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Abstract: Using the sterile insect technique,in which sterile mosquitoes are released to reduce or eradicate the wild mosquito population,is an effective weapon to prevent transmission of mosquito-borne diseases.To study the impact of the sterile insect technique on the disease transmissions,we formulate stage-structured discrete-time mathematical models,based on difference equations,for the interactive dynamics of the wild and sterile mosquitoes.We incorporate different strategies for releasing sterile mosquitoes,investigate the model dynamics,and compare the impact of the different release strategies.Numerical examples are also provided to demonstrate dynamical features of the models.
Key words: Mathematical modeling; Ricker-type nonlinearity; stage structure; sterile mosquitoes; thresholds; vector-borne diseases
CLC number: TQ 018, Q 241.84, O 415.6 Document code: A
Article ID: 1000-5137(2014)05-0511-12
AMS Subject Classification: 39A30, 39A60, 92B05, 92D25, 92D30, 92D40
Malaria and other mosquito-borne diseases are a considerable public health concern worldwide.These diseases are transmitted between humans by blood-feeding mosquitoes.No vaccines are available and an effective way to prevent these mosquito-borne diseases is to control mosquitoes.Among the mosquitoes control measures,the sterile insect technique (SIT) has been applied to reducing or eradicating the wild mosquitoes.SIT is a method of biological control in which the natural reproductive process of mosquitoes is disrupted.Utilizing radical or other chemical or physical methods,male mosquitoes are genetically modified to be sterile which are incapable of producing offspring despite being sexually active.These sterile mosquitoes are then released into the environment to mate with wild mosquitoes that are present in the environment.A wild female that mates with a sterile male will either not reproduce,or produce eggs but the eggs will not hatch.Repeated releases of sterile mosquitoes or releasing a significantly large number of sterile mosquitoes may eventually wipe out or control a wild mosquito population ［1-3］.
While SIT brings an effective weapons to fight vector-borne diseases,and has shown promising in laboratory research,the assessment of the impact of releasing sterile mosquitoes on the wild mosquitoes controlling remains a challenging task.
Mathematical models have proven useful in getting insights to such challenging questions in population dynamics and epidemiology.There are mathematical models in the literature formulated to study the interactive dynamics of mosquito populations or the control of mosquitoes ［4-10］.In particular,models incorporate different strategies in releasing sterile mosquitoes have been formulated and the studied in ［11,12］.However,the mosquito population has been assumed to be homogeneous without distinguishing the metamorphic stages of mosquitoes. Mosquitoes undergo complete metamorphosis going through four distinct stages of development during a lifetime: egg,pupa,larva,and adult ［13］.While interspecific competition and predation are rather rare events and could be discounted as major causes of larval mortality,intraspecific competition could represent a major density dependent source for the population dynamics,and hence the effect of crowding could be an important factor in the population dynamics of mosquitoes ［14-16］.Hence,to have a better understanding of the impact of the releases of sterile mosquitoes,the metamorphosis stage structure needs to be included ［17］.Nevertheless,to keep our mathematical modeling as simple as possible,due to the fact that the first three stages in a mosquito′s life cycle are aquatic,we follow a line similar to the stage-structured models for transgenic mosquitoes in ［18-20］ where the three aquatic metamorphic stages are combined as one group,we group the three aquatic stages of mosquitoes into one class,called larvae,and divide the mosquito population into only two classes.We still simplify our models such that no male and female individuals are distinguished,and assume that the mosquito populations follow the nonlinearity of Ricker-type ［21］.We first give general modeling descriptions in Section 2.We then formulate a model,similar to that in ［4,5,11,12］,where the number of releases of sterile mosquitoes is constant in Section 3.Complete mathematical analysis for the model dynamics is given.We then formulate a model for the case where the number of sterile mosquito releases is proportional to the wild mosquito population size in Section 4.Mathematical analysis and numerical examples are provided to demonstrate the complexity of the model dynamics.To provide a different releasing strategy,we assume,in Section 5,that releases are of Holling-II type such that the number of sterile mosquitoes is proportional to the wild mosquito population size when the wild mosquito population size is small but is saturated and approaches a constant as the wild mosquito population size is sufficiently large.We also provide complete mathematical analysis for the model dynamics.We finally provide brief discussions on our findings,particularly the impact of the three different strategies on the mosquito control measures in Section 6.
Stage-structured models for interacting wild and sterile mosquitoes 2 The model basis
Now suppose sterile mosquitoes are released into the field of wild mosquitoes and we let Bn be the number of sterile mosquitoes released at generation n.Since sterile mosquitoes do not reproduce,their population size at generation n+1 has no input from their size at n.Hence Bn only depends on the size of the releases of the sterile mosquitoes.After the sterile mosquitoes are released,the mating interaction between the wild and sterile mosquitoes takes place.Following the line of the homogeneous population models in ［22,23］,we assume harmonic means for matings such that the per capita birth rate is given bybn=C(Nn) aynyn+Bn=C(Nn) aynNn,where C(Nn) is the number of matings per mosquitoes with Nn= yn+Bn,the total adult mosquito population size,and a is the number of wild larvae produced per wild mosquito.The interactive dynamics of wild and sterile mosquitoes are then described by the following system:xn+1=C(Nn) aynyn+Bnyne-k1 xn, yn+1=βxne-k2xn.
4 Releases proportional to the wild mosquito population size
To have a more optimal and economically effective strategy for releasing sterile mosquitoes in an area where the population size of wild mosquitoes is relatively small,instead of releasing sterile mosquitoes constantly,we may consider to keep closely sampling or surveillance of the wild mosquitoes and let the releases be proportional to the population size of the wild mosquitoes such that the number of releases is B(?)=by where b is a constant.
Figure 2 The parameters are given in (19).The two threshold values are bc=31 and bs=0.1416.When b=0.13,there exists a unique positive fixed point,which is unstable as shown in the upper left figure.For bs＜b=1＜bc,there is a stable positive fixed point as shown in the upper right figure.With b increased to b=10,there is a stable positive fixed point which has smaller magnitudes x and y compared to those for b=0.13 and b=1 as shown in the lower left figure.When b=32＞bc,there exists no positive fixed point,and the origin is asymptotically stable as shown in the lower left figure.
5 Proportional releases with saturation
The proportional releases introduced in Section 4,compared to the constat releases,may have advantage when the size of the wild mosquito population is small since the the size of releases is also small.However,if the wild mosquito population size is big,the release size is supposed to be big also that may exceed our affordability.Then we propose a new strategy such that the number of releases is proportional to the wild adult mosquito population size when it is small,but is saturated and approaches a constant when the wild adult mosquito population size is sufficiently large.To this end,we let the releases be of Holling-II type such that B=by1+y.
6 Concluding remarks
We introduced the metamorphic stage structure of mosquitoes into dynamical models for the interactive wild and sterile mosquitoes to study the impact of the releases of sterile mosquitoes in this paper.We simplify the models by combing the three aquatic metamorphic stages into one group,called larvae,and assume that the density-dependence,due to intraspecific competition,is only on the larvae.We considered three different strategies for the releases in model systems (6),(16),and (20),respectively.We determined the threshold value of the releases,bc,and the stability threshold value for positive fixed points,bs,for each of the model systems.If b＞bc,there exists no positive fixed point for all of the three model systems,in which case the wild mosquito population will be wiped out if the origin is stable,or oscillates.If b＜bc,there exist two positive fixed points for systems (6) and (20),and unique positive fixed point for system (16).A positive fixed point is asymptotically stable if b＜bs,and is unstable if b＞bs.When the positive fixed point is unstable,a period-doubling bifurcation occurs. While the biological outcomes from the model systems in this paper are similar to those in ［12］,particularly as the density-dependence is assumed to be only based on the larvae,as the stage structure is included in the three model systems,the mathematical analysis becomes more challenging.We have managed to obtain fundamental results from our model systems,but some of the mathematical analysis is not complete.Further research is planned in the near future.
References:
［1］ L.Alphey,M.Benedict,R.Bellini,G.G.Clark,D.A.Dame,M.W.Service,S.L.Dobson.Sterile-insect methods for control of mosquito-borne diseases: An analysis［J］.Vector-Borne and Zoonotic Diseases,2010,10:295-311.
［2］ A.C.Bartlett,R.T.Staten.Sterile Insect Release Method and other Genetic Control Strategies ［M/OL］ Radcliffe′s IPM World Textbook,1996,http://ipmworld.umn.edu/chapters/bartlett.htm.
［3］ Wikipedia.Sterile insect technique［J/OL］ 2014,http://en.wikipedia.org/wiki/Sterile_insect_technique.
［4］ H.J.Barclay.The sterile insect release method for species with two-stage life cycles ［M］.Researches on Population Ecology,1980,21:165-180.
［5］ H.J.Barclay.Pest population stability under sterile releases ［J］.Res.Popul.Ecol.,1982,24:405-416.
［6］ H.J.Barclay.Modeling incomplete sterility in a sterile release program:interactions with other factors ［J］.Popul Ecol,2001,43:197-206.
［7］ H.J.Barclay.Mathematical models for the use of sterile insects ［C］//Sterile Insect Technique.Principles and Practice in Area-Wide Integrated Pest Management,(V.A.Dyck,J.Hendrichs,and A.S.Robinson,Eds.),Heidelberg,Springer,2005:147-174.
［8］ H.J.Barclay,M.Mackuer.The sterile insect release method for pest control:a density dependent model ［J］.Environ.Entomol.,1980,9:810-817.
［9］ K.R.Fister,M.L.McCarthy,S.F.Oppenheimer,Craig Collins.Optimal control of insects through sterile insect release and habitat modification ［J］.Math.Biosci.,2013,244:201-212.
［10］ J.C.Floresa.A mathematical model for wild and sterile species in competition: immigration ［J］.Physica A,2003,328:214-224.
［11］ Liming Cai,Shangbing Ai,Jia Li.Dynamics of mosquitoes populations with different strategies of releasing sterile mosquitoes,(preprint).
［12］ Jia Li,Zhiling Yuan.Modeling of releasing sterile mosquitoes with different strategies,(preprint).
［13］ N.Becker.Mosquitoes and Their Control ［M］.New York:Kluwer Academic/Plenum,2003.
［14］ C.Dye.Intraspecific competition amongst larval aedes aegypti: Food exploitation or chemical interference ［J］.Ecological Entomology,1982,7:39-46. ［15］ R.M.Gleiser,J.Urrutia,D.E.Gorla.Effects of crowding on populations of aedes albifasciatus larvae under laboratory conditions ［J］.Entomologia Experimentalis et Applicata,2000,95:135-140.
［16］ M.Otero,H.G.Solari,N.Schweigmann.A stochastic population dynamics model for Aedes aegypti: formulation and application to a city with temperate climate ［J］.Bull.Math.Biol.,2006,68:1945-1974.
［17］ Junliang Lu,Jia Li.Dynamics of stage-structured discrete mosquito population models ［J］.J.Appl.Anal.Compt.,2011,1:53-67.
［18］ Jia Li.Simple stage-structured models for wild and transgenic mosquito populations ［J］.J.Diff.Eqns.Appl.,2009,17:327-347.
［19］ Jia Li.Malaria model with stage-structured mosquitoes ［J］.Math.Biol.Eng.,2011,8:753-768.
［20］ Jia Li.Discrete-time models with mosquitoes carrying genetically-modified bacteria ［J］.Math.Biosci.,2012,240:35-44.
［21］ W.E.Ricker.Stock and recruitment ［J］.Journal of the Fisheries Research Board of Canada,1954,11:559-623.
［22］ Jia Li.Simple mathematical models for interacting wild and transgenic mosquito populations ［J］.Math.Biosci.,2004,189:39-59.
［23］ Jia Li. Modeling of mosquitoes with dominant or recessive transgenes and Allee effects ［J］.Math.Biosci.Eng.,2010,7:101-123.
(Zhenzhen Feng,Zhenyu Bao)
Key words: Mathematical modeling; Ricker-type nonlinearity; stage structure; sterile mosquitoes; thresholds; vector-borne diseases
CLC number: TQ 018, Q 241.84, O 415.6 Document code: A
Article ID: 1000-5137(2014)05-0511-12
AMS Subject Classification: 39A30, 39A60, 92B05, 92D25, 92D30, 92D40
Malaria and other mosquito-borne diseases are a considerable public health concern worldwide.These diseases are transmitted between humans by blood-feeding mosquitoes.No vaccines are available and an effective way to prevent these mosquito-borne diseases is to control mosquitoes.Among the mosquitoes control measures,the sterile insect technique (SIT) has been applied to reducing or eradicating the wild mosquitoes.SIT is a method of biological control in which the natural reproductive process of mosquitoes is disrupted.Utilizing radical or other chemical or physical methods,male mosquitoes are genetically modified to be sterile which are incapable of producing offspring despite being sexually active.These sterile mosquitoes are then released into the environment to mate with wild mosquitoes that are present in the environment.A wild female that mates with a sterile male will either not reproduce,or produce eggs but the eggs will not hatch.Repeated releases of sterile mosquitoes or releasing a significantly large number of sterile mosquitoes may eventually wipe out or control a wild mosquito population ［1-3］.
While SIT brings an effective weapons to fight vector-borne diseases,and has shown promising in laboratory research,the assessment of the impact of releasing sterile mosquitoes on the wild mosquitoes controlling remains a challenging task.
Mathematical models have proven useful in getting insights to such challenging questions in population dynamics and epidemiology.There are mathematical models in the literature formulated to study the interactive dynamics of mosquito populations or the control of mosquitoes ［4-10］.In particular,models incorporate different strategies in releasing sterile mosquitoes have been formulated and the studied in ［11,12］.However,the mosquito population has been assumed to be homogeneous without distinguishing the metamorphic stages of mosquitoes. Mosquitoes undergo complete metamorphosis going through four distinct stages of development during a lifetime: egg,pupa,larva,and adult ［13］.While interspecific competition and predation are rather rare events and could be discounted as major causes of larval mortality,intraspecific competition could represent a major density dependent source for the population dynamics,and hence the effect of crowding could be an important factor in the population dynamics of mosquitoes ［14-16］.Hence,to have a better understanding of the impact of the releases of sterile mosquitoes,the metamorphosis stage structure needs to be included ［17］.Nevertheless,to keep our mathematical modeling as simple as possible,due to the fact that the first three stages in a mosquito′s life cycle are aquatic,we follow a line similar to the stage-structured models for transgenic mosquitoes in ［18-20］ where the three aquatic metamorphic stages are combined as one group,we group the three aquatic stages of mosquitoes into one class,called larvae,and divide the mosquito population into only two classes.We still simplify our models such that no male and female individuals are distinguished,and assume that the mosquito populations follow the nonlinearity of Ricker-type ［21］.We first give general modeling descriptions in Section 2.We then formulate a model,similar to that in ［4,5,11,12］,where the number of releases of sterile mosquitoes is constant in Section 3.Complete mathematical analysis for the model dynamics is given.We then formulate a model for the case where the number of sterile mosquito releases is proportional to the wild mosquito population size in Section 4.Mathematical analysis and numerical examples are provided to demonstrate the complexity of the model dynamics.To provide a different releasing strategy,we assume,in Section 5,that releases are of Holling-II type such that the number of sterile mosquitoes is proportional to the wild mosquito population size when the wild mosquito population size is small but is saturated and approaches a constant as the wild mosquito population size is sufficiently large.We also provide complete mathematical analysis for the model dynamics.We finally provide brief discussions on our findings,particularly the impact of the three different strategies on the mosquito control measures in Section 6.
Stage-structured models for interacting wild and sterile mosquitoes 2 The model basis
Now suppose sterile mosquitoes are released into the field of wild mosquitoes and we let Bn be the number of sterile mosquitoes released at generation n.Since sterile mosquitoes do not reproduce,their population size at generation n+1 has no input from their size at n.Hence Bn only depends on the size of the releases of the sterile mosquitoes.After the sterile mosquitoes are released,the mating interaction between the wild and sterile mosquitoes takes place.Following the line of the homogeneous population models in ［22,23］,we assume harmonic means for matings such that the per capita birth rate is given bybn=C(Nn) aynyn+Bn=C(Nn) aynNn,where C(Nn) is the number of matings per mosquitoes with Nn= yn+Bn,the total adult mosquito population size,and a is the number of wild larvae produced per wild mosquito.The interactive dynamics of wild and sterile mosquitoes are then described by the following system:xn+1=C(Nn) aynyn+Bnyne-k1 xn, yn+1=βxne-k2xn.
4 Releases proportional to the wild mosquito population size
To have a more optimal and economically effective strategy for releasing sterile mosquitoes in an area where the population size of wild mosquitoes is relatively small,instead of releasing sterile mosquitoes constantly,we may consider to keep closely sampling or surveillance of the wild mosquitoes and let the releases be proportional to the population size of the wild mosquitoes such that the number of releases is B(?)=by where b is a constant.
Figure 2 The parameters are given in (19).The two threshold values are bc=31 and bs=0.1416.When b=0.13,there exists a unique positive fixed point,which is unstable as shown in the upper left figure.For bs＜b=1＜bc,there is a stable positive fixed point as shown in the upper right figure.With b increased to b=10,there is a stable positive fixed point which has smaller magnitudes x and y compared to those for b=0.13 and b=1 as shown in the lower left figure.When b=32＞bc,there exists no positive fixed point,and the origin is asymptotically stable as shown in the lower left figure.
5 Proportional releases with saturation
The proportional releases introduced in Section 4,compared to the constat releases,may have advantage when the size of the wild mosquito population is small since the the size of releases is also small.However,if the wild mosquito population size is big,the release size is supposed to be big also that may exceed our affordability.Then we propose a new strategy such that the number of releases is proportional to the wild adult mosquito population size when it is small,but is saturated and approaches a constant when the wild adult mosquito population size is sufficiently large.To this end,we let the releases be of Holling-II type such that B=by1+y.
6 Concluding remarks
We introduced the metamorphic stage structure of mosquitoes into dynamical models for the interactive wild and sterile mosquitoes to study the impact of the releases of sterile mosquitoes in this paper.We simplify the models by combing the three aquatic metamorphic stages into one group,called larvae,and assume that the density-dependence,due to intraspecific competition,is only on the larvae.We considered three different strategies for the releases in model systems (6),(16),and (20),respectively.We determined the threshold value of the releases,bc,and the stability threshold value for positive fixed points,bs,for each of the model systems.If b＞bc,there exists no positive fixed point for all of the three model systems,in which case the wild mosquito population will be wiped out if the origin is stable,or oscillates.If b＜bc,there exist two positive fixed points for systems (6) and (20),and unique positive fixed point for system (16).A positive fixed point is asymptotically stable if b＜bs,and is unstable if b＞bs.When the positive fixed point is unstable,a period-doubling bifurcation occurs. While the biological outcomes from the model systems in this paper are similar to those in ［12］,particularly as the density-dependence is assumed to be only based on the larvae,as the stage structure is included in the three model systems,the mathematical analysis becomes more challenging.We have managed to obtain fundamental results from our model systems,but some of the mathematical analysis is not complete.Further research is planned in the near future.
References:
［1］ L.Alphey,M.Benedict,R.Bellini,G.G.Clark,D.A.Dame,M.W.Service,S.L.Dobson.Sterile-insect methods for control of mosquito-borne diseases: An analysis［J］.Vector-Borne and Zoonotic Diseases,2010,10:295-311.
［2］ A.C.Bartlett,R.T.Staten.Sterile Insect Release Method and other Genetic Control Strategies ［M/OL］ Radcliffe′s IPM World Textbook,1996,http://ipmworld.umn.edu/chapters/bartlett.htm.
［3］ Wikipedia.Sterile insect technique［J/OL］ 2014,http://en.wikipedia.org/wiki/Sterile_insect_technique.
［4］ H.J.Barclay.The sterile insect release method for species with two-stage life cycles ［M］.Researches on Population Ecology,1980,21:165-180.
［5］ H.J.Barclay.Pest population stability under sterile releases ［J］.Res.Popul.Ecol.,1982,24:405-416.
［6］ H.J.Barclay.Modeling incomplete sterility in a sterile release program:interactions with other factors ［J］.Popul Ecol,2001,43:197-206.
［7］ H.J.Barclay.Mathematical models for the use of sterile insects ［C］//Sterile Insect Technique.Principles and Practice in Area-Wide Integrated Pest Management,(V.A.Dyck,J.Hendrichs,and A.S.Robinson,Eds.),Heidelberg,Springer,2005:147-174.
［8］ H.J.Barclay,M.Mackuer.The sterile insect release method for pest control:a density dependent model ［J］.Environ.Entomol.,1980,9:810-817.
［9］ K.R.Fister,M.L.McCarthy,S.F.Oppenheimer,Craig Collins.Optimal control of insects through sterile insect release and habitat modification ［J］.Math.Biosci.,2013,244:201-212.
［10］ J.C.Floresa.A mathematical model for wild and sterile species in competition: immigration ［J］.Physica A,2003,328:214-224.
［11］ Liming Cai,Shangbing Ai,Jia Li.Dynamics of mosquitoes populations with different strategies of releasing sterile mosquitoes,(preprint).
［12］ Jia Li,Zhiling Yuan.Modeling of releasing sterile mosquitoes with different strategies,(preprint).
［13］ N.Becker.Mosquitoes and Their Control ［M］.New York:Kluwer Academic/Plenum,2003.
［14］ C.Dye.Intraspecific competition amongst larval aedes aegypti: Food exploitation or chemical interference ［J］.Ecological Entomology,1982,7:39-46. ［15］ R.M.Gleiser,J.Urrutia,D.E.Gorla.Effects of crowding on populations of aedes albifasciatus larvae under laboratory conditions ［J］.Entomologia Experimentalis et Applicata,2000,95:135-140.
［16］ M.Otero,H.G.Solari,N.Schweigmann.A stochastic population dynamics model for Aedes aegypti: formulation and application to a city with temperate climate ［J］.Bull.Math.Biol.,2006,68:1945-1974.
［17］ Junliang Lu,Jia Li.Dynamics of stage-structured discrete mosquito population models ［J］.J.Appl.Anal.Compt.,2011,1:53-67.
［18］ Jia Li.Simple stage-structured models for wild and transgenic mosquito populations ［J］.J.Diff.Eqns.Appl.,2009,17:327-347.
［19］ Jia Li.Malaria model with stage-structured mosquitoes ［J］.Math.Biol.Eng.,2011,8:753-768.
［20］ Jia Li.Discrete-time models with mosquitoes carrying genetically-modified bacteria ［J］.Math.Biosci.,2012,240:35-44.
［21］ W.E.Ricker.Stock and recruitment ［J］.Journal of the Fisheries Research Board of Canada,1954,11:559-623.
［22］ Jia Li.Simple mathematical models for interacting wild and transgenic mosquito populations ［J］.Math.Biosci.,2004,189:39-59.
［23］ Jia Li. Modeling of mosquitoes with dominant or recessive transgenes and Allee effects ［J］.Math.Biosci.Eng.,2010,7:101-123.
(Zhenzhen Feng,Zhenyu Bao)