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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. Church∗and 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 −)bw−dw)Mw+ (1 −)bwNw,

˙

Mm= (bm−dm)Mm+bmNm+bwMw+bwNw,

˙

Nw=−dwNw,

˙

Nm=−dmNm,

3

where 0 < 1 is the mutation rate. The birth and death rates are

bw=b0

w−Kbww Vw−KbwmVm, dw=d0

w+KdwwVw+KdwmVm,

bm=b0

m−KbmmVm−Kbmw 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

w−d0

w, rm=b0

m−d0

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

rw≥rm, Kw≥Km, 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=π−β(P)S+hI +δR −µHS, Θ6= Θ

˙

I=β(P)S−hI −αI −(µH+γ)I, Θ6= Θ

˙

R=αI −δR −µHR, Θ6= Θ

˙

Mw= ((1 −)bw−dw)Mw+ (1 −)bwNw−bwMw−βM(P)Mw,Θ6= Θ

˙

Mm= (bm−dm)Mm+bw(Mw+Nw) + bmNm−βM(P)Mm,Θ6= Θ

˙

Nw=βM(P)Mw−dwNw,Θ6= Θ

˙

Nm=βM(P)Mm−dmNm,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=rwVw1−Vw+αwmVm

Kw−Vwb0

w−KbwwVw−KbwmVm,

˙

Vm=rmVm1−Vm+αmwVw

Km+Vwb0

w−Kbww Vw−KbwmVm.

(6)

Suppose the set

Ωbw=(Vw, Vm)∈R2

+:Vw≤b0

w−KbwmVm

Kbww

is positively invariant under the ﬂow of (6). Then the set

Ω∗

bw=(P, Θ) ∈R8

+: Θ ≤Θ, S +I+R≤π

µH

,(Mw+Nw, Mm+Nm)∈Ωbw

is positively invariant under the ﬂow of (5), and all solutions that begin in

Ω∗

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 k∈Z,T∈R+and

phase space Ω×R+, where Ω⊂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:Rn→R

satisfying u(ϕ(t)) >0, the autonomous tracking system,

dx

dt =g(x),∆x|θ=θ=a(x−),

dθ

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 θ∈R+, 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=rwVw1−Vw+αwmVm

Kw−Vwb0

w−Kbww Vw−KbwmVm, t 6=kT

˙

Vm=rmVm1−Vm+αmwVw

Km+Vwb0

w−KbwwVw−KbwmVm, 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=rwVw1−Vw+αwmVm

Kw−Vwb0

w−Kbww Vw−KbwmVm,

˙

Vm=rmVm1−Vm+αmwVw

Km+Vwb0

w−KbwwVw−KbwmVm.

(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 Ωbe 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, Ωexists 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

w−KbwwVw−KbwmVmappearing 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 Ωbwof 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 Ω\ {(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:U⊂R→R, with Uan open set

containing 0, satisfying g(0) = 1 and

g0(0)

rw−Kw

<αmwαw m

Kwrw(rw−Kw)Kw(Kbwmrw−Kbwwαwmrm)

rw

+KbwwKw−b0

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 q≈q∗

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

w−KbwmKm)−c,

c=rw1−αwmKm

Kw.

Suppose the inequality

rm(ecT −e−rmT)

αwm(rm+c)T< ecT ZT

0

ecs 1 + αwmαmwrm

rm+c(ecs −e−rms)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+ (erwt−1)w0(q),

w0(q) = Kw(erwT(1 −q)−1)

erwT−1.

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 −rwT1−Km

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(erwT−1)(1 −q∗

2)2erwT(1 −q∗

2)·(erwT(1 −q∗

2)−1)

q

rmαmw

rwKm

×

F×(1 −q∗

2)rwαwm

rmKw

(ermT−1) −q∗

1

Kw−2

Kw

(ermT−1) 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 q∈N. 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 q∈N.

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 q≈0.7495. 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={0≤Vw+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= ˜q≈0.7406. 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(1−q)< 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)

1−qlie 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

1−q% larger than B0, in

the sense that

µ(Bq)

µ(B0)≥ZV+

m

0

b(m)

1−qdm

ZV+

m

0

b(m)dm

= 1 + q

1−q,

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 q≈0.7406, the action of the im-

pulse failed to bring solutions across the basin boundary, let alone the lower

boundary, Vw=b(Vm)

1−q.

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=π−β(P)S+hI +δR −µHS, t 6=kT,

˙

I=β(P)S−hI −αI −(µH+γ)I, t 6=kT,

˙

R=αI −δR −µHR, t 6=kT,

˙

Ψ = rwΨ1−Ψ

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 βH∈R+. Then the system with

impulses at ﬁxed times (21) has a nontrivial, nonnegative periodic solution,

provided T > T∗≡ − log(1−q)

rw.If T≤T∗, 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(1−q)

rw.If T≤T∗, 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 T∗is 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=π−β(P)S+hI +δR −µHS, Θ6= Θ,

˙

I=β(P)S−hI −αI −(µH+γ)I, Θ6= Θ,

˙

R=αI −δR −µHR, Θ6= Θ,

˙

Ψ = rwΨ1−Ψ

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 T∗is 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 T≈T∗), 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≡

supP∈Ωβ(P) exists. But then, for all (P, Θ) ∈Ω, we have Θ0≤ηξπ/µH≡C,

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

w−KbwwVw−KbwmVm, where Vw=Mw+Nwand Vm=Mm+Nm,

the nonnegativity of bwis determined by the dynamics of system (6). Specif-

ically, the condition bw≥0 is equivalent to the positive invariance of Ωbw

under the ﬂow of (6). The condition bw≥0 accounts for the restriction

on Mw, Mm, Nwand Nmin the expression for Ω∗

bw. Finally, the diﬀerential

inequality (S+I+R)0≤π−µH(S+I+R) accounts for the other restriction,

0≤S+I+R≤π

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∈Ω×R+≡X. The left limit of the forward ﬂow

φ:X×R+→Xof (9) at time Tis

φ(z+

0, T −) = lim

t→T−φ(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

t→T+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· · · 0−1

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

4rmαmw

Km

+Kbwmrmb0

w−Kbwmr2

m−rm

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

Km−Kbwm

b0

wb0

w

Km−Kbww.

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

−4rmKbww

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

w1−αmwb0

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

w−KmKbwm)<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 1−b0

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 bw≥0, contained in R2

+.

This is the region deﬁned by

Vw≤b0

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 M0≤W0. 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 M0≤W0to 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 W0≤M0for > 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 1−b0

w

KbwwKw<0,

by the hypothesis b0

w> KbwwKw. Also,

V0

m(W(0),0) = W (0)(b0

w−KbwwW(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 ≤Vm≤E1

0, W (Vm, )≤Vw≤M(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

rw1−αwmKm

Kw−(b0

w−KbwmKm),

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=Kw−αwmKm

1−αwmαmw,Km−αmw Kw

1−αwmαmw .

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, rm1−αmwKw

Km, σ(LM0) = rw1−αwmKm

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="rm1−αmwKw

Km· · ·

0−rw#,

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

w−Kbww Kw)

rwrm1−αmwKw

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+W2Kbw 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=rm1−Kw

rw,

∂G

∂ Kw

αwm

,0=−g0(0)K2

wrm

αwm

+rmαmwKw

rwKw(Kbwmrw−Kbwwαwmrm)

αwmrw

+kbwwKw−b0

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)T−1 0

ξ ermT−1,,

ξ=b0

w(ermT−e(rw−b0

w)T)

(rm−rw+b0

w)T.

Consequently, det DxN(q∗,0) = 0. Let us deﬁne the variable q=q−

q∗and 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) = ermT−1.

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

ermT−1

∂M2

∂x1

=−ξ

ermT−1

∂s

∂q =−∂f2

∂s

−1∂f2

∂q =−1

ermT−1

∂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

ermT−1= 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(q−q∗,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·(2∂x2x1N1+∂x2x2N1),

(35)

where ∂x2N1= 0 is known by previous calculation. Notice, however, that

since ∂rs=−ξ/(ermT−1) 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

dt∂x1x1z= (rw−b0

w)∂x1x1z+h(1 −q∗)e(rw−b0

w)ti22Kbww −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)T−1)

rw−b0

wKbww −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+∂x2N1∂qr 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)T−1. 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

2∂r2g(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(q−q∗).

Therefore the initial condition of our nontrivial periodic solution can be

written, locally, as the approximation

p(T, q)≈−B

A(q−q∗), s q−q∗,−B

A(q−q∗).

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

A∂rs(0)=−B

A,Bξ

A(ermT−1),

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

w−KbwmKm)T−1 0

χ e−rmT−1,#,

χ=χ() =

(−rmαmw +b0

w)he−rmT−exp rw(1 −αwmKm

Kw)−(b0

w−KbwmKm)Ti

−rm+rw(1 −αwmKm

Kw)−(b0

w−KbwmKm)T

.

Consequently, det DxN(q∗, M0) = 0.

As before, deﬁne q=q−q∗and deﬁne the function f= (f1, f2) : R3→R2

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 =e−rmT−1,∂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

1−e−rmT.

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(1−αwmKm

Kw)−(b0

w−KbwmKm))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) + (2∂rs)·∂x1x2N1(q∗, M0).

The mixed partial derivative can be found by solving the diﬀerential equa-

tion

d

dt∂x2x1z=A∂x1x2z+ (1 −q∗)·Kbwm −rwαwm

Kwe(A−rm)t,

A=A() = rw(1 −αwmKm

Kw

)−(b0

w−KbwmKm),

∂x2x1z(0) = 0.

Speciﬁcally, we have ∂x2x1z(T) = ∂x2x1φ(T, q∗, M0) = ∂x2x1N1(q∗, M0).

Computing, we ﬁnd, since eA()T=1

1−q∗, that

∂x2x1N1(q∗, M0) = (1 −e−rmT)Kbwm −rwαwm

Kw

rm

.

The double partial derivative in x1can calculated similarly: when = 0,

∂x1x1N1(q∗, M0) = ∂x1x1w(T), where

d

dt∂x1x1w=A(0)∂x1x1w−2rw

Kw

e2A(0)t1 + αwmαmwrm

rm+A(0) (eA(0)t−e−rmt)

∂x1x1w(0) = 0.

Therefore,

∂x1x1N1(q∗, M0) = −2rw

Kw

eA(0)TZT

0

eA(0)s1 + αwmαmwrm

rm+A(0) (eA(0)s−e−rms)ds

46

In summary, when = 0, we have the following conclusions:

∂2g

∂qr =−1

1−q∗

M(0) <0,

∂2g

∂q2= 0,

∂2g

∂r2=−2rw

Kw

eA(0)TZT

0

eA(0)s1 + αwmαmwrm

rm+A(0) (eA(0)s−e−rms)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)]−2−1−αwm erwT[Z(xw)]−2(1−[Z(xw)]−2−Y)

2+Y

0ermT[Z(xw)]−Y−1#

Z(xw) = Kw+xw(erwT−1)

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

w−Kbww ˜w(t, q∗

2)],

so that d

dt

∂φ2

∂ (t, w0(q∗

2), q∗

2,0) = ˜w(t, q∗

2)(b0

w−Kbww ˜w(t, q∗

2)).

From this, we conclude that

E=∂N2

∂ (w0(q∗

2), q∗

2,0) = ZT

0

˜w(t, q∗

2)(b0

w−Kbww ˜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

dVw◦DxN(w0(q∗

2), q∗

2,0)

tr adj(L1)d

dVm◦DxN(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

M≡d

dVw◦DxN(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

D−M22B6= 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))]−Y−1=1

Kw

e−rwT(erwT−1) = 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

0−q∗

2Z, t =kT

∆S=0 0

0−q∗

2S, t =kT

Z(0) = 0,

S(0) = I,

Q(t) =

rw1−2 ˜w(t,q∗

2)

Kw−rwαwm

Kw˜w(t, q∗

2)

0rm1−αmw ˜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,

H≡d

dVm

Dxφ(T, w0(q∗

2), q∗

2,0) = S(T)ZT

0

S−1(t)R(t)1−q∗

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 0−B

0A·H= tr −BH21 · · ·

· · · AH22 =−BH21 +AH22.

An elementary calculation shows that H21 =−(1 −q∗

2)αmw

Km(ermT