The Game-Theoretical Model of Using Insecticide-Treated Bed-Nets to Fight Malaria

Transcription

1 Te Game-Teoretical Model of Using Insecticide-Treated Bed-Nets to Figt Malaria By: Mark Broom, Jan Ryctář, Tracy Spears-Gill Broom, M., Ryctář, J., & Spears-Gill, T. (6). Te Game-Teoretical Model of Using Insecticide-Treated Bed-Nets to Figt Malaria. Applied Matematics, 7, Made aailable courtesy of Scientific Researc Publising: ttp://dx.doi.org/.46/am Copyrigt 6 by autors and Scientific Researc Publising Inc. Tis work is licensed under te Creatie Commons Attribution International License (CC BY). ttp://creatiecommons.org/licenses/by/4./ Abstract: Malaria infection is a major problem in many countries. Te use of te Insecticide-Treated Bed- Nets (ITNs) as been sown to significantly reduce te number of malaria infections; oweer, te effectieness is often jeopardized by improper andling or uman beaior suc as inconsistent usage. In tis paper, we present a game-teoretical model for ITN usage in communities wit malaria infections. We sow tat it is in te indiidual s self-interest to use te ITNs as long as te malaria is present in te community. Suc an optimal ITN usage will significantly decrease te malaria prealence and under some conditions may een lead to complete eradication of te disease. Keywords: Game Teory Malaria Preention Optimal Strategy Epidemic Modelling SIS Model Article: ***Note: Full text of article below

2 Applied Matematics, 6, 7, Publised Online May 6 in SciRes. ttp:// ttp://dx.doi.org/.46/am Te Game-Teoretical Model of Using Insecticide-Treated Bed-Nets to Figt Malaria Mark Broom, Jan Ryctář, Tracy Spears-Gill Department of Matematics, City Uniersity London, Nortampton Square, London, UK Department of Matematics and Statistics, Te Uniersity of Nort Carolina at Greensboro, Greensboro, USA Department of Biostatistics, Uniersity of Nort Carolina at Capel Hill, Capel Hill, USA Receied April 6; accepted May 6; publised 6 May 6 Copyrigt 6 by autors and Scientific Researc Publising Inc. Tis work is licensed under te Creatie Commons Attribution International License (CC BY). ttp://creatiecommons.org/licenses/by/4./ Abstract Malaria infection is a major problem in many countries. Te use of te Insecticide-Treated Bed- Nets (ITNs) as been sown to significantly reduce te number of malaria infections; oweer, te effectieness is often jeopardized by improper andling or uman beaior suc as inconsistent usage. In tis paper, we present a game-teoretical model for ITN usage in communities wit malaria infections. We sow tat it is in te indiidual s self interest to use te ITNs as long as te malaria is present in te community. Suc an optimal ITN usage will significantly decrease te malaria prealence and under some conditions may een lead to complete eradication of te disease. Keywords Game Teory, Malaria Preention, Optimal Strategy, Epidemic Modelling, SIS Model. Introduction Malaria is a ector born disease tat can be transmitted to people troug mosquito bites, prealent in tropical and subtropical regions of te world. Te World Healt Organisation estimates tat eery year nearly one million people die as a result of malaria []. A lot of effort as been put into treatment and preention strategies, and an ambitious goal of organisations suc as te World Healt Organisation and te Bill and Melinda Gates foundation is te complete eliation of malaria. One of te key metods of malaria preention is te use of Insecticide-Treated Bed-Nets (ITNs) tat kill and How to cite tis paper: Broom, M., Ryctář, J. and Spears-Gill, T. (6) Te Game-Teoretical Model of Using Insecticide- Treated Bed-Nets to Figt Malaria. Applied Matematics, 7, ttp://dx.doi.org/.46/am

3 also repel mosquitoes []. [] found tat te use of ITNs can reduce malaria cases by 5% and deat in cildren by %. Te use of ITNs was demonstrated to be ery cost-effectie, see e.g. [4]-[6]. Tis is important gien bot te scale of te problem (about million people are at a ery ig risk [7]), and te fact tat most of te areas of ig malaria prealence are relatiely poor. Howeer, uman beaior suc as inconsistent usage due to ot weater diises te effectieness of te ITNs. Te impact of ITN usage on te spread of te malaria infection as been recently studied in a series of matematical models suc as [8]-[]. In particular, [] deeloped an ODE model for te effects of ITN on te malaria transmission dynamics and called for a better understanding of te impact of uman beaior on te usage. Game teory is a well suited tool to study uman beaior in a quantitatie way []. Starting by [], game teory as been long used in biology [], and [4] used it to study accination decisions in situation were indiiduals face a potentially costly preentie action (suc as to use an ITN) tat can reduce a risk of contracting te disease. Game teory as been applied to accination against major public ealt treats [5]-[] and it is particularly useful because te outcome of te indiidual decision (to accinate or to use an ITN) depends on te decisions of oter members of te population because te oerall accination or ITN usage yields te disease prealence in te community []. In particular, game teoretical models can be used to understand wy in many cases disease eradication can be ery ard [4] een wen te cost of preentie action is ery low []. Tere is often a great incentie to use preentatie measures wen a disease is at ig leels, but wen brougt below a certain leel te incremental benefit of taking te presentatie action declines, and so indiiduals ae less incentie to use it. Tus preention can sometimes (but not always) become ineffectie before total eradication occurs. In tis paper we adapt a non-game teoretical model of ITN usage [] to study te impact of indiidual decisions on te effectieness of ITN usage.. Te Model.. Model of Malaria Transmission We will use te basic model for te transmission dynamics of malaria infection as presented in []. Te ost population (umans) is diided into two compartments (susceptible and infectious), wic are denoted by S, I, respectiely. A total ost population is gien by N = S + I. Te ector population (mosquitoes) is also diided into two compartments (susceptible and infectious) wic are denoted S, I, respectiely. Te total ector population is gien by N = S + I. Wen a mosquito bites a uman, te susceptible uman is infected by an infected mosquito wit probability p and a susceptible mosquito is infected by an infected uman wit probability p. Hosts die wit natural mortality rates being µ and can also die because of malaria wit rate δ. Hosts recoer wit rate γ. Vectors can die wit rate µ b = µ + µ b () were µ is te natural mortality rate of osts and µ b is te mortality rate due to ITN use in te population. Here, b [,] denotes te aerage ITN usage in te population. Eac indiidual cooses its own indiidual ITN usage b {,}. We assume tat all indiiduals are born as susceptible (wit recruitment rates Λ and Λ respectiely) and no infected indiiduals come from outside. All parameters and teir alues are summarized in Table. Let λ ( bb, ) be te force of infection for te focal susceptible ost. [] gie te formula β N I β I λ ( bb, ) = p = p () N b N b N b were p is te transmission probability per bite from infectious mosquitoes to umans, number of bites per mosquito per unit of time, N N β b is te aerage is te aerage number of mosquitoes per ost, and tus N β b is te aerage number of bites a susceptible ost gets per unit time, and finally N b I N is te 85

4 Table. Parameters, teir description and alue. Te cost of malaria infection aries by te country an te used treatment, see for example [8]-[]. Oer te life of te ITN, te total cost of infections is more tan te cost of te ITN. Moreoer, ITNs are often subsidized, so te actual cost of ITN to te indiiduals is only around $ per ITN [6]. Description Value Reference µ Natural mortality for osts in, [] [4] δ Diseased induced mortality in osts in, 4. 4 [4] γ Hosts recoery rate /4 [4] [5] µ Natural mortality rate for ectors 8, [] [4] Λ Recruitment rate in osts in 6 µ, µ [8] [] 4 Λ Recruitment rate in ectors in µ, 5 [9] [] β Minimal possible contact rate [] β Maximal possible contact rate in [.9, ] [] [5] µ Net related ector mortality rate µ [] p Prob. of ector to ost transmission [] p Prob. of ost to ector transmission [] c MAL Cost of malaria infection >$5 per case See caption c ITN Cost of net usage <$ per - 5 years [6] [7] b Indiidual ITN usage in {, } b Aerage ITN usage in [, ] proportion of infectious mosquitoes. [] considers wit β =. and tection). Also, te aerage force of te infection is and tus β = β ( β β ) β = wic yields β =. and b b () λ = p λ ( bb) β I N β, = λ. β ( b ) In te state wen malaria reaces equilibrium, [] ealuates λ ( b ) as a solution to were λ β = (i.e. ITNs proide complete pro- a λ b + b b + c =, (6) ( ) a = µ b Λ p β b + µ b (7) b = µ b Λ Lp β b + µ b Λ L p p β b Λ µ + δ (8) ( µ ) c = b L Λ R (9) (4) (5) 854

5 L = µ + γ + δ () ( b ) ( b ) β Λµ Λ pp = µ L R In [] it is sown tat tere are potentially two stable steady states of infections and if te malaria was originally common, te stable states yields () b + b 4ac, if b 4ac > λ ( b ) = a () ; oterwise. λ b eentually becomes (quite abruptly as seen in Figure (a)) as b increases. Tis is caused by te fact tat for large b, te disease can be eradicated according to te model presented in []. Figure sows alues of λ ( b ) for specific parameter alues. Note tat.. Game-Teoretical Model Let (, ) E bb be te payoff for a focal indiidual using ( b = ) or not using ( b = ) te ITN in a population b,. We sall find te ITN usage equilibrium usage leel b s satisfying were te aerage use of ITNs is [ ] E, b E, b if and only if b b. () Suc a usage leel is stable, because if > s E, b < E, b and tus indiiduals are better off not using ITNs, and te aerage population usage leel will tus go down (proided te indiiduals acts rationally). On te oter and, if b < b s, ten E(, b) > E(, b ) and tus indiiduals are better off if tey use te ITN and te population usage will tus go up. Here we assume tat wile indiiduals cannot accurately assess te oerall state of te population (in particular te prealence of malaria) instantaneously, tey can cange teir beaior at a faster rate tan te state of te population canges, in particular wen te disease prealence is eiter ery ig or ery low. Tis seems reasonable, since indiiduals ae a complete control oer weter tey use an ITN and can adapt quickly (toug ow tey act will depend upon te information tat tey ae aailable, wic we consider in te Discussion). Following [4], we will ealuate b b, ten (, ) = π (, ) s E bb cb bb (4) Figure. Values of λ ( b ) for different ector recruitment rates. (a) parameters are Λ = µ, (b) 4 Λ = µ. Values of te remaining µ = 8, µ = µ, β =., β =, µ = 55 65, p =, p =, Λ = µ, δ = 4., γ =

6 π bb is te probability of te focal indiidual getting infected in a population wit aerage ITN usage b and c = c ITN c MAL is te relatie cost of te ITN and te cost of getting infected by malaria. To ealuate π ( bb, ), we will use and adapt te model deeloped in []. Te probability of a ost getting infected is ten gien by te ratio of te rate of getting infected to te rate of leaing te susceptible state for any reason. Tus, were (, ). Analysis π ( bb) λ ( bb, ) ( bb, ) +, =. λ µ Gien a relatiely small alue of µ and a relatiely large alue of λ ( b ), we get from (5) and (5) β ( b ) µ, if λ ( b ) >, π (, b ) β λ ( b ) =, if λ ( b ) =, π (, ) =. Tere will of course be alues of λ (5) (6) b (7) b just aboe wic are not small compared to µ, but for te parameters in our model (see Table ) tis appens for a ery narrow range of alues of b, and so te aboe is a good approximation. It follows from (4) tat and tus, by (6), (7), (, ) π (, ) = π E b = c b = c (8) E, b, b, (9) π c λ E(, b) E(, b) π (, b) c c λ Because by (6), π (, b ) for most b suc tat λ ( b ) >, we get tat [ ] ITN MAL, if >, = =, if =. E, b E, b <, for all b, if c = c c > () and oterwise tere is a unique b s satisfying te equilibrium condition (). Te condition c > is oweer not realistic for real populations, and tus in practice we will always get a unique b s. Te plots of E(, b) E(, b ) are sown in Figure. Moreoer, te unique b s is gien exactly as a solution of b 4ac = () were a, b and c are gien by (7)-(9). As in [], () appens for a critical alue b Rc = 4 a Λ µ ( L) R c gien by weneer te term under. is nonnegatie. Te critical ITN usage is ten gien as a unique solution of a b = b b were fixed point problem s c( s) were = Λ Λ µ Q L pp. Te plots of c () () β µ QR c, if Rc is defined, bc = β β + QR cµ (4), oterwise, b b for different parameter alues are sown at Figure. 856

7 Figure. Values of (, ) (, ) E b E b for different ector recruitment rates. (a) Λ = µ Λ = µ. Values of, (b) 4 te remaining parameters are γ = 4, Results µ = 8, µ = µ, β =., β =, µ = 55 65, p =, p =, Λ = µ, δ = 4. It follows from our analysis tat it is in te indiidual s best interest to use te ITN as long as tere is malaria present in te population. For te parameter alues used aboe, namely β =, te optimal usage (from te indiidual s perspectie) coincide wit te aerage usage in te population tat eradicates te disease. Howeer, te model used sows particular sensitiity to te parameter alue β. Tis is unfortunate because te effectieness of te insecticide sprayed on te ITN deteriorates oer - 5 years [], te net cannot effectiely proide protection for te wole 4 ours, and te mosquitoes may adapt teir biting beaior in areas of ig ITN coerage []. Consequently, te realistic alue of β is positie and can be relatiely close to β. As sown in Figure 4 wen β =.75 (or een β =.5 for larger alues of Λ ), te malaria does not get eradicated. Still, te force of infection, λ ( b ), decreases wit b ; and te decrease is significant. Te quantity E(, b) E(, b ) is also always positie (close to ), tus it is in te indiidual s interest to keep using te ITN. Consequently, te ITN usage does not eradicate te disease, but elps to significantly reduce its force. Howeer, in all of te aboe, we considered tat malaria was transmitted between ectors and ost during eery bite. If we relax tis assumption and consider p, p <, ten again malaria can be eradicated een for larger alues of β as in Figure 4(c) and Figure 4(d). 5. Discussion In tis paper we ae considered a game-teoretical model of ITN usage, allowing indiiduals in te population to decide weter (and ow often) tey use ITNs. We ae sown tat optimal beaior from te indiidual s perspectie leads to te decrease (or een eradication) of te disease, i.e. an optimal outcome from te population perspectie. In particular if it is not possible to eradicate te disease troug ITN usage, indiiduals use ITNs all te time, wic leads to te imum possible leel of malaria. If eradication is possible, indiiduals select a strategy wic is approximately at te leel tat leads to eradication. Moreoer, te optimal beaior does not depend on te exact cost of te ITN use (as long as te cost is smaller tan te cost of te disease). Tis type of outcome is not always te case; see for example [4]. We ae seen tat indiidual strategies lead to te eradication leel effectiely independently of te cost in our case, because indiiduals ae a ery ig likeliood of getting infected by te disease weneer te disease is present in te population, due to te ig rate of infection ( λ ), wic is muc iger tan te rate of natural deat, meaning tat people become infected wit probability almost equal to. We ae assumed tat indiiduals are able to adjust teir ITN usage strategy faster tan canges in te underlying state of te population, suc as te prealence of te disease, to always obtain te optimal leel of 857

8 Figure. Plots of bc b for different ector recruitment rates. (a) parameters are Λ = µ, (b) 4 Λ = µ. Values of te remaining µ = 8, µ = µ, β =., β =, µ = 55 65, p =, p =, Λ = µ, δ = 4., γ = 4. 4 Figure 4. Values of λ ( b ) for different ector recruitment rates and imal contact rates. (a) β (d) =.75, p =, p =, (b) 4,.5,.75,.75 Λ = = p = p =, (c) 4 µ, β.5,, Λ = µ β = p = p =. Values of te remaining parameters are 4 µ = 55 65, Λ = µ, δ = 4., γ = 4. Λ = µ, Λ = = p = p =, µ, β.75,.75,.75 µ = 8, µ = µ, β =., 858

9 usage. It is clear tat tey are pysically able to do tis, because to put up or remoe an ITN takes a relatiely sort time. Howeer, tey clearly need sufficient information about te state of te population to make a rational decision to cange. Wat information would tey ae on wic to base teir decision? It is clear from Figure tat except for b witin a ery narrow range te disease is eiter absent or effectiely eeryone as it; tis information is obious, and so any indiidual wit te wrong strategy would quickly cange it. Tus te population usage would quickly eole to a leel close to te critical leel tat we ae assumed. We note tat because of te ig infection rate, a leel of ITN usage close to te eradication leel but still below it could formally still lead to ig leels of disease prealence. Tus if indiiduals ad perfect information about te state of te population tey could coose teir strategies to aciee te precise leel of te equilibrium strategy, wic would lead to some intermediate leel of infection between and. It is our assumption in tis paper tat tey cannot do tis precisely. In any case natural ariations witin te real population parameters, would mean tat te exact alue of tis critical leel would be subject to some fluctuation. Te consequence is tat te cosen strategy is sufficiently close to te eradication leel, tat tere is a good cance of suc eradication occurring, on te assumption tat indiiduals play a fixed bed-net usage strategy. In reality, it may be tat te leel of malaria will not ae enoug time to tend to before indiiduals in te population notice te low leel and gie up te use of ITNs. Tus if te malaria leel is significantly different to te equilibrium leel from our analysis, indiidual beaiour will preent it from being eliated. Tus a more complex dynamically coupled model of te eolution of ITN usage and malaria leel may be needed to decide te precise beaiour at suc intermediate leels of infection. Acknowledgements Te researc was supported by te Simons Foundation grant #454 (Jan Ryctář). References [] WHO (5) World Malaria Report. [] Ragaendra, K., Barik, T.K., Niranjan Reddy, B.P., Sarma, P. and Das, A.P. () Malaria Vector Control: From Past to Future. Parasitology Researc, 8, ttp://dx.doi.org/.7/s46--- [] Lengeler, C. (4) Insecticide-Treated Bed Nets and Curtains for Preenting Malaria. Cocrane Database of Systematic Reiews,, Article ID: CD6. ttp://dx.doi.org/./ cd6.pub [4] Goodman, C.A. and Mills, A.J. (999) Te Eidence Base on te Cost-Effectieness of Malaria Control Measures in Africa. Healt Policy and Planning, 4, -. ttp://dx.doi.org/.9/eapol/4.4. [5] Goodman, C.A., Coleman, P.G. and Mills, A.J. (999) Cost-Effectieness of Malaria Control in Sub-Saaran Africa. Te Lancet, 54, ttp://dx.doi.org/.6/s4-676(99)4-8 [6] Wite, M.T., Conte, L., Cibulskis, R. and Gani, A.C. () Costs and Cost-Effectieness of Malaria Control Interentions A Systematic Reiew. Malaria Journal,, ttp://dx.doi.org/.86/ [7] Miller, J.M., Korenromp, E.L., Nalen, B.L. and Steketee, R.W. (7) Estimating te Number of Insecticide-Treated Nets Required by African Houseolds to Reac Continent-Wide Malaria Coerage Targets. JAMA, 97, 4-5. ttp://dx.doi.org/./jama [8] Wite, L.J., Maude, R.J., Pongtaornpinyo, W., Saralamba, S., Aguas, R., Van Effelterre, T., Day, N.P.J. and Wite, N.J. (9) Te Role of Simple Matematical Models in Malaria Eliation Strategy Design. Malaria Journal, 8,. ttp://dx.doi.org/.86/ [9] Citnis, N., Scapira, A., Smit, T. and Steketee, R. () Comparing te Effectieness of Malaria Vector-Control Interentions troug a Matematical Model. Te American Journal of Tropical Medicine and Hygiene, 8,. ttp://dx.doi.org/.469/ajtm [] Agusto, F.B., Del Valle, S.Y., Blayne, K.W., Ngongala, C.N., Goncales, M.J., Li, N.P., Zao, R.J. and Gong, H.F. () Te Impact of Bed-Net Use on Malaria Prealence. Journal of Teoretical Biology,, ttp://dx.doi.org/.6/j.jtbi...7 [] Von Neumann, J. and Morgenstern, O. (7) Teory of Games and Economic Beaior (Commemoratie Edition). Princeton Uniersity Press. [] Smit, J.M. and Price, G.R. (97) Te Logic of Animal Conflict. Nature, 46, 5-8. ttp://dx.doi.org/.8/465a [] Broom, M. and Ryctář, J. () Game-Teoretical Models in Biology. CRC Press. 859

10 [4] Bauc, C.T. and Earn, D.J.D. (4) Vaccination and te Teory of Games. Proceedings of te National Academy of Sciences of te United States of America,, ttp://dx.doi.org/.7/pnas.48 [5] Bauc, C.T., Galani, A.P. and Earn, D.J.D. () Group Interest ersus Self-Interest in Smallpox Vaccination Policy. Proceedings of te National Academy of Sciences of te United States of America,, ttp://dx.doi.org/.7/pnas.74 [6] Bauc, C.T. (5) Imitation Dynamics Predict Vaccinating Beaiour. Proceedings of te Royal Society B: Biological Sciences, 7, ttp://dx.doi.org/.98/rspb.5.5 [7] Galani, A.P., Reluga, T.C. and Capman, G.B. (7) Long-Standing Influenza Vaccination Policy Is in Accord wit Indiidual Self-Interest but Not wit te Utilitarian Optimum. Proceedings of te National Academy of Sciences, 4, ttp://dx.doi.org/.7/pnas [8] Sim, E., Kocin, B. and Galani, A. (9) Insigts from Epidemiological Game Teory into Gender-Specific Vaccination against Rubella. Matematical Biosciences and Engineering: MBE, 6, ttp://dx.doi.org/.94/mbe [9] Sim, E., Grefenstette, J.J., Albert, S.M., Cakouros, B.E. and Burke, D.S. () A Game Dynamic Model for Vaccine Skeptics and Vaccine Belieers: Measles as an Example. Journal of Teoretical Biology, 95, 94-. ttp://dx.doi.org/.6/j.jtbi...5 [] Crawford, K., Lancaster, A., O, H. and Ryctář, J. (5) A Voluntary Use of Insecticide-Treated Cattle Can Eliate African Sleeping Sickness. Letters in Biomatematics,, 9-. ttp://dx.doi.org/.8/ [] Sykes, D. and Ryctář, J. (5) A Game-Teoretic Approac to Valuating Toxoplasmosis Vaccination Strategies. Teoretical Population Biology, 5, -8. ttp://dx.doi.org/.6/j.tpb.5.8. [] Sim, E., Capman, G.B., Townsend, J.P. and Galani, A.P. () Te Influence of Altruism on Influenza Vaccination Decisions. Journal of te Royal Society Interface, 9, 4-4. [] Geoffard, P.-Y. and Pilipson, T. (997) Disease Eradication: Priate ersus Public Vaccination. Te American Economic Reiew, 87, -. [4] Tebo-Ewungkem, M.I., Podder, C.N. and Gumel, A.B. () Matematical Study of te Role of Gametocytes and an Imperfect Vaccine on Malaria Transmission Dynamics. Bulletin of Matematical Biology, 7, 6-9. ttp://dx.doi.org/.7/s [5] Bowman, C., Gumel, A.B., an den Driessce, P., Wu, J. and Zu, H. (5) A Matematical Model for Assessing Control Strategies against West Nile Virus. Bulletin of Matematical Biology, 67, 7-. ttp://dx.doi.org/.6/j.bulm.5.. [6] Guyatt, H.L., Kinnear, J., Burini, M. and Snow, R.W. () A Comparatie Cost Analysis of Insecticide-Treated Nets and Indoor Residual Spraying in Higland Kenya. Healt Policy and Planning, 7, ttp://dx.doi.org/.9/eapol/7..44 [7] Mulligan, J.-A., Yukic, J. and Hanson, K. (8) Costs and Effects of te Tanzanian National Voucer Sceme for Insecticide-Treated Nets. Malaria Journal, 7,. ttp://dx.doi.org/.86/ [8] Asenso-Okyere, W.K. and Dzator, J.A. (997) Houseold Cost of Seeking Malaria Care. A Retrospectie Study of Two Districts in Gana. Social Science & Medicine, 45, ttp://dx.doi.org/.6/s77-956(96)8-8 [9] Akazili, J., Aikins, M. and Binka, F.N. (8) Malaria Treatment in Nortern Gana: Wat Is te Treatment Cost per Case to Houseolds? African Journal of Healt Sciences, 4, ttp://dx.doi.org/.44/ajs.4i.849 [] Mutabingwa, T.K. (5) Artemisinin-Based Combination Terapies (ACTs): Best Hope for Malaria Treatment but Inaccessible to te Needy! Acta Tropica, 95, 5-5. ttp://dx.doi.org/.6/j.actatropica [] Witty, C.J.M., Candler, C., Ansa, E., Leslie, T. and Staedke, S.G. (8) Deployment of ACT Antimalarials for Treatment of Malaria: Callenges and Opportunities. Malaria Journal, 7, S7. ttp://dx.doi.org/.86/ s-s7 [] Lubell, Y., Reyburn, H., Mbakilwa, H., Mwangi, R., Conya, S., Witty, C.J.M. and Mills, A. (8) Te Impact of Response to te Results of Diagnostic Tests for Malaria: Cost-Benefit Analysis. BMJ, 6, -5. ttp://dx.doi.org/.6/bmj [] Killeen, G.F., Kionda, J., Lyimo, E., Oketc, F.R., Kotas, M.E., Matenge, E., Scellenberg, J.A., Lengeler, C., Smit, T.A. and Drakeley, C.J. (6) Quantifying Beaioural Interactions between Humans and Mosquitoes: Ealuating te Protectie Efficacy of Insecticidal Nets against Malaria Transmission in Rural Tanzania. BMC Infectious Diseases, 6, 6. ttp://dx.doi.org/.86/