Bull Math BiolDOI 10.1007/s11538-011-9628-6

O R I G I NA L A RT I C L E

Modelling Malaria Control by Introductionof Larvivorous Fish

Yijun Lou · Xiao-Qiang Zhao

Received: 21 May 2010 / Accepted: 7 January 2011© Society for Mathematical Biology 2011

Abstract Malaria creates serious health and economic problems which call for inte-grated management strategies to disrupt interactions among mosquitoes, the parasiteand humans. In order to reduce the intensity of malaria transmission, malaria vectorcontrol may be implemented to protect individuals against infective mosquito bites.As a sustainable larval control method, the use of larvivorous fish is promoted in somecircumstances. To evaluate the potential impacts of this biological control measure onmalaria transmission, we propose and investigate a mathematical model describingthe linked dynamics between the host–vector interaction and the predator–prey inter-action. The model, which consists of five ordinary differential equations, is rigorouslyanalysed via theories and methods of dynamical systems. We derive four biologicallyplausible and insightful quantities (reproduction numbers) that completely determinethe community composition. Our results suggest that the introduction of larvivorousfish can, in principle, have important consequences for malaria dynamics, but alsoindicate that this would require strong predators on larval mosquitoes. Integratedstrategies of malaria control are analysed to demonstrate the biological applicationof our developed theory.

Keywords Vector borne diseases · Disease control · Malaria transmission ·Predator–prey model · Larvivorous fish · Global attractivity

Supported in part by the NSERC of Canada and the MITACS of Canada.

Y. Lou (�) · X.-Q. ZhaoDepartment of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s,NL A1C 5S7, Canadae-mail: ylou@mun.ca

X.-Q. Zhaoe-mail: zhao@mun.ca

Present address:Y. LouMITACS Centre for Disease Modelling, York University, 4700 Keele Street, Toronto, ON M3J 1P3,Canada

Y. Lou, X.-Q. Zhao

1 Introduction

Human malaria is a parasitic disease which is caused by the genus Plasmodium, and itis spread among humans by female Anopheles mosquitoes. Currently, approximatelytwo billion people are at risk of Plasmodium falciparum malaria (Snow et al. 2005),which primarily affects poor populations in tropical and subtropical areas. Malaria isalso a major hindrance to economic development. The economic impacts of malariainclude costs of health care, decreased school attendance and work productivity dueto the disease, loss of investment and tourism (Greenwood et al. 2005).

The Global Malaria Eradication Programme, using two key tools: chloroquine fortreatments and prevention and dichloro-diphenyl-trichloroethane (DDT) for vectorcontrol, was launched by WHO in 1955 (World Health Organization 2000). Withthe emergence of chloroquine-resistant Plasmodium parasites and DDT-resistantAnopheles mosquitoes, the goal for global eradication of malaria was officially aban-doned in 1972 (Brito 2001). After that, more effective interventions, including drugcombinations with an artemisinin derivative and anti-vector measures, were deliv-ered (Greenwood et al. 2008). However, global warming, drug resistance and insec-ticide resistance may deteriorate malaria burden. Although malaria vaccine researchhas progressed rapidly over the past few years, it is likely to be at least a decadebefore an efficacious vaccine is available for widespread use in malaria-endemiccountries (Greenwood et al. 2005). Without this kind of prospect vaccines in thenear future, vector control becomes an essential component. Contemporary vectorcontrol strategies include indoor residual spraying (IRS) and insecticide treated nets(ITNs). These programmes target the adult female mosquito population to shortenthe length of its life stage rather than reduce the vector population. However, harm-ful effects of chemicals on mosquitoes and non-target populations, emerging resis-tance to insecticides, environmental pollution, logistical difficulties and prohibitivecosts call for other vector-targeted malaria control strategies (Ghosh and Dash 2007;Walker and Lynch 2007). On the contrary, unlike adult mosquitoes, immature mos-quitoes cannot escape from their breeding sites until the adult stage and hence, theycannot easily avoid control measures (Killeen et al. 2002). Larval control of Anophe-les mosquitoes is a well-proven effective tool that has been neglected, but deservesrenewed consideration for malaria control in the twenty first century (Walker andLynch 2007).

Manipulating an auto-reproducing predator into the ecosystem may provide sus-tained biological control of the vector population (Chandra et al. 2008). Larvivorousfish which feeds on immature stages of mosquitoes has been introduced as a success-ful intervention tool against Anopheles mosquitoes (Chandra et al. 2008). In orderto control malaria, introduction of larvivorous fish has been shown to be an efficientand suitable intervention tool for larval control in many circumstances (see Chan-dra et al. 2008; Ghosh and Dash 2007; Greenwood et al. 2005; Howard et al. 2007;Singh et al. 2006; Walker and Lynch 2007 and references therein). For example, oneyear after the release of larvivorous fish, malaria in three villages in India (Puram,Bodapatt and Banganatham) was eradicated (Ghosh et al. 2005). According to WorldHealth Report 2009 (World Health Organization 2010), introduction of larvivorousGambusia fish is promoted by almost all affected countries in rice-growing areas

Modelling Malaria Control by Introduction of Larvivorous Fish

in the European region. Integrated malaria management, such as a combination ofIRS, early detection, prompt treatments and larvivorous fish, can successfully bringmalaria under control (Singh et al. 2006). In order to achieve an optimal strategy,sound insights into the dynamics among larvivorous fish, the vector and human pop-ulations are desirable. Our main object is to model and investigate this interaction.

In the larvivorous fish–mosquitoes–humans interaction, there exist two sub-interactions: the predator–prey (larvivorous fish and larval mosquitoes) and vector–host (adult mosquitoes and humans) interactions. To model this process, we can usethe eco-epidemiological modelling idea. Eco-epidemiological models have been in-vestigated for a long period, and some mathematical modelling works about eco-epidemiological interactions have been published recently (see, e.g. Auger et al.2009; Hilker and Schmitz 2008; Moore et al. 2010; Oliveira and Hilker 2010 andreferences therein). However, there is just one paper (Moore et al. 2010), to the au-thors’ knowledge, studying the impact of predator–prey interactions on the transmis-sion of vector-borne diseases. Moreover, the models in Moore et al. (2010) assumethe predation on the adult mosquitoes and the analysis mainly focuses on the basicreproduction number and the local stability of equilibria. Until now, there has beenlittle systematic analysis on the consequences of the larvivorous fish to the malariatransmission.

To gain insights into the transmission dynamics of malaria with the introduc-tion of larvivorous fish, we couple the Ross–Macdonald model with an age struc-tured prey–predator model. The population dynamics is governed by a system offive ordinary differential equations. In order to get qualitative dynamics, we firststudy the subsystem describing the predator–prey interaction between larvivorousfish and mosquitoes. Then we use the limiting system to obtain the information forthe whole model. We derive biologically plausible and insightful quantities (the mos-quito, predator and disease reproduction numbers) that allow us to completely deter-mine community compositions. In comparison with the study of Moore, Borer andHosseini (Moore et al. 2010), our model focuses on the impact of the predator–larval(immature) vector interaction on the transmission of vector-borne pathogens. More-over, since larviciding is an important tool to contain malaria transmission, the modelallows us to study the integrated use of larvicides and larvivorous fish. On the otherhand, we rigorously analyse the global dynamics of this system.

The article is organised as follows. The next section presents the eco-epidemiolo-gical model. We first study the predator free system, and then investigate the dynam-ics of the model with predators. Section 3 provides numerical simulations and the lastsection offers a discussion.

2 The Model and Mathematical Results

In this section, we introduce and investigate an eco-epidemiological model, whichcombines two submodels, one describing the age-structured prey–predator dynamicsof mosquitoes and larvivorous fish, and the other describing disease spread betweenmosquitoes and humans.

Y. Lou, X.-Q. Zhao

2.1 Model Formulation

The model is based on monitoring the temporal dynamics of the host (humans), vector(mosquitoes), and predator (larvivorous fish) population sizes.

Following (Li 2009; Wonham et al. 2004), we simplify the biological cycle offemale mosquitoes into two stages: the larval state (Lv) and adult one (Nv), withbirth into the larval stage and natural death in each stage. From the viewpoint of theepidemiology, adult mosquitoes contribute to the transmission of disease, and larvalcontrol is an important measure for vector control. We suppose that the infectious-ness does not affect vector’s egg-laying rate g and death rate dv . For the immaturemosquitoes, the natural death rate and the maturation rate are dL and λv , respectively.As noted in Li (2009), Reiskind and Lounibos (2009), larval crowding or competi-tion is common in container-breeding mosquitoes. We use α to denote the density-dependent development mortality of larvae. To account for the predation of immaturemosquitoes by predators, we take the linear response form with a constant rate γ , sim-ilar to that in the Lotka–Volterra model. Following the modelling idea of the classicalRoss–Macdonald model (Aron and May 1982), the adult female mosquito popula-tion and the human population are divided into two epidemiological categories: thesusceptible class and infectious class. Letting Sv and Iv be the population sizes ofsusceptible and infectious mosquitoes, we have Nv = Sv + Iv . Assume β is the mos-quito biting rate, that is, β is the average number of bites per mosquito per unit time.The force of infection for susceptible mosquitoes can be represented as

cβSv

Ih

Nh

,

where c is the transmission probability from infectious humans to mosquitoes, Ih andNh are population sizes of infectious and total humans, respectively.

For the predator (larvivorous fish) population P , we assume that k is the tropicalconvention efficiency and dp is the mortality rate.

Due to the fact that the total human population for a specified region remainsalmost constant, we suppose that the total human population is unchanged, beingNh, to keep our mathematical modelling as simple as possible. Letting Sh and Ih

be the population sizes of susceptible and infectious humans, we get Nh = Sh + Ih.Hence, we only need to keep track of the population size of infectious humans. Basedon the conservation of bites, that is, the total number of bites made by mosquitoesequals to the number of bites received by humans (Bowman et al. 2005), the averagenumber of bites per human receives per unit time is β Nv

Nh. Suppose the transmission

probability per bite from infectious mosquitoes to humans is b, then the infection rateper susceptible human is given by

bβNv

Nh

Iv

Nv

= bβIv

Nh

.

This cross-infection between the host and vector populations is modelled as mass-action mechanism normalized by the host density, see, e.g. (Wonham et al. 2006).Let dh be the average mortality rate of humans and ρ the recovery rate, i.e. 1/ρ is thehuman infection period.

Modelling Malaria Control by Introduction of Larvivorous Fish

Based on the above assumptions, the community dynamics can be described bythe following system:

dLv(t)

dt= gNv − dLLv − αL2

v − λvLv − γLvP,

dSv(t)

dt= λvLv − cβSv

Ih

Nh

− dvSv,

dIv(t)

dt= cβSv

Ih

Nh

− dvIv,

dP (t)

dt= kγLvP − dpP,

dIh(t)

dt= bβ(Nh − Ih)

Iv

Nh

− (dh + ρ)Ih.

(1)

For the sake of simplicity, it is convenient to use a change of variable Nv = Sv + Iv .It is easy to see that system (1) is equivalent to the following one:

dLv(t)

dt= gNv − dLLv − αL2

v − λvLv − γLvP,

dNv(t)

dt= λvLv − dvNv,

dIv(t)

dt= cβ(Nv − Iv)

Ih

Nh

− dvIv,

dP (t)

dt= kγLvP − dpP,

dIh(t)

dt= bβ(Nh − Ih)

Iv

Nh

− (dh + ρ)Ih.

(2)

Our first task is to show that the mathematical model is biologically meaningful.

Theorem 2.1 System (2) has a unique and bounded solution with the initial value(L0

v,N0v , I 0

v ,P 0, I 0h

) ∈ X := {(Lv,Nv, Iv,P, Ih) ∈ R

5+ : Iv ≤ Nv, Ih ≤ Nh

}.

Moreover, the compact set

� :={(Lv,Nv, Iv,P, Ih) ∈ X : Lv ≤ gλv

αdv

,Nv ≤ g

α

(λv

dv

)2

,

P ≤ k(gλv)2

α(dv)2 min{dL + λv, dp}}

attracts all positive solutions in X.

Proof It follows from Smith (1995, Theorem 5.2.1) that system (2) admits aunique nonnegative solution (Lv(t),Nv(t), Iv(t),P (t), Ih(t)) through an initial value

Y. Lou, X.-Q. Zhao

(L0v,N

0v , I 0

v ,P 0, I 0h ) ∈ X with the maximal interval of existence [0, σ ) for some

σ > 0. Note that for any (Lv,Nv, Iv,P, Ih) ∈ X, if Ih = Nh, dIh

dt≤ 0. It then follows

from Smith (1995, Remark 5.2.1) that Ih(t) ≤ Nh for all t ∈ [0, σ ).According to Zhao and Jing (1996, Corollary 3.2), it is easy to observe that the

following system

du1(t)

dt= gu2 − αu2

1,

du2(t)

dt= λvu1 − dvu2,

admits a globally asymptotically stable equilibrium (gλv

αdv,

gα(λv

dv)2) with respect to all

initial values in R2+ \ {(0,0)}. Since

dLv(t)

dt≤ gNv − αL2

v,

dNv(t)

dt= λvLv − dvNv,

according to the comparison principle (see, e.g. Smith and Waltman 1995, TheoremB.1), there exist M1 and M2 such that

Lv(t) ≤ M1, Nv(t) ≤ M2, ∀t ∈ [0, σ ).

Note that

d(kLv(t) + P(t))

dt≤ kgNv − min{dL + λv, dp}(kLv + P)

≤ kgM2 − min{dL + λv, dp}(kLv + P), ∀t ∈ [0, σ ).

By the comparison principle, we obtain that P(t) is bounded on [0, σ ). Thus, we seethat σ = ∞ and the solution exists globally.

From the previous arguments, we can see that lim supt→∞(Lv(t),Nv(t)) ≤(gλv

αdv,

gα(λv

dv)2) and lim supt→∞ P(t) ≤ k(gλv)

2

α(dv)2 min{dL+λv,dp} . Hence, � is globally at-

tractive. �

The above lemma shows that solutions starting in the region X remain there for alltime t ≥ 0. Biologically, we may restrict our attention to this closed set X. Before weinvestigate the dynamics of our model with larvivorous fish, we would like to studythe case where the predator is not introduced. There are two reasons to start withthis situation. Firstly, it facilitates us to compare with the case where the predatoris introduced. Secondly, it is mathematically easier to deal with the predator-freesystem, while still enables us to clearly illustrate the main mathematical ideas usedin this paper.

Modelling Malaria Control by Introduction of Larvivorous Fish

2.2 Malaria Transmission Without the Predator

In this subsection, we consider the situation where there is no predator, that is P = 0.The predator-free model is given by

dLv(t)

dt= gNv − dLLv − αL2

v − λvLv,

dNv(t)

dt= λvLv − dvNv,

dIv(t)

dt= cβ(Nv − Iv)

Ih

Nh

− dvIv,

dIh(t)

dt= bβ(Nh − Ih)

Iv

Nh

− (dh + ρ)Ih.

(3)

Denote X1 := {(Lv,Nv, Iv, Ih) ∈ R4+ : Iv ≤ Nv, Ih ≤ Nh}. Using the same argument

as that in Theorem 2.1, we can show that X1 is positively invariant for system (3).In the following, we first investigate the dynamics of the mosquito population whenthere is no predator on larvae. It is easy to see from system (3) that the mosquitopopulation is described by the following age-structured growth equations:

dLv(t)

dt= gNv − dLLv − αL2

v − λvLv,

dNv(t)

dt= λvLv − dvNv.

(4)

Since one female mosquito can produce an average of g larvae per unit time whichwill survive to the adult stage with the probability of λv

dL+λv, and 1

dvgives the aver-

age lifespan of mosquitoes, an adult mosquito can produce gλv

dv(dL+λv)adults in its

life time. Denote Rv = gλv

dv(dL+λv). Borrowing the concept of the basic reproduction

number (Anderson and May 1991; Diekmann et al. 2009), we call Rv the vector re-production number. It is clear that when Rv ≤ 1, system (4) has only one equilibrium(0,0). However, system (4) admits a positive equilibrium (L∗

v,N∗v ) when Rv > 1,

where

L∗v = 1

α

(g

λv

dv

−(dL +λv)

)= dL + λv

α(Rv −1) and N∗

v = λv

dv

dL + λv

α(Rv −1).

Let f (x) = (gx2−dLx1−αx2

1−λvx1

λvx1−dvx2

), then f : R

2+ → R2 is a continuously differen-

tiable map. Moreover, f admits the following properties:

(1) f is cooperative on R2+ and Df (x) = (

∂fi

∂xj)1≤i,j≤2 is irreducible for every x ∈

R2+;

(2) f (0) = 0 and fi(x) ≥ 0 for all x ∈ R2+ with xi = 0, i = 1,2;

(3) f is strictly sublinear on R2+, i.e. f (px) > pf (x) for any p ∈ (0,1) and

(x1, x2) ∈ Int(R2+).

Y. Lou, X.-Q. Zhao

Since Df (0) = ( −dL−λv g

λv −dv

), the spectral bound of Df (0), s(Df (0)) :=

max{Reλ : det(λ − Df (0)) = 0}, has the same sign as Rv − 1. Hence, there existsa positive equilibrium (L∗

v,N∗v ) for system (4) when s(Df (0)) > 0. It then follows

from Zhao and Jing (1996, Corollary 3.2) that the subsequent result holds.

Lemma 2.1 The following statements are valid:

(1) If Rv ≤ 1, the trivial equilibrium (0,0) is globally asymptotically stable for sys-tem (4) in R

2+;(ii) If Rv > 1, the positive equilibrium (L∗

v,N∗v ) is globally asymptotically stable for

system (4) in R2+ \ {(0,0)}.

The previous result indicates that the mosquito population will die out if the vectorreproduction number is less than or equal to unity, while the mosquito populationwill eventually stabilize at a positive equilibrium (L∗

v,N∗v ) if the vector reproduction

number is greater than one. If (Lv(t),Nv(t)) → (L∗v,N

∗v ) as t → ∞, the last two

equations in system (3) give rise to the following limiting system:

dIv(t)

dt= cβ(N∗

v − Iv)Ih

Nh

− dvIv,

dIh(t)

dt= bβ(Nh − Ih)

Iv

Nh

− (dh + ρ)Ih.

(5)

It is clear that system (5) has the same form as the classical Ross–Macdonaldmodel (Aron and May 1982). We can define the basic reproduction ratio of this sys-tem by the next generation operator approach (Diekmann et al. 2009; van den Driess-che and Watmough 2002). This reproduction number turns out to be

R0 =√

bβ1

dv

1

(dh + ρ)cβ

1

Nh

λv

dv

1

α

(g

λv

dv

− (dL + λv)

).

Biologically, (R0)2 gives the average number of infected mosquitoes (humans)

caused by one typical infected mosquito (human) in completely susceptible popu-lations. It is easy to observe that when R0 ≤ 1, the limiting system (5) has only onetrivial equilibrium (0,0). If R0 > 1, system (5) has a positive equilibrium (I ∗

v , I ∗h )

with

I ∗v = (dh + ρ)dvNh(R2

0 − 1)

bcβ2 + bβdv

,

and

I ∗h = bcβ2N∗

v Nh − dv(dh + ρ)N2h

bcβ2N∗v + cβ(dh + ρ)Nh

= dv(dh + ρ)N2h(R2

0 − 1)

bcβ2N∗v + cβ(dh + ρ)Nh

.

Let B := [0,N∗v ] × [0,Nh]. It then follows that ω((Iv(0), Ih(0))) ⊂ B , where ω

((Iv(0), Ih(0))) is the omega limit set of (Iv(0), Ih(0)) ∈ R2+ for the solution semiflow

of system (5). Moreover, B is positively invariant and system (5) is cooperative in B .By using Zhao and Jing (1996, Corollary 3.2) on the subset B , as argued in the proof

Modelling Malaria Control by Introduction of Larvivorous Fish

of Lemma 2.1, we have the following threshold-type result for the classical Ross–Macdonald model.

Lemma 2.2 The following statements are valid:

(i) If R0 ≤ 1, the trivial equilibrium (0,0) is globally asymptotically stable for sys-tem (5) in R

2+;(ii) If R0 > 1, the limiting system (5) admits a unique positive equilibrium (I ∗

v , I ∗h ),

and it is globally asymptotically stable for system (5) in R2+ \ {(0,0)}.

Lemmas 2.1 and 2.2 show that Rv and R0 are threshold parameters for vectorsustainment and disease persistence, respectively. Our next task is to show that thesetwo reproduction numbers also serve as important indexes for the global dynamics ofthe predator-free system (3). We refer the reader to Appendix A for a detailed proof.

Theorem 2.2 Let (Lv(t), Nv(t), Iv(t), Ih(t)) be the solution of system (3) through(Lv(0), Nv(0), Iv(0), Ih(0)). The following statements are valid:

(i) If Rv ≤ 1, (0,0,0,0) is globally attractive for system (3) in X1;(ii) If Rv > 1 and R0 ≤ 1, limt→∞(Lv(t),Nv(t), Iv(t), Ih(t)) = (L∗

v,N∗v ,0,0) for

all initial values in Int(X1);(iii) If Rv > 1 and R0 > 1, there exists a positive equilibrium (L∗

v , N∗v , I ∗

v , I ∗h ), and

it is globally attractive for system (3) in Int(X1).

The previous theorem indicates that the long-time behaviour of solutions can becompletely determined by two reproduction numbers, and that the parameter spacecan be divided into three regions to have different global attractors for system (3).

2.3 Malaria Transmission with Predators

In this subsection, we study the malaria transmission when larvivorous fish is intro-duced as a control measure and investigate the dynamics of system (2). We first studythe subsystem for the mosquito and predator populations. It is easy to see that popu-lation sizes of mosquitoes and their predators are governed by the following system:

dLv(t)

dt= gNv − dLLv − αL2

v − λvLv − γLvP,

dNv(t)

dt= λvLv − dvNv,

dP (t)

dt= kγLvP − dpP.

(6)

Since P(t) ≥ 0, ∀t ≥ 0, we have the following comparison relation:

dLv(t)

dt≤ gNv − dLLv − αL2

v − λvLv,

dNv(t)

dt= λvLv − dvNv.

Y. Lou, X.-Q. Zhao

By the comparison principle and Lemma 2.1, we obtain that Lv(t) → 0 and Nv(t) →0 as t → ∞ when Rv ≤ 1. Using the theory of internally chain transitive sets (see,e.g. Hirsch et al. 2001; Zhao 2003), as argued in the proof of Theorem 2.2, we havethe following result.

Theorem 2.3 Let (Lv(t), Nv(t), P(t)) be the solution of system (6) through (Lv(0),Nv(0), P(0))∈ R

3+. If Rv ≤ 1, limt→∞(Lv(t),Nv(t),P (t)) = (0,0,0).

Biologically, it is easy to see that kγ [ 1α(g λv

dv− dL − λv)] accounts for the mean

number of the offspring which can be reproduced by index larvivorous fish when thelarval mosquito population stabilizes at L∗

v (recall that L∗v = 1

α(g λv

dv− dL − λv)), and

that 1dp

is the average lifespan of larvivorous fish. Hence, the product kγdp

1α(g λv

dv−

dL − λv) gives the expected number of larvivorous fish produced by typical larvivo-rous fish in its lifetime. Set Rp = kγ

dp

1α(g λv

dv− dL − λv). Similar to the concept of the

basic reproduction number, we call Rp the predator reproduction number. The subse-quent result shows that the predator cannot sustain itself if the predator reproductionnumber is not greater than unity.

Theorem 2.4 Let (Lv(t), Nv(t), P(t)) be the solution of system (6) through (Lv(0),Nv(0), P(0))∈ R

3+ with Lv(0) > 0 and Nv(0) > 0. If Rv > 1 and Rp ≤ 1,

limt→∞

(Lv(t),Nv(t),P (t)

) = (L∗

v,N∗v ,0

).

On the other hand, if the predator reproduction number exceeds one, that iskγα

(g λv

dv− dv − dL) > dp , there exists a positive equilibrium (Lv , Nv , P ), where

Lv = dp

kγ, Nv = λv

dv

Lv and P = αdp

kγ 2(Rp − 1).

Our next goal is to show that when Rp > 1, the solutions of system (6) will convergeto the unique positive equilibrium.

Theorem 2.5 Let (Lv(t), Nv(t), P(t)) be the solution of system (6) through(Lv(0),Nv(0),P (0)) ∈ R

3+. If Rp > 1, limt→∞(Lv(t),Nv(t),P (t)) = (Lv, Nv, P )

for all Lv(0) > 0, Nv(0) > 0 and P(0) > 0.

The key ideas to prove Theorems 2.4 and 2.5 are similar. We first show that thesystem is uniformly persistent with respect to a specific set. By choosing suitableLyapunov functions and using the LaSalle’s invariance principle (LaSalle 1976), wethen obtain the global attractivity results. In our proof, we take a widely used typeof Lyapunov functions (see, e.g. Guo and Li 2006; Korobeinikov and Maini 2004;Ma et al. 2003; Zhang et al. 2000 and references therein). For detailed proofs of The-orems 2.4 and 2.5, we refer the reader to Appendix B and Appendix C, respectively.We should point out that a similar result is given in Zhang et al. (2000) when there isa density-dependent death rate for the predator population.

Modelling Malaria Control by Introduction of Larvivorous Fish

Theorem 2.5 implies that the adult mosquito population will stabilize at Nv if thepredator reproduction number (Rp) is greater than one. In this situation, if we denote

Rc0 =

√

bcβ2 Nv

(dh + ρ)dvNh

=√

bcβ2 1

(dh + ρ)dvNh

λv

dv

dp

kγ,

the value of (Rc0)

2 gives the average number of infectious mosquitoes (humans)reproduced by a typical infectious mosquito (human) in its infection period, withthe introduction of larvivorous fish (a control measure). We call Rc

0 the control re-production ratio. It then follows from Lemma 2.2 that when Rc

0 > 1, the classical

Ross–Macdonald model (5) with N∗v replaced by Nv has a positive equilibrium (Iv ,

Ih), where Iv = (dh+ρ)dvNh

bcβ2+bβdv((Rc

0)2 − 1) and Ih = dv(dh+ρ)N2

h

bcβ2Nv+cβ(dh+ρ)Nh

((Rc0)

2 − 1).

Using the theory of internally chain transitive sets (see, e.g. Hirsch et al. 2001;Zhao 2003), as argued in the proof of Theorem 2.2, we then have the following result.

Theorem 2.6 Let (Lv(t), Nv(t), Iv(t), P(t), Ih(t)) be the solution of system (2)through (Lv(0), Nv(0), Iv(0), P(0), Ih(0)). The following statements are valid:

(i) If Rv ≤ 1, (0,0,0,0,0) is globally attractive for system (2) in X;(ii) If Rv > 1, Rp ≤ 1 and R0 ≤ 1,

limt→∞

(Lv(t),Nv(t), Iv(t),P (t), Ih(t)

) = (L∗

v,N∗v ,0,0,0

)

for all initial values in Int(X);(iii) If Rv > 1, Rp ≤ 1 and R0 > 1,

limt→∞

(Lv(t),Nv(t), Iv(t),P (t), Ih(t)

) = (L∗

v,N∗v , I ∗

v ,0, I ∗h

)

for all initial values in Int(X);(iv) If Rv > 1, Rp > 1 and Rc

0 ≤ 1,

limt→∞

(Lv(t),Nv(t), Iv(t),P (t), Ih(t)

) = (Lv, Nv,0, P ,0)

for all initial values in Int(X);(v) If Rv > 1, Rp > 1 and Rc

0 > 1,

limt→∞

(Lv(t),Nv(t), Iv(t),P (t), Ih(t)

) = (Lv, Nv, Iv, P , Ih)

for all initial values in Int(X).

Theorem 2.6 implies that the global dynamics of our model system can be com-pletely determined by four threshold parameters: the vector reproduction number Rv ,the basic reproduction number R0, the predator reproduction number Rp and thecontrol reproduction number Rc

0, and there are five possible community composi-tions.

Y. Lou, X.-Q. Zhao

3 Numerical Simulations

In this section, we first present some simulations which illustrate our analytic results.

3.1 Different Community Components

Almost 200 fish species are known to feed on mosquito larvae (Jenkins 1964) anddifferent species have different efficacy in controlling the disease in different ecolog-ical settings. Introduction of different fish species may give rise to different potentialoutcomes of our model (as shown in Theorem 2.6). To do this, we first choose a set ofdefault parameters which are biologically accepted. By changing some coefficientsto satisfy different conditions in Theorem 2.6, we simulate the long-time popula-tion sizes of the larval mosquitoes, infectious mosquitoes, infectious humans and thepredator.

We adopt some parameter values from Hancock and Godfray (2007) and refer-ences therein except dh, Nh, γ , k, and dp . These values are roughly consistent with P.falciparum transmitted by Anopheles Gambiae. We suppose that the life expectancyof humans is 70 years, then dh = 1

70×365 d−1. For illustration, we choose γ = 0.01,

k = 0.0005, dp = 1365 , and Nh = 1000. The default parameters are summarized in

Table 1.To get potential community compositions of the model, we change the reproduc-

tion numbers by varying some parameters, while keeping others as default parame-ters. The observed population dynamics of four components (larval mosquitoes, in-fectious mosquitoes, infectious humans and the predator) are shown in Fig. 1.

In Fig. 2, we simulate the impact of larvivorous fish on the disease control. Herewe assume that the maximum lifespan of an adult mosquito is 16 days (that is,1dv

≤ 16) and the human biting rate of mosquitoes is between 0.1 and 0.3 (that is,β ∈ [0.1,0.3]). It is easy to see from Fig. 2 that the high biting rate (correspondingto large values of β) and low efficacy of adulticiding (corresponding to small valuesof mosquito death rate dv) make R0 > 1 and Rc

0 > 1. This indicates that the diseasewill eventually stabilize at a positive state (Theorem 2.6) and persist in the humanpopulation. When the human biting rate is 0.1, the disease will be eradicated if theadulticiding can ensure that the mean lifespan of mosquitoes is less than 12 days (thisstrategy results in R0 < 1 in the top picture of Fig. 2). If the larvivorous fish is intro-duced and the adulticiding can ensure the mean lifespan of mosquitoes to be less than15.7 days, the strategies combined with the larvivorous fish and the adulticiding caneradicate the disease (these combined strategies make Rc

0 < 1 in the bottom pictureof Fig. 2). On the other hand, when the mean lifespan of adult mosquitoes is 10 days,

Table 1 Default parameters for model (2)

Parameter dL dv g b c β ρ

Value 0.1 0.1 30 0.5 0.5 0.3 0.01

Parameter dh Nh α λv γ k dp

Value 170×365 1000 0.05 1

16 0.01 0.0005 1365

Modelling Malaria Control by Introduction of Larvivorous Fish

Fig. 1 Potential community compositions for model (2). (a) dL = 1, dv = 2, β = 0.1, ρ = 0.1,dp = 10/365 with other parameters shown in Table 1. In this case, Rv = 0.88 < 1. (b) β = 0.1 with otherparameters shown in Table 1. In this case, Rv = 115.38, R0 = 0.76 and Rp = 0.68. (c) Parameters areshown in Table 1. In this case, Rv = 115.38, R0 = 2.28 and Rp = 0.68. (d) Nh = 3000, k = 0.0025 withother parameters shown in Table 1. In this case, Rv = 115.38, R0 = 1.32, Rp = 3.39 and Rc

0 = 0.72.(e) k = 0.001 with other parameters shown in Table 1. In this case, Rv = 115.38, R0 = 2.28, Rp = 1.36and Rc

0 = 1.96

the disease will be eradicated if people protect themselves to make the human bitingrate of mosquitoes to be less than 0.14 (R0 < 1 in the top picture of Fig. 2). If thelarvivorous fish is introduced and the human biting rate of mosquitoes is less than

Y. Lou, X.-Q. Zhao

Fig. 2 Two (basic/control)reproduction numbers asfunctions of β (representing theuse of bed nets) and 1

dv(representing the use ofadulticides). Parameters arechosen as those in Table 1except k, β and dv . Herek = 0.001, β and dv are variable

0.16, the disease will die out (Rc0 < 1 in the bottom picture of Fig. 2). Hence, in-

troduction of the larvae predator can be incorporated into an integrated strategy fordisease control.

Next, we implement a case study for two fish species in specific ecological set-tings.

3.2 A Case Study

In this subsection, we investigate the efficacy of two different fish species, Orechromisniloticus L. and Oreochromis spilurus, on malaria transmission. We use same para-meters as those in Table 1 except k, γ and dp , which represent the fish predationpotential.

In Howard et al. (2007), Howard, Zhou and Omlin reported that Orechromis niloti-cus L. can dramatically cause a 94% reduction in the population size of larval mos-quitoes Anopheles gambiae s.l. in Kisii Central District of western Kenya. From our

Modelling Malaria Control by Introduction of Larvivorous Fish

theoretical results, we can see that the equilibria for the population sizes of larvalmosquitoes, L∗

v and Lv (corresponding to two different settings without and withfish, respectively) can be expressed by:

L∗v = 1

α

(g

λv

dv

− (dL + λv)

)and Lv = dp

kγ.

Hence, we have Lv = (1 − 94%)L∗v , that is

dp

kγ= 6% × 1

α

(g

λv

dv

− (dL + λv)

)= 22.26.

In this case, we can compute the reproduction numbers R0 = 2.28 and Rc0 = 0.56.

Thus, the community components are similar to Fig. 1(d), implying that O. niloticuscan indirectly control malaria in this setting.

However, introducing Oreochromis spilurus into the water storage container inSomalia produces a mean reduction of 52.8% in larval mosquitoes (Mohamed 2003).For the predation efficacy of O. spilurus, we have this relation: Lv = (1 − 52.8%)L∗

v ,implying that the population size of larval mosquitoes is reduced to 47.2% of itssaturated value by fish introduction. Hence, dp

kγ= 175.47. In this case, R0 = 2.28

and Rc0 = 1.57. The final outcome of community components should be similar to

Fig. 1(e). In this setting, the population sizes of infectious humans without and withfish are I ∗

h = 627.3 and Ih = 367.6, respectively. Although the disease remains en-demic in the case of O. spilurus introduction, the fish can drastically reduce the sizeof the infectious human population.

4 Discussion

In this paper, we discussed a mathematical model designed to describe vital dynam-ics of malaria transmission with vector predators, aiming to highlight practical pro-cedures for the introduction of predators as a biological control method.

We coupled an age-structured prey–predator model with the Ross–Macdonaldmodel. The age-structured prey–predator system describes the vector–larvivorousfish interaction with the structured mosquito population, while the Ross–Macdonaldmodel describes the vital cross-transmission of malaria parasites between humansand mosquitoes.

In our model, the equations for mosquitoes and their predators can be decoupledand completely separated. We systematically used the behavioural process for sub-models to investigate the population changes on the long term. By utilizing the theo-ries of monotone dynamical systems, Lyapunov functions, and internally chain tran-sitive sets, we completely investigated the global dynamics of this model. We derivedfull characterization of the global behaviour of the model, and showed that the para-meter space can be divided into five parts according to global attractors of the system.

Our main result (Theorem 2.6) indicates that three scenarios exist for disease erad-ication: (1) Rv ≤ 1; (2) R0 ≤ 1; (3) Rc

0 ≤ 1. In order to control the disease, we canuse the larvicides or adulticides to reduce the vector reproduction number to be less

Y. Lou, X.-Q. Zhao

than unity (Scenario (1)). The integrated strategy combined with larviciding, adulti-ciding, bed nets, prompt treatments and vaccination can be used to reduce the basicreproduction number to be less than one (Scenario (2)). If the larvivorous fish is suc-cessfully introduced (Rp > 1), the integrated strategy combined with adulticiding,prompt treatments, bed nets, vaccination can be implemented to let Scenario (3) hap-pen.

If the predator is aggressive enough (for example, the fish Orechromis niloticus L.),Rc

0 can be reduced to be less than unity. In this case, the infection can be eradicated.Even though the fish is not so aggressive (for example, the fish Oreochromis spilurus)but can sustain itself, it is easy to observe that Rc

0 < R0 and Ih < I ∗h (see Sect. 3).

Hence, the introduction of larvivorous fish has an important effect on malaria control.It may increase the extinction probability of malaria and reduce the prevalence ofinfection. On the other hand, larvivorous fish provides an opportunity to eradicatemalaria locally in combination with other methods such as insecticide-treated nets(ITNs) and prompt treatments. Although the use of predators can not control thedisease solely, it may dramatically decrease the population size of infectious humansand slow down the initial speed of disease spread, which in turn, will earn importanttime for vaccine and drug treatment development for other emerging mosquito bornediseases.

It is easy to see that the larval death rate dL is not incorporated in the expressionsof the control reproduction number Rc

0 and the population size of infectious humans

Ih at the positive equilibrium. Thus, increasing the larval death rate may not haveeffect on malaria transmission. As pointed out in Jacob et al. (1982), larviciding, astandard method to increase the larval death rate, may also increase the death rate oflarvivorous fish dp , which will subsequently increase Rc

0 and Ih. In this sense, usingthe larvicides, a common method for malaria control, may have negative effects onmalaria containing. Therefore, we should carefully design the integrated strategies,which calls for more field work.

In this work, the dynamics of a malaria model with larvivorous fish is fully inves-tigated in the absence of the disease-induced death rate. Extension of our frameworkto include a disease-induced death rate is an interesting future work. Incorporatingage-structured culling for malaria is also another interesting topic in the future. Formore details about culling strategies for the vector population, we refer the readerto reference (Gourley et al. 2007). Note that we did not consider seasonal effectson mosquito dynamics in this paper. In the seasonally forced case, although we canget a similar result (see Lou and Zhao 2010; Zhao and Jing 1996) for the periodicRoss–Macdonald model via the theory of monotone dynamical systems, the Lya-punov function method may fail to work for the periodic prey–predator system. Thisis another challenging problem for future investigation.

We hope that this paper would lead to a better understanding of the vector-bornediseases with introduction of larval vector predators and provide new conceptual toolsfor the advancement of our knowledge to this important subject.

Acknowledgement We are grateful to anonymous referees for their careful reading and helpful sugges-tions which led to an improvement of our original manuscript.

Modelling Malaria Control by Introduction of Larvivorous Fish

Appendix A: Proof of Theorem 2.2

Our key idea to prove Theorem 2.2 is to use the theory of internally chain transitivesets, see, e.g. (Hirsch et al. 2001; Zhao 2003).

Let Φ(t) be the solution semiflow of system (3) on X1, that is,

Φ(t)(Lv(0),Nv(0), Iv(0), Ih(0)

) = (Lv(t),Nv(t), Iv(t), Ih(t)

).

Then Φ(t) is compact for each t > 0. Let ω = ω(Lv(0),Nv(0), Iv(0), Ih(0)) be theomega limit set of Φ(t)(Lv(0),Nv(0), Iv(0), Ih(0)). It then follows from Hirsch etal. (2001, Lemma 2.1′) (see also Zhao 2003, Lemma 1.2.1′) that ω is an internallychain transitive set for Φ(t).

(i) In the case where Rv ≤ 1, we have Lv(t) → 0, Nv(t) → 0 and Iv(t) → 0 ast → ∞. Hence, we have ω = {(0,0,0)} × ω1 for some ω1 ⊂ R. It is easy to see that

Φ(t)|ω(0,0,0, Ih(0)

) = (0,0,0,Φ1(t)

(Ih(0)

)),

where Φ1(t) is the solution semiflow associated with the following equation:

dIh

dt= −(dh + ρ)Ih. (7)

Since ω is an internally chain transitive set for Φ(t), it easily follows that ω1 is aninternally chain transitive set for Φ1(t). Since {0} is globally asymptotically stablefor (7), Hirsch et al. (2001, Theorem 3.2 and Remark 4.6) implies that ω1 = {0}.Thus, we have ω = {(0,0,0,0)}, and hence

limt→∞

(Lv(t),Nv(t), Iv(t), Ih(t)

) = (0,0,0,0).

This proves the statement (i).(ii) In the case where Rv > 1, we have Lv(t) → L∗

v , Nv(t) → N∗v as t → ∞ for

any Lv(0) > 0 and Nv(0) > 0. Thus, ω = {(L∗v,N

∗v )} × ω2 for some ω2 ⊂ R

2, and

Φ(t)|ω(L∗

v,N∗v , Iv(0), Ih(0)

) = (L∗

v,N∗v ,Φ2(t)

(Iv(0), Ih(0)

)),

where Φ2(t) is the solution semiflow associated with the following system:

dIv(t)

dt= cβ

(N∗

v − Iv

) Ih

Nh

− dvIv,

dIh(t)

dt= bβ(Nh − Ih)

Iv

Nh

− (dh + ρ)Ih.

(8)

Since ω is an internally chain transitive set for Φ(t), it is easy to see that ω2 is aninternally chain transitive set for Φ2(t). Since R0 ≤ 1, the trivial equilibrium {(0,0)}is globally asymptotically stable for system (8) according to Lemma 2.2. It then fol-lows from Hirsch et al. (2001, Theorem 3.2 and Remark 4.6) that ω2 = {(0,0)}. Thisproves ω = {(L∗

v,N∗v ,0,0)}, and hence, the statement (ii) holds.

Y. Lou, X.-Q. Zhao

(iii) In the case where Rv > 1 and R0 > 1, we have Lv(t) → L∗v , Nv(t) → N∗

v

as t → ∞. We have ω = {(L∗v,N

∗v )} × ω3 for some ω3 ⊂ R

2, and

Φ(t)|ω(L∗

v,N∗v , Iv(0), Ih(0)

) = (L∗

v,N∗v ,Φ2(t)

(Iv(0), Ih(0)

)),

where Φ2(t) is the solution semiflow of system (8). Since ω is an internally chaintransitive set for Φ(t), it follows that ω3 is an internally chain transitive set for Φ2(t).We claim that ω3 �= {(0,0)} for all Iv(0) > 0 and Ih(0) > 0. Assume that, by contra-diction, ω = {(L∗

v,N∗v ,0,0)} for some Iv(0) > 0 and Ih(0) > 0. Then, we have

limt→∞

(Lv(t),Nv(t), Iv(t), Ih(t)

) = (L∗

v,N∗v ,0,0

).

Since R0 > 1, there exists some δ > 0 such that

bcβ2 N∗v − δ

(dh + ρ)dvNh

> 1.

Moreover, there exists some T0 > 1 such that∣∣(Lv(t),Nv(t), Iv(t), Ih(t)

) − (L∗

v,N∗v ,0,0

)∣∣ < δ, ∀t > T0.

Hence, we have

dIv(t)

dt≥ cβ

(N∗

v − δ − Iv

) Ih

Nh

− dvIv,

dIh(t)

dt= bβ(Nh − Ih)

Iv

Nh

− (dh + ρ)Ih,

for all t > T0. It then follows from Lemma 2.2 that the following system

du1(t)

dt= cβ

(N∗

v − δ − u1) u2

Nh

− dvu1,

du2(t)

dt= bβ(Nh − u2)

u1

Nh

− (dh + ρ)u2

admits a positive equilibrium (u∗1, u

∗2) such that

limt→∞

(u1(t), u2(t)

) = (u∗

1, u∗2

), ∀(

u1(0), u2(0)) ∈ R

2+ \ {(0,0)

}.

By the comparison principle, we have

lim inft→∞

(Iv(t), Ih(t)

) ≥ (u∗

1, u∗2

),

a contradiction. Since ω3 �= {(0,0)} and (I ∗v , I ∗

h ) is globally asymptotically stable forsystem (8) in R

2+\{(0,0)}, it follows that ω3 ∩Ws((I ∗v , I ∗

h )) �= ∅, where Ws((I ∗v , I ∗

h ))

is the stable set for (I ∗v , I ∗

h ). By Hirsch et al. (2001, Theorem 3.1 and Remark 4.6),we then get ω3 = {(I ∗

v , I ∗h )}. Thus, ω = {(L∗

v,N∗v , I ∗

v , I ∗h )}, and hence, the statement

(iii) is valid.

Modelling Malaria Control by Introduction of Larvivorous Fish

Appendix B: Proof of Theorem 2.4

Denote X0 := {(Lv,Nv,P ) ∈ R3+ : Lv > 0,Nv > 0}, ∂X0 := R

3+ \ X0 ={(Lv,Nv,P ) ∈ R

3+ : Lv = 0 or Nv = 0} and X∂ := {(Lv(0),Nv(0),P (0)) ∈∂X0 : (Lv(t),Nv(t),P (t)) ∈ ∂X0,∀t ≥ 0}. It is clear that X∂ = {(Lv,Nv,P ) ∈R

3+ : Lv = 0,Nv = 0}.By the form of system (6), it is easy to see that both R

3+ and X0 are posi-tively invariant. We first show that when Rv > 1, system (6) is uniformly persis-tent with respect to (X0, ∂X0) in the sense that there exists some ε > 0 such thatlim inft→∞ Lv(t) > ε and lim inft→∞ Nv(t) > ε for all (Lv(0),Nv(0),P (0)) ∈ X0.Since Rv > 1, there exists some δ > 0 such that gλv

dv(dL+λv+γ δ)> 1 and 1

α(g λv

dv− dL −

λv − γ δ) > δ. Then we have the following clam, which says that (0,0,0) is a weakerrepeller for X0.

Claim lim supt→∞ ‖(Lv(t),Nv(t),P (t))‖ ≥ δ for any (Lv(0),Nv(0),P (0)) ∈ X0.Suppose, by contradiction, that

lim supt→∞

∥∥(Lv(t),Nv(t),P (t)

)∥∥ < δ for some(Lv(0),Nv(0),P (0)

) ∈ X0.

Then there exists a T0 > 0 such that Lv(t) < δ, Nv(t) < δ and P(t) < δ for t ≥ T0.Thus, we have

dLv(t)

dt≥ gNv − dLLv − αL2

v − λvLv − γ δLv,

dNv(t)

dt= λvLv − dvNv,

for all t ≥ T0. By Lemma 2.1, there exists a positive globally asymptotically stableequilibrium (u∗

1, u∗2) for the following system:

du1(t)

dt= gu2 − dLu1 − αu2

1 − λvu1 − γ δu1,

du2(t)

dt= λvu1 − dvu2.

Moreover, u∗1 = 1

α(g λv

dv−dL −λv −γ δ) and u∗

2 = λv

dvu∗

1. Hence, lim supt→∞ Lv(t) ≥limt→∞ u1(t) = u∗

1 > δ, a contradiction.Let Ψ (t) be the solution semiflow associated with system (6). Then (0,0,0) is a

compact and isolated invariant set for Ψ (t) in ∂X0, and �(X∂) := ⋃x∈X∂

ω(x) =(0,0,0). Furthermore, no subset of (0,0,0) forms a cycle in ∂X0. In view of theclaim above, (0,0,0) is an isolated invariant set for Ψ (t) in X and Ws((0,0,0)) ∩X0 = ∅. By (Thieme 1993, Theorem 4.6), Ψ (t) is uniformly persistent with respectto (X0, ∂X0). Thus, we have ω((Lv,Nv,P )) ⊂ X0 for any (Lv,Nv,P ) ∈ X0.

Define a function V1 : X0 → R by

Y. Lou, X.-Q. Zhao

V1 =(

Lv − L∗v − L∗

v lnLv

L∗v

)

+ g

dv

(Nv − N∗

v − N∗v ln

Nv

N∗v

)+ 1

kP, ∀(Lv,Nv,P ) ∈ X0.

It then follows that

V ′1(Lv,Nv,P )

=(

1 − L∗v

Lv

)(gNv − (dL + λv)Lv − αL2

v − γLvP)

+ g

dv

(1 − N∗

v

Nv

)(λvLv − dvNv)

+ 1

k(kγLvP − dpP )

= gNv − gL∗v

Nv

Lv

− (dL + λv)Lv − αL2v − γLvP + (dL + λv)L

∗v + αL∗

vLv

+ γL∗vP + g

dv

λvLv − gNv − gN∗v

dv

λv

Lv

Nv

+ gN∗v + γLvP − dp

kP.

Since γL∗v = dp

k, N∗

v = λv

dvL∗

v and gN∗v = (dL + λv)L

∗v + α(L∗

v)2, we further have

V ′1(Lv,Nv,P )

= −gL∗v

Nv

Lv

− (dL + λv)Lv − αL2v + (dL + λv)L

∗v + αL∗

vLv

+ g

dv

λvLv − gN∗v

dv

λv

Lv

Nv

+ gN∗v

≤ −2gN∗v − (dL + λv)Lv − αL2

v + (dL + λv)L∗v + αL∗

vLv + gλv

dv

Lv + gN∗v

= −gN∗v − (dL + λv)Lv − αL2

v + (dL + λv)L∗v + αL∗

vLv + gλv

dv

Lv

= −(dL + λv)L∗v − α(L∗

v)2 − (dL + λv)Lv − αL2

v + (dL + λv)L∗v

+ αL∗vLv + gλv

dv

Lv

= −α(L∗

v

)2 − (dL + λv)Lv − αL2v + αL∗

vLv + gλv

dv

Lv

≤(

−αL∗v − (dL + λv) + g

λv

dv

)Lv = 0.

Now we find the invariant set of Ψ (t) that is contained in the set{(Lv,Nv,P ) ∈ X0 : V ′

1(Lv,Nv,P ) = 0}.

Modelling Malaria Control by Introduction of Larvivorous Fish

If V ′1 = 0, we have Lv = L∗

v and Nv = N∗v . Consequently, dLv(t)

dt= −γL∗

vP . Thisimplies that P = 0. Thus, the only compact invariant subset of the set where V ′

1 = 0is the singleton {(L∗

v,N∗v ,0)}. By LaSalle’s invariance principle (LaSalle 1976), we

obtain that (L∗v,N

∗v ,0) is globally attractive in X0.

Appendix C: Proof of Theorem 2.5

Denote Y0 := {(Lv,Nv,P ) ∈ R3+ : Lv > 0,Nv > 0,P > 0}, ∂Y0 := R

3+ \ Y0 ={(Lv,Nv,P ) ∈ R

3+ : Lv = 0 or Nv = 0 or P = 0} and Y∂ := {(Lv(0),Nv(0),P (0)) ∈∂Y0 : (Lv(t),Nv(t),P (t)) ∈ ∂Y0, t ≥ 0}. It is easy to see that Y∂ = {(Lv,Nv,P ) ∈R

3+ : Lv = Nv = 0 or P = 0}. By a similar argument as that in the proof of Theo-rem 2.4, we can show that if Rp > 1, system (6) is uniformly persistent with respectto (Y0, ∂Y0) in the sense that there exists some ε > 0 such that lim inft→∞ Lv(t) > ε,lim inft→∞ Nv(t) > ε and lim inft→∞ P(t) > ε for any (Lv(0),Nv(0),P (0)) ∈ Y0.Thus, we have ω((Lv,Nv,P )) ⊂ Y0 for any (Lv,Nv,P ) ∈ Y0.

Define a function V2 : Y0 → R by

V2(Lv,Nv,P ) =(

Lv − Lv − Lv lnLv

Lv

)

+ g

dv

(Nv − Nv − Nv ln

Nv

Nv

)+ 1

k

(P − P − P ln

P

P

).

We then have

V ′2(Lv,Nv,P )

=(

1 − Lv

Lv

)(gNv − (dL + λv)Lv − αL2

v − γLvP)

+ g

dv

(1 − Nv

Nv

)(λvLv − dvNv)

+ 1

k

(1 − P

P

)(kγLvP − dpP )

= gNv − (dL + λv)Lv − αL2v − γLvP − gLv

Nv

Lv

+ (dL + λv)Lv + αLvLv

+ γ LvP + g

dv

λvLv − gNv − gNvλv

dv

Lv

Nv

+ gNv

+ γLvP − dp

kP − γ PLv + dpP

k.

Since γ Lv = dp

k, (dL + λv)Lv + α(Lv)

2 + dp

kP = gNv and dL + λv + αLv + γ P =

gNv

Lv, it follows that

Y. Lou, X.-Q. Zhao

V ′2(Lv,Nv,P )

= −(dL + λv)Lv − α(Lv)2 − gLv

Nv

Lv

+ gNv − α(Lv)2

+ αLvLv + g

dv

λvLv − gNvλv

dv

Lv

Nv

+ gNv − γ PLv

= −gNv

Lv

Lv + αLvLv − α(Lv)2 − α(Lv)

2 + αLvLv + gλv

dv

Lv

−(

gLv

Nv

Lv

− 2gNv + gNvλv

dv

Lv

Nv

).

Note that Nv

Lv= λv

dv, we further obtain

V ′2(Lv,Nv,P ) = −α

((Lv)

2 − 2LvLv + (Lv)2) − gNv

(dv

λv

Nv

Lv

− 2 + λv

dv

Lv

Nv

)≤ 0.

Now we find the invariant set of Ψ (t) that is contained in the set{(Lv,Nv,P ) ∈ Y0 : V ′

2(Lv,Nv,P ) = 0}.

If V ′2 = 0, we have Lv = Lv and Nv = Nv . Consequently, dLv(t)

dt= γ Lv(P − P).

This implies P = P , and the largest compact invariant subset of the set where V ′2 = 0

is {(Lv, Nv, P )}. It then follows from LaSalle’s invariance principle that (Lv, Nv, P )

is globally attractive in Y0.

References

Anderson, R. M., & May, R. M. (1991). Infectious diseases of humans: dynamics and control. Oxford:Oxford University Press.

Aron, J. L., & May, R. M. (1982). The population dynamics of malaria. In R. M. Anderson (Ed.), The pop-ulation dynamics of infectious diseases: theory and applications (pp. 139–179). London: Chapmanand Hall.

Auger, P., McHich, R., Chowdhury, T., Sallet, G., Tchuente, M., & Chattopadhyay, J. (2009). Effects of adisease affecting a predator on the dynamics of a predator–prey system. J. Theor. Biol., 258, 344–351.

Bowman, C., Gumel, A. B., van den Driessche, P., Wu, J., & Zhu, H. (2005). A mathematical model forassessing control strategies against West Nile virus. Bull. Math. Biol., 67, 1107–1133.

Brito, I. (2001). Eradicating malaria: high hopes or a tangible goal? Health Policy Harvard, 2, 61–66.Chandra, G., Bhattacharjee, I., Chatterjee, S. N., & Ghosh, A. (2008). Mosquito control by larvivorous

fish. Indian J. Med. Res., 127, 13–27.Diekmann, O., Heesterbeek, J. A. P., & Roberts, M. G. (2009). The construction of next-generation matri-

ces for compartmental epidemic models. J. R. Soc. Interface, 5, 1–13.Ghosh, S., & Dash, A. (2007). Larvivorous fish against malaria vectors: a new outlook. Trans. R. Soc.

Trop. Med. Hyg., 101, 1063–1064.Ghosh, S., Tiwari, S., Sathyanarayan, T., Sampath, T., Sharma, V., Nanda, N., Joshi, H., Adak, T., &

Subbarao, S. (2005). Larvivorous fish in wells target the malaria vector sibling species of the complexin villages in Karnataka, India. Trans. R. Soc. Trop. Med. Hyg., 99, 101–105.

Gourley, S. A., Liu, R., & Wu, J. (2007). Eradicating vector-borne diseases via age-structured culling.J. Math. Biol., 54, 309–335.

Modelling Malaria Control by Introduction of Larvivorous Fish

Greenwood, B. M., Bojang, K., Whitty, C. J., & Targett, G. A. (2005). Malaria. Lancet, 365, 1487–1498.Greenwood, B. M., Fidock, D. A., Kyle, D. E., Kappe, S. H. I., Alonso, P. L., Collins, F. H., & Duffy, P. E.

(2008). Malaria: progress, perils, and prospects for eradication. J. Clin. Invest., 118, 1266–1276.Guo, H., & Li, M. Y. (2006). Global stability in a mathematical model of tuberculosis. Can. Appl. Math.

Q., 14, 185–197.Hancock, P. A., & Godfray, H. C. J. (2007). Application of the lumped age-class technique to studying the

dynamics of malaria–mosquito–human interactions. Malar. J., 6, 98.Hilker, F., & Schmitz, K. (2008). Disease-induced stabilization of predator–prey oscillations. J. Theor.

Biol., 255, 299–306.Hirsch, M. W., Smith, H. L., & Zhao, X.-Q. (2001). Chain transitivity, attractivity, and strong repellors for

semidynamical systems. J. Dyn. Differ. Equ., 13, 107–131.Howard, A. F., Zhou, G., & Omlin, F. X. (2007). Malaria mosquito control using edible fish in western

Kenya: preliminary findings of a controlled study. BMC Public Health, 7, 199.Jacob, S. S., Nair, N. B., & Balasubramanian, N. K. (1982). Toxicity of certain mosquito larvicides to the

larvivorous fishes Aplocheilus lineatus (cuv. & val.) and Macropodus cupanus (cuv. & val.). Environ.Pollut. A, 28, 7–13.

Jenkins, D. W. (1964). Pathogens, parasites and predators of medically important arthropods. Annotatedlist and bibliography. Bull. World Health Organ., 30(Suppl), 1–150.

Killeen, G. F., Fillinger, U., & Knols, B. G. (2002). Advantages of larval control for African malaria vec-tors: low mobility and behavioural responsiveness of immature mosquito stages allow high effectivecoverage. Malar. J., 1, 8.

Korobeinikov, A., & Maini, K. P. (2004). A Lyapunov function and global properties for SIR and SEIRepidemiological models with nonlinear incidence. Math. Biosci. Eng., 1, 57–60.

LaSalle, J. P. (1976). The stability of dynamical systems. In Regional conference series in applied mathe-matics. Philadelphia: SIAM.

Li, J. (2009). Simple stage-structured models for wild and transgenic mosquito populations. J. Differ. Equ.Appl., 15, 327–347.

Lou, Y., & Zhao, X.-Q. (2010). Periodic Ross–Macdonald model with diffusion and advection. Appl. Anal.,89, 1067–1089.

Ma, Z., Liu, J., & Li, J. (2003). Stability analysis for differential infectivity epidemic models. NonlinearAnal., Real World Appl., 4, 841–856.

Mohamed, A. A. (2003). Study of larvivorous fish for malaria vector control in Somalia, 2002. East.Mediterr. Health J., 9, 618–626.

Moore, S. M., Borer, E. T., & Hosseini, P. R. (2010). Predators indirectly control vector-borne disease:linking predator–prey and host–pathogen models. J. R. Soc. Interface, 42, 161–176.

Oliveira, N. M., & Hilker, F. M. (2010). Modelling disease introduction as biological control of invasivepredators to preserve endangered prey. Bull. Math. Biol., 72, 444–468.

Reiskind, M. H., & Lounibos, L. P. (2009). Effects of intraspecific larval competition on adult longevity inthe mosquitoes Aedes aegypti and Aedes albopictus. Med. Vet. Entomol., 23, 62–68.

Singh, N., Shukla, M. M., Mishra, A. K., Singh, M. P., Paliwal, J. C., & Dash, A. P. (2006). Malaria controlusing indoor residual spraying and larvivorous fish: a case study in Betul, central India. Trop. Med.Int. Health, 11, 1512–1520.

Smith, H. L. (1995). Monotone Dynamical Systems: An Introduction to the Theory of Competitive andCooperative Systems. Amer. Math. Soc. Math. Surveys and Monographs.

Smith, H. L., & Waltman, P. (1995). The theory of the chemostat. Cambridge: Cambridge University Press.Snow, R. W., Guerra, C. A., Noor, A. M., Myint, H. Y., & Hay, S. I. (2005). The global distribution of

clinical episodes of Plasmodium falciparum malaria. Nature, 434, 214–217.Thieme, H. (1993). Persistence under relaxed point-dissipativity (with application to an endemic model).

SIAM J. Math. Anal., 24, 407–435.van den Driessche, P., & Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilib-

ria for compartmental models of disease transmission. Math. Biosci., 180, 29–48.Walker, K., & Lynch, M. (2007). Contributions of Anopheles larval control to malaria suppression in

tropical Africa: review of achievements and potential. Med. Vet. Entomol., 21, 2–21.Wonham, M. J., de Camino-Beck, T., & Lewis, M. A. (2004). An epidemiological model for West Nile

Virus: Invasion analysis and control applications. Proc. R. Soc. Lond. B, Biol. Sci., 271, 501–507.Wonham, M. J., Lewis, M. A., Renclawowicz, J., & van den Driessche, P. (2006). Transmission assump-

tions generate conflicting predictions in host-vector disease models: a case study in West Nile virus.Ecol. Lett., 9, 706–725.

Y. Lou, X.-Q. Zhao

World Health Organization (2000). The world health report 1999, WHO.World Health Organization (2010). The world health report 2009, WHO.Zhang, X., Chen, L., & Neumann, A. U. (2000). The stage-structured predator–prey model and optimal

harvesting policy. Math. Biosci., 168, 201–210.Zhao, X.-Q. (2003). Dynamical systems in population biology. New York: Springer.Zhao, X.-Q., & Jing, Z. (1996). Global asymptotic behavior in some cooperative systems of functional-

differential equations. Can. Appl. Math. Q., 4, 421–444.