High and patchy resistance to pyrethroid insecticide has been confirmed in the endemic region of western Kenya, leaving the government with limited option but to seek other control measures [7] .

In this study, we develop a mathematical model with two types of vectors, one which is sensitive to the insecticides and a resistant type which adapts easily and survives despite the two types of intervention. We assume that the An. fambiae and the An. fenestus mosquito species are either sensitive or resistant to insecticides.

In section 2 we develop the model and equations. In section 3 the basic properties of the model are shown for positive invariance and computation of the basic reproduction number is also done. Section 4, we show the local and global stability of the Disease Free Equilibrium and section 6 is the conclusion.

We shall subdivide the mosquito population in Western Kenya into the traditional (non resistant) group and the new resistant group. This new resistance type has been termed as a “super mosquito”, but for the sake of terminology, we shall refer to them generally as “resistant” mosquito vectors. We shall use the subscripts “n” to represent non resistant traditional vectors, while, “r” represents the resistant vectors.

In this model will represent the susceptible human hosts while will represent the infectious human population. The variable representing the total human population will be given by. The non resistant susceptible (infectious) vectors will be represented by (), respectively, while the resistant vector population will be represented likewise as (), for the susceptible (infectious) population respectively. The total non resistant vector, therefore is given by and the resistant vector population by. We shall use, reservedly for the total resistant and non resistant vector populations respectively, hence.

When there are no malaria deaths, the host population dynamics is given by, and the total

human and mosquito population size approaches a carrying capacity for any non zero initial pop-

ulation size.

The non-resistant vector population defined by approaches a carrying capacity, while the resistant vector population approaches a carrying capacity.

Model assupmtions:

Ÿ the two types of vectors have different biting rates hence differentiated infectivity,

Ÿ the two types coexist and no vector changes status during the entire life span, i.e. not resistance vector becomes non-resistant or vice versa,

Ÿ The total vector and human populations are constant.

The following parameter symbols will be used in the equations:

Ÿ : The per capita rate of human birth,

Ÿ : The per capita rate birth rate of traditional vector and resistant vector respectively,

Ÿ : The proportion of infectious bites on hosts that produce a patent infection,

Ÿ : The proportion of bites by susceptible vectors on infectious hosts that produce a patent infection,

Ÿ : The per capita death rate for the human, traditional and resistant vectors, respectively,

Ÿ : Hosts rate of recovery,

Ÿ : The man biting rates of traditional and resistant vector, respectively.

The dynamics of our model will be governed by the following set of equations:

The term in the susceptible host’s compartment corresponds to a constant recruitment of susceptible hosts by natural birth.

The transmission term corresponds to frequency dependent infection of susceptible hosts by infectious non resistant mosquitoes, on infection they move to the infectious compartment.

The transmission term corresponds to frequency dependent infection of susceptible hosts by infectious resistant mosquitoes, on infection they also move to the infectious compartment.

The infected hosts who recover become susceptible again as malaria has no permanent immunity.

The last terms represents per capita deaths of the susceptible, infected hosts respectively.

In the susceptible vectors, represent the recruitment of susceptible non-resistant, (resistant) mosquitoes, respectively, by birth.

The term corresponds to the transmission of malaria to an susceptible non-resistant, (resistant) vectors, respectively, by an infected host.

Natural deaths affects all the groups as denoted by the parameter for the susceptible non- resistant, (resistant) vectors respectively, and for the infectious non-resistant, (resistant) vectors respectively.

Both resistant and non resistant vectors, once infected, are assumed to remain infected till death as mosquitoes do not recover or develop immunity from the parasite [22] [23] .

All the parameters in the model are non negative and the model equations are well posed.

Equation (1) is defined in feasible region

where denotes the non-negative cone of including its lower dimensional faces. It is clear that Ω is positively invariant with respect to (1). We denote the boundary and the interior of Ω by and respectively.

We use the relation and, and to reduce Equation (1), and therefore study the system

Jacobian evaluated at disease-free equilibrium solution:

Characteristic polynomial:

where

it is easy to show that

which is positive if.

This means all the roots of the polynomial equations are negative, hence the system is locally asymptotically stable.

The local dynamics of a general SIS and SI model is determined by the reproduction number. If, then each infected individual in its entire period of infectiousness will produce less than one infected individual on average. This means that the disease will be wiped out of the population. If, then each infected individual in its entire infectious period having contact with susceptible individuals will produce more than one infected individual implying that the disease persists in the population. If, and this is defined as the disease threshold, then one individual infects one more individual. For the disease free equilibrium is locally asymptotically stable while for the disease free equilibrium becomes unstable. By using the theory of Lasalle-Lyapunov function V, we will show the global asymptotic stability. The disease free equilibrium point is.

Theorem

If, then the disease-free equilibrium of the system is globally asymptotically stable on.

Proof

We construct the following Lasalle-Lyapunov function on the positively invariant compact set. Thus on, is continuous and non negative.

We define

The system of ordinary differential equations given by Equation (2) can be written as

This can be written as where

and.

If we define

then the derivative along the trajectories is given by as

We define the set. The largest invariant set is contained in the set E

for which or or. Thus when and. If or and, then. Thus by Lasalle’s invariance principle the disease free equilibrium is globally asymptotically stable on.

In this study, we formulated a malaria model representing the transmission of malaria by two types of vectors; the traditional mosquito which is sensitive to insecticides in ITNS and IRS, and a resistant type which is able to survive despite the control measures aimed at shortening their life span and limiting the biting rate. The basic reproduction number is determined as a contribution of the two types of vectors. The model is shown to be positively invariant, hence well posed. The Disease Free Equilibrium and the Endemic equilibrium are shown to be locally and globally asymptotically stable when and, respectively. The development of resistance in sensitive mosquitoes and the loss of resistance in resistant mosquitoes will be done in an upcoming article.

We wish to thank Calistus Ngonghala for the contribution and advise he gave during the formulation of this model.

Josephine Wairimu,Marilyn Ronoh, (2016) Modeling Insecticide Resistance in Endemic Regions of Kenya. Applied Mathematics,07,542-555. doi: 10.4236/am.2016.76050