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Comparing malaria surveillance with periodic spraying in the presence of insecticide-resistant mosquitoes: Should we spray regularly or based on human infections?

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There is an urgent need for more understanding of the effects of surveillance on malaria control. Indoor residual spraying has had beneficial effects on global malaria reduction, but resistance to the insecticide poses a threat to eradication. We develop a model of impulsive differential equations to account for a resistant strain of mosquitoes that is entirely immune to the insecticide. The impulse is triggered either due to periodic spraying or when a critical number of malaria cases are detected. For small mutation rates, the mosquito-only submodel exhibits either a single mutant-only equilibrium, a mutant-only equilibrium and a single coexistence equilibrium, or a mutant-only equilibrium and a pair of coexistence equilibria. Bistability is a likely outcome, while the effect of impulses is to introduce a saddle-node bifurcation, resulting in persistence of malaria in the form of impulsive periodic orbits. If certain parameters are small, triggering the insecticide based on number of malaria cases is asymptotically equivalent to spraying periodically.
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Comparing malaria surveillance with periodic
spraying in the presence of insecticide-resistant
mosquitoes: should we spray regularly or
based on human infections?
Kevin E.M. Churchand Robert J. Smith?
1 Introduction
It has been estimated that one in two humans who ever lived has been killed
by malaria [7]. Three billion people — almost half the world’s population
— are at risk of malaria [13, 21, 23]. 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 [17]; 90% of deaths due to malaria occur in sub-
Saharan Africa [18], 75% of whom are African children [6]. In 2015, it caused
more than 214 million acute illnesses and 438,000 deaths [27]. This represents
a 37% reduction in cases over the previous 15 years [27].
This reduction has been largely driven by vector-control methods, pro-
marily insecticide-treated bednets and indoor residual spraying (IRS) [9, 26];
both are known to be highly effective [15]. The latter involves spraying houses
and structures with insecticides, thereby killing mosquitoes after they have
fed, in an effort to stop transmission of the disease. Recent data reconfirm
the efficacy and effectiveness of IRS in malaria control in countries where it
Department of Applied Mathematics, The University of Waterloo, Waterloo ON N2L
3G1 Canada
Department of Mathematics and Faculty of Medicine, The University of Ottawa, Ot-
tawa ON K1N 6N5 Canada
1
was implemented well [26]. Since many malaria vectors are endophilic, rest-
ing 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. 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 for timely, good-quality spraying is
greater than six months [26].
Using these methods, malaria was eradicated or greatly reduced in many
countries in the world between the 1940s and 1960s. Due to its success, DDT
was rapidly introduced into public-health and malaria-control campaigns,
and was the main insecticide used in the malaria-eradication campaign car-
ried out between 1955 and 1969 [25]. There is evidence of resistance, but
spraying with multiple insecticides has been successful in controlling Anophe-
les funestus,Anopheles gambiae and Anopheles melas in Equatorial Guinea,
for example [20].
Surveillance is an important tool in disease management [10]; this is par-
ticularly true of malaria, which is spatially heterogeneous [2]. The World
Health Organization has identified effective surveillance as a critical compo-
nent in malaria elimination and has called for stronger surveillance systems to
track and prevent outbreaks in endemic regions [27]. However, the majority
of scientific and surveillance efforts are focused on countries that are unlikely
to be the location of important emerging infectious diseases [10]. Further-
more, many countries with a high burden of malaria have weak surveillance
systems and are not in a position to assess disease distribution and trends
[27].
We assume that spraying occurs at times tk. The effect of the insec-
ticide is assumed to be instantaneous, resulting in a system of impulsive
differential equations. Impulsive differential equations consist of a system of
ordinary differential equations (ODEs), together with difference equations.
Between impulses, 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 spatio-temporal conditions are met [3, 4, 5, 12]. This
is related to the use of pulse vaccinations [1], seasonal skipping in recurrent
epidemics [24], antiretroviral drug treatment [16] and birth pulses in animals
[19]. Impulse times may be fixed or non-fixed [22] and may be either time-
or state-dependent.
This paper is organized as follows. In Section 2, we introduce the impul-
2
sive model in its general form, detailing key assumptions. In Section 3, we
develop preliminary results, such as existence and uniqueness properties. In
Section 4, we analyze a non-impulsive submodel consisting only of mosquito
dynamics. In Section 5, we analyze the mosquito-only submodel with impul-
sive effects. In Section 6 we illustrate our theoretical results with numerical
simulations. In Section 7, we determine global results for periodic orbits
under a simplifying assumption. We conclude with a discussion and relegate
all proofs to the appendix.
2 The model
All humans are either susceptible (S), infected (I) or partially immune (R).
Humans are born susceptible at a constant background birth rate π, indepen-
dent of the population size. The background death rate is µH. Susceptible
humans can become infected after being bitten by infected mosquitoes at rate
β. Infected humans die from the disease at rate γ, recover without immunity
at rate hor acquire immunity at rate α. Humans with temporary immunity
lose their immunity at rate δ. All rates are per capita rates, unless otherwise
mentioned.
In the absence of humans (and therefore new infections), mosquitoes un-
dergo logistic growth with competition and a small probability of unidirec-
tional mutation at birth. There is no competitive advantage or disadvantage
to being infected with malaria. Malaria infection has a negligible effect on
the lifespan of mosquitoes. Therefore, if Mwand Mmdenote the population
sizes of wild-type and mutant susceptible mosquitoes, Nwand Nmdenote the
corresponding infected mosquitoes, and Vw=Mw+Nwand Vm=Mm+Nm
denote the total populations of wild-type and mutant mosquitoes (susceptible
and infected), then we assume
˙
Mw= ((1 )bwdw)Mw+ (1 )bwNw,
˙
Mm= (bmdm)Mm+bmNm+bwMw+bwNw,
˙
Nw=dwNw,
˙
Nm=dmNm,
3
where 0 <  1 is the mutation rate. The birth and death rates are
bw=b0
wKbww VwKbwmVm, dw=d0
w+KdwwVw+KdwmVm,
bm=b0
mKbmmVmKbmw Vw, dm=d0
m+KdmmVm+Kdmw Vw,
where all parameters are assumed to be positive. There is no vertical malaria
transmission among mosquitoes. Notice that the above is not the usual, ele-
gant definition of logistic growth. However, this parameter-heavy definition
is necessary to take into account the correct mutation rate.
Wild-type mosquitoes are more evolutionarily fit in the absence of insec-
ticide: Define the intrinsic growth rates
rw=b0
wd0
w, rm=b0
md0
m,(1)
carrying capacities
Kw=rw
Kbww +Kdww
, Km=rm
Kbmm +Kdmm
,(2)
and competition coefficients
αwm =Kbwm +Kdwm
Kbww +Kdww
, αmw =Kbmw +Kdmw
Kbmm +Kdmm
.(3)
It is assumed that these bulk parameters satisfy the inequalities
rwrm, KwKm, αmw αwm,(4)
and at least one of the inequalities is strict.
Susceptible mosquitoes can become infected by biting an infectious hu-
man at rate βM, which may depend on the sizes of human and mosquito
subpopulations. The infection rate of humans by mosquitoes, β, is positively
correlated to the population of infected mosquitoes. The infection rate of
mosquitoes by humans, βM, is positively correlated with the population of
infected humans. Infection rates are assumed to be smooth functions of the
population variables. All infection rates are nonnegative.
The probability of passive surveillance efforts detecting a given human
malaria infection is given by η. Insecticide is sprayed if the number of re-
ported malaria cases since the previous insecticide application reaches a crit-
ical level, Θ. Application of the insecticide instantaneously decreases the
4
population of wild-type mosquitoes by a factor of q(0,1). The insecticide
has no effect on the mutant strain.
With these assumptions in place, we obtain the following system of im-
pulsive differential equations:
˙
S=πβ(P)S+hI +δR µHS, Θ6= Θ
˙
I=β(P)ShI αI (µH+γ)I, Θ6= Θ
˙
R=αI δR µHR, Θ6= Θ
˙
Mw= ((1 )bwdw)Mw+ (1 )bwNwbwMwβM(P)Mw,Θ6= Θ
˙
Mm= (bmdm)Mm+bw(Mw+Nw) + bmNmβM(P)Mm,Θ6= Θ
˙
Nw=βM(P)MwdwNw,Θ6= Θ
˙
Nm=βM(P)MmdmNm,Θ6= Θ
˙
Θ = ηβ(P)S, Θ6= Θ
Mw=qMw,Θ = Θ
Nw=qNw,Θ = Θ
∆Θ = Θ,Θ = Θ
(5)
Here S, I and Rrepresent the number of susceptible, infected and temporarily
immune humans, and P= (S, I, R, Mw, Nw, Mm, Nm).
Due to the model assumptions, we may assume the infection rates satisfy
∂β
∂Nw
>0,∂β
∂Nm
>0,∂βM
∂I >0,
β(S, I, R, Mw, Mm,0,0) = βM(S, 0, R, Mw, Mm, Nw, Nm)=0.
3 Preliminary results
3.1 Existence and uniqueness of solutions and the bi-
ological domain
To properly discuss existence, uniqueness and boundedness of solutions of
(5), it is necessary to adequately describe the “biological domain”. Specif-
ically, we must deal with the lack of positive invariance of the nonnegative
cone. This lack of invariance is the result of incorporating mutation at birth
5
with the logistic growth model. Indeed, it is possible for mosquito subpop-
ulations to “become negative” if initial conditions are chosen improperly.
A general invariance theorem is difficult to state; we provide a sufficient,
implicit condition based on properties of a vector-only two-dimensional sub-
model. This submodel is considered in Sections 2–3, where a more explicit
invariance theorem is given.
Theorem 1. Consider the two-dimensional system of ordinary differential
equations,
˙
Vw=rwVw1Vw+αwmVm
KwVwb0
wKbwwVwKbwmVm,
˙
Vm=rmVm1Vm+αmwVw
Km+Vwb0
wKbww VwKbwmVm.
(6)
Suppose the set
bw=(Vw, Vm)R2
+:Vwb0
wKbwmVm
Kbww
is positively invariant under the flow of (6). Then the set
bw=(P, Θ) R8
+: Θ Θ, S +I+Rπ
µH
,(Mw+Nw, Mm+Nm)bw
is positively invariant under the flow of (5), and all solutions that begin in
bwexist for all positive time and are unique.
3.2 Equivalence and stability of periodic solutions of
fixed-time and autonomous models
Let us denote G= (S, I, R, Mw, Mm, Nw, Nm), ˙
G=F(G) and ∆G=LG,
where the vector field Fand linear impulse Lare obtained from (5) by
simply ignoring the Θ equations. Consider the following impulsive differential
equation with impulses at fixed times kT , for integers kand a fixed spraying
period T:
˙
G=F(G), t 6=kT,
G=LG, t =kT. (7)
6
It is fairly obvious that periodic solutions of the autonomous model (5) give
rise to periodic solutions of the model with fixed-time spraying events (7),
provided the correct spraying period Tis chosen. It turns out that the
converse of this statement holds as well. Additionally, certain uniqueness
results are transferable. This equivalence of periodic orbits is applicable to
many impulsive vector-control models, so we state and prove the following
proposition in full generality. In the following, we assume that all differential
equations in question admit unique, globally defined solutions.
Proposition 1. Consider the following impulsive differential equation un-
dergoing impulse effects at fixed times tk=kT with kZ,TR+and
phase space ×R+, where Rn
+, the nonnegative orthant:
dx
dt =g(x), t 6=kT,
x=a(x), t =kT.
(8)
Suppose (8) has a unique, non-trivial T-periodic solution with one impulse
per cycle. Denote said periodic solution by ϕ(t). Then, for any u:RnR
satisfying u(ϕ(t)) >0, the autonomous tracking system,
dx
dt =g(x),x|θ=θ=a(x),
dt =u(x),θ|θ=θ=θ,
(9)
has a unique T-periodic solution up to phase shift, with one impulse per cycle,
given by (ϕ(t), θ(t)), whenever
θ=ZT
0
u(ϕ(t))dt. (10)
Conversely, if (9) has a non-trivial T0-periodic solution with one impulse per
cycle for some θR+, then (8) has a T0-periodic solution when T=T0.
Moreover, θsatisfies equation (10).
From now on, when referring to an autonomous model, uniqueness of
periodic solutions will always be taken to mean uniqueness up to phase shift.
The next lemma states that stability of periodic orbits in the system with
impulses at fixed times is equivalent to the stability of the corresponding
periodic orbit in the autonomous tracking system.
7
Proposition 2. Let ϕ(t)be a T-periodic solution of the system with fixed
impulses (8) and let ˜ϕ(t)denote the corresponding periodic solution of the
autonomous tracking system in (9). Suppose the solution operators of sys-
tems (8) and (9) are smooth with respect to initial conditions. Then ϕ(t)is
exponentially stable if and only if ˜ϕ(t)is orbitally asymptotically stable with
asymptotic phase.
The previous two results clearly apply to our model (5) and the associated
fixed-time model (7). As such, we have the following theorem.
Theorem 2. Suppose the model with spraying at fixed times, (7), has a T-
periodic solution ϕ(t). If
Θ = ZT
0
ηβ(P(ϕ))S(ϕ)dt,
then the function ˜ϕ(t) = (ϕ(t),Θ(t)) is a T-periodic solution of the au-
tonomous model (5), where Θ(t) = ηβ(P(ϕ))S(ϕ).ϕ(t)is exponentially
stable if and only if ˜ϕ(t)is orbitally asymptotically stable with asymptotic
phase. If ϕ(t)is the only T-periodic solution of (7), then ˜ϕ(t)is the unique
T-periodic solution of (5).
Corollary 2.1. Suppose the model with spraying at fixed times, (7), has a
unique hyperbolic endemic periodic orbit, ϕ(t;T),for all T(0, T +)for
some T+>0. Then, there exists Θ+>0such that, for all Θ(0,Θ+), the
autonomous model (5) has a periodic solution.
The above is largely a consequence of Theorem 2 and the Lebesgue dom-
inated convergence theorem; it can be shown that the map
T7→ ZT
0
ηβ(P(ϕ(t, T )))S(ϕ(t, T ))dt
is continuous and vanishes at T= 0 using the hyperbolicity assumption; the
result follows immediately.
We can conclude that if the model with spraying at fixed times has a
unique stable periodic solution of period T, then the autonomous tracking
model does too, provided the spraying threshold Θ is chosen appropriately.
In a certain sense, the autonomous and fixed-time spraying strategies are
“asymptotically equivalent” for initial conditions sufficiently close to the pe-
riodic orbit. However, we cannot rule out the existence of periodic solutions
8
of the autonomous model (5) with a period that is different from T, and we
know nothing of the global stability of the periodic solutions. Statements
regarding these properties can be made for simplified versions of the afore-
mentioned models, and this is considered in Section 4.
3.3 Introducing the mosquito-only submodel
To understand the dynamics of the full models (5) and (7), we first consider
the two-dimensional system of impulsive differential equations with impulses
at fixed times that describes the dynamics of the vector populations:
˙
Vw=rwVw1Vw+αwmVm
KwVwb0
wKbww VwKbwmVm, t 6=kT
˙
Vm=rmVm1Vm+αmwVw
Km+Vwb0
wKbwwVwKbwmVm, t 6=kT
Vw=qVw, t =kT.
(11)
This system can be easily derived from the fixed-time model (7) by defining
Vw=Mw+Nwand Vm=Mm+Nmand noticing that the derivatives
and impulse conditions decouple from the human (S, I, R) components. As
previously illustrated in Theorem 2, not much information is lost by working
with fixed-time spraying instead of autonomous spraying.
It turns out that a wealth of qualitatively different dynamics can be seen
in this simplistic two-dimensional model (11). We will not attempt to classify
all of them but will instead focus on the effect of adding the insecticide
spraying in the case where the system without impulses exhibits one of two
phase portraits: global stability of a coexistence equilibrium or bistability of
a coexistence equilibrium with a mutant-only equilibrium.
9
4 Analysis of the mosquito-only submodel with-
out impulses
In this section, we consider the system (11) without impulses
˙
Vw=rwVw1Vw+αwmVm
KwVwb0
wKbww VwKbwmVm,
˙
Vm=rmVm1Vm+αmwVw
Km+Vwb0
wKbwwVwKbwmVm.
(12)
4.1 Nullcines
Lemmas 1–2 characterize the Vmnullcline. Lemma 3 describes the Vwnull-
clines. Theorem 3 is a summary.
Lemma 1. Suppose at most one of the equalities
b0
wαmw
Km
=Kbww,b0
w
Km
=Kbwm (13)
holds. There exists 0>0such that, for 0<  < 0, the Vmnullcline is a
non-degenerate hyperbola.
Lemma 2. Suppose the inequality
Kbww
b0
w
αmw
Km
(14)
holds. There exists >0such that if 0<<, then the conclusions of
Lemma 1 hold, one branch of the hyperbola is nonpositive and intersects the
origin, while the other branch intersects R2
+in a curve and can be described
by the graph of a strictly decreasing convex function M:Vm7→ M(Vm) = Vw.
Msatisfies M(0) = b0
w/kbww and M(Km) = 0.
Lemma 3. The Vwnullclines consist of the line Vw= 0 and the parameter-
ized line
W(Vm, ) = αwmrwKbw mKw
rwKbwwKwVm+Kw(rwb0
w)
rwKbwwKw
.(15)
If ||is sufficiently small, W(·, )is decreasing and Vw(0, )>0.
10
With the previous three Lemma in hand, we have the following qualitative
description of the equilibrium points of the submodel (12).
Theorem 3. Suppose inequality (14) holds and at most one of the equal-
ities of (13) is satisfied. There exists 0>0such that, for 0<  < 0,
the mosquito-only submodel without impulses, (12), has the trivial extinction
equilibrium (0,0), in addition to exactly one of the following:
1. A mutant-only equilibrium, M0.
2. A mutant-only equilibrium, M0, and a single coexistence equilibrium,
E0.
3. A mutant-only equilibrium, M0, and two coexistence equilibria, C0and
C1.
The mutant-only equilibrium has coordinates (0, Km), and the coexistence
equilibria are formed by intersections of the nonnegative branch of the hyper-
bolic Vmnullcline, M, with the parameterized Vwnullcline, W(Vm, ).
Though certainly possible, it will not be our goal to determine the stability
of the equilibria in all of the above cases, nor to completely classify each case
by constraints on the model parameters. One reason is that the resulting
expressions are very complicated. Secondly, there are sufficient conditions
that guarantee that, for example, Case 2 occurs, which will be discussed in
Section 2.3. Finally, since will always be assumed to be small, it is much
more beneficial to simply consider perturbations from = 0 by techniques of
bifurcation theory, in the event we wish to study Case 3, which is the most
difficult to classify. This will be the subject of Section 2.4. For the moment,
we will briefly comment on Case 1.
In the subsequent sections, we will regularly make reference to the hyper-
bolic criteria; these will consist of the hypotheses of Theorem 3. Specifically,
the hyperbolic criteria are satisfied if inequality (14) and at most one equality
of (13) hold.
4.2 The feasible and biologically relevant domains
We first describe a nonnegative domain that is positively invariant, yet whose
biological interpretation is inappropriate.
11
Lemma 4. Suppose the hyperbolic criteria are satisfied and, additionally,
that b0
w> KbwwKw. Let γdenote the backward orbit through the point
(b0
w/Kbww ,0). Let be the domain in the nonnegative quadrant whose bound-
ary consists of γ, the line segment connecting (b0
w/Kbww ,0) to the origin and
the positive Vmaxis. If  > 0is sufficiently small, exists and is the largest
positively invariant set contained in R2
+.
The domain defined in Lemma 4 is not biologically “correct”, because
the term bw=b0
wKbwwVwKbwmVmappearing in the differential equa-
tion (12) represents the birth rate of wild-type mosquitoes, which should be
nonnegative. Since the domain Ω described in Lemma 4 is unbounded, there
are points in Ω where bwis negative. The most straightforward (though not
necessarily optimal) way to fix this problem is as follows.
Theorem 4. Suppose the following inequalities hold, in addition to the hy-
perbolic criteria.
b0
w> KbwwKw,max Kw
αwm
, Km<b0
w
Kbwm
.(16)
Then, for  > 0sufficiently small, the set bwof Theorem 1 is positively
invariant under the flow of (12).
4.3 The case of no coexistence equilibria
In this section, we will justify our claim that the case of no coexistence
equilibria is not of biological interest.
Proposition 3. Suppose the hypotheses of Lemma 4 are satisfied. If there
are no coexistence equilibria, then the mutant-only equilibrium is globally
attracting in \ {(0,0)}.
Therefore if there are no coexistence equilibria, the mutant-only equilib-
rium attracts all nonzero trajectories. Since it was assumed that the mutant
strain of mosquito was evolutionarily weaker in the absense of insecticide
than the wild-type, we can safely ignore this case.
4.4 The case of a single coexistence equilibrium
When is sufficiently small, so that Theorem 3 holds, there are two pos-
sible ways in which there can be only one coexistence equilibrium. The
12
non-degenerate case is when the hyperbolic Vmnullcline intersects the pa-
rameterized Vwnullcline transversally; the degenerate case is where they
intersect tangentially. Completely classifying the relationships between pa-
rameters in the degenerate case is tedious but certainly not impossible. For
the non-degenerate case, we can provide the following classification when
is small.
Theorem 5. Let the hyperbolic criteria be satisfied, and suppose b0
w> KbwwKw.
Then there exists 0>0such that, for <0, there is a single coexistence
equilibrium formed by the transversal intersections of two nullclines, provided
one of the following holds:
A1.
Km<Kw
αwm
Km< αmwKw
A2.
Km=Kw
αwm
, Km< αmwKw, αwmb0
w> KbwmKw, αmw >1.
In this case, the coexistence equilibrium is globally asymptotically stable on
R+
2\ {Vw= 0}if and only if the mutant-only equilibrium is unstable. This
will be guaranteed if Condition A1 holds.
4.5 The case of bistability and multiple coexistence
equilibria
In this section, we establish two conditions under which bistability can occur.
In the first, bistability is already present when = 0, and it is preserved for
 > 0 small. In the second, the coexistence equilibrium is globally stable when
= 0, and bistability occurs when  > 0 is small, due to a bifurcation of the
mutant-only equilibrium. In both cases, we demonstrate that a heteroclinic
orbit exists, connecting the extinction equilibrium to one of the coexistence
equilibria. This orbit is essentially the “bistability boundary”.
Theorem 6 (Preservation of bistability).Suppose the inequalities
Kw< αwmKm, Km< αmw Kw,1< αwmαmw , b0
w> KbwwKw(17)
13
are satisfied, in addition to the hyperbolic criteria. Then there exists 0>
0such that, for 0<  < 0, there is a pair of coexistence equilibria: the
mutant-only equilibrium is a sink, and the extinction equilibrium is a saddle.
There are no periodic orbits, and there is a heteroclinic orbit connecting the
extinction equilibrium to the saddle coexistence equilibrium.
Theorem 7 (Bifurcation from wild-type-only global stability at = 0).
Suppose the hyperbolic criteria are satisfied, in addition to the following.
B1. The inequalities
αmwαw m >1, b0
w> KbwmKm, Kw6=rw
are satisfied.
B2. There exists a C1function g:URR, with Uan open set
containing 0, satisfying g(0) = 1 and
g0(0)
rwKw
<αmwαw m
Kwrw(rwKw)Kw(KbwmrwKbwwαwmrm)
rw
+KbwwKwb0
w,
(18)
where Kmis a function of and can be written Km=Kw
αwm
g().
Then there exists 0>0such that, for 0<<0, there is a pair of coex-
istence equilibria: the mutant-only equilibrium is a sink, and the extinction
equilibrium is a saddle. There are no periodic orbits, and there is a hete-
roclinic orbit connecting the extinction equilibrium to the saddle coexistence
equilibrium.
5 Analysis of the mosquito-only submodel with
impulse effects
In this section, we investigate the effect of incorporating impulsive vector
control on the mosquito-only submodel. We assume we are working in a
feasible domain, such as the one described in Lemma 4.
To begin, we describe what happens to the extinction equilibrium at
arbitrary (but small) mutation rates, as a function of spraying efficacy.
14
Theorem 8 (Bifurcation at extinction).Define the quantity q
0()as follows:
q
0() = 1 e(rwb0
w)T.
If is sufficiently small, a biologically irrelevant (non-positive) periodic orbit
collides with the extinction equilibrium, resulting in a transcritical bifurcation
(in the one-dimensional centre dynamics) when q=q
0(). The extinction
equilibrium transforms from a sink into a saddle, while the periodic orbit
transforms from a saddle to a sink, as qincreases through q
0(). The periodic
orbit remains biologically irrelevant for qq
0().
Next, we state the critical spraying threshold where the mutant-only equi-
librium undergoes a saddle-node bifurcation.
Theorem 9 (Mutant-only saddle-node bifurcation).Define the quantity q
M()
as follows:
q
M() = 1 exp T(b0
wKbwmKm)c,
c=rw1αwmKm
Kw.
Suppose the inequality
rm(ecT ermT)
αwm(rm+c)T< ecT ZT
0
ecs 1 + αwmαmwrm
rm+c(ecs erms)ds (19)
holds. Then, for sufficiently small, we have 0< q
M()<1, and the mutant-
only equilibrium for system (11) undergoes a saddle-node bifurcation at pa-
rameter q=q
M(). Specifically, a locally stable, nonnegative periodic or-
bit collides with the mutant-only equilibrium, losing stability and becoming
nonpositive, while the mutant-only equilibrium becomes locally stable, as q
increases through q
M().
When = 0, the wild-type equilibrium is replaced with a wild-type pe-
riodic orbit, when q > 0 (that is, when the impulse effect is included). An
explicit formula for this periodic orbit for t(0, T ] is as follows:
˜w(t;q) = Kwerwtw0(q)
Kw+ (erwt1)w0(q),
w0(q) = Kw(erwT(1 q)1)
erwT1.
15
It is worthwhile determining the critical value of q > 0 at which the wild-
type periodic orbit loses its stability when there is no mutation (that is, when
= 0). There are two critical control efficacies.
Proposition 4 (Critical control efficacies pertinent to stability of the wild–
type-only periodic orbit with no mutation).Define the quantities q
1and q
2
as follows:
q
1= 1 exp (rwT),
q
2= 1 exp rwT1Km
αmwKw.
The wild-type-only periodic orbit for system (11) with no mutation (= 0)
undergoes a transcritical bifurcation, colliding with the the extinction equilib-
rium, losing stability and becoming nonpositive, as the parameter qincreases
through q
1, while the extinction equilibrium becomes a saddle point. The lin-
earization of the wild-type-only periodic orbit has a simple unit eigenvalue
when q=q
2, and this periodic orbit loses stability as qincreases through q
2.
The inequality 0< q
2< q
1<1holds, provided Km< αmwKw.
We do not prove the above proposition, since its correctness can be in-
ferred from the proof of the following theorem, which states that, under cer-
tain genericity assumptions, there is a bifurcation curve c(q) = (V(q), (q)),
defined in a neighbourhood of q
2, satisfying c(q
2)=(w0(q
2),0) and for which
V(q) is corresponds to a periodic orbit of system (11) with spraying efficacy
qand mutation rate (q).
Theorem 10 (Existence of a bifurcation curve near the wild-type-only pe-
riodic orbit with no mutation).Suppose the following inequality is satisfied:
ermTrwαwm
erwT(erwT1)(1 q
2)2erwT(1 q
2)·(erwT(1 q
2)1)
q
rmαmw
rwKm
×
F×(1 q
2)rwαwm
rmKw
(ermT1) q
1
Kw2
Kw
(ermT1) 6= 0,
(20)
where
F=2F1rmαmw
rwKm
+ 3,rm
rw
;rm+rw
rw
;erwT(1 q
2)1
q
2,
and 2F1is the Gauss hpergeometric function. Define N(V, q, ) = φ(T;V, q, )
V, where t7→ φ(t;V, q, )is the solution map of (11) with initial condition
16
φ(0; V, q, ) = V, initialized from time t= 0 in the model time coordinates.
There exists a unique C1curve c:q7→ (V(q), (q)), defined in a neighbour-
hood Nof q
2, with the following properties.
1. The function c= (V, )satisfies the equalities V(q
2) = w0(q
2),(q
2) =
0,∂
∂q (q
2)=0.
2. N(V(q), (q), q)=0for qN. That is, V(q)corresponds to the
initial condition at time t= 0 of a periodic solution of system (11)
with mutation rate (q)and spraying efficacy q.
3. DVN(V(q), (q), q)is non-invertible for qN.
6 Numerical Simulations
In this section, we provide graphical representations of the bifurcations that
can occur in the mosquito-only submodel by simulating the model numeri-
cally. We also approximate the relative sizes of the basins of attraction for
the various equilibria and periodic orbits of the model, under the assumption
that, in the absence of insecticide, the model exhibits bistability. The results
suggest that the basin of attraction of the mutant-only state increases as the
efficacy of spraying increases, and that another bifurcation may occur involv-
ing two coexistence periodic orbits. This is further supported by a readily
obtained analytical lower bound on the size of the basin of attraction.
In Figure 1, plots are displayed of solutions curves with two different
intial conditions: (10,10) and (10,90), for initial populations of wild-type
and mutant mosquitoes, respectively. The curve from the smaller initial
condition approaches a coexistence state, while the curve from the initial
condition with a large initial mutant population approaches the mutant-only
equilibrium. This is as predicted by Theorem 6; one can verify that the
illustrative parameters satisfy the bistability conditions.
When the spraying is included on a quarterly basis (T= 365/4) with a
mild efficacy (q= 0.2), the bistability is preserved. This is shown in Figure 2,
in which different initial conditions clearly yield qualitively different results
when the differential equations are simulated. These trajectories persist with
perturbed initial conditions.
The size of basin of attraction of the mutant-only equilibrium is essentially
constant, up until q0.7495. At this point, another bifurcation occurs; the
17
0 0.5 1 1.5 2 2.5 3 3.5 4
10
20
30
40
50
60
70
Time (days)
Mosquitoes
0 0.5 1 1.5 2 2.5 3 3.5 4
0
20
40
60
80
100
120
Time (days)
Mosquitoes
Figure 1: Plots of wild-type (solid line) and mutant (dashed line) mosquito
populations for four days, with no spraying (q= 0). Illustrative parameters
are chosen as b0
w= 100, rw= 80, rm= 75, Kbww = 0.5, Kdww = 0.5,
Kw= 100, Km= 100, αwm = 1, αmw = 1.2 and = 0.05. Top: initial
condition (10,10). Bottom: initial condition (10,90). Initial conditions are
ordered pairs of initial susceptible and infected populations.
0 10 20 30 40 50 60 70 80 90 100
0
10
20
30
40
50
60
70
Mutant mosquitoes
Wild-type mosquitoes
Figure 2: Plots of mosquito populations in the phase plane with spraying
efficacy q= 0.2 and spraying period T= 0.25 ×365 days; arrows indicate
direction of forward time. Four initial conditions are chosen on the line
Vw= 10; notice that half of the chosen solutions converge to a coexistence
periodic orbit, while the others converge to the mutant-only equilibrium.
18
0.7 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Spraying efficacy (q)
Proportional basin of attraction [0,1]
Figure 3: Proportion of trajectories in the set U={0Vw+Vm<110}that
converge to the mutant-only equilibrium (dots) and the stable coexistence
equilibrium (stars), as a function of spraying efficacy, q, ranging from 0.7
to 0.8. Initial conditions of the form X(0) = X(0+) were used for this
figure. Notice that the proportions, and hence basins of attraction, appear
numerically constant, until the mutant-only equilibrium begins attracting
almost all points in Uat approximately q= ˜q0.7406. The computations
required to produce this figure are numerically expensive; the (somewhat
stiff) vector field is integrated more than 4000 times to produce only twenty
samples for qin the interval [0.7,0,8].
19
mutant-only equilibrium abruptly becomes globally stable. See Figure 3.
This change in stability has nothing to do with a bifurcation of the mutant-
only equilibrium, since that equilibrium is locally stable and hyperbolic for all
q[0,1]. We conjecture that, at some critical spraying efficacy, ˜q, the stable
coexistence periodic orbit collides with an unstable coexistence periodic orbit,
with the result being that they annihilate; that is, neither orbit persists for
q > ˜q. This is certainly possible, for there is indeed an unstable coexistence
equilibrium present when q= 0, which lies on the bistability boundary. This
equilibrium generically persists for qsufficiently close to q= 0, as a periodic
orbit. It should be mentioned that the size of the basin of attraction depends
on the initial time coordinate, and in this case, initial conditions of the form
X(0) = X(0+) were used. This means that spraying does not occur at time
t= 0, but only begins at time T
The presence of such a bifurcation is more difficult to detect analytically,
however. If the unstable coexistence equilibrium is hyperbolic when q=
0, then the bifurcation cannot be analytically detected using only a local
analysis. If the associated equilibrium is non-hyperbolic when q= 0, then
it likely coincides with the other coexistence equilibrium. When = 0, a
codimension-two bifurcation can occur at this equilibrium for a specific value
of q, as illustrated by Theorem 10, which would complicate the analysis.
Conversely, when q= 0 and 6= 0, it is difficult to analytically express the
relevant equilibrium point.
It should be mentioned that the conclusions on the size of the basin of
attraction of the above numerical simulations will differ if initial conditions of
the form X(0) = X(0) are used instead. However, with the initial conditions
as chosen, beginning spraying at time t=Tmay be a more biologically
appropriate interpretation, since in reality, one would expect the mosquito
population to be closer to its natural coexistence equilibrium than to, say,
the extinction equilibrium. Allowing Tunits of time to pass before beginning
to spray allows for the “pre-calibration” of the system.
For the sake of mathematical precision, we will discuss what may be
observed if spraying begins at time t= 0. From the phase portrait im-
plied by Theorem 6, the stable coexistence equilibrium is separated from
the mutant-only equilibrium by a heteroclinic orbit through the extinction
state and another, unstable coexistence equilibrium. The heteroclinic orbit
can be interpreted as the stable manifold of the “middle” unstable coexis-
tence equilibrium. Due to the orientation of the nullclines, the global stable
manifold, restricted to the positive quadrant, can be identified with a single
20
smooth function, b(Vm). It follows that when q= 0, the basin of attraction of
the mutant-only equilibrium, denoted M0, contains the interior of the region
bounded by the curves Vw= 0 and Vw=b(Vm).
Now, if q > 0, any initial condition X= (Vm, Vw) satisfying the inequality
Vw(1q)< b(Vm) will be mapped immediately mapped into the (continuous)
basin of attraction of M0at time t= 0+. As such, all initial conditions lying
below the curve Vw=b(Vm)
1qlie in the (impulsive) basin of attraction. If we
denote Bqto be (impulsive) basin of attraction of M0at spraying efficacy q,
the previous discussion implies that Bqis at least q
1q% larger than B0, in
the sense that
µ(Bq)
µ(B0)ZV+
m
0
b(m)
1qdm
ZV+
m
0
b(m)dm
= 1 + q
1q,
and V+
mis any given prescribed upper bound on the number of mutant
mosquitoes.
The above analysis provides another possible explanation of the theorized,
numerically-motivated bifurcation discused earlier. The boundary of the im-
pulsive basin of attraction of M0dominates a function that is everywhere
monotone increasing in q, so it seems reasonable that the true boundary may
interact with the stable coexistence periodic orbit at some critical value of
q, resulting in a bifurcation. This also serves to explain the appeared con-
stancy of the basin of attraction exhibited in the numerical simulations, prior
to the bifurcation point. The growth of the impulsive basin of attraction of
M0might not have been observed because the stable coexistence periodic
orbit attracted points outside of the continuous basin of attraction of M0
very quickly. Solutions numerically converged to the periodic orbit within
the time interval (0, T ) and then, until q0.7406, the action of the im-
pulse failed to bring solutions across the basin boundary, let alone the lower
boundary, Vw=b(Vm)
1q.
7 Global existence and uniqueness results for
periodic orbits in a simplified model
The model from Section 1 can be simplified — and much more information
obtained about the nature of its solutions — if we make the simplifying
21
assumption that there are no mutant mosquitoes and that all mosquitoes are
infectious. That is, we set Mm=Nm== 0 and define Ψ = Mw+Nw, so
that Ψ0=rw(1 Ψ
Kw). Under this assumption, we can say much more about
the endemic periodic orbits. The impulsive differential equations are
˙
S=πβ(P)S+hI +δR µHS, t 6=kT,
˙
I=β(P)ShI αI (µH+γ)I, t 6=kT,
˙
R=αI δR µHR, t 6=kT,
˙
Ψ = rwΨ1Ψ
Kw, t 6=kT,
∆Ψ = qΨ, t =kT,
(21)
7.1 Existence and stability of periodic orbits for the
system with impulses at fixed times
In contrast to the full model, the reduced model with spraying at fixed times
is much more amenable to analytical techniques. Our first result pertains
to the existence of periodic orbits under the assumption that the transmis-
sion rate is a linear function of the mosquito population (i.e., mass-action
transmission).
Lemma 5. Suppose β(P) = βHΨ,for some βHR+. Then the system with
impulses at fixed times (21) has a nontrivial, nonnegative periodic solution,
provided T > T≡ − log(1q)
rw.If TT, the disease-free equilibrium is a
global attractor.
A similar result can be obtained for standard incidence, with the caveat
that we must assume that the death rate due to malaria is absent.
Lemma 6. Suppose β(P) = βHΨ
S+I+R, and γ= 0. Then the system with
impulses at fixed times (21) has a nontrivial, nonnegative periodic solution
provided T > T≡ − log(1q)
rw.If TT, the disease-free equilibrium is a
global attractor.
The final result of this section is a perturbation result, applicable to ei-
ther mass-action or standard-incidence transmission. It states that if certain
parameters are sufficiently small, a unique, positive, exponentially stable pe-
riodic orbit exists.
22
Theorem 11. There exist 1, . . . , 4>0such that, for
γ < 1, δ < 2, h < 3, α < 4
the system with impulses at fixed times (21) has a unique, nonnegative, hy-
perbolic periodic solution, provided either β(P) = βHΨor β(P) = βHΨ
S+I+R
and T > T, where Tis defined as in Lemma 6. If T > T, the periodic
solution is exponentially stable.
7.2 Properties of periodic orbits in the autonomous
tracking model
We will now characterize the periodic orbits of the simplified autonomous
tracking model,
˙
S=πβ(P)S+hI +δR µHS, Θ6= Θ,
˙
I=β(P)ShI αI (µH+γ)I, Θ6= Θ,
˙
R=αI δR µHR, Θ6= Θ,
˙
Ψ = rwΨ1Ψ
Kw,Θ6= Θ,
˙
Θ = ηβ(P)S, Θ6= Θ,
∆Ψ = qΨ,Θ = Θ,
∆Θ = Θ,Θ = Θ.
(22)
Theorem 12. Suppose one of the following conditions is satisfied.
1. The transmission is by mass action, so that β(P) = βHΨ. For all
T > 0, the system (21) has a unique nonnegative branch of periodic
solutions, ϕT, that depend continuously on T.
2. There is no disease-associated death and the transmission is by standard
incidence, so that β(P) = βHΨ
S+I+Rand γ= 0. For T > T, system
(21) has a unique nonnegative branch of periodic solutions, ϕT, that
depend continuously on T, where Tis defined as in Lemma 6.
The system with autonomous spraying (22) has a periodic solution for every
Θ>0. If Θis sufficiently small and the branch ϕTis hyperbolic for T
sufficiently small (or, for Condition 2, for TT), there is a unique periodic
solution.
23
Corollary 12.1. Suppose the malaria transmission is modelled either by
mass action or standard incidence. Then there exist 1, . . . , 4>0and Θ+>
0such that, for
γ < 1, δ < 2, h < 3, α < 4,Θ<Θ+,
the following are true.
1. The system with spraying at fixed times (21) has a unique, nonnegative
periodic solution that is asymptotically stable.
2. The system with incidence-based spraying (22) has a unique, nonnega-
tive periodic solution that is orbitally asymptotically stable and enjoys
the property of asymptotic phase.
8 Discussion
A mathematical model of malaria with insecticide effect has been proposed,
where there is a mutant strain of mosquito that has complete immunity to the
insecticide. The insecticide control is triggered when some critical number
of new human malaria cases is detected. It was shown that this control
strategy is asymptotically equivalent to spraying at periodic times, in that
both systems share the same common endemic periodic orbits, and stability
of the orbit in one system implies its stability in the other. A general version
of this result is provided by Propositions 1–2.
The positive invariance of the nonnegative cone does not hold in general
for this model, due to how the mutation is modelled. An implicit condition
for the existence of a nonnegative, convex, positively invariant domain was
provided by Theorem 1. An explicit condition is also available and is given
by Theorem 4, assuming the mutation rate, , is sufficiently small.
Following this, we consider the mosquito-only submodel. Theorem 3 out-
lines conditions under which, for small mutation rates, the mosquito sub-
model exhibits either a single mutant-only equilibrium, a mutant-only equi-
librium and a single coexistence equilibrium, or a mutant-only equilibrium
and a pair of coexistence equilibria; in all cases, the extinction equilibrium is
also present. Numerous results pertaining to the persistence or development
of bistability, when the mutation rate is small and positive, are presented in
Theorems 6–7.
24
The impulse effects are then re-introduced. Theorem 8 provides condi-
tions under which the extinction equilibrium undergoes a saddle-node bifur-
cation. This bifurcation involves a biologically irrelevant periodic orbit and
is not of great interest.
Theorem 9 demonstrates that, under certain conditions, the mutant-only
equilibrium undergoes a saddle-node bifurcation. In this case, as the spray-
ing efficacy qincreases and passes through the q
M(), a stable nonnegative
coexistence equilibrium collides with the mutant-only equilibrium, becoming
nonnegative and losing stability. The mutant-only equilbrium changes from
a saddle to a sink.
The final explicit bifurcation result is Theorem 10. This theorem pro-
vides a somewhat complicated condition under which a codimension-two
bifurcation can occur at the wild-type-only equilibrium, when = 0 and
q=q
2= 1 exp(rwT(1 K m
αmwK w )).
Numerical simulations were provided to reinforce the theoretical results
on bistability of the coexistence periodic orbit and the mutant-only equilib-
rium. It was suggested that another bifurcation can occur when the spraying
efficacy is significantly large. We conjecture that, under certain conditions,
the stable coexistence periodic orbit collides with an unstable coexistence pe-
riodic orbit and both vanish, leaving room for the mutant-only equilibrium
to become globally stable, when the spraying efficacy qreaches some critical
threshold.
From a policy perspective, this result is important. There is essentially a
one-to-one correspondence between critical spraying thresholds and spraying
periods (the defining equations for every critical spraying threshold can be
inverted to solve for T), the latter of which are typically controllable. As
such, another interpretation for the consequence of the above conjecture is
that if spraying occurs too frequently, the mutant allele may become very
prevalent in the mosquito population.
Finally, we returned to the human-mosquito dynamics by considering a
simplified model in which all mosquitoes are assumed to be infectious. This
simplification makes the model far more amenable to analysis. We are able
to prove that, provided certain parameters are small, the correspondence
between spraying at fixed times and spraying according to the autonomous
tracking model is even more strict. Specifically, the period Tand infec-
tion threshold Θ uniquely determine each other, provided each is sufficiently
small. These results hold for mass-action or standard-incidence infection
rates and are summarized in Theorem 11–12 and Corollary 12.1.
25
In conclusion, the mathematical model (5) and its various submodels ex-
hibit a wealth of different qualitative dynamics, including bistability, several
bifurcations of fixed points and periodic orbits. We have shown that spraying
at fixed times is asymptotically equivalent to spraying when a critical number
of new human infections are detected. As such, one strategy may be more
or less costly to implement yet yield the same long-term result as the other.
A sufficiently powerful insecticide, sprayed too frequently, could result in the
mutant allele becoming very common in the mosquito population, thereby
reducing its efficacy as a control method. Care must therefore be taken to
not spray too often. How frequently to spray to avoid this problem can be
informed using the results from Section 3, although the picture is incom-
plete, as discussed in this final section, as well as in Section 4. Finally, by
appealing to a simpler model, we see that whether spraying at fixed times or
according to an infection threshold, there can be only one endemic periodic
orbit, provided the period or threshold is small enough.
Acknowledgements
KEC is supported by an Ontario Graduate Scholarship. RJS? is supported
by an NSERC Discovery Grant. For citation purposes, please note that the
question mark in “Smith?” is part of his name.
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Appendix: Proofs
Proof of Theorem 1
The impulse effect clearly maps Ω into itself. Second, since Ω is compact, ξ
supPβ(P) exists. But then, for all (P, Θ) Ω, we have Θ0ηξπHC,
so that if an impulse effect occurs at some time, tk, then the next impulse
effect can occur only as early as time tk+ Θ/C. Consequently, there is a
finite amount of time between each impulse effect, so that no solution can
have an accumulation point on the hypersurface Θ = Θ. We may therefore
conclude that if Ω
bwis positively invariant, then solutions are defined for all
(positive) time [4].
To see the invariance, note that it is only necessary to examine the in-
variance of the Pcomponent of (5). If P0is a point on the boundary of
the nonnegative cone, it is clear that, provided the state-dependent coeffi-
cient bwis nonnegative at P0, the forward flow from P0will remain non-
negative for some small positive amount of time. It is thus sufficient to
determine a domain on which bwremains nonnegative for all time. Since
bw=b0
wKbwwVwKbwmVm, where Vw=Mw+Nwand Vm=Mm+Nm,
the nonnegativity of bwis determined by the dynamics of system (6). Specif-
ically, the condition bw0 is equivalent to the positive invariance of Ωbw
under the flow of (6). The condition bw0 accounts for the restriction
on Mw, Mm, Nwand Nmin the expression for Ω
bw. Finally, the differential
inequality (S+I+R)0πµH(S+I+R) accounts for the other restriction,
0S+I+Rπ
µH, present in Ω
bw. Uniqueness is guaranteed by standard
results [4].
29
Proof of Proposition 1
Let (8) have a unique T-periodic solution, ϕ(t), and let (10) hold. Consider
the point (ϕ(0+),0) z+
0×R+X. The left limit of the forward flow
φ:X×R+Xof (9) at time Tis
φ(z+
0, T ) = lim
tTφ(z+
0, t) = ϕ(T),ZT
0
u(ϕ(t))dt
=ϕ(T), θ.
Hence an impulse occurs along the forward orbit from z+
0at time t=T. By
periodicity of ϕ(t) and the impulse condition, we have
φ(z+
0, T +) = lim
tT+φ(z+
0, t)=(ϕ(0),0) = z+
0.
Note also that θ6= 0, since uis strictly positive along ϕ. It follows that
there is always a finite amount of time between impulses. We may therefore
conclude that equation (9) has a T-periodic solution. Moreover, it is unique
up to phase shift. If (9) had another T-periodic solution, say (x1(t), θ1(t)),
which had an impulse effect at t=T, then x1(t+T) would be a T-periodic
solution of (8). By uniqueness of periodic solutions of the fixed-time equation,
we would then have x1(t+T) = ϕ(t). By the impulse condition, we have
˜
θ(0+) = 0 = θ1(T+). Finally, for t(0, T ),
θ(t) = θ(0+) + Zt
0
u(ϕ(s))ds
=θ1(T+) + Zt
0
u(x1(T+s))ds
=θ1(T+) + ZT+t
T
u(x1(s))ds
=θ1(T+t).
Therefore (x1, θ1) is a phase shift of (ϕ, ˜
θ). There can thus be only one
T-periodic solution of the autonomous equation (9) up to phase shift equiv-
alence. The converse statement is obvious.
Proof of Proposition 2
Without loss of generality, we have ˜ϕ= (ϕ, θ) for a T-periodic function θ
satisfying θ(0+) = 0. The variational equation associated to the periodic
30
orbit ϕ(t) is
˙z=dg
dx(ϕ(t))z, t 6=kT,
z=da
dxz, t =kT.
(23)
Conversely, the variational equation at ˜ϕ(t) is
˙w=
0
dg
dx(ϕ(t)) .
.
.
0
u(ϕ(t)) 0
w, t 6=kT,
w=
da
dx ξ
0· · · 01
w, t =kT,
(24)
where
ξ=g(ϕ(T+)) g(ϕ(T)) da
dxg(ϕ(T)),
and w= (w1:n, wn+1). It follows by classical results [4] that the above equa-
tion (24) has a nontrivial T-periodic solution, namely ˜ϕ0(t). Without loss
of generality, we may assume ˜ϕ0(0) = en+1. If ϕ(t) is exponentially stable,
then there exist nlinearly independent solutions v1, . . . , vnof (23) satisfying
vj(0) = ejfor j= 1, . . . , n. It is then easy to check that (vj,0) are lin-
early independent solutions of (24). Therefore a monodromy matrix for the
linearized tracking system is
M=v1(T)· · · vn(T)ϕ0(T)
0· · · 0θ0(T).
The matrix Mhas a block structure; the floquet multipliers are precisely
θ0(T) = θ0(0) = 1 and the multipliers of ϕ(t), which are, by hypothesis,
within the unit disc, since this solution is exponentially stable. Therefore
˜ϕ(t) is orbitally asymptotically stable and has the property of asymptotic
phase [4]. The converse follows by similar reasoning.
31
Proof of Lemma 1
The Vmnullcline is the solution of the second-degree equation
N(Vw, Vm) = V2
mrm
Km+VmVwrmαmw
Km
+Kbwm+V2
w(Kbww)
+Vm(rm) + Vw(b0
w)=0.
(25)
Its determinant, which we consider as a function of , is
det() = 1
4rmαmw
Km
+Kbwmrmb0
wKbwmr2
mrm
Km
2(b0
w)2
=rm
4rmb0
wαmw
Km
Kbww b0
wb0
w
Km
Kbwm.
Thus, the determinant vanishes if = 0 or if = ˆ, where
ˆ
rmb0
wαmw
KmKbwm
b0
wb0
w
KmKbww.
By the hypothesis on the equalities (13), ˆis either well-defined and nonzero,
zero, or is undefined. If ˆis undefined, we will formally write ˆ=. The
case ˆ=occurs when b0
w
Km=Kbwm, so that the determinant reduces to
det() = r2
m
4b0
wαmw
Km
Kbww.
The determinant is nonzero when 6= 0. In summary, if we formally write
1=ˆ, ˆ6= 0,
,ˆ= 0,
then det()6= 0 provided 0 <  < 1.
Next, the discriminant of (25) is
∆() = rmαmw
Km
+Kbwm2
4rmKbww
Km
=rmαmw
Km2
+2rm
Km
(αmwKbw m 2Kbww ) + K2
bwm.
Since ∆() is continuous and ∆(0) >0, it follows that there exists 2>0
such that if 0 <  < 2, we have ∆ >0. If we define 0= min{1, 2}, then
we have ∆ >0 and det 6= 0 whenever 0 <  < 0. This indicates that (25)
describes a non-degenerate hyperbola.
32
Proof of Lemma 2
Let 0 <  < 0, so that the Vmnullcline is a hyperbola. Notice that (0,0),
(0, Km) and (b0
w/Kbww,0) are all solutions of the equation N(Vw, Vm) = 0,
and the first two constitute the only solutions of the equation N(0, Vm) = 0.
Implicitly differentiating both sides N(Vw, Vm) = 0 with respect to Vmat
(0,0), we obtain the equation rm+b0
wV0
w(0,0) = 0,which gives
V0
w(0,0) = rm
b0
w
<0.(26)
Implicitly differentiating with respect to Vmat (b0
w/Kbww ,0) results in
V0
w(b0
w/Kbww,0) = rm
b0
w1αmwb0
w
KmKbww Kbwm
Kbww
<0 (27)
where the inequality follows by the assumption in equation (14). Finally,
implicitly differentiating at (0, Km) gives
V0
w(0, Km) = rm
rmαmw +(b0
wKmKbwm)<0 (28)
provided is sufficiently small; say, <0. Inequalities (26) and (28) indicate
that the points (0,0) and (0, Km) must lie on distinct branches of the hy-
perbola; otherwise, there would need to be a branch with two critical points.
Similarly, inequalities (26) and (27) guarantee that the points (0,0) and
(b0
w/Kbww,0) must lie on different branches. We conclude that (b0
w/Kbww ,0)
and (0, Km) lie on the same branch of the hyperbola, while (0,0) is on the
other. From the above analysis, we can also conclude that the portion of the
first branch contained in the nonnegative quadrant can be identified with the
graph of a strictly decreasing function M: [0, Km][0, b0
w/Kbww ].
Proof of Lemma 4
Assume  > 0 is sufficiently small to guarantee the conditions of Theorem
3 are satisfied. Let W(Vm) and M(Vm) denote, respectively, the Vwand Vm
nullclines (15) and (25) in the nonnegative quadrant. Let V0
wand V0
mdenote,
respectively, the intersections of Wand Mwith the line Vw= 0. A quick
33
calculation reveals
W0=Kw(rwb0
w)
αwmrwKbw mKw
, M0=Km,
W(0) = Kw(rwb0
w)
rwKbwwKw
, M(0) = b0
w
Kbww
.
We have
V0
w(M(0),0) = b0
w
Kbww 1b0
w
KbwwKw<0,(29)
by the hypothesis b0
w> KbwwKw. Also
∂V 0
m
∂Vw
(M(0),0) = b0
w<0.(30)
By inequalities (29)–(30), if γis given a parameterization γ: [0,)R2
+
such that γ(0) = (b0
w/Kbww ,0), then both components of γare initially in-
creasing. The condition b0
w> KbwwKwimplies that W(0) is a decreasing func-
tion of , so that if 0 <  rw
Kbww Kw, we will have 0 < W (0) < M(0). Con-
sequently, there are no nullclines with Vwcoordinate greater than b0
w/Kbww
in the nonnegative quadrant. We conclude that γis increasing for all time.
The inequality (30) implies that V0
m>0 along the line connecting γ(0) to
the origin (except at γ(0), where V0
m= 0). Finally, the line Vw= 0 is a
Vwnullcline, so is itself positively invariant since ∂ V 0
m
∂Vm>0 at the origin. It
follows that Ω is positively invariant. See Figure 4 for a visualization.
That Ω is the largest nonnegative positively invariant set can be explained
as follows. Any trajectory that is initially positive but not contained in Ω
must satisfy V0
m<0 and V0
w<0 until it traverses a nullcline. However, since
Ω contains all portions of all nullclines contained in the nonnegative quadrant,
and the “upper” boundary of Ω is a solution curve, the proposed trajectory
cannot cross it. Since the upper boundary of Ω contains no equilibrium point,
the proposed trajectory cannot have a limit point on the upper boundary of
Ω. Therefore the trajectory must eventually leave the nonnegative quadrant.
Proof of Theorem 4
Verification that inequalities (16) imply the hyperbolic criteria is straight-
forward. Next, as in the proof of Lemma 4, the first condition of (16) guar-
antees that W(0) is a decreasing function of and, when is small, that
34
W(0) < M(0). The second inequality of (16) is equivalent to
max{M0, W 0}<b0
w
Kbwm
(31)
when = 0. Since M0is constant and W0is continuous with respect to ,
inequlity (31) persists for  > 0 small.
Consider the region defined by the inequality bw0, contained in R2
+.
This is the region defined by
Vwb0
w
Kbww
Kbwm
Kbww
Vm.(32)
The Vwintercept is b0
w
Kbww =M(0), while the Vmintercept is the right-hand
side of (31). We conclude from the above analysis that the graph of the
boundary of (32) lies above the the graph of both Wand M, in the non-
negative quadrant. Consequently, the signs of ˙
Vmand ˙
Vware constant above
both the graph of max{W, M }and below the graph of the boundary of (32).
Both of these derivatives are nonpositive in this region, since ˙
Vw(M(0),0) <0
by (29) and ˙
Vm(0, b0
w/Kbwm)<˙
Vm(0, M0) = 0. This allows us to conclude
that (32) is positively invariant, since this set is a strict subset of the posi-
tively invariant set Ω from Theorem 4, and the boundaries of (32) consist of
boundaries of Ω, together with the line Vw=b0
w
Kbww Kbwm
Kbww Vm, on which both
˙
Vwand ˙
Vmare nonpositive (in fact negative, except ˙
Vm(M(0),0), which has
been discussed in Theorem 4). See Figure 4 for a visualization. The result
follows if one recalls that Ωbwof Theorem 1 is defined by inequality (32).
Proof of Proposition 3
Since the only biologically relevant equilibrium points lie on the the line
Vw= 0, which is a Vwnullcline, it follows that there are no periodic orbits.
The extinction equilibrium’s eigenvalues are λ0
1=rwb0
wand λ0
2=rm,
so that (0,0) is unstable for all R; in particular, when is small, both
its eigenvalues are positive. Consequently, there are no heteroclinic circuits
between the equilibria. By the Poincar´e–Bendixson theorem, it follows that
the mutant-only equilibrium, (0, Km), is globally attracting on Ω \ {(0,0)}.
35
0
0
V
m
V
w
b
0
w
K
bwm
.
W
0
M(0)
W(0)
M
0
Figure 4: A typical phase portrait of the system (12). In this case, there
is a single coexistence equilibrium that appears to be asymptotically stable.
Nullclines are indicated by bold lines. The backward orbit from (M(0),0),
denoted γ(see Lemma 4), is present in the figure. The dashed line indicates
the “upper” boundary of the region (32) of Theorem 4. Equilibrium points
are indicated by dots, and arrow directions indicate approximate directions
of solution velocity in forward time.
36
Proof of Theorem 5
Let W(Vm) and M(Vm) denote, respectively, the Vwand Vmnullclines (15)
and (25) in the nonnegative quadrant. Let W0and M0denote, respectively,
the intersections of Wand Mwith the line Vw= 0. We have
W0=Kw(rwb0
w)
αwmrwKbw mKw
, M0=Km,
W(0) = Kw(rwb0
w)
rwKbwwKw
, M(0) = b0
w
Kbww
.
The condition Km< αmwKwimplies that Wand M, when = 0, cannot
intersect on the interior of R2
+(see the proof of Theorem 6 for the explicit
intersection point), so that there can be at most three equilibrium points, all
of which must be on the boundary. This fact holds true for  > 0 sufficiently
small, by continuity of Wand Mwith respect to .
The condition b0
w> Kbww Kwimplies that W(0) is a decreasing function
of , so that if 0 <  rw
Kbww Kw, we will have 0 < W (0) < M(0). But
then, due to the hyperbolic criteria, Lemma 2 implies that if there is a single
coexistence equilibrium at which the nullclines (15) and (25) intersect, then
we must have M0W0. One will notice that, when = 0, the inequality
M0< W 0is equivalent to condition A1, but since M0and W0are continuous
functions of at = 0, the inequality must persist for small. On the other
hand, when = 0, the condition M0=W0is equivalent to the first equality
of condition A2. The other two inequalities of condition A2 are necessary
for the condition M0W0to persist when  > 0 is small. This establishes
necessity.
If condition A1 holds, so that we have M0< W 0and W(0) < M (0) for
 > 0 small, the intermediate value theorem guarantees that the nullclines
must intersect at least once, and there can be no more than one intersection
in the positive quadrant because Mis convex there. Conversely, if condition
A2 holds, then W0M0for  > 0 small, and one will find that the third
inequality of condition A2 implies that W0(M0)< M0(M0), provided  > 0
is small, so that for 0 v < Vm, we have M(v)< W (v). Again, since
W(0) < M(0), the intermediate value theorem guarantees that Wand M
intersect in the interval (0, v), and there can be only one intersection in the
positive quadrant, due to the convexity of M. We have proven sufficiency of
conditions A1 and A2.
37
Next we discuss attractivity. It has already been demonstrated in Propo-
sition 3 that the extinction equilibrium is unstable. If both it and the mutant-
only equilibrium are unstable, then, by similar reasoning as that of the proof
of Proposition 3, E0will either be globally attracting, or there will be a stable
periodic orbit that encloses it. We will now rule out the existence of periodic
orbits.
Suppose conditions A1 or A2 of the theorem hold. We have
V0
w(M(0),0) = b0
w
Kbww 1b0
w
KbwwKw<0,
by the hypothesis b0
w> KbwwKw. Also,
V0
m(W(0),0) = W (0)(b0
wKbwwW(0)) >0,
provided b0
w> KbwwW(0) >0. That is, to maintain the above inequality, we
require
b0
w> KbwwKwrwb0
w
rwKbwwKw>0.(33)
When = 0, the inequality holds due to the hypothesis b0
w> Kbww Kw. By
continuity, (33) holds for sufficiently small. Finally, we have
∂V 0
m
∂Vw
(M(0),0) = b0
w<0.
Therefore it follows that, when is sufficiently small, the region
Ω = {(Vw, Vm)R+
2: 0 VmE1
0, W (Vm, )VwM(Vm)}
is postively invariant. However, any periodic orbit enclosing E0must inter-
sect Ω. Hence there can be no periodic orbit.
As for the sufficient condition for instability of the mutant-only equilib-
rium, we observe that one of the eigenvalues of its linearization is
rw1αwmKm
Kw(b0
wKbwmKm),
so that the equilibrium will be unstable if the above is positive. When con-
diton A1 holds, the above is positive when = 0, as stated in the theorem.
38
Proof of Theorem 6
Part 1: Existence of equilibria and their stability. When = 0 and the in-
equalities (17) are satisfied, there are four equilibria: the extinction equilib-
rium, ¯
0 = (0,0); the wild-type-only equilibrium, W0= (Kw,0); the mutant-
only equilibrium, M0= (0, Km); and a coexistence equilibrium,
E0=KwαwmKm
1αwmαmw,Kmαmw Kw
1αwmαmw .
The linearizations, Lx, at these equilibria satisfy the following:
σ(L¯
0) = {rw, rm},det(LE0) = rwrmE1
0E2
0
KwKm
(1 αwmαmw ),
σ(LW0) = rw, rm1αmwKw
Km, σ(LM0) = rw1αwmKm
Kw,rm.
From the above, we conclude that ¯
0 is a source, W0and M0are sinks, E0
is a saddle and their linearizations are invertible. By the implicit function
theorem, these equilibria persist for  > 0 small, and they have the same
stability. The extinction and mutant-only equilibria do not depend on ,
and the coexistence equilibrium is strictly positive, which means that these
equilibria will all remain nonnegative when is small. Also, we note that
LW0="rm1αmwKw
Km· · ·
0rw#,
which implies that the implicit function W0() = (W(), M()) describing
the perturbation of the wild-type-only equilibrium satisfies, by the implicit
function theorem,
M0(0) = rwKw(b0
wKbww Kw)
rwrm1αmwKw
Km>0,
where positivity is guaranteed by inequalities (17). It follows that W0() is
nonnegative for  > 0 small. In conclusion, all equilibria described in the
theorem exist and their stability is as stated.
Part 2: Nonexistence of periodic orbits. An argument similar to the proof
39
of Theorem 5 can be used to show that no periodic orbit can enclose W0().
Since ¯
0 and M0() both lie on the invariant Wnullcline, Vw= 0, neither of
these can be enclosed within a periodic orbit. From the linearizations in Part
1, all equilibria are hyperbolic. Since the only other equilibrium (E0) is a
saddle, by index theory, we conclude that there are no periodic orbits.
Part 3: Existence of the heteroclinic orbit. To exhibit the heteroclinic or-
bit, we consider the α-limit set of a point, x0, on the stable manifold of the
saddle-type coexistence equilibrium, E0, lying below the Vmnullcline. By the
Poincar´e–Bendixson theorem, this limit set must be an (unstable) equilibrium
point, a periodic orbit or a connected set of homoclinic/heteroclinic orbits.
Part 2 rules out the possibility of a periodic orbit. The only possible ho-
moclinic orbit would need to be based at the saddle coexistence equilibrium.
However, the unstable manifold of E0is contained within two disjoint (except
for one point: E0itself) positively invariant sets, each of which contains a
single, stable equilibrium point (W0() and M0respectively). Consequently,
α(x0) cannot contain a point on the unstable manifold of E0, since any point
on this manifold has either W0() or M0as its ω-limit set. It follows that
α(x0) does not contain a homoclinic orbit. Moreover, all equilibria excluding
E0have all of the real parts of their eigenvalues strictly negative or strictly
positive, indicating that α(x0) does not consist of a circuit of heteroclinic or-
bits connecting these equilibria. The above discussion allows us to conclude
that α(x0) = 0. Therefore there is a heteroclinic orbit connecting E0with
the extinction equilibrium.
Proof of Theorem 7
First, we consider the stability of equilibria of system (12) when = 0, under
the assumptions B1–B2 of Theorem 7. Due to assumption B2, there are
no coexistence equilibria. The equilibria are ¯
0 = (0,0), W0= (Kw,0) and
M0= (0, Km), as in the proof of Theorem 6. W0and ¯
0 are seen to be stable
and hyperbolic, by considering the linearizations appearing in the proof of
Theorem 6. Consequently, W0persists under small perturbations of and
remains a sink. M0, however, is non-hyperbolic, because of condition B2.
When  > 0, there is a mutant-only equilibrium at
M0() = (0, Km) = 0,Kw
αwm
g().
40
This particular equilibrium is formed by the nullclines Vw= 0 and Vw=
M(Vm) (see the proof of Theorem 5). The non-hyperbolicity occurs because,
at = 0, the nullclines Vw=W(Vm) and Vw=M(Vm) both intersect at
M0(0). The nullcline Vw=M(Vm) is the solution Vwof equation (25). Re-
stricting that equation to the nullcline Vw=W(Vm) and applying assumption
B2 results in
V2
mrmαwm
g()+VmW·rmαmwαwm
Kwg()+Kbwm+W2Kbw m b0
wW= 0.
Let us denote the left-hand side of the above equation by G(Vm, ). Recall
now that we can express Was
W=αwmrmKbwmKw
rwKbwwKwVm+Kw(rwb0
w)
rwKbwwKw).
It follows that Gis C1and satisfies GKw
αwm ,0= 0. The partial derivatives
are
∂G
∂VmKw
αwm
,0=rm1Kw
rw,
∂G
∂ Kw
αwm
,0=g0(0)K2
wrm
αwm
+rmαmwKw
rwKw(KbwmrwKbwwαwmrm)
αwmrw
+kbwwKwb0
w.
(34)
By the implicit function theorem, there exists a unique C1function Vm:
7→ Vm() satisfying Vm(0) = Kw
αwm and G(Vm(), ) = 0 for ||sufficiently
small. It follows that (W(Vm()), Vm()) is a coexistence equilibrium (i.e.,
is nonnegative in both components) for  > 0 sufficiently small, provided
∂Vm
∂ (0) <0. By the implicit function theorem, we have
∂Vm
∂ (0) = G
∂VmKw
αwm
,01∂G
∂ Kw
αwm
,0.
Equation (34), along with some straightforward algebra, shows that Vm
∂ (0) <
0 is equivalent to inequality (18).
The rest of the proof proceeds similarly to that of Theorem 6 and is thus
omitted.
41
Proof of Theorem 8
For brevity, we will write q=q
0(). Let φ(t, q, x) denote the flow from time
t= 0 of the solution of the impulsive system (12), with initial condition
φ(0, q, x) = xand spraying efficacy q. All other parameters are assumed to
be fixed. Denote N(q, x) = φ(T , q, x)x.DxN(q, 0) is easily found to be
DxN(q, 0) = (1 q)e(rwb0
w)T1 0
ξ ermT1,,
ξ=b0
w(ermTe(rwb0
w)T)
(rmrw+b0
w)T.
Consequently, det DxN(q,0) = 0. Let us define the variable q=q
qand define M(q, x) = N(q+q, x), so that we have M(0,0) = 0 and
det DxM(0,0) = 0. Our objective will now be to obtain a nontrivial solution
of the equation M(q, x) = 0.
Write M= [ M1M2]Tin component form, and define f:R×R×R
R2by f(q, r, s) = M(q, re1+se2), where e1and e2are the standard basis
vectors in R2. Then, with f= [ f1f2]T, we can readily calculate the
partial derivative of f2with respect to sat 0. We have
∂f2
∂s (0) = M2
∂x2
(0) = ermT1.
Therefore ∂f2
∂s (0) 6= 0 and, by the implicit function theorem, there exists a
unique smooth function s: (q, r)7→ s(q, r) such that f2(q, r, s(q, r)) = 0 and
s(0,0) = 0, which is defined in some neighbourhood of (0,0). We also have
the partial derivatives
∂s
∂r =f2
∂s
1∂f2
∂r =1
ermT1
∂M2
∂x1
=ξ
ermT1
∂s
∂q =f2
∂s
1∂f2
∂q =1
ermT1
∂M2
∂q = 0,
with all partial derivatives evaluated at zero, and the final equality follows
from the fact that N(q, 0) = 0, so that M(q, 0) = 0.
Our problem is now reduced to solving the equation f1(q, r, s(q, r)) = 0.
This will be accomplished by a second-order application of Taylor’s theo-
rem. To simplify notation, write g(q, r) = f1(q, r, s(q, r)), so we must solve
42
g(q, r) = 0. The first-order partial derivatives of gat zero are
∂r g=
∂r f1(0, r, s(0, r))r=0
=∂f1
∂r (0) + ∂f1
∂s (0) s
∂q (0)
=DxM(0,0)11 +DxM(0,0)12 ·ξ
ermT1= 0,
∂q g=N1
∂q +N1
∂x2
∂s
∂q = 0.
We must now calculate the second-order partial derivatives of g. First,
since N1(q, se2) = 0 for all q, s R, we readily find that 2g
∂q2is zero. Indeed,
2g
∂q2=2
∂q2N1(q, s(qq,0)e2)q=q
= 0.
The other second-order partial derivatives require a bit more effort. We
find
2g
∂r2=x1x1N1+x2x1N1·rs+x2x1N1·rs+x2x2N1·rs+x2N1·rr s
=x1x1N1+rs·(2x2x1N1+x2x2N1),
(35)
where x2N1= 0 is known by previous calculation. Notice, however, that
since rs=ξ/(ermT1) and ξ=C() is continuous at = 0, it follows
that rs= 0 when = 0. Therefore the sign of 2g
∂r2near = 0 is determined
primarily by the sign of x1x1N1. We therefore compute only this partial
derivative.
The partial derivatives of Nseen above can be determined by solving
certain differential equations. The following are true about the function
z(t, x) = φ1(t, q, x):
d
dtx1x1z= (rwb0
w)x1x1z+h(1 q)e(rwb0
w)ti22Kbww 2rw
Kw,
x1x1z(0) = 0.
The above can be computed analytically, and we find
x1x1z(T) = 2e(rwb0
w)T(e(rwb0
w)T1)
rwb0
wKbww rw
Kw<0,
43
where the inequality holds provided ||is sufficiently small. By (35), it follows
that 2g
∂r2(0) <0 when ||is small enough.
Next we calculate the mixed partial derivative in the variables qand r.
We find
2g
∂qr =qx1N1+x2x1N1·qs+qx2N1·rs+x2x2N1·(rs)2+x2N1qr s
=qx1N1,(36)
as almost all partial derivatives vanish.
As before, qx2N1can be found by solving a particular differential equa-
tion. However, we do not have to, since it is already known that x1N1(q, 0) =
(1 q)e(rwb0
w)T1. Therefore we have
qx1N1=
∂q x1N1(q, 0)q=q
=e(rwb0
w)T<0.
Hence, by (36), we have 2g
∂qr <0.
From the above calculations and Taylor’s theorem, we have
g(q, r) = Ar2+Bqr +o(q2+r2)
as (q, r)0, for real numbers A=1
2r2g(0) and B=qr g(0) satisfying
A, B < 0. Near (q, r) = 0, there is a nontrivial branch of solutions, written
approximately as
r(q)≈ −B
Aq(q) = B
A(qq).
Therefore the initial condition of our nontrivial periodic solution can be
written, locally, as the approximation
p(T, q)B
A(qq), s qq,B
A(qq).
To determine the location of this point as qvaries near q, we calculate
partial derivatives, considering, for this purpose, the above approximation to
be exact (as it is essentially a linear approximation). We have
∂p
∂q (T , q) = B
A, ∂qs(0) B
Ars(0)=B
A,
A(ermT1),
whose components have sign (,+). It follows that p(T , q) is nonpositive for
all qsufficiently close to q, while p(T, q) = 0. This proves the theorem.
44
Proof of Theorem 9
The proof will follow much the same format as that of Theorem 8. As before,
let φ(t, q, x) denote the flow from time t= 0 of the solution of the impulsive
system (12), with initial condition φ(0, q, x) = xand spraying efficacy q. All
other parameters are assumed to be fixed. Denote N(q, x) = φ(T , q, x)x,
M0= (0, Km) and q=q
M(). DxN(q, M0) is easily found to be
DxN(q, M0) = "(1 q) exp rw(1 αwm Km
Kw)(b0
wKbwmKm)T1 0
χ ermT1,#,
χ=χ() =
(rmαmw +b0
w)hermTexp rw(1 αwmKm
Kw)(b0
wKbwmKm)Ti
rm+rw(1 αwmKm
Kw)(b0
wKbwmKm)T
.
Consequently, det DxN(q, M0) = 0.
As before, define q=qqand define the function f= (f1, f2) : R3R2
by
f(q, r, s) = N(q+q, re1+ (s+M0)e2).
The first-order partial derivatives of fare very similar to those appearing
in the proof of Theorem 8; we state the results without proof. All partial
derivatives are calculated at zero.
∂f2
∂s =ermT1,f2
∂q = 0,f2
∂r =χ.
Therefore, as before, we apply the implicit function theorem to write s=
s(q, r). If we define g(q, r) = f1(q, r, s(q, r)), then the partial derivatives at
zero satisfy ∂g
∂q =g
∂r = 0.
Also, the implicit function ssatisfies
∂s
∂q (0) = 0,s
∂r (0) = χ
1ermT.
As for the second-order partial derivatives at zero, we find 2g
∂q2(0) = 0,
since N1(q, M0) = 0 for all q. The symbolic calculation of the other partial
45
derivatives are the same as in the previous theorem; in particular, all partial
derivatives appearing in 2g
∂qr vanish except for one:
2g
∂qr =qx1N(q, M0) =
∂q DxN(q, M0)11q=q
=e(rw(1αwmKm
Kw)(b0
wKbwmKm))T<0.
For the double partial derivative in the variable r, not as many terms
are able to be ignored as in the previous proof, since χdoes not become
negligible as becomes small. Comparing to the first line of (35), many terms
do indeed vanish (any term involving only N1without partial derivatives in
x1will vanish). We find
2g
∂r2=x1x1N1(q, M0) + (2rs)·x1x2N1(q, M0).
The mixed partial derivative can be found by solving the differential equa-
tion
d
dtx2x1z=A∂x1x2z+ (1 q)·Kbwm rwαwm
Kwe(Arm)t,
A=A() = rw(1 αwmKm
Kw
)(b0
wKbwmKm),
x2x1z(0) = 0.
Specifically, we have x2x1z(T) = x2x1φ(T, q, M0) = x2x1N1(q, M0).
Computing, we find, since eA()T=1
1q, that
x2x1N1(q, M0) = (1 ermT)Kbwm rwαwm
Kw
rm
.
The double partial derivative in x1can calculated similarly: when = 0,
x1x1N1(q, M0) = x1x1w(T), where
d
dtx1x1w=A(0)x1x1w2rw
Kw
e2A(0)t1 + αwmαmwrm
rm+A(0) (eA(0)termt)
x1x1w(0) = 0.
Therefore,
x1x1N1(q, M0) = 2rw
Kw
eA(0)TZT
0
eA(0)s1 + αwmαmwrm
rm+A(0) (eA(0)serms)ds
46
In summary, when = 0, we have the following conclusions:
2g
∂qr =1
1q
M(0) <0,
2g
∂q2= 0,
2g
∂r2=2rw
Kw
eA(0)TZT
0
eA(0)s1 + αwmαmwrm
rm+A(0) (eA(0)serms)ds 2χ(0)rwαwm
rmKw
=2rw
Kw
(Y+χ(0)Z).
Under the assumptions of the theorem, we have Y+χ(0)Z > 0. The rest of
the proof is now essentially identical to the proof of Theorem 8 and is hence
omitted.
Proof of Theorem 10
Let φ(t, x, q, ) denote the flow from time t= 0 of the solution of the impulsive
system (12), with initial condition φ(0, x, q, ) = x, spraying efficacy qand
mutation rate . All other parameters are assumed to be fixed. Denote
N(x, q, ) = φ(T , x, q, )x.
A straightforward calculation shows that, for an initial condition of the
form xw= (Vw, Vm)=(xw,0), the differential DxN(xw, q, 0) can be written
DxN(xw, q, 0) = "(1 q)erwT[Z(xw)]21αwm erwT[Z(xw)]2(1[Z(xw)]2Y)
2+Y
0ermT[Z(xw)]Y1#
Z(xw) = Kw+xw(erwT1)
Kw
Y=rmαmwKw
rmKm
.
(37)
It is easy to verify that Dx(w0(q
2), q
2,0) is non-invertible and has the form
DxN(w0(q
2), q
2,0) = A B
0 0 . Note that A6= 0.
Define the map L:R3+1 R3by L(x, q;) = N(x, q, ) det DxN(x, q, )T.
47
From the above, we know that Dx,L(w0(q
2), q
2,0) has the structure
Dx,L(w0(q
2), q
2,0) =
A B · · ·
0 0 E
C D · · ·
,
[A B ] = DxN(w0(q
2), q
2,0)e1,
[C D ] = [xdet Dx]N(w0(q
2), q
2,0),
E=∂N2
∂ (w0(q
2), q
2,0).
It follows that Dx,L(w0(q
2), q
2,0) is invertible, provided E6= 0 and AD
BC 6= 0. If this is the case, the implicit function theorem will guarantee the
existence of a function satisfying all the conditions of Theorem 10 except for
the condition on the partial derivative of (q). We will prove this part later.
Calculation of E.We have
d
dt
∂φ2
∂ (t, w0(q
2), q
2, ) =
∂ [˜w(t, q2)(b0
wKbww ˜w(t, q
2)],
so that d
dt
∂φ2
∂ (t, w0(q
2), q
2,0) = ˜w(t, q
2)(b0
wKbww ˜w(t, q
2)).
From this, we conclude that
E=∂N2
∂ (w0(q
2), q
2,0) = ZT
0
˜w(t, q
2)(b0
wKbww ˜w(t, q
2))dt.
Since we are working in one of the biologically relevant domains described
in Theorem 4, we know that b0
w> Kbww ˜w(t, q
2) for all t[0, T ] because
˜w(t, q
2)>0. Consequently, the integrand above is strictly positive, and we
conclude that E > 0.
Simplification of AD BC.By Jacobi’s formula, [ C D ] can be written
[C D ] = xdet Dx(w0(q
2), q
2,0) =
tr adj(L1)d
dVwDxN(w0(q
2), q
2,0)
tr adj(L1)d
dVmDxN(w0(q
2), q
2,0)
T
,
48
where L1=DxN(w0(q
2), q
2,0) = A B
0 0 . However, it can be shown that
Md
dVwDxN(w0(q
2), q
2,0) is upper triangular. Consequently,
C= tr (adj(L1)M) = A·M22 .
Taking into account that A6= 0, we have AD BC 6= 0 if and only if
DM22B6= 0.
Calculation of M22.We have M22 =d
dxwDx(xw, q
2,0)22xw=w0(q
2).The matrix
Dx(xw, q, 0) appears in equation (37). We have
M22 =Y ermTZ0(w0(q
2))[Z(w0(q
2))]Y1=1
Kw
erwT(erwT1) = q
1
Kw
,
where the final two equalities result from straightforward algebra.
Calculation of D.It can be shown that Z(t)d
dVmDxφ(t, w0(q
2), q
2,0) sat-
isfies the set of matrix initial-value problems
Z0=Q(t)Z+R(t)S(t), t 6=kT
S0=Q(t)S, t 6=kT
Z=0 0
0q
2Z, t =kT
S=0 0
0q
2S, t =kT
Z(0) = 0,
S(0) = I,
Q(t) =
rw12 ˜w(t,q
2)
Kwrwαwm
Kw˜w(t, q
2)
0rm1αmw ˜w(t,q
2)
Km
,
R(t) = ermtrwαwm
Kw0
rmαmw
Km
2rm
Km
As such, if we denote by S(t) the fundamental matrix solution of the system
S0=Q(t)Ssatisfying S(0) = I, we can write, by the variation of constants
formula,
Hd
dVm
Dxφ(T, w0(q
2), q
2,0) = S(T)ZT
0
S1(t)R(t)1q
20
0 1 S(t)dt.
49
Now, since d
dVmN(x, q, ) = d
dVm(Dxφ(T, x, q, )I) = d
dVmDxφ(T, x, q, ),
we have
D= tr  0B
0A·H= tr BH21 · · ·
· · · AH22 =BH21 +AH22.
An elementary calculation shows that H21 =(1 q
2)αmw
Km(ermT1) and
H22 =2
Km(ermT1).
Conclusion. The value of Bcannot be written in terms of elementary
functions unless rw=rm. The condition that BH21 +AH22 M22B=
AH22 B(H21 +M22)6= 0 is equivalent to condition (20) of the theorem.
Proof of Lemma 5
The mosquito compartment has a unique nontrivial periodic solution,
˜
Ψ(t;T) = Kwerwtw0(T)
Kw+ (erwt1)w0(T),
w0(T) = Kw(erwT(1 q)1)
erwT1.
It follows that if T > T, then ˜
Ψ>0. Therefore, to search for nontrivial
periodic solutions of (21), we may take Ψ = ˜
Ψ. We must prove the existence
of positive periodic solutions of
˙
S=πβHS˜
Ψ + hI +δR µHS,
˙
I=βHS˜
ΨhI αI γI µhI ,
˙
R=αI δR µHR.
(38)
However, this ordinary differential equation is linear, R3
+is positively invari-
ant, and the sum H=S+I+Rsatisfies
˙
HπµHH
for positive initial conditions, implying that this linear system has a bounded
solution. By the Massera theorem [11], it has a periodic solution.
Conversely, if TT, the only nonnegative recurrent state of the mosquito
dynamics is the extinction equilibrium, which is globally asymptotically sta-
ble for nonnegative initial population. A comparison principle argument then
shows that the disease-free equilibrium is globally attracting.
50
Proof of Lemma 6
As before, we may take Ψ = ˜
Ψ. Under the assumptions of the lemma,
H=S+I+Rsatisfies the differential equation