Stability Analysis of a Model with Integrated Control for Population Growth of the Aedes Aegypti Mosquito

Transcription

1 Appl. Mat. Inf. Sci. 11, No. 5, ) 139 Applied Matematics & Information Sciences An International Journal ttp://dx.doi.org/ /amis/1158 Staility Analysis of a Model wit Integrated Control for Population Growt of te Aedes Aegypti Mosquito Ana M. Pulecio 1,, Anial Muñoz 2 and Gerard Olivar 3 1 asic Sciences Department, University CESMAG, Pasto, Colomia. 2 Faculty of Education, University of te Quindío, Armenia, Colomia. 3 Matematics Department, National University of Colomia, Manizales, Colomia. Received: 5 May 217, Revised: 2 Aug. 217, Accepted: 8 Aug. 217 Pulised online: 1 Sep. 217 Astract: We present a system of ordinary nonlinear differential equations descriing te population growt dynamics of te Aedes aegypti mosquito, te main transmitter of te dengue virus in Colomia. Tis model incorporates te tree types of known control for mosquito eradication: mecanical, iological and cemical, focusing on iological control troug te use of te Wolacia acterium, wic is te new ope for te control of te diseases transmitted y tis mosquito. A local staility analysis of te model is performed on te tree equilirium points tat are found, determining te conditions under wic tose points ecome stale or unstale. Finally, we present numerical simulations implemented in Matla, were te numerical results are otained using ypotetical values of te parameters otained from te literature. Keywords: Aedes aegypti; integrated control; Wolacia; staility. 1 Introduction Aedes aegypti is a mosquito tat mainly lives close to uman populations. It flies only sort distances and requires lood primarily uman), to reproduce [1], [7]. During teir lifetime te mosquitoes go troug two stages: immature and mature. In te immature pase, te mosquito is aquatic and undergoes a metamorposis from egg to an adult. It feeds mainly on residues in te water were tey were laid y te female. Adult mosquitoes are airorne and wile te males feed on plant nectar, te females feed on lood. [4]. y feeding on lood, in order to mature and deposit er eggs, te female mosquito promotes te transmission of viruses and patogens tat cause various diseases including Dengue fever. For tis reason gloal campaigns ave een founded to eradicate te mosquito. So far, te struggle as een unsuccessful ecause altoug some countries ave acieved temporary te extinction, te mosquito soon returns due to te infestation of neigouring countries [2], [7]. Tere are tree main mecanisms to control te propagation of mosquitoes, namely: mecanical control, focused on preventing te reproduction of te mosquito using traps, destroying reeding grounds, etc.; cemical control ased on insecticides or larvicides; and iological control tat makes use of oter living organisms suc as te Wolacia acterium wic reduces te life span of te mosquito and also, in te case of dengue, almost eliminates te proaility of transmitting te virus to umans [5], [7], [8]. Te present article demonstrates a matematical model tat descries te population growt of te female mosquito in te adult pase. Te model incorporates all tree control mecanisms for te mosquito. A staility analysis is performed and we sow ow te population growt dynamics cange in response to a program of iological control via te introduction of te Wolacia acterium into te population. In tis way, te model will serve as a tool for te tose wo wis to determine te way in wic mecanical, cemical and iological controls sould e applied to diminis te reeding of mosquitoes and, terefore, te propagation of diseases like dengue tat are transmitted y tem. Corresponding autor ampuleciom@unal.edu.co c 217 NSP Natural Sciences Pulising Cor.

2 131 A. Pulecio et al.: Staility analysis of a model for population growt... 2 Te Model Te following ypoteses are considered in te creation of te model: Te population of interest is tat of te adult female Aedes aegypti mosquitoes. Tere are two populations of adult female mosquitoes: tose tat are infected y te Wolacia acterium and tose tat are not. Te deat rate of an adult mosquito infected wit te acterium is greater tan te deat rate of an uninfected mosquito. Tree controls on te mosquito population are used: traps tat prevent te eggs from reacing adultood mecanical control); in te immature state a proportion of te eggs are infected y te Wolacia acterium, wic genetically manipulates te mosquito and is transmitted vertically iological control); a proportion of adult mosquitoes die from te use of insecticides cemical control). Taking tese ypoteses into account, let us consider, te total numer of adult female mosquitoes tat are not infected wit te acterium at time t; and, te total numer of adult female mosquitoes tat are infected wit te acterium in a time t. We also consider te parameters in Tale 1. In tis analysis we use te week as te period of time ecause it is sort enoug tat a single female mosquito will lay eggs at most once during te period. Tale 1: Parameters of te model Parameter Description ξ Rate of development of a mosquito from te immature pase to te adult pase f Proportion of immature mosquitoes tat develop into adult females φ Te proaility tat an adult female mosquito will lay eggs in te week δ Te average numer of eggs laid at a time y an adult female mosquito π Te natural deat rate of immature mosquitoes κ Te maximum numer of mosquitoes tat te environment can support ε Te natural deat rate of te mosquito witout Wolacia ν Te deat rate of mosquitoes infected y Wolacia u 1 Te proportion of immature mosquito deats caused y traps u 2 Te proportion of immature mosquito deats caused y insecticides u 3 Te proportion of eggs infected y te Wolacia acterium troug micro-injection If we consider te variation of wit over time, we recognise tat tis population grows continuously as a result of te development of immature mosquitoes. It also decreases as a result of te natural deat of mosquitoes or te use of insecticides. Terefore, in terms of te parameters sown in Tale 1, we see tat te expression 1 u 3 ) ξ f φδ π+u 1 represents, for eac adult female tat lays eggs in te week, te average numer of eggs tat survive to adultood witout eing infected y te Wolacia acterium. Furtermore, te expression 1 + κ represents te proaility tat a mosquito tat develops to te mature pase finds space availale in te environment. From tese, we ave te numer of mosquitoes tat enter te population of adult females witout eing infected y te acterium is given y 1 u 3 ) ξ f φδ ) π+u κ. Similarly, te numer of female mosquitoes in tis state tat die in eac instant is given y ε + u 2 ). Tus we ave tat: d dt =1 u 3) ξ f φδ 1 + ) ε+ u 2 ). π+ u 1 κ Now, te mosquitoes tat enter te population of adult females infected y te Wolacia acterium are tose tat develop from eggs tat ave een laid y an infected female mosquito ecause te acterium is transmitted vertically) and tose tat develop from eggs tat ave een laid y uninfected females ut are infected troug micro-injection Terefore te cange in tis population is given y: d dt = ξ f φδ 1 + ) + u 3ξ f φδ 1 + ) π+ u 1 κ π+ u 1 κ ν+ u 2 ). Tus te system of ordinary non-linear equations representing te growt dynamic of te population of adult female mosquitoes ot wit and witout Wolacia infection is given y: d dt =1 u 3) ξ f φδ π+ u 1 d dt = ξ f φδ π+ u κ 3 Points of Equilirium 1 + κ ) ε+ u 2 ) 1) ) +u 3 ) ν+ u 2 ). Setting te rigt and side of te system s differential equations to zero, we find tat te model as tree points of equilirium: )) H 1 P 1 =,), P 2 =,k H and P 3 = 1, 1 ) c 217 NSP Natural Sciences Pulising Cor.

3 Appl. Mat. Inf. Sci. 11, No. 5, ) / H = were wit 1 = k 1 1 u ) 3ε+ u 2 ), )1 u 3 ) 1 = 1 ξ f φδ ν+ u 2 )π+ u 1 ) u 3 kε+ u 2 ) )1 u 3 ), and = 1 u 3)ξ f φδ π+ u 1 )ε+ u 2 ), wic represent te tresolds of te growt of te populations of adult female mosquitoes wit and witout Wolacia infection respectively. We can furter see tat, wen u 3 =, te tird point of equilirium can e written as: P 3 = k 1 ), 4 Staility Analysis Teorem 1. Te local staility of te system can e summarised as: 1.If H, <1 and H < 1, P 1 is asymptotically stale, and te points P 2 and P 3 are unstale. 2.If =H and <1, P 1 is asymptotically stale and te points P 2 y P 3 are not yperolic. 3.If = H = 1, none of te tree points of equilirium are yperolic. 4.If =1 and H 1, P 1 and P 3 are not yperolic, ut P 2 is asymptotically stale wen H > 1 and unstale wen H < 1. 5.If H = 1 and 1, P 1 y P 2 are not yperolic, ut P 3 is asymptotically stale wen >1 and unstale wen <1. 6.If < H, 1 y H > 1, P 2 is asymptotically stale and te points P 1 and P 3 are unstale. 7.If > H, > 1 y H 1, P 3 is asymptotically stale and te points P 1 and P 2 are unstale. 8.If =H, >1, P 1 is unstale and te points P 2 y P 3 are not yperolic. Proof. We can see tat te Jacoian matrix associated wit te linearised system aout P 1 is given y: ) ε+ u2 ) 1) JP 1 )= ν+ u 2 )u 3 H ν+ u 2 )H 1) and so it s caracteristic equation is λ ε+ u 2 ) 1))λ ν+ u 2 )H 1))=. 2) For te linearisation aout te point P 2, we ave te matrix: ) H JP 2 )= ε+ u 2) H ν+ u 2 )u 3 H 1)) ν+ u 2 )1 H) and te caracteristic equation of tis system is were ) H λ 1 =ε+ u 2 ) H λ λ 1 )λ λ 2 )=. 3) and λ 2 =ν+ u 2 )1 H). Finally, te matrix for te linearised system around P 3 is: ) a11 a JP 3 )= 12 a 21 a 22 were [ )] u3 ε+ u 2 ) a 11 =ε+ u 2 ) 1) )1 u 3 ) 1 [ a 12 = ε+ u 2 ) 1) 1 u ] 3ε+ u 2 ) )1 u 3 ) a 21 = u 3 ν+ u 2 ) H [ ) ] ν+ u2 1) 1 [ H a 22 =ν+ u 2 ) 1 u ) ] 3 1)ν+ u 2 ) 1 and it s caracteristic equation is: λ 2 + ν+ u 2 [H 1)+ H)]λ+ 4) +ε+ u 2 ) 1)) 1 u ) 3ν+ u 2 ) H =. Tus we ave: 1.If < 1, H < 1, y equation 2) te eigenvalues of JP 1 ) are real and negative, from wic te equilirium point P 1 is asymptotically stale at te local level. If we consider furter tat H, from equation 3) we can see tat tere are no null eigenvalues and tat, ecause 1 H >, tere exists an eigenvalue tat is real and positive. Tis leads us to conclude tat P 2 is an unstale point of equilirium. Finally, under tis ypotesis, we ave tat <H < 1 or H < < 1. If H < < 1 and 2H H+1, ten H 1)+ H) and H < 1. Tat is to say tat c 217 NSP Natural Sciences Pulising Cor.

4 1312 A. Pulecio et al.: Staility analysis of a model for population growt... ε+ u 2 < 1 and, applying algeraic ν+ u 2 )1 u 3 ) operations to tis equation, we arrive at u 3 ν+ u 2 ) < 1. From tis we get: ε+ u 2 ) 1)) 1 u 3ν+ u 2 ) ) H <, from wic we conclude, y te Rout-Hurwitz criterion, tat equation 4) as at least one root wit a positive real part and so P 3 is unstale. Similarly, wen H < <1 y = H+1 2H, te roots of equation 4) are real and of opposite signs wic implies tat P 3 is an unstale point of equilirium. On te oter and, if si < H < 1, ten H 1)+ H) < and ε + u 2 ) 1)) 1 u ) 3ν+ u 2 ) H, and y te Rout-Hurwitz criterion, equation 4) as at least one root wit a positive real part wic implies tat te point P 3 is unstale. 2.Analogously, if we ave tat = H, < 1, te point of equilirium P 1 is asymptotically stale at te local level. However, wen = H, equation 3) as an eigenvector of zero, wic implies tat P 2 is not a H yperolic point. Similarly, = 1 gives us u 3 ν+ u 2 ) = 1 and, from equation 4), we ave tat te matrix JP 3 ) as a zero eigenvalue; and so P 3 is not yperolic eiter. 3.We oserve tat wen =1and H = 1, we find a zero eigenvalue in eac of te tree caracteristic equations and terefore none of te tree points of equilirium are yperolic. 4.Analogously to te previous case, wen = 1, te caracteristic equations for P 1 and P 3 give eigenvalues of zero, and tese points are terefore not yperolic. However, wen H < 1 in equation 3) we get a positive real eigenvalue and anoter non-zero. Tus te point P 2 is unstale. In te case tat H > 1, from equation 2) we get negative real eigenvalues wic means tat te point P 2 is asymptotically stale. 5.If H = 1, te caracteristic equations for P 1 and P 2 we get eigenvalues of zero wic implies tat neiter point is yperolic. Wen > 1, H 1)+ H)> and u 3ν+ u 2 ) H < 1, and so all te coefficients in te caracteristic equation for P 3 are positive. Tus, y te Rout-Hurwitz criterion, te eigenvalues of JP 3 ) ave a negative real part and P 3 is asymptotically stale. However, wen < 1, H 1) + H) < and ε + u 2 ) 1)) 1 u ) 3ν+ u 2 ) H, and y te Rout-Hurwitz criterion tere exists an eigenvalue wit a positive real part and tus P 3 is unstale. 6.If H > 1, we can see tat from equation 2) we otain an positive real eigenvalue and, as 1, tere are no zero eigenvalues. Terefore, te point of equilirium P 1 is unstale. If we also ave tat < H, equation 3) gives us negative real eigenvalues from wic we conclude tat P 2 is asymptotically stale at te local level. On te oter and, if 1, we must ave eiter <1 or > 1. ut if < 1 and < H, we ave tat H 1) + H) < and ε + u 2 ) 1)) 1 u ) 3ν+ u 2 ) H, and terefore, y te Rout-Hurwitz criterion, equation 4) as at least one root wit a positive real part wic means tat P 3 is unstale. Similarly, if >1 y <H, ten ε+ u 2 ) 1)) 1 u ) 3ν+ u 2 ) H <, wic leads us to te same conclusion. 7.If >1, equation 2) gives us a positive real eigenvalue and, since H 1, tere are no zero eigenvalues wic implies tat te equilirium point P 1 is unstale. If, in addition, >H, equation 3) as no zero roots and also we get a positive real eigenvalue, so P 2 is unstale. We also oserve tat under tese ypoteses, ν+ u 2 [H 1)+ H)] > and ε + u 2 ) 1)ν ε) 1 u ) 3ν+ u 2 ) H >, wic, y te Rout-Hurwitz criterion, guarantees tat equation 4) gives eigenvalues aving a negative real part. Tus te point P 3 is asymptotically stale a te local level. 8.If = H and > 1, from equation 2) we get JP 1 ) as two positive real eigenvalues and terefore P 1 is unstale. However, from equations 3) and 4) we ave zero eigenvalues wic implies tat te equilirium points P 2 and P 3 and not yperolic. 5 Numeric Results For te numerical results, ypotetical values ave een considered for eac of te parameters in te model. In Tales 2 and 3 we can see te values tat ave een given to tese parameters for te different scenarios and also te values of te tresolds and H. Te simulations corroorate te analytic results. However, of special interest are te cases in wic te local staility analysis doesn t determine te asymptotic values of te system, as is te case were = H = 1. In tis case, according to Figure 3 wic uses te conditions given in Scenario 3 of Tale 2, te stale solution is te point P 1. Tis result is also otained wen = 1 and H < 1 under te conditions given in scenario 5 of Tale 2, or wen H = 1 and < 1 under te conditions given in scenario 7 of Tale 3. Tese results can e seen Figures 5 and 7, respectively. On te oter and, wen = H > 1, c 217 NSP Natural Sciences Pulising Cor.

5 Appl. Mat. Inf. Sci. 11, No. 5, ) / Tale 2: Values of te parameters in scenario 1-5 Parameter Sce. 1 Sce. 2 Sce. 3 Sce. 4 Sce. 5 ξ f φ δ π k ε ν u u u H P Fig. 2: Scenario 2. Tale 3: Values of te parameters in scenario 6-1 Parameter Sce. 6 Sce. 7 Sce. 8 Sce. 9 Sce. 1 ξ f φ δ π k ε ν u u u , H P Fig. 3: Scenario P P P Fig. 1: Scenario 1. Fig. 4: Scenario 4. according to Figure 1 wic uses te conditions given in scenario 1 of Tale 3, te stale solution is te equilirium P 2. c 217 NSP Natural Sciences Pulising Cor.

6 1314 A. Pulecio et al.: Staility analysis of a model for population growt P P Fig. 5: Scenario Fig. 8: Scenario P 3 P Fig. 6: Scenario P P1 4 Fig. 9: Scenario Fig. 7: Scenario 7. c 217 NSP Natural Sciences Pulising Cor Fig. 1: Scenario 1.

7 Appl. Mat. Inf. Sci. 11, No. 5, ) / Conclusions In te proposed model, as in all matematical models ased on ordinary differential equations tat descrie population growt, te so-called Growt Tresold was determined. Tis, like te asic Reproduction Numer descried in [9], determines te numer of new individuals of a species tat are generated during te lifetime of one of tem, and wic are capale reproducing wen introduced into a free environment of tat species. For tis model two growt tresolds ave een found, since adult female mosquitoes are classified into two groups: tose tat are carriers of te Wolacia acterium and tose tat aren t. Tus H represents te numer of infected adult female mosquitoes tat can e generated y a single mosquito wit tese caracteristics during its life span. Similarly, represents te numer of adult female mosquitoes witout te acterium tat can e generated y a single mosquito during its lifetime. Te staility analysis of te system, determined tat tere are tree equilirium points. iologically, te point of equilirium P 1 represents te free equilirium of te two populations, i.e. were neiter of te two groups of adult female mosquitoes exist. Te equilirium point P 2 represents te situation were all adult female mosquito population are carriers, wile non-carriers disappear from te environment. On te oter and, te equilirium P 3 is te point at wic te two groups of adult female mosquitoes coexist. In tis case it is necessary tat te control u 3 e practised permanently, oterwise, te adult female mosquito population witout Wolacia continues and tose wo are infected wit te acterium disappear. Analytically we ave estalised tat wen < H and H > 1, under initial conditions close to P 2, te population of adult female mosquitoes wit te Wolacia acterium will persist in te environment and te oter population will e extinguised, as demonstrated y conditions 4 and 6 of Teorem 1. Tis also seems to e te result in te case were = H as sown in Figure 1, ut te local staility analysis does not prove tis. Wen H < and > 1, under initial conditions close to point P 3, te two groups of mosquitoes will coexist in te medium provided tat u 3 ), as demonstrated y conditions 5 and 7 of Teorem 1. Finally, wen ot tresolds are less tan one, under initial conditions close to P 1, ot populations will disappear, as demonstrated y conditions 1 and 2 of Teorem 1. Tis condition seems to persist wen te tresolds are equal to one, as sown in Figure 3, and also wen = 1 and H < 1 as sown in Figure 5, or in te case were H = 1 and H < 1 as sown in Figure 7. function. Tis function is ased on te Runge-Kutta metod, wic is defined in [6] and is useful for solving ordinary differential equations wit initial conditions. Acknowledgement Te autors tank te GMME and ACDynamics researc groups. Te autors are grateful to te anonymous referee for a careful cecking of te details and for elpful comments tat improved tis paper. References [1]. Adams, D. Kapan, Man ites mosquito: understanding te contriution of uman movement to vector-orne disease dynamics. PloS one, 48), 29). [2] M. Eiman, V. Introini, C. Ripoll, Directrices para la prevencin y control de Aedes aegypti. Direccin de Enfermedades Transmitidas por Vectores. uenos Aires: Ministerio de Salud de la Nacin, 21). [3] C. Favier, D. Scmit, C. Mller-Graf,. Cazelles, N. Degallier,. Mondet, M. Duois, Influence of spatial eterogeneity on an emerging infectious disease: te case of dengue epidemics. Proceedings of te Royal Society of London : iological Sciences, ), ). [4] C. Ferreira, H. Yang, Estudo Dinmico da populaao de mosquitos Aedes aegypti. Trends in Applied and Computational Matematics 42), ). [5] H. Huges, N. ritton, Modelling te use of Wolacia to control dengue fever transmission. ulletin of matematical iology 755), ). [6] J. Matews, K. Fink, Mtodos numricos con Matla, Vol.4, Prentice Hall, Madrid, Espaa, ). [7] M. Rafikov, E. Rafikova, H. Yang, Optimization of te Aedes aegypti control strategies for integrated vector management. Journal of Applied Matematics 215). [8] R. Tom, H. Yang, L. Esteva, Optimal control of Aedes aegypti mosquitoes y te sterile insect tecnique and insecticide. Matematical iosciences, 2231), ). [9] P. Van den Driessce, J. Watmoug, Reproduction numers and su-tresold endemic equiliria for compartmental models of disease transmission. Matematical iosciences181), ). 7 Materials and Metods Te numerical solutions of te proposed model were derived using te software Matla 215a and its ODE45 c 217 NSP Natural Sciences Pulising Cor.

8 1316 A. Pulecio et al.: Staility analysis of a model for population growt... Ana María Pulecio Montoya received te title of Matematician from te National University of Colomia in 211. Se received te title of Magister in Sciences - Applied Matematics from te National University of Colomia in 213. At present, se is PD student of Automatic engineering in te National University of Colomia. Her sujects of interest are in Matematical Epidemiology and Applied Matematics. Anial Muñoz Loaiza, Specialization in iomatematics, University of Quindo, Colomia; Dr. in Matematical Sciences, FCFM-UAP, Mexico; Researcer at te Faculty of Education, Universidad del Quindo, Colomia and director of te Matematical Modeling in Epidemiology Group GMME). 45 articles in different journal, numerous presentations as speaker and lecturer in events in different countries. At present, e is developing a postdoctoral project on Matematical Modeling wit new tecnologies and inclusion of limiting factors to control vectors of arovirosis, in te Federal Center of Tecnological Education of Minas Gerais, Department of Pysics and Matematics DFM), razil. Wit Dr. Rodrigo Toms Nogueira Cardoso and is doctoral students. Gerard Olivar received te title of Matematician from te University of arcelona arcelona, Spain) in 1987 and te Doctor of Sciences - Matematics y te Universitat Politcnica de Catalunya arcelona, Spain) in 1997 Cum Laude). From 1987 to 25 e was assigned to te Department of Applied Matematics IV at te Universitat Politcnica de Catalunya, were e was Professor. Since 25 e as een linked to te National University of Colomia, were e currently works as a Full Professor. His sujects of interest are in Matematical Engineering and Applied Matematics. Specifically, in applications to science and engineering modeling and simulation, nonlinear dynamics and complex systems. Since 211, e as eld te Presidency of te Colomian Section of te Society for Industrial and Applied Matematics SIAM). c 217 NSP Natural Sciences Pulising Cor.