Modeling Insecticide Resistance in Endemic Regions of Kenya

In this study, we develop an SIS model for two types of mosquitoes, a traditional one and one that is resistant to IRS and ITNs. The resistant mosquito develops behavioral adaptation to control measures put in place to reduce their biting rate. They also bite early before dusk and later after dark when people are outside the houses and nets. We determine the effect of the two types of mosquitoes on malaria transmission in Kenya. The basic reproduction number 0  is established as a sharp threshold that determines whether the disease dies out or persists in the population. Precisely, if 0  ≤ 1, the disease-free equilibrium is globally asymptotically stable and the disease always dies out and if 0  > 1, there exists a unique endemic equilibrium which is globally stable and the disease persists. The contribution of the two types of mosquitoes to the basic reproduction number and to the level of the endemic equilibrium is analyzed.


Introduction
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.
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 voltagegated 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.

The Model Formulation and Equations
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 h S will represent the susceptible human hosts while h I will represent the infectious human population.The variable h N representing the total human population will be given by h h S I + .The non resistant susceptible (infectious) vectors will be represented by n S ( n I ), respectively, while the resistant vector population will be represented likewise as h S ( h I ), for the susceptible (infectious) population respectively.The total non resistant vector, therefore is given by n n n N S I = + and the resistant vector population by r r S I + .We shall use v N , reservedly for the total resistant and non resistant vector populations respectively, hence

Model assupmtions:
 the two types of vectors have different biting rates hence differentiated infectivity,  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,  The total vector and human populations are constant.
The following parameter symbols will be used in the equations:  a a : The man biting rates of traditional and resistant vector, respectively.The dynamics of our model will be governed by the following set of equations:

h h m n h m r h h h h h h h n r h m n h m r h h h h h h h h n n h n n n n h h n h n n n n h h r r h n r r r h
The term h Λ in the susceptible host's compartment corresponds to a constant recruitment of susceptible hosts by natural birth.Λ Λ represent the recruitment of susceptible non-resistant, (resistant) mos- quitoes, 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 nonresistant, (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 8 +  denotes the non-negative cone of 8   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 ∂Ω and Ω  respectively.1), and therefore study the system

We use the relation
.

A Compact Positively Invariant Set
In this section we prove that the following set ( ) , , , , 0 ,0 ,0 ,  is a positively invariant compact set for system (2) by barrier theorems (e.g.[24] [25]).Moreover  is a global attractor on the nonnegative orthant 5 +  Now we show that the vector field induced by the system is either tangent or entering  on the boundary  .
we have ( ) The Equation for the Traditional non reistant mosquito n N is given by d d , which can be written as Integrating both sides we and applying the intial conditions ( 0 t = ) we have Finally, the equation for the resistant mosquito given by d d Integrating both sides we and applying the intial conditions ( 0 t = ) we have Thus the feasible set for the model system ( 1) is given by  , which is a positively invariant set.Hence the model is well posed and biologically meaningful.

Basic Reproduction Number
( ) which can be simplified as The expression 0  is caled the basic reproduction number, with a biological meaning that is can be inter- represents the number of secondary infections to a resistant mosquito vector by a human host.The square root sign represent the two generations that the disease has to undergo from a mosquito to a human being and to a mosquito again or vice versa for the infection to take place.It is a number that determines the threshold for disease spread, as well as a control tool that whose parameters can be targeted for control.

Stability of Disease-Free Equilibrium Solution
Jacobian evaluated at disease-free equilibrium solution: ) ,

, r h n n h m r n h r n h h h h r n h
( ) ( ) This means all the roots of the polynomial equations are negative, hence the system is locally asymptotically stable.

Global Stability of the DFE
The local dynamics of a general SIS and SI model is determined by the reproduction number 0  .If 0 1 ≤  , 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 0 1 >  , 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 0 1 =  , and this is defined as the disease threshold, then one individual infects one more individual.For 0 1  ≤  the disease free equilibrium is locally asymptotically stable while for 0 1  >  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 We define , , .
The system of ordinary differential equations given by Equation ( 2) can be written as .

n r h m n h h m r h h h h h h h h h n h n n n n n n
This can be written as ( ) , , , then the derivative along the trajectories is given by ( )  .Thus by Lasalle's invariance principle the disease free equilibrium is globally asymptotically stable on  .

Theorem
The endemic equilibrium * h I , * n I and * r I is locally asymptotically stable on  .

Proof
The system of equations 5 can also be expressed as follows when we let ( ) .
The Jacobian computed at the endemic equilibrium using the relations given by Equation ( 6) can be expressed as:

m n n m r r h h h h h h n r n n h n h h n r h n n h r h r r
To determine the stability of the endemic equilibrium ( )            The requirements of Routh-Hurwitz stability criteria are satisfied hence this proves that the endemic equilibrium is locally asymptotically stable  .

Theorem
The endemic equilibrium is globally asymptotically stable on  if 0 Then the derivative of 1 V , obtained by direct calculation along the solution of ( ) Substituting the expressions of the model system ( )   . .Therefore the largest compact invariant set is the singleton set 1 E which is the endemic equilibrium.Lasalle Invariance principle 1 E is globally asymptotically stable on  .

m n n h m r h r m n n h m r r r h n n h h h h h h h n h n h n h r r h h r h r
NB: In an upcoming article, we include a human protection factor and the development of mosquito resistance during their life time.Wa also allow some resistant vectors to become sensitive to insecticides.

Conclusion
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 0 1 <  and 0 1 >  , respectively.The development of resistance in sensitive mosquitoes and the loss of resistance in resistant mosquitoes will be done in an upcoming article.
there are no malaria deaths, the host population dynamics is given by h

hΛ:
The per capita rate of human birth,  , n r Λ Λ : The per capita rate birth rate of traditional vector and resistant vector respectively,  m b : The proportion of infectious bites on hosts that produce a patent infection,  h b : The proportion of bites by susceptible vectors on infectious hosts that produce a patent infection, : The per capita death rate for the human, traditional and resistant vectors, respectively, The Integrating factor for this linear differential equation is given by d The Integrating factor for this linear differential equation is given by d preted from terms under the square root sign.The first term n m h b b Λ, represents the number of secondary human infections caused by one infected resistant and one none resistant mosquito vector.The term ( ) of secondary mosquito infections caused by one infected human to an non-resistant vector, = of the system is globally asymptotically stable on  .ProofWe construct the following Lasalle-Lyapunov function ( ) on the positively invariant compact set  .Thus on  , ( ) is continuous and non negative.
The largest invariant set is contained in the set E , we use the Routh-Hurwitz stability criteria on the characteristic equation of a third degree polynomial given by ( )

F
represents the positive terms of the equation above and G represents the negative terms of the said equation.The expression of F and G are as follows: prove the Routh-Hurwitz stability criteria we compute 1 2 3