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Assessing the optimal virulence of malaria‐targeting mosquito pathogens: a mathematical study of engineered… Konrad, Bernhard P; Lindstrom, Michael; Gumpinger, Anja; Zhu, Jielin; Coombs, Daniel Jan 8, 2014

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Konrad et al. Malaria Journal 2014, 13:11http://www.malariajournal.com/content/13/1/11RESEARCH Open AccessAssessing the optimal virulence ofmalaria-targeting mosquito pathogens: amathematical study of engineeredMetarhizium anisopliaeBernhard P Konrad1, Michael Lindstrom1, Anja Gumpinger2, Jielin Zhu1 and Daniel Coombs1*AbstractBackground: Metarhizium anisopliae is a naturally occurring fungal pathogen of mosquitoes. Recently,Metarhiziumhas been engineered to act against malaria by directly killing the disease agent within mosquito vectors and alsoeffectively blocking onward transmission. It has been proposed that efforts should be made to minimize the virulenceof the fungal pathogen, in order to slow the development of resistant mosquitoes following an actual deployment.Results: Two mathematical models were developed and analysed to examine the efficacy of the fungal pathogen.It was found that, in many plausible scenarios, the best effects are achieved with a reduced or minimal pathogenvirulence, even if the likelihood of resistance to the fungus is negligible. The results for both models depend on theinterplay between two main effects: the ability of the fungus to reduce the mosquito population, and the ability offungus-infected mosquitoes to compete for resources with non-fungus-infected mosquitoes.Conclusions: The results indicate that there is no obvious choice of virulence for engineeredMetarhizium or similarpathogens, and that all available information regarding the population ecology of the combined mosquito-fungussystem should be carefully considered. The models provide a basic framework for examination of anti-malarialmosquito pathogens that should be extended and improved as new laboratory and field data become available.Keywords: Metarhizium anisopliae, Mathematical modelling, Fungal insecticide, Malaria control, Vector controlBackgroundThe major route of malaria transmission to humans isvia blood feeding of female Anophelesmosquitoes (princi-pally Anopheles gambiae and Anopheles funestus). There-fore, major efforts have been made to control mosquitopopulations in areas where malaria is prevalent. Whenfirst introduced, chemical insecticides were very efficientat reducing malaria prevalence in humans, although notwithout environmental damage. However, resistance hasbeen observed to develop rapidly to broadly-used insec-ticides and there is a lack of new chemical agents [1,2].For this reason, fungal entomopathogens have been under*Correspondence: coombs@math.ubc.ca1Institute of Applied Mathematics and Department of Mathematics, Universityof British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, CanadaFull list of author information is available at the end of the articleextensive investigation as alternatives for mosquito con-trol [3,4]. Diverse fungal pathogens of mosquitoes existin nature and additionally can be genetically modified togenerate desirable properties.The focus of this paper is the application of engineeredfungal pathogens of mosquitoes that can neutralize or killmalarial sporozoites in themosquito vector itself, prevent-ing onward transmission to humans. This is motivatedby the recent development of an engineered (Metarhiz-ium anisopliae) fungus strain [5].Metarhizium is a naturalparasite of mosquitoes that infects through direct con-tact with the insect cuticle, and therefore is appropriatefor control strategies based on local spraying indoors orbaited traps. Recombinant strains of Metarhizium havebeen designed to (i) block attachment of malarial sporo-zoites to salivary glands of the mosquito; and (ii) neu-tralize or kill Plasmodium falciparum directly within the© 2014 Konrad et al.; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the CreativeCommons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.Konrad et al. Malaria Journal 2014, 13:11 Page 2 of 10http://www.malariajournal.com/content/13/1/11mosquito hemolymph. Under laboratory conditions, theseengineered pathogens were found to substantially reducesporozoite counts in the salivary glands of mosquitoes,compared to both fungus-uninfected mosquitoes andwild-type Metarhizium [5]. This raises the possibility ofproducing a biological agent which targets the malariaparasite within the mosquito, and is thus able to dis-rupt the transmission cycle and reduce the prevalence ofmalaria in humans.Interestingly, because fungal pathogens such asMetarhizium don’t kill infected mosquitoes until theirlater life-stages, after the majority of mosquito reproduc-tion has occurred, it is believed that the selection pressurefor mosquito resistance is quite low and therefore resis-tance should develop slowly even under widespreaddeployment [6]. Mosquito resistance to fungal biopes-ticides of this type has not been reported to date.Nonetheless, the concern of emerging resistance leadsFang et al. [5] to argue against engineering Metarhiziumto kill mosquitoes faster. This argument stands in contrastto a strategy where fungal species are applied in con-junction with chemical pesticides to reduce the overallnumbers of mosquitoes (discussed in [3]).However, even in the absence of developing resistance,an interesting question remains about how to optimizeMetarhizium or similar agents in terms of virulenceagainst mosquitoes: should one expect a high virulenceagent to outperform an alternative low-virulence strain?Indeed, if a high-virulence mosquito pathogen strain hasworse performance than a low-virulence strain even inthe absence of resistance, the threat of resistance addi-tionally counts against it. (In this context, performanceis measured in terms of reducing human malaria preva-lence.) This paper presents simple mathematical modelsto investigate this question. It will be shown that theoptimal choice of virulence level is not obvious, anddepends quite sensitively on the details of the complexecological system. In certain circumstances, it is expectedto be preferable to apply a low-virulence agent whichpenetrates the mosquito population, and is thus able toreduce the prevalence of malaria parasites in mosquitoesand humans, even though the overall population level ofmosquitoes is not strongly impacted. Under other cir-cumstances, a high-virulence fungus that more effectivelyreduces the total mosquito population is expected to bepreferable.Mathematical modelling has been used in the studyof malaria and as a tool for evaluating possible con-trol strategies for over a century. The first mathemati-cal models for malaria transmission were pioneered byRoss in 1911 [7] and developed further by Macdonald[8] and Anderson and May [9]. More recent mathemat-ical models distinguish between exposed and infectioushumans and mosquitoes reflect the fact that humanscan become (temporarily) immune, treated or vaccinated,allow spatial heterogeneity, or include time-dependentparameters to account for environmental factors such asrainfall and humidity (see for example [10-13]). The cur-rent paper presents results from simple models for thisnew agent and potential improvements for future work aredescribed.MethodsTwo prototype mathematical models are proposed thattake into account the most fundamental properties of themalaria parasite, transmission between human host andmosquito vector, and the fungal mosquito pathogen. Toderive the models in a simple form, it is necessary to makea number of simplifying assumptions. These assump-tions could be relaxed in future versions of the model,at the expense of analytical and intuitive understandingof the model results. Some possibilities for this work aredescribed in the conclusions.Simplified model set-up and assumptions1. The total human population, H, is taken to beconstant. This is roughly equivalent to supposingthat the population dynamics of mosquitoes, malariaparasites and fungal pathogens equilibrate rapidlycompared to the human demographic timescale.The (time-dependent) fraction of humans that areable to infect mosquitoes with the malaria parasite isdenoted by h(t) and these patients leave the infectedgroup at rate ρ (for instance due to treatment,recovery or death). The human population is taken tobe homogeneous in all other ways and the incubationperiod in the human host is assumed to be negligiblecompared to the duration of infectiousness.2. The mosquito population is represented in threeparts: uninfected (susceptible), infected with themalaria parasite, and fungus-infected. These threepopulations are denoted by S(t), I(t), and F(t),respectively. It is necessary to make some simplifyingassumptions concerning the population dynamics ofmosquitoes, with and without the malaria parasite orthe fungal pathogen. A commonly simplified modelfor biological populations is that of logistic growth[14]. This model has been applied in several previousworks on malaria [12,15-17]. Here, it is assumed that,in the absence of the malaria parasite and the fungus,the mosquito population experiences logistic growthwith innate growth rate κ˜ and carrying capacity P˜.All mosquitoes also suffer a natural backgrounddeath with rate denoted by μ, which is assumed to beindependent of infection with the malaria parasite[18]. An alternative model of mosquito populationdynamics, that includes larval and adult stages, isdescribed below.Konrad et al. Malaria Journal 2014, 13:11 Page 3 of 10http://www.malariajournal.com/content/13/1/113. It is assumed that every mosquito has a human bitingrate of β , irrespective of its infection status, and thatif the human host is infectious, the bite alwaystransmits the malaria parasite to the mosquito. Onthe other hand if an infected mosquito bites anuninfected human, the human will developtransmissible malaria with probability γ .4. The fungal pathogen is constantly applied to theenvironment, and causes mosquitoes to transitionfrom the S and I classes to the F class continuously,at constant rate α.5. The fungal virulence (additional death of mosquitosdue to fungal infection) is denoted by σ . To explicitlycompare the natural mosquito death rate μ with thefungal virulence σ , one may rewrite the classicallogistic growth equation dS/dt = κ˜S(1 − S/P˜) in theform presented below, where κ = κ˜ + μ andP = P˜κ/(κ − μ).6. The fungus is assumed to be perfectly, permanentlyand immediately effective at blocking malariaparasite transmission to and from fungus-infectedmosquitoes.Under these assumptions, the following system of ordi-nary differential equations can be put forward (t denotestime):dhdt = βγIH (1 − h) − ρh,dSdt = κ (S + I + F)(1 − S + I + FP)− βSh − (μ + α)S,dIdt = βSh − (μ + α)I,dFdt = α(S + I) − (μ + σ)F .(1)A model schematic is given in Figure 1a.Life-stage-structured mosquito model and assumptionsRecent field studies strongly indicate that mosquito pop-ulations are controlled via density-dependent regulationat the larval stage of development [19,20]. The simplifiedmodel above was derived under the assumption that theseeffects can be captured via a logistic model for the adultmosquito population, since the larval population is notmodelled explicitly. This simplification allows for fairlyclean analytical results, but the biological system mayhave been oversimplified. Additionally, vertical transmis-sion is a feature of certain fungal symbionts of insects [21].Therefore, a possible improvement of the engineered fun-gus would be a vertically transmissible variant, that couldbe passed from mosquitoes to their offspring. To accom-modate the additional realism of larval competition forHumansββγμ+σκμμκκMosquitoesSIFh1-hραsusceptiblemalaria-infectedfungus-infectedsusceptibleinfectedHumansββγμ+σμμκcMosquitoesSIFh1-hραsusceptibleinfectedLarvaeLFLSααξκcmmκL(1−ξ)κcabFigure 1Model schematics. a. Simplified model. The fraction ofinfected humans in the population is given by h. Infection ofsusceptible mosquitoes S occurs at mass-action rate β whileremoval/recovery occurs at a rate ρ . Infected mosquitoes I infectsusceptible humans with probability γ , given an encounter. Malariainfection does not significantly alter the mosquito death rate μ, butfungus infection, occurring with rate α reflecting the intensity offungal application, increases the death rate by σ . Since neithermalaria nor the fungus is transmitted vertically in the simplifiedmodel all new-born mosquitoes (indicated by κ) are susceptible.b. Life-stage structured model. In this model mosquitoes producelarvae at rate κc . Larvae may be fungal carriers (LF ) or fungus-free (LS)and undergo density-dependent competition (see text) with intensityparameter κL . Larvae mature to produce adult mosquitoes at ratem.The parameter ξ determines the degree of vertical transmissibility ofthe fungal pathogen.resources, and to consider vertical transmission, an alter-nativemodel is now put forward. Thismodel is built underthe following set of additional assumptions:1. Adult mosquitoes produce new larvae at a constantrate κc, irrespective of infections with fungus ormalaria parasites. The offspring of fungal-infectedmosquitoes are themselves infected with probabilityξ , representing the probability of verticaltransmission of the fungal pathogen. Thepopulations of fungus-uninfected andfungus-infected larvae are denoted by LS(t) and LF(t)Konrad et al. Malaria Journal 2014, 13:11 Page 4 of 10http://www.malariajournal.com/content/13/1/11respectively. In the absence of information aboutlarval infectibility, it is assumed that larvae are notdirectly infected by the fungus (although they may beinfected via vertical transmission), and that thefungus has no detrimental effects in the larval stage.2. Larvae compete for resources in a density-dependentmanner, with the intensity of competitiondetermined by the parameter κL.3. Larvae mature into adult mosquitoes at rate m. Incontrast to the simplified model, adult mosquitoes donot directly compete for resources.Under these additional assumptions, a new life-stage-structured model can be put forward:dhdt = βγ I(1 − h)H − ρhdLSdt = κc(S + I + (1 − ξ)F) − κL (LS + LF) LS − mLSdLFdt = κcξF − κL (LS + LF) LF − mLFdSdt = mLS − βSh − (μ + α)SdIdt = βSh − (μ + α)IdFdt = mLF + α(S + I) − (μ + σ)F(2)A model schematic is given in Figure 1b.Results and discussionThe goal is now to study the level of malaria in humans(h) as a function of the fungal parameters: the sprayingrate α, fungal virulence σ and fungal vertical transmissi-bility ξ . For the simplified model, analytical results will bepresented and applied to find the optimal values of α andσ in terms of reducing malaria in humans. For the life-stage-structuredmodel, the situation is more complex andresults will be shown only for particular numerical valuesof the parameters. In both models, it is found that appli-cation of a highly virulent fungus as biopesticide may notbe the best strategy.The simplified model with no fungusIf there is no fungus present in the system, the simplifiedmodel simplifies to the following:dhdt = βγIH (1 − h) − ρh,dSdt = κ (S + I)(1 − S + IP)− βSh − μS,dIdt = βSh − μI.(3)This system has three relevant equilibria: (1) the triv-ial equilibrium (h, S, I) = (0, 0, 0) (no mosquitoes, nomalaria); (2) the malaria-free equilibrium (h, S, I) =(0, (1 − μκ)P, 0) (mosquitoes but no malaria) and (3) theendemic equilibrium(h, S, I)=(β2γP(κ−μ)−Hκμρβ(Hκρ+βγP(κ−μ)) ,μ(Hκρ+βγP(κ−μ))βγ κ(β+μ) ,β2γP(κ − μ) − Hκμρβγ κ(β + μ)).(4)From the malaria-free equilibrium it is observed thatκ > μ is the basic requirement for there to be mos-quitoes in the system. This corresponds to the basicmosquito growth rate κ being large enough to outweighmosquito death (and with no mosquitoes there can be nomalaria).The key quantity determining whether malaria will existin the system is the basic reproduction number for theno-fungus system, R(NF)0 = β√γ Sm/(ρμH), where Sm =P(1 − μκ) is the mosquito population at the malaria-free steady state. This quantity can be calculated usingthe next-generation method [22]. The square root occursin this formula because there are two steps per roundsof transmission, mosquito to human to mosquito, andthe next-generation R0 is, by definition, a per-step quan-tity. The parameter β is not included in the square rootbecause it determines the transmissibility human-vectorand vector-human. It can also be shown that if R(NF)0 < 1then the malaria-free equilibrium is locally stable and theendemic equilibrium does not exist, while if R(NF)0 > 1then the endemic equilibrium exists and is locally stable.Equilibrium analysis of the simplified modelThe simplified model including the fungus (1) supportsthree distinctive equilibria, which can be defined as thetrivial,malaria-free and endemic equilibria. The followingresults on the local stability of these equilibria can thenbe proven (see Additional file 1 for the proofs, which arelengthy but straightforward):1. The trivial equilibrium (h, S, I, F) = (0, 0, 0, 0) islocally asymptotically stable if and only ifκ <(μ+α)(μ+σ)μ+α+σ . This condition is analogous to thecondition κ > μ for the model with no fungus,described above.2. The malaria-free equilibrium(hm, Sm, Im, Fm)=(0, P(μ+σ)μ+α+σ(1− (μ+α)(μ+σ)κ(μ+α+σ)),0, Pαμ+α+σ(1− (μ+α)(μ+σ)κ(μ+α+σ)))Konrad et al. Malaria Journal 2014, 13:11 Page 5 of 10http://www.malariajournal.com/content/13/1/11exists in the positive plane and is locallyasymptotically stable if and only if both of thefollowing conditions are fulfilled:κ >(μ + α)(μ + σ)μ + α + σ and R0 = β√γ Smρ(μ + α)H < 1.The first condition corresponds to the survival of themosquito population (exactly as for the trivialequilibrium). The second condition states that thebasic reproductive number for malaria, R0, must besubcritical.3. The endemic equilibrium(he, Se, Ie, Fe)=(R20−1R20+β/(μ+α),(1− βμ+α+βR20−1R20)×Sm, βμ + α + βR20 − 1R20Sm, Fm)exists in the positive plane and is locallyasymptotically stable if and only if both of thefollowing conditions are fulfilled:κ >(μ + α)(μ + σ)μ + α + σ and R0 = β√γ Smρ(μ + α)H > 1.By comparing R0 for the simplified model to the equiva-lent no-fungus quantity (R(NF)0 ), it is possible to see the roleof the spraying rate α in reducing the reproductive num-ber of the malaria parasite. The virulence of the fungus(parameter σ ) does not affect R0, because it is assumedthat mosquitoes that are infected with the fungus are notable to transmit the malaria parasite.Optimizing the virulence and application of the fungalpathogen in the simplified modelThe two parameters of the model that are in principleunder control in a real application are α (the applica-tion/spraying rate) and σ (the parameter controlling fun-gal virulence). Our goal is to choose α and σ such that theendemic malaria prevalence in humanshe = he(α, σ) = R20(α, σ) − 1R20(α, σ) + β/(μ + α),is minimized. The first and most intuitive result is thathe is a monotonic decreasing function of α, which meansthat the more the mosquitoes are exposed to the fungus,the lower the endemic malaria prevalence is in humans.Increasing the fungus application rate is therefore alwaysbeneficial. This result is numerically illustrated in Figure 2.It can further be shown (see Additional file 1) that the totalnumber of mosquitoes (S + I + F) also decreases when α0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.20.40.60.81Relative malaria prevalence in humansFungus exposure rate α0.750.80.850.90.951Relative total number of mosquitoeshumansmosquitoesFigure 2 Prevalence of malaria in humans and total number ofmosquitoes for varying fungus deployment rates. The steadystate human malaria prevalence and the total mosquito population,both relative to baseline, are plotted against the fungus exposureintensity α, with fixed fungal pathogen virulence σ = 0.1/day. Asexpected, quantities decrease when α increases. Note that, once α islarge enough so that R0 < 1 and malaria is eradicated, he remainsconstant at 0. Full details of all other chosen parameter values aregiven in Additional file 2.increases, unless the fungal virulence σ is zero, in whichcase the total number of mosquitoes is independent of α.There are two possible effects of continual fungus appli-cation that have to be considered in answering the ques-tion of how to find the optimal fungal virulence σ :1. The biopesticide effect: Increasing the fungalvirulence will lead to a decrease in the total numberof mosquitoes (S + I + F). This can be confirmedmathematically for the presented models (seeAdditional file 1).2. The competition effect: Fungus-infected mosquitoesF do not contribute directly to the transmission ofthe malaria parasite to humans. However, they docompete for resources with all thenon-fungus-infected mosquitoes. Therefore, if thefungal virulence is low, then the population will havea higher fraction of non-malaria-carrying mosquitoesand the force of infection of the malaria parasite onhumans could plausibly be reduced.Hence there is a tradeoff between the two mechanismsabove: the total number of mosquitoes could be mini-mized by a high-virulence fungus (biopesticide effect), butwhat is critical for the prevalence in humans is the totalnumber of mosquitoes infected with the malaria pathogen- and this might be minimized (via the competition effect)by a low-virulence fungus.In order to determine the balance of these two effects,one must carefully analyse the mathematical model. It isKonrad et al. Malaria Journal 2014, 13:11 Page 6 of 10http://www.malariajournal.com/content/13/1/11then possible to establish a critical threshold for the rela-tion between the mosquito innate growth rate κ and thenatural mosquito death rate μ that makes one or the otherargument stronger. Specifically, it can be shown that themalaria prevalence in the human population he(α, σ) andthe total number of infected mosquitoes Ie(α, σ) are bothmaximized ifσ = σ ∗ ≡ (κ − 2μ)2 − κ/(μ + α) .This result indicates how to best design the fungus: Toavoid the worst case σ = σ ∗, the virulence σ must bechosen far away from σ ∗.Note that while σ = σ ∗ maximizes the malaria preva-lence in humans as well as the total number of malaria-infected mosquitoes, it does not maximize or minimizethe total number of mosquitoes. This result reveals thecomplexity of the tradeoff between the biopesticide andthe competition effect, and indicates the importance of agood understanding of the mosquito population as well asthe fungus-mosquito interaction, in designing the optimalpathogen for deployment.Optimal virulence depends on backgroundmosquitogrowth rateContinuing the analysis of the previous section, two inter-esting cases can be distinguished: (i) If σ ∗ is negative,then σ is necessarily greater than σ ∗ and hence vir-ulence should always be maximized to reduce malariaprevalence in humans (indicating that the biopesticideeffect is stronger than the competition effect); (ii) If σ ∗is positive, however, then a very large σ or a very lowσ is desirable, but not an intermediate value. In the sec-ond case, this shows that the choice is either to use thefungal pathogen as a biopesticide (high virulence) or adistributed anti-Plasmodium agent (low virulence) in themosquito population.The choice depends on the relation between themosquito growth rate κ , the mosquito death rate μ andthe fungal exposure rate α. If the growth rate κ is smallcompared to the mosquito death rate μ (specifically, ifκ < 2μ) then σ ∗ < 0 for any value of α, and hence alarger fungal virulence σ is always desirable. This meansthat the fungus should be used as a biopesticide for slow-growing mosquito populations. On the other hand, if themosquito population is relatively fast-growing (κ > 2μ),then σ ∗ is negative for 0 < α < (κ − 2μ)/2, while σ ∗ ispositive for α > (κ − 2μ)/2. Parameter estimation (seeAdditional file 2) yields that κ ≈ 5μ, implying that in mostreal mosquito populations, the latter scenario is muchmore realistic. Hence, if the fungus can be applied at asufficient rate, then minimizing the fungal virulence σhas the most beneficial effect of malaria prevalence inhumans.Figure 3 illustrates these principles for particular param-eter choices. The human malaria prevalence he(α, σ) isplotted, relative to the baseline prevalence when no fun-gus is applied, and hence indicating how malaria man-agement can be optimized. The left column shows theslow-growing result (κ < 2μ) where the optimal strat-egy is to use the fungal pathogen as a biopesticide. Theright column shows the more surprising result where, ifthe fungus deployment rate is sufficient, it is preferable toselect a low-virulence fungus strain. In all cases, verifyingthe first result of this section, it can be seen that a higherfungus-exposure rate α is always desirable.Equilibrium analysis of the life-stage-structured modelThe investigation of the simplified model has shown thatdepending on the ecology of the mosquito population, itcan be expected that there are circumstances where theoptimal virulence of the fungal pathogen can be zero.However, competition between mosquitoes in the sim-plified model takes place in the adult stage, but recentevidence indicates that competition occurs at the larvalstage. To examine this effect it is necessary to use themorecomplex, life-stage-structured model (2), which includesthe larval stage. In this section the effects of verticaltransmission of fungus will also be examined.Life-stage-structured model without vertical transmissionof fungusBy setting ξ = 0 in the second model, it is possible toinvestigate the model without vertical transmission. Sim-ilar to the simplified model, it can be shown that thismodel supports three easily explained equilibria, corre-sponding to (i) the trivial equilibrium with no mosquitoesand no malaria, (ii) a malaria-free equilibrium withmosquitoes but no malaria, and (iii) an endemic equi-librium state. Provided that the malaria-free equilibriumwith mosquitoes exists, it can be shown that the endemicequilibrium state exists and is stable given the followingreproductive number condition:R0 =√β2γm2((κc − μ)(μ + α + σ) − ασ)κLρ(μ + α)3(μ + σ)HR0 = β√γ S0ρ(μ + α)H > 1,whereS0 = m2 ((κc − μ)(μ + α + σ) − ασ)κL(μ + α)2(μ + σ)is the equilibrium number of mosquitoes in the popula-tion in the absence of the malaria parasite. Under thiscondition, the fraction of the human population infectedwith malaria, at the endemic equilibrium, is given by theKonrad et al. Malaria Journal 2014, 13:11 Page 7 of 10http://www.malariajournal.com/content/13/1/11Fungal virulenceExposurerateα  0 0.25 0.5 0.7500.10.20.30.40.500.20.40.60.810 0.15 0.3 0.45 0.6 0.7500.20.40.60.8Fungal virulenceRelprevinhumansFungal virulenceExposurerateα  σ ∗0 0.25 0.5 0.7500.250.50.75100.20.40.60.810.50.550.60.650 0.15 σ* 0.3 0.45 0.6 0.75Fungal virulenceRelprevinhumansFigure 3Malaria prevalence in humans varying over fungal pathogen virulence and deployment rate. The heat maps (top row) indicate theendemic malaria prevalence in humans he , relative to baseline where no fungus is applied, for the given parameters, the graphs (bottom row) showa one-dimensional projection of the heat maps, for a fixed fungal application (spraying) rate. Two distinct cases can be distinguished: (i) Ifκ − 2μ < 0 (left column) then human malaria prevalence is a decreasing function of fungal virulence σ and deployment rate α. (ii) If κ − 2μ > 0(right column), the curve describing the worst-case fungal virulence σ ∗ is superimposed, indicating a non-monotonic relationship between humanmalaria prevalence and σ . Top row: The dark blue region indicates R0 < 1 and hence he = 0. Bottom row: one-dimensional slice through the heatmap above for a fixed value of α = 0.1/day (left) and α = 0.5/day (right), as indicated by the black line in the heat map above. Here, μ = 0.1/day,while κ = 0.18/day (left) and κ = 0.48/day (right). All other parameter values are as given in Additional file 2.same function of R0 as before (reflecting the very similarstructures of the two models),h(σ ) = R20 − 1R20 + β/(μ + α).The derivative of this function with respect to σ is foundto have no zeroes and is always negative for R0 > 1. Thisindicates that the human prevalence of malaria is a strictlydecreasing function of σ and therefore the best strategyfor the life-stage-structured model with no vertical trans-mission of fungus is to increase the fungal virulence asmuch as possible. This result (illustrated in Figure 4, topleft panel where ξ = 0) stands in contrast to the previoussection wheremore nuanced conclusions were drawn, andis a consequence of the fact that the competition effect isoccurring at the larval stage, while the biopesticide effectis occurring at the adult stage. Since it is adult mosquitoes,not larvae, that act as vectors for the malaria parasite, onlythe biopesticide effect can reducemalaria incidence in thisversion of the model.Life-stage-structured model with vertical transmission offungusVertical transmission is a feature of fungal symbionts ofinsects [21] and it seems natural to consider this in thecontext of the model. The life-stage-structured model,where the parameter ξ is the vertical transmission frac-tion, reflects this possibility and allows the previous anal-ysis to be repeated. In this case, the analytical expressionsfor the steady states of the full model can be obtainedusing a computer algebra system, such as Mathematica,but are too long to usefully give here. However, the basicanalytical results remain: there are three potential equi-libria of the model, but only one has mosquitoes andmalaria. Further, although the model is resistant to analyt-ical exploration, it is easy to choose parameters and worknumerically.Figure 4 shows the equilibrium fraction of infectedhumans as a function of the fungal virulence σ andapplication rate α, across a range of possible verticaltransmission probabilities ξ . It is observed that if verti-cal transmission is unlikely then the benefit of the fungusis maximized by high virulence, but when vertical trans-mission becomes more likely, an intermediate or lowvirulence fungus would be favourable. In fact, in somecases a highly vertically transmissible fungus (ξ ≥ 0.85 inFigure 4) is predicted to be able to eliminate malaria alto-gether, provided the virulence is low, but not too low. In allcases it should be noted that, as is to be expected, increas-ing the fungal application rate α is always beneficial.Konrad et al. Malaria Journal 2014, 13:11 Page 8 of 10http://www.malariajournal.com/content/13/1/110.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Fungal Virulence σSteadystatehζ=00.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Fungal Virulence σSteadystatehζ=0.250.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Fungal Virulence σSteadystatehζ=0.50.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Fungal Virulence σSteadystatehζ=0.750.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Fungal Virulence σSteadystatehζ=0.80.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Fungal Virulence σSteadystatehζ=0.850.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Fungal Virulence σSteadystatehζ=0.90.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Fungal Virulence σSteadystatehζ=0.950.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Fungal Virulence σSteadystatehζ=1.0Figure 4Malaria prevalence in humans varying over vertical transmission, deployment rate and fungal virulence. Each subfigure showsthe endemic malaria prevalence in humans, relative to the no-fungus baseline, plotted against the fungal virulence σ , for five different levels of thefungal application (spraying) rate α. The different values of α in each subfigure are {0.05, 0.1, 0.15, 0.2, 0.25} days−1, corresponding to curves fromtop to bottom, and the length of the dashing increases with α. Each subfigure represents a different value of the vertical transmissibility ξ fromξ = 0 (no vertical transmission) to ξ = 1 (perfect vertical transmission). All other parameter values are as given in Additional file 2.These results indicate how the biopesticide effect ofhigh virulence, acting at the adult stage, can be dominatedby the competition effect at the larval stage, provided theadult and larval stages are coupled sufficiently strongly byvertical transmission of the fungus. In this case, highly vir-ulent fungus are found to be less effective because theyprevent the fungus-infected mosquitoes from generatingfungus-carrying larvae. This reduces competitive inhibi-tion of fungus-uninfected larvae, thus reducing the overalleffectiveness of the fungus in preventing malaria parasiteinfection of mosquitoes, and so ultimately increasing theprevalence of malaria in humans.ConclusionsIn this paper, mathematical models have been employedto investigate the effectiveness of a fungal pathogen thatblocks malaria transmission in mosquitoes to reducemalaria prevalence in humans. Unsurprisingly, all mod-els indicate that malaria prevalence in humans could bereduced substantially by such a counter measure, andthat the mosquito exposure rate to the fungus shouldbe maximized to reduce malaria prevalence. However,the main interest of the results is to demonstrate thatthe optimal design of an agent that can simultaneouslykill mosquitoes, and malaria parasites within mosquitoesdepends quite sensitively on the details of a complexecological system.The first model, in which competition betweenmosquitoes occurred at the adult level, showed that infast growing mosquito populations the fungal pathogenshould be engineered to have low virulence. This resultis independent of the possibility of mosquito resistancedeveloping to the fungal pathogen and can be under-stood in very simple terms: fungus-infected mosquitoesKonrad et al. Malaria Journal 2014, 13:11 Page 9 of 10http://www.malariajournal.com/content/13/1/11do not directly contribute to the malaria epidemic, butcompetitively hamper the introduction/survival of newsusceptible mosquitoes. If mosquito resistance to fun-gal biopesticides arises in the field, this argument wouldadditionally be strengthened.In the second model, competition occurs at the larvalstage of development. This model predicted that, in theabsence of vertical transmission of the fungal pathogen, ahighly virulent (biopesticidal) fungus would be desirable.However, the addition of reliable vertical transmission tothis model significantly alters the predictions. If the fun-gal pathogen could be engineered in this way, then a highvirulence would be highly detrimental to its efficacy as ananti-malarial strategy.The key observation that should be drawn is that find-ing the optimal properties of agents that can reduce themalaria parasite incidence in mosquitoes is not an easytask, since several direct and indirect effects need to beparameterized and balanced, leading to conclusions thatdepend on the details of the mosquito population, fungalpathogen, and environment.In order to obtain a straightforward analytical treat-ment, several simplifying assumptions were madethroughout. It was assumed that the fungus could com-pletely block onwardmalaria transmission.Modifying thisassumption in the simplified model leads to an increasedfavourability of virulence, but the authors believe thatthis issue will best be resolved in a future model wherethe time dependence of the fungus-mosquito interac-tion is explicitly analysed. Also, by assuming a constantmass-action biting coefficient, human reactions to avoidmosquito bites (bed nets or indoor residual spraying)were neglected. This was done to study the pure effectof the fungus interaction, but is likely not a realisticrepresentation of a real-life setting. Furthermore, thehuman population was modelled as a homogeneouspopulation, neglecting for example co-infections, agestructure, and previous malaria history. All of the abovehave an important impact on the malaria epidemic:co-infections increase the severity of each disease, chil-dren are much more vulnerable to malaria, and previousmalaria infections can lead to temporal immunity. Theincubation period of the malaria parasite and of the fun-gus in mosquitos were also neglected. This simplificationparticularly affects the mosquito population, where theincubation period is of about the same order as the lifeexpectancy. Future work will include extending the cur-rent model to add more details of mosquito and malariaparasite life history in an age- and life-stage-dependentmodel of the mosquito.Horizontal fungal transfer betweenmosquitoes was alsonot considered here. This effect has been observed fora number of fungal symbionts of insects and could playa role similar to vertical transmission in enhancing theeffectiveness of the fungal pathogen. Horizontal trans-fer (possibly mediated through the environment) betweenmosquitoes would also be expected to increase the effec-tiveness of fungal spread, and reduce the necessary levelof fungal application/spraying in the environment, whichmight otherwise have to be very intense. Further, inassuming a fairly simple logistic growth model, seasonaleffects such as rainfall and humidity were neglected.These effects would lead to temporal variations in themosquito population growth rate and carrying capacity, aswell as unknown possible effects on the fungal pathogen.A final topic for possible future work is to consider spa-tial heterogeneities such as breeding sites and humanhabitat.Despite these limitations, the models are useful in defin-ing an argument for a minimal virulence of the anti-malarial fungal pathogens. In future work, as field studiesof Metarhizium or similar agents are completed, theparameterization of the model can be improved, and newmodels that allow insights into the potential of a large-scale deployment of such controls for malaria and othermosquito-borne diseases can be developed.Additional filesAdditional file 1: Contains full details of mathematical proofs forsome statements made in the main text.Additional file 2: Includes details of parameter estimation used ingenerating the figures.Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsEstablished the study: BPK, ML, JZ, DC. Developed and analysed the models:BPK, ML, JZ, AG, DC. Wrote the paper: BPK, DC. All authors have read andapproved the final version.AcknowledgementsThis work was supported by the Natural Science and Engineering ResearchCouncil of Canada and the Pacific Institute for Mathematical Sciences throughthe International Graduate Training Centre in Mathematical Biology. Theauthors thank Fred Brauer, Malte Peter and Raymond St. Leger for helpfuldiscussions.Author details1Institute of Applied Mathematics and Department of Mathematics, Universityof British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada.2TU München, Fakultät für Mathematik, Boltzmannstraße 3, 85748 Garching(b. München), Germany.Received: 11 June 2013 Accepted: 31 December 2013Published: 8 January 2014References1. Enayati A, Hemingway J:Malaria management: past, present, andfuture. Annu Rev Entomol 2010, 55:569–591.2. 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Princeton NJ, USA: PrincetonUniversity Press; 2012.doi:10.1186/1475-2875-13-11Cite this article as: Konrad et al.: Assessing the optimal virulence ofmalaria-targeting mosquito pathogens: a mathematical study of engineeredMetarhizium anisopliae.Malaria Journal 2014 13:11.Submit your next manuscript to BioMed Centraland take full advantage of: • Convenient online submission• Thorough peer review• No space constraints or color figure charges• Immediate publication on acceptance• Inclusion in PubMed, CAS, Scopus and Google Scholar• Research which is freely available for redistributionSubmit your manuscript at www.biomedcentral.com/submit

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