# Determining Effective Spraying Periods to Control Malaria via Indoor Residual Spraying in Sub-Saharan Africa

Authors: Robert J. Smith?; Senelani D. Hove-Musekwa
Journal: Journal of Applied Mathematics and Decision Sciences
 (2008)

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## Abstract

Indoor residual spraying—spraying insecticide inside houses to kill mosquitoes—is an important method
for controlling malaria vectors in sub-Saharan Africa. We propose a mathematical model for both regular
and non-fixed spraying, using impulsive differential equations. First, we determine the stability properties
of the nonimpulsive system. Next, we derive minimal effective spraying intervals and the degree of
spraying effectiveness required to control mosquitoes when spraying occurs at regular intervals. If
spraying is not fixed, then we determine the “next best” spraying times. We also consider the effects of
climate change on the prevalence of mosquitoes. We show that both regular and nonfixed spraying will
result in a significant reduction in the overall number of mosquitoes, as well as the number of malaria
cases in humans. We thus recommend that the use of indoor spraying be re-examined for widespread
application in malaria-endemic areas.


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## Body

## 1. Introduction

Malaria causes more than 300 million acute
illnesses and at least one million deaths annually, and remains one of the
most important human diseases throughout the tropical and subtropical regions
of the world [1]. It
is a leading cause of death and disease in many
developing countries, where young children and pregnant women are the groups
most affected. 40% of the world's population live in
malaria-endemic areas [2]; 90% of deaths due to malaria occur in sub-Saharan
Africa [3], 75%
of whom are African children [4].Control of malaria is largely through vector control
and chemoprophylaxis. Vector control is an intervention targeted to reducing
vector population density and survival, which aims as an end product to reduce
malaria transmission. Indoor residual spraying (IRS) is one of the primary
vector control interventions for reducing and interrupting malaria
transmission. In recent years, however, it has received relatively little
attention. Recent data reconfirm the efficacy and effectiveness of IRS in
malaria control in countries where it was implemented well [5]. Since many malaria vectors
are endophilic, resting inside houses after taking a blood meal, they are
particularly susceptible to be controlled through IRS. This method kills the
mosquitoes after they have fed, thereby stopping transmission of the disease.
IRS resulted in the suppression of An. funestus, which is no longer an
important vector for transmission of malaria, in some areas of the subregion
[6]. An. gambiae s.s. was also well controlled
[5]. The user is able
to spray the whole house or dwelling on the inside, and under the eaves on the
outside. The duration of effective action ranges from two to greater than six
months [7].Malaria eradication projects in the 1950's through
1970's in Benin, Brukina Faso, Burundi, Cameroon, Kenya, Liberia, Madagascar,
Nigeria, Rwanda, Senegal, Uganda, and the United Republic of Tanzania
demonstrated that malaria was highly responsive to control by IRS, with a
significant reduction of anopheline vector mosquitoes and malaria. The
application of IRS consistently over time in large areas has altered the vector
distribution and subsequently the epidemiological pattern of malaria in
Botswana, Namibia, South Africa, Swaziland, and Zimbabwe [8–11]. IRS has commonly been the intervention of choice in
areas of particular economic interest (e.g., tourism, mining, oil extraction,
and agricultural schemes) that require a rapid and effective prevention, where
financial and logistic constraints do not prevail [5].We have developed a mathematical model to account for
IRS using impulsive differential equations, in order to determine the minimal
effective spraying period, as well as the amount by which mosquitoes should be
reduced at each spraying event. However, the spraying may not happen at fixed
intervals, due to limitations in resources and unforeseen events. If the
spraying times are not fixed, then the optimal solution for the next spraying
event can be calculated, but it depends on the entire history of spraying
events, which may not be known. However, a suboptimal solution can be found,
using partial information: the spraying effectiveness and the time of the last
two spraying events.This paper is organised as follows. In Section2, we
introduce the mathematical model. In Section 3, we analyse the nonimpulsive
version of the model and determine the basic reproductive ratio. In Section 4,
we analyse the model with impulses and determine minimal effective spraying
times and spraying effectiveness, for both regular and nonfixed spraying. In
Section 5, we examine the effects of climate change on the results. In Section 6, we illustrate the results with numerical simulations.
Finally, in Section 7, we discuss the implications of the results.


## 2. The Model

It can be assumed that mosquitoes are either
susceptible (M) or infected (N), have birth rate Λ,
and their death rate (μ) does not vary significantly if they are
infected. Thus, we assume that the infective period of the vector ends with its
death, and therefore the vector does not recover from being infective [12]. Individuals who have
experienced infection may recover (without substantial gain in immunity) at
recovery rate h or may become temporarily immune at acquired
immunity rate α.
See [13–17] for further details.
Temporarily immune individuals will become susceptible again at rate δ.
The rate of infection of a susceptible individual is βH,
and the rate of infecting a mosquito is βM.
The birth rate for humans is π,
the background death rate is μH,
and γ is the death rate due to malaria. Humans may
be susceptible (S), infected (I), or temporarily immune (R). See Figure 1.Figure 1
The model
consists of susceptible (S), infected (I), and recovered (R) humans, as well as susceptible (M) and infected (N) mosquitoes. Humans may recover
without immunity at rate h, or may become temporarily immune at rate α. Such individuals will later
become susceptible again at rate δ. The rate of infection of a
susceptible individual is βH, and the rate of infecting a mosquito is
βM. The birth
and death rates are not drawn in, for conciseness.We assume that spraying reduces both susceptible and
infected mosquitoes by the same proportionr (satisfying 0≤r≤1), and that it occurs at distinct times tk (k=0,1,2,…). These times may be fixed or variable. We
model the effect of spraying by a system of impulsive differential equations.
Impulsive differential equations consist of a system of ordinary differential
equations (ODEs), together with difference equations. Between
“impulses” tk,
the system is continuous, behaving as a system of ODEs. At the impulse points,
there is an instantaneous change in state in some or all of the variables. This
instantaneous change can occur when certain spatial, temporal, or
spatiotemporal conditions are met. This has the advantage of capturing the
dynamics between spraying events, while ignoring the short-term transient
behaviour during the spraying itself. We refer the interested reader to
[18–21] for more details on the
theory of impulsive differential equations.Thus, the model is(2.1)dSdt=π−βHSN+hI+δR−μHS,dIdt=βHSN−hI−αI−(μH+γ)I,dRdt=αI−δR−μHR,dMdt=Λ−μM−βMMI,dNdt=βMMI−μNfor t≠tk,
with impulsive conditions given by(2.2)ΔM=−rM−,ΔN=−rN−for t=tk,
where ΔM=M+−M−, M−≡M(tk−),
and, equivalently, M+≡M(tk+).Hence, we are modelling the situation where IRS occurs
simultaneously in multiple households, as occurs in areas in several countries [5]. Our
model assumes that both humans and mosquitoes are well mixed in these areas.
However, our results do not depend upon the form of the model for humans and
only rely on certain aspects of the equations for mosquitoes. Further
implications are taken up in Section 7.


## 3. Analysis of the Nonimpulsive System

First, we will analyse the system without impulses; that is, without spraying. The disease-free equilibrium for the nonimpulsive
model is given by(3.1)E0=(S¯,I¯,R¯,M¯,N¯)=(πμH,0,0,Λμ,0).The endemic equilibrium is given by(3.2)E1=(S*,I*,R*,M*,N*),where(3.3)S*=πμH+δαI*μH(δ+muH)−α+μH+γμHI*,R*=αI*δ+μH,M*=Λμ+βMI*,N*=βMΛI*μ(μ+βMI*),I*=[βHβMΛπ−(h+α+μH+γ)μ2μH](δ+μH)βM[(μH+γ)(βHΛ+μ)(δ+μH)+(βMΛ+μ)αμH+μh(δ+μH)+μαδ].It can be seen thatE0 attracts the region(3.4)Ω0={(S,I,R,M,N):I=R=N=0}.Theorem 3.1.
The
basic reproductive ratio [22] for model (2.1) is given by(3.5)R0=βHβMΛπμ2μH(μH+α+γ+h).The disease-free equilibrium is
stable if and only if R0<1.
Furthermore, the endemic equilibrium is positive if and only if R0>1.Proof.
The Jacobian matrix for model (2.1)
is(3.6)J=[−βHN−μHhδ0−βHSβHN−(h+α+μH+γ)00βHS0α−(δ+μH)000−βMM0−μ−βMI00βMM0βMI−μ].At the disease-free
equilibrium,(3.7)J|I=N=0=[−μHhδ0−βHS¯0−(h+α+μH+γ)00βHS¯0α−(δ+μH)000−βMM¯0−μ00βMM¯00−μ].The eigenvalues of this matrix
satisfy the characteristic equation(3.8)(−μH−λ)(−δ−μH−λ)(−μ−λ)det⁡[−(h+α+μH+γ)−λβHS¯βMM¯−μ−λ]=0.The only change in sign from the
eigenvalues can occur from this last determinant, which
satisfies(3.9)λ2+λ(μ+h+α+μH+γ)+μ(h+α+μH+γ)−βHβMS¯M¯=0.This equation will have negative
roots if μ(h+α+μH+γ)−βHβMS¯M¯>0,
or, equivalently, if and only if(3.10)R0≡βHβMΛπμ2μH(μH+α+γ+h)<1.
Finally,I* is clearly positive if and only if R0>1.Remark 3.2.
It follows that there is a
transcritical bifurcation atR0=1.
Thus, R0 is the average number of mosquitoes infected by
a single human (βMπ/μ2) multiplied by the average number of humans
infected by a single mosquito (βHΛ/μH(μH+α+γ+h)).


## 4. Analysis of the Impulsive System

When spraying events are included, the system will
undergo an instantaneous jump when IRS is applied. We thus analyse model (2.1)
when impulses are included. However, the mosquito dynamics will prove to be far
more important in the model than those of humans.If we define the total mosquito population
by(4.1)Ψ=M+N,then we have the decoupled
impulsive differential equation(4.2)dΨdt=Λ−μΨ,t⁢≠tk,ΔΨ=−rΨ,t=tk.Thus,(4.3)Ψ+−Ψ−=−rΨ−,Ψ+=(1−r)Ψ−.Hence, for tk≤t<tk+1,(4.4)Ψ′(t)+μΨ(t)=Λ,ddt(eμtΨ)=Λeμt,eμtΨ(t)−eμtkΨ(tk+)=Λμeμt−Λμeμtk,Ψ(t)=Λμ(1−eμ(tk−t))+Ψ(tk+)eμ(tk−t).It follows that(4.5)Ψk+1−=Λμ(1−e−μ(tk+1−tk))+Ψk+e−μ(tk+1−tk)=Λμ(1−e−μ(tk+1−tk))+(1−r)Ψk−e−μ(tk+1−tk),using (4.3).We thus have a recurrence relation for the total
number of mosquitoes immediately before spraying. This relation depends on the
birth and death rates of mosquitoes, the spraying times, and the spraying
effectiveness.Theorem 4.1.
If spraying occurs at fixed times, satisfyingtk+1−tk=τ,
then(4.6)Ψ−(r)=Λμ⋅1−e−μτ1+(r−1)e−μτis a globally asymptotically
stable fixed point of the recurrence relation(4.7)Ψk+1−=Λμ(1−e−μ(tk+1−tk))+(1−r)Ψk−e−μ(tk+1−tk).Proof.
For completeness, defineΨ0 to be the preimage of Ψ(0) under the impulsive condition. That is, Ψ0=(1/(1−r))Ψ(0).
Then, we have(4.8)Ψ1−=Λμ(1−e−μ(t1−t0))+(1−r)Ψ0e−μ(t1−t0),Ψ2−=Λμ(1−e−μ(t2−t1))+(1−r)Ψ1−e−μ(t2−t1)=Λμ(1−e−μ(t2−t1))+(1−r)Λμ(1−e−μ(t1−t0))e−μ(t2−t1)+(1−r)2Ψ0e−μ(t1−t0)e−μ(t2−t1)=Λμ(1−re−μ(t2−t1)−(1−r)e−μ(t2−t0))+(1−r)2Ψ0e−μ(t2−t0),Ψ3−=Λμ(1−e−μ(t3−t2))+(1−r)Ψ2−e−μ(t3−t2)=Λμ(1−e−μ(t3−t2))+(1−r)Λμ(1−re−μ(t2−t1)−(1−r)e−μ(t2−t0))e−μ(t3−t2)+(1−r)3Ψ0e−μ(t2−t0)e−μ(t3−t2)=Λμ(1−re−μ(t3−t2)−r(1−r)e−μ(t3−t1)−(1−r)2e−μ(t3−t0))+(1−r)3Ψ0e−μ(t3−t0),Ψ4−=Λμ(1−re−μ(t4−t3)−r(1−r)e−μ(t4−t2)−r(1−r)2e−μ(t4−t1)−(1−r)3e−μ(t4−t0))+(1−r)4Ψ0e−μ(t4−t0)⋮Ψn−=Λμ(1−∑i=1n−1r(1−r)n−i−1e−μ(tn−ti)−(1−r)n−1e−μ(tn−t0))+(1−r)nΨ0e−μ(tn−t0).
For regular spraying,tn−ti=(n−i)τ,
so we have(4.9)Ψn−=Λμ(1−re−μτ−(1−r)n−1re−μτ1−(1−r)e−μτ−(1−r)n−1e−μnτ)+(1−r)nΨ0e−μnτ⟶Λμ(1−re−μτ1−(1−r)e−μτ)as n→∞, since 0<r<1.Remarks 4.2.
(1) Note that(4.10)⁡limτ→0n→∞⁡Ψn−=0.Thus, the total mosquito
population shrinks to zero as spraying period decreases.
(2) It follows from Theorem4.1 that the impulsive
periodic orbit given by (4.4), with endpoints Ψ− and (1−r)Ψ−,
where Ψ− satisfies (4.6), is asymptotically
stable.Corollary 4.3.
(1) To reduce the total mosquito
population below a desired threshold Ψ,
the minimum spraying effectiveness satisfies(4.11)r=1−[1−ΛμΨ(1−e−μτ)]eμτ.(2) To reduce the mosquito
population below a desired threshold Ψ,
the minimum spraying period satisfies(4.12)τ=−1μln⁡[Λ−μΨΛ+μΨ(r−1)].Proof.
(1) SinceΨ(t)≤Ψ− for tk≤t≤tk+1,
the maximum within each cycle occurs immediately before spraying is
undertaken, so we can set Ψ=Ψ−.
By Theorem 4.1, we have(4.13)Ψ=Λμ⋅1−e−μτ1+(r−1)e−μτ,1+(r−1)e−μτ=ΛμΨ(1−e−μτ),r=1−[1−ΛμΨ(1−e−μτ)]eμτ.
(2) Similarly, we have(4.14)(r−1+ΛμΨ)e−μτ=ΛμΨ−1,τ=−1μln⁡[Λ−μΨΛ+μΨ(r−1)].It follows that we can find the minimal spraying
effectiveness or the minimal spraying period, in terms of the birth and death
rates of mosquitoes and the spraying effectiveness.Theorem 4.4.
If spraying occurs at nonfixed times, then, assuming
the two previous spraying events are known, the population of mosquitoes can be
reduced below the thresholdΨ if the next spraying event
satisfies(4.15)tn+1≤tn−1μln⁡[2−r−μΨ/Λ1+r(1−r)e−μ(tn−tn−1)].Proof.
Forn large,(4.16)Ψn−≈Λμ(1−∑i=1n−1r(1−r)n−i−1e−μ(tn−ti))since (1−r)n−1≈0 and e−μ(tn−t0)≈0.
If we assume e−μ(tn−tn−2) is small, then, using (4.5), we
have(4.17)Ψn−<Λμ(1−e−μ(tn−tn−1)),Ψn+1+<Λμ(1−e−μ(tn+1−tn))+(1−r)Ψn−e−μ(tn+1−tn)<Λμ(1−e−μ(tn+1−tn))+(1−r)Λμ(1−re−μ(tn−tn−1))e−μ(tn+1−tn).Define(4.18)Ψ≡Λμ(1−e−μ(tn+1−tn))+(1−r)Λμ(1−re−μ(tn−tn−1))e−μ(tn+1−tn).Thus,(4.19)Λμ(1+(1−r))−Ψ=e−μ(tn+1−tn)Λμ(1+r(1−r)e−μ(tn−tn−1)),e−μ(tn+1−tn)=2−r−μΨ/Λ1+r(1−r)e−μ(tn−tn−1),tn+1=tn−1μln⁡[2−r−μΨ/Λ1+r(1−r)e−μ(tn−tn−1)].Hence, if spraying occurs at tn+1 or earlier, then the number of mosquitoes will
be less than or equal to Ψ,
immediately after the (n+1)th spraying event.Thus, we can derive the “next best” spraying events
for nonfixed spraying, by assuming that the time between the current spraying and two sprayings events previously is sufficiently large.Theorem 4.5.
If nonfixed spraying occurs indefinitely, then there exists a minimum spraying effectivenessr0,
satisfying 0<r0<1,
such that variable spraying is only effective for r0≤r≤1.
Furthermore, on this interval, the minimum spraying interval for indefinite
nonfixed spraying is always less than the minimum spraying interval for regular
spraying.Proof.
First, note that, for regular
spraying, we have(4.20)τ=−1μln⁡[Λ−μΨΛ+μΨ(r−1)],τ|r=0=−1μln⁡[Λ−μΨΛ−μΨ]=0,(4.21)τ|r=1=−1μln⁡[Λ−μΨΛ]=−1μln⁡[1−μΨΛ].
However, Ψ=M+N.
So, if there is no impulse, then, from (4.2), limt→∞Ψ(t)=Λ/μ.
Thus, we can assume that Ψ<Λ/μ.
Hence,(4.22)0<1−μΨΛ<1,and thus τ|r=1>0.
If nonfixed spraying occurs indefinitely, then letτnf≡tn+1−tn=tn−tn−1.
The minimum spraying effectiveness then satisfies(4.23)τnf=−1μln⁡[2−r−μΨ/Λ1+r(1−r)e−μτnf].If τnf=0,
then(4.24)−1μln⁡[2−r−μΨ/Λ1+r(1−r)]=0,2−r−μΨΛ=1+r(1−r),r2−2r+1−μΨΛ=0,r=1±μΨΛ.Clearly, the larger root exceeds
unity and can hence be discounted. The smaller root, r0≡1−μΨ/Λ,
satisfies 0<r0<1 by (4.22). It follows that spraying is only
effective in the range r0≤r≤1.
Next, we have(4.25)τnf|r=1=−1μln⁡[1−μΨ/Λ1]=τ|r=1,from (4.21).
Note thatΛ+μΨ(r−1) and (2−r)Λ−μΨ are both positive on 0<r<1,
since Λ−μΨ>0. Since e−μτnf<1, we have(4.26)Λ−μΨΛ+μΨ(r−1)⋅1+r(1−r)e−μτnf2−r−μΨ/Λ<Λ(Λ−μΨ)Λ+μΨ(r−1)⋅1+r(1−r)(2−r)Λ−μΨ.Thus,(4.27)Λ(Λ−μΨ)Λ+μΨ(r−1)⋅1+r(1−r)(2−r)Λ−μΨ−1=γ[Λ+μΨ(r−1)][(2−r)Λ−μΨ],
where
(4.28)γ=−Λ2(r−1)2+2μΨΛ(r−1)2+μ2Ψ2(r−1)=−(r−1)[Λ2(r−1)−2μΨΛ(r−1)−μ2Ψ2].For 0<r<1, r−1<0. Furthermore, if Λ−2μΨ>0, then the quantity in the square brackets is increasing and hence the maximum value it attains on the interval 0<r<1 is −μ2Ψ2 at r=1. Conversely, if Λ−2μΨ<0, then the quantity in the square brackets is decreasing and hence the maximum value it attains on the interval 0<r<1 is −(Λ−μΨ)2 at r=0. In either case, γ<0 on the interval 0<r<1.
Consequently,(4.29)Λ−μΨΛ+μΨ(r−1)⋅1+r(1−r)e−μτnf2−r−μΨ/Λ<1,and hence(4.30)τ−τnf=−1μln⁡[Λ−μΨΛ+μΨ(r−1)⋅1+r(1−r)e−μτnf2−r−μΨ/Λ]>0.Thus, τ>τnf for 0<r<1.It follows that nonfixed spraying is always worse than
regular spraying—even in the best-case scenario where such spraying is applied
at regular intervals—and is only defined for a sufficiently effective
insecticide.


## 5. The Impact of Climate Change

As global temperatures increase, one of the major
impacts will be an increase in the birth rate of mosquitoes [23, 24]. Consequently, we examine
the impact of increasing the birth rate on the minimal effective period of IRS
required to maintain mosquitoes at given thresholds.If the mosquito birth rate is increased fromΛ to Λ+Λ1,
then the recursion relation (4.5), with regular spraying,
becomes(5.1)Ψk+1−=Λ+Λ1μ(1−e−μτ)+(1−r)Ψk−e−μτ.This has
solution(5.2)Ψ−=Λ+Λ1μ⋅1−e−μτ1+(r−1)e−μτ.Rearranging, we
have(5.3)τ=−1μln⁡[Λ+Λ1−μΨΛ+Λ1+μΨ(r−1)].It follows that(5.4)∂τ∂Λ1=−rΨ(Λ+Λ1−μΨ)(Λ+Λ1−μΨ+rμΨ)<0since Ψ<Λ/μ.
Thus, as the mosquito birth rate increases, the minimal effective spraying
period will always be reduced, for a fixed mosquito threshold Ψ.
In particular, we have(5.5)lim⁡Λ1→0τ=0.


## 6. Numerical Simulations

The average lifespan of a mosquito is of the order of
days to weeks [25]; we
chose an intermediate value of 7 days. The birth rate of mosquitoes is the
carrying capacity divided by the lifespan [26]. With a lifespan of 7 days and a carrying capacity of
20 000 [26], this
results in 1400 females per year. Correcting for those not reached by spraying
(e.g., those who feed away from houses), we assumed 1000 females per year. The
probability of infection is the product of the biting rate times the
probability that a bite is infectious. The former value is 0.7 per day and the
latter is 0.75 [27],
resulting in an infection probability for humans of 0.5 per day. The value for
mosquito infection is assumed to be one tenth of the value for humans. The
total duration of malaria infection in humans is 3–7 days [28]. We chose recovery,
immunity, and mortality rates so that the total duration of infection was 3
days.The dependency of the mosquito population upon the
spraying effectiveness is illustrated in Figure2. The two curves indicate the
maximum and minimum mosquito populations if an insecticide is used which
reduces mosquitoes by the percentage on the x-axis, when sprayed every three months. These
are the long-term outcomes of fixed spraying. The greater the spraying
effectiveness is, the more variation in the overall mosquito population exists,
but the lower the average mosquito population will be. We chose parameters to
simulate a small spraying region, since mosquito spraying may occur at different
times.Figure 2
The relative reduction of
mosquitoes as a function of the spraying effectiveness. The two
curves indicate the maximum and minimum numbers of mosquitoes,
given byΨ− and Ψ+=(1−r)Ψ−,
respectively. Parameters used were Λ=1000mosquitoes⋅years−1, μ=1/7.3days−1, and spraying that occurred every
three months. Note that there is a discontinuity at the right
endpoint due to the impulsive nature of the
solutions; if spraying is 100% effective, then the mosquito population will be
zero.Varying the period of spraying will result in a change
of strategy, as shown in Figure3. A mosquito control program aiming to reduce
the maximum number of mosquitoes by 15% would require an insecticide that
reduced mosquitoes by 92% per spraying if spraying occurred three times a year,
or by 54% if spraying occurred 2.3 times a year.Figure 3
Two spraying options: every three
months (solid curve) and every 2.3 months (dashed curve). A
mosquito-control program
aiming to reduce the number of mosquitoes by 15% would require a 92% effective
insecticide if spraying occurred every three months, or 54% if spraying
occurred every 2.3 months. Note that these curves illustrate the
maximum number of mosquitoes in each case, showing the worst-case
scenario.The dependency of the mosquito population upon the
spraying effectiveness, for both regular and nonfixed spraying, is illustrated
in Figure4. If spraying is fixed, then any spraying effectiveness may
theoretically be used, when the insecticide is applied with sufficient
frequency. If spraying is not fixed, then there is minimum spraying
effectiveness that must be satisfied. A 90% effective insecticide should be
sprayed at  three-month intervals for regular spraying, or every 2.3 months for
nonfixed spraying, to reduce the overall mosquito population to 85% of that of
the mosquito population without spraying.Figure 4
The minimum spraying intervals, for both regular
spraying and nonfixed spraying, to reduce the overall mosquito
population by 15% of the levels without spraying. While regular spraying can
theoretically be applied for any spraying effectiveness, nonfixed
spraying is
only applicable if the spraying is 8% effective or greater. An insecticide that
reduced mosquitoes by 90% at each spraying would have to be applied every three
months, if it were applied regularly, but not more than every 2.3
months if spraying was not fixed.To illustrate this, model (2.1) was simulated, over a
period of 100 years. Regular spraying occurred every three months, for an
insecticide that was 85% effective. Regular spraying significantly reduced the
number of malaria cases in humans (Figure 5(a)) and the number of infected mosquitoes (Figure 5(b)). Nonfixed spraying was also illustrated, for a spraying
program with random spraying events chosen from a normal distribution with a
mean of 4 months and a standard deviation of 1.2 months. In this case, the
number of malaria cases in both humans and mosquitoes was also reduced
significantly (Figure 6). However, during some periods where the gap between
spraying events was excessive, the peaks of
infection matched the number of infections without spraying.(a)
Number of infected humans, over a ten year period, for no spraying
(dashed curve) and regular, three-monthly spraying (solid curve)
Inset: Mean number of infected humans over a 100 year period. (b)
Number of infected mosquitoes over a ten year period, for no
spraying (dashed curve) and regular, three-monthly spraying (solid
curve). Inset: Mean number of infected mosquitoes over a 100 year
period. Data used wereΛ=1000mosquitoes⋅years−1, βM=0.05 mosquitoes−1 days−1, h=1/9 days−1, δ=1/30 days−1, μH=1/30 years−1, α=1/8 days−1, γ=1/20 days−1, μ=1/7.3 days−1, π=100 humans ⋅ years−1, βH=0.5 humans−1 days−1, r=0.85 and τ=0.25 years.
(a)(b)(a) Number of
infected humans, over a 25-year period, for no spraying (dashed
curve) and nonfixed spraying (solid curve). (b) Number of infected
mosquitoes over a 25-year period, for no spraying (dashed curve)
and nonfixed spraying (solid curve). Data used were identical to
those in Figure5, except for the time of spraying. These times
were randomly generated from a normal distribution, with a mean of
4 months and a standard deviation of 1.2 months.
(a)(b)The effects of increasing the mosquito birth rate are
illustrated in Figure7. The minimal effective spraying period for regular
spraying will always decrease as the mosquito birth rate increases; however,
even a small increase in the mosquito birth rate has a significant effect on
the reduction of the minimal effective spraying period.Figure 7
The amount of IRS
required to maintain present mosquito levels, as the mosquito
birth rate increases. If the mosquito
birth rate is increased by 25%, then the minimal effective spraying period for
regular spraying decreases by about half (dashed red lines). If
the mosquito birth rate doubles, then the minimal effective
spraying period is reduced by three quarters. As the mosquito
birth rate continues to increase, the minimal effective spraying
period is driven to zero. In this case, identical parameters were used to Figure2.Finally, sensitivity to the other significant
parameter, the mosquito death rate, is illustrated in Figure8. As the death
rate increases, the minimal effective spraying period decreases. There is a
vertical asymptote at μ=Λ/Ψ,
since Ψ≤Λ/μ,
the equilibrium level from the nonimpulsive system. That is, if μ>Λ/Ψ,
then dΨ/dt<0 and thus the number of mosquitoes would never
increase.Figure 8
Sensitivity of IRS to changes in the
mosquito death rate. Parameters used are identical to Figure2 and
the median case illustrated by the dashed red line. As the death
rate decreases, the minimal effective spraying period decreases,
but it is bounded below by τmin=0.085 years. Thus, if regular spraying occurs every month,
then mosquitoes would be eradicated even if they never died due to
any other cause. Conversely, as the death rate increases, the
minimal effective IRS period is increased. There is a vertical
asymptote at μ=Λ/Ψ; above this level, mosquitoes would be
dying faster than they were born—an unrealistic scenario.


## 7. Discussion

We derived minimal effective spraying times for either
fixed or variable spraying. Once the birth and death rates of mosquitoes and
spraying effectiveness of the insecticide are known, the minimal effective
spraying period can be determined, using (4.12). This is a simple formula that
can be easily calculated by policy makers and health officials.If spraying occurs at regular, known intervals (e.g., every
six months), then the minimal insecticide effectiveness or spraying period can
be derived (Corollary4.3). If spraying does not occur at fixed intervals, then
the optimal result would depend on knowing the entire history of spraying in
the area. Since this is not possible, we assume that only the previous two
spraying events are known. In this case, the next best spraying is given by
Theorem 4.4. While this provides a recipe for coping with the “next best”
outcome, it should be noted (from Theorem 4.5) that (a) nonfixed spraying is
always less optimal than regular spraying and (b) only applies for a
sufficiently effective insecticide.These thresholds are analytical, so their application may vary, depending on the region in which they are applied. However, we
provide an illustrative example: an insecticide which reduces mosquitoes by 90%
at each spraying will ultimately result in a 15% reduction in mosquitoes if
sprayed every three months. If the same insecticide is used, but with nonfixed
spraying, then the insecticide should be applied at 2.3-month intervals to
achieve a 15% reduction.The mosquito birth rate may increase due to a variety
of factors; one of those factors will be the impact of climate change, as
global temperatures increase. The effect of global warming will have an
increasingly heavy burden on the resulting change in strategy; if the mosquito
birth rate increased by one quarter, as a result of temperate changes, then the
minimal effective IRS period would be reduced by roughly half. If the mosquito
birth rate doubled, then the minimal effective IRS period would be reduced by
about three quarters. Since the spraying of insecticide consumes valuable and
limited resources [29],
we therefore conclude that global warming will have a disproportionately
detrimental effect in malaria-endemic countries. However, it should be noted
that the effects of climate change are likely to be significantly more
complicated than considered here.The dependence of the minimal effective spraying
period upon mosquito birth rates is also a measure of the sensitivity of the
results to changes in the latter. Since the thresholds for the insecticide
effectiveness and spraying period also serve as sensitivity analyses to their
respective parameters, we thus performed a sensitivity analysis on the only
remaining significant parameter, the death rate of mosquitoes. The result is
reasonably sensitive to changes in the death rate (Figure8), but this is
unsurprising; many models are sensitive to changes in death rates (see
[30] for more
discussion on this topic), but we do not expect the death rate to vary
enormously.We use a simple SIR model for humans, with mass action
terms, but the bulk of the analysis only depends on the form of the mosquito
interactions. Thus, the results are independent of the mass-action condition,
and will be similar for other models, as long as the total mosquito population
satisfies (4.1). In particular, the model could easily accommodate a separate,
exposed, class, and specific biting rates of mosquitoes, with the ODEs for
mosquitoes satisfying(7.1)dMdt=Λ−μM−βMbMIΣ,dEdt=βMbMIΣ−θE,dNdt=θE−μN,where E is the exposed (but noninfectious) class, b is the biting rate of mosquitoes, θ is the duration of exposure, and Σ is the total human population. These more
complicated dynamics for mosquitoes still satisfy (4.1), and thus our results
still apply. Similarly, if only a single household were modelled, the dynamics
for humans would not be well approximated by ordinary differential equations,
whereas the dynamics of mosquitoes still might be, if sufficiently prevalent. In
this case, the human interactions might take other forms, such as network
models.Future work will examine the impact of spatial
variation on the implementation of IRS, including the reemergence of disease
from point sources missed from the previous spraying. More complex criteria for
nonfixed spraying will also be considered.In conclusion, regular spraying is clearly superior to
nonfixed spraying, but either will result in a significant reduction in the
overall number of mosquitoes, as well as the number of malaria cases in humans.
We thus recommend that the use of indoor spraying be reexamined for widespread
application in malaria-endemic areas.



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*Source: 745463-2008-09-21.xml*