zbMATH — the first resource for mathematics

Optimal control approach for establishing wMelpop Wolbachia infection among wild Aedes aegypti populations. (English) Zbl 1390.92133
Summary: Wolbachia-based biocontrol has recently emerged as a potential method for prevention and control of dengue and other vector-borne diseases. Major vector species, such as Aedes aegypti females, when deliberately infected with Wolbachia become less capable of getting viral infections and transmitting the virus to human hosts. In this paper, we propose an explicit sex-structured population model that describes an interaction of uninfected (wild) male and female mosquitoes and those deliberately infected with wMelPop strain of Wolbachia in the same locality. This particular strain of Wolbachia is regarded as the best blocker of dengue and other arboviral infections. However, wMelPop strain of Wolbachia also causes the loss of individual fitness in Aedes aegypti mosquitoes. Our model allows for natural introduction of the decision (or control) variable, and we apply the optimal control approach to simulate wMelPop Wolbachia infestation of wild Aedes aegypti populations. The control action consists in continuous periodic releases of mosquitoes previously infected with wMelPop strain of Wolbachia in laboratory conditions. The ultimate purpose of control is to find a tradeoff between reaching the population replacement in minimum time and with minimum cost of the control effort. This approach also allows us to estimate the number of Wolbachia-carrying mosquitoes to be released in day-by-day control action. The proposed method of biological control is safe to human health, does not contaminate the environment, does not make harm to non-target species, and preserves their interaction with mosquitoes in the ecosystem.

92D30 Epidemiology
92D25 Population dynamics (general)
49K15 Optimality conditions for problems involving ordinary differential equations
Full Text: DOI
[1] Ascher UM, Mattheij RMM, Russell RD (1988) Numerical solution of boundary value problems for ordinary differential equations. Prentice Hall series in computational mathematics. Prentice Hall Inc., Englewood Cliffs · Zbl 0671.65063
[2] Barton, N; Turelli, M, Spatial waves of advance with bistable dynamics: cytoplasmic and genetic analogues of allee effects, Am Nat, 178, e48-e75, (2011)
[3] Bian, G; Xu, Y; Lu, P; Xie, Y; Xi, Z, The endosymbiotic bacterium wolbachia induces resistance to dengue virus in aedes aegypti, PLoS Pathog, 6, e1000,833, (2010)
[4] Blayneh, K; Cao, Y; Kwon, HD, Optimal control of vector-borne diseases: treatment and prevention, Discrete Contin Dyn Syst B, 11, 587-611, (2009) · Zbl 1162.92034
[5] Bliman PA, Aronna MS, da Silva MA, et al (2015) Global stabilizing feedback law for a problem of biological control of mosquito-borne diseases. In: 2015 54th IEEE conference on decision and control (CDC). IEEE, pp 3206-3211
[6] Brauer, F; Castillo-Chávez, C, Mathematical models in population biology and epidemiology, Texts Appl Math, (2012) · Zbl 1302.92001
[7] Brown, JE; McBride, CS; Johnson, P; Ritchie, S; Paupy, C; Bossin, H; Lutomiah, J; Fernandez-Salas, I; Ponlawat, A; Cornel, AJ; Black, WC; Gorrochotegui-Escalante, N; Urdaneta-Marquez, L; Sylla, M; Slotman, M; Murray, KO; Walker, C; Powell, JR, Worldwide patterns of genetic differentiation imply multiple “domestications” of aedes aegypti, a major vector of human diseases, Proc R Soc B Biol Sci, 278, 2446-2454, (2011)
[8] Bryson A, Ho YC (1975) Applied optimal control: optimization, estimation and control. Halsted Press Book, New York
[9] Bull, JJ; Turelli, M, wolbachia versus dengue evolutionary forecasts, Evol Med Public Health, 2013, 197-207, (2013)
[10] Campo-Duarte, DE; Cardona-Salgado, D; Vasilieva, O, Establishing wmelpop wolbachia infection among wild aedes aegypti females by optimal control approach, Appl Math Inf Sci, 11, 1011-1027, (2017)
[11] Campo-Duarte, DE; Vasilieva, O; Cardona-Salgado, D, Optimal control for enhancement of wolbachia frequency among aedes aegypti females, Int J Pure Appl Math, 112, 219-238, (2017)
[12] Castillo-Chávez, C; Feng, Z; Huang, W; Castillo-Chávez, C (ed.); Blower, S (ed.); Driessche, P (ed.); Kirschner, D (ed.); Yakubu, A (ed.), On the computation of ro and its role on global stability, No. 125, 229-250, (2002), Berlin
[13] Caswell, H; Weeks, DE, Two-sex models: chaos, extinction, and other dynamic consequences of sex, Am Nat, 128, 707-735, (1986)
[14] Chan, M; Johansson, MA, The incubation periods of dengue viruses, PloS ONE, 7, e50,972, (2012)
[15] Coelho, FC; Codeço, CT; Gomes, MGM, A Bayesian framework for parameter estimation in dynamical models, PloS ONE, 6, e19,616, (2011)
[16] Denlinger, DL; Armbruster, PA, Mosquito diapause, Ann Rev Entomol, 59, 73-93, (2014)
[17] Diekmann O, Heesterbeek J (2000) Mathematical epidemiology of infectious diseases: model building, analysis and interpretation. Wiley, Hoboken Wiley Series in Mathematical & Computational Biology · Zbl 0997.92505
[18] Diekmann, O; Heesterbeek, J; Metz, J, On the definition and the computation of the basic reproduction ratio ro in models for infectious diseases in heterogeneous populations, J Math Biol, 28, 365-382, (1990) · Zbl 0726.92018
[19] Domínguez, MC; Ludueña-Almeida, FF; Almirón, WR, Dinámica poblacional de aedes aegypti (diptera: culicidae) en Córdoba capital [population dynamics of aedes aegypti (diptera: culicidae) in cordoba capital], Revista de la Sociedad Entomológica Argentina, 59, 41-50, (2000)
[20] Dutra, HLC; Rocha, MN; Dias, FBS; Mansur, SB; Caragata, EP; Moreira, LA, wolbachia blocks currently circulating zika virus isolates in Brazilian aedes aegypti mosquitoes, Cell Host Microbe, 19, 771-774, (2016)
[21] Dye, C, Models for the population dynamics of the yellow fever mosquito, aedes aegypti, J Anim Ecol, 53, 247-268, (1984)
[22] Farkas, JZ; Hinow, P, Structured and unstructured continuous models for wolbachia infections, Bull Math Biol, 72, 2067-2088, (2010) · Zbl 1201.92044
[23] Farkas, JZ; Gourley, SA; Liu, R; Yakubu, AA, Modelling wolbachia infection in a sex-structured mosquito population carrying west nile virus, J Math Biol, 75, 621-647, (2017) · Zbl 1387.92082
[24] Farkas M (2001) Dynamical models in biology. Elsevier, Amsterdam · Zbl 0972.92001
[25] Ferguson, NM; Kien, DTH; Clapham, H; Aguas, R; Trung, VT; Chau, TNB; Popovici, J; Ryan, PA; O’Neill, SL; McGraw, EA; Long, VT; Dui, LT; Nguyen, HL; Vinh Chau, NV; Wills, B; Simmons, CP, Modeling the impact on virus transmission of wolbachia-mediated blocking of dengue virus infection of aedes aegypti, Sci Transl Med, 7, 279ra37-279ra37, (2015)
[26] Fleming W, Rishel R (1975) Deterministic and stochastic optimal control. Springer, New York · Zbl 0323.49001
[27] Frentiu, FD; Walker, T; O’Neill, SL; Gubler, D (ed.); Ooi, E (ed.); Vasudevan, S (ed.); Farrar, J (ed.), Biological control of dengue and wolbachia-based strategies, (2014), CABI
[28] Frentiu, FD; Zakir, T; Walker, T; Popovici, J; Pyke, AT; Hurk, A; McGraw, EA; O’Neill, SL, Limited dengue virus replication in field-collected aedes aegypti mosquitoes infected with wolbachia, PLoS Negl Trop Dis, 8, e2688, (2014)
[29] Garg D, Patterson MA, Darby CL, Francolin C, Huntington GT, Hager WW, Rao AV (2009) Direct trajectory optimization and costate estimation of general optimal control problems using a Radau pseudospectral method. In: Proceedings of the AIAA guidance, navigation, and control conference and exhibit
[30] Hancock, PA; Godfray, HCJ, Modelling the spread of wolbachia in spatially heterogeneous environments, J R Soc Interface, 9, 3045-3054, (2012)
[31] Hancock, PA; Sinkins, SP; Godfray, HCJ, Population dynamic models of the spread of wolbachia, Am Nat, 177, 323-333, (2011)
[32] Hancock, PA; Sinkins, SP; Godfray, HCJ, Strategies for introducing wolbachia to reduce transmission of mosquito-borne diseases, PLoS Negl Trop Dis, 5, e1024, (2011)
[33] Hilgenboecker, K; Hammerstein, P; Schlattmann, P; Telschow, A; Werren, JH, How many species are infected with wolbachia?—A statistical analysis of current data, FEMS Microbiol Lett, 281, 215-220, (2008)
[34] Hoffmann, A; Montgomery, B; Popovici, J; Iturbe-Ormaetxe, I; Johnson, P; Muzzi, F; Greenfield, M; Durkan, M; Leong, Y; Dong, Y; Cook, H; Axford, J; Callahan, A; Kenny, N; Omodei, C; McGraw, E; Ryan, P; Ritchie, S; Turelli, M; O’Neill, S, Successful establishment of wolbachia in aedes populations to suppress dengue transmission, Nature, 476, 454-457, (2011)
[35] Hoffmann, AA, Facilitating wolbachia invasions, Austral Entomol, 53, 125-132, (2014)
[36] Hoffmann, AA; Turelli, M, Facilitating wolbachia introductions into mosquito populations through insecticide-resistance selection, Proc R Soc Lond B Biol Sci, 280, 20130371, (2013)
[37] Hughes, H; Britton, NF, Modelling the use of wolbachia to control dengue fever transmission, Bull Math Biol, 75, 796-818, (2013) · Zbl 1273.92034
[38] Hurst, TP; Pittman, G; O’Neill, SL; Ryan, PA; Nguyen, H; Kay, BH, Impacts of wolbachia infection on predator prey relationships: evaluating survival and horizontal transfer between wmelpop infected aedes aegypti and its predators, J Med Entomol, 49, 624-630, (2012)
[39] Jansen, CC; Beebe, NW, The dengue vector aedes aegypti: what comes next, Microbes Infect, 12, 272-279, (2010)
[40] Keyfitz N (1972) The mathematics of sex and marriage. In: Proceedings of the sixth Berkeley symposium on mathematical statistics and probability, vol 4. University of California Press, Berkeley, pp 89-108
[41] Kobayashi, Y; Telschow, A, Cytoplasmic feminizing elements in a two-population model: infection dynamics, gene flow modification, and the spread of autosomal suppressors, J Evolut Biol, 23, 2558-2568, (2010)
[42] Koiller J, Da Silva M, Souza M, Codeço C, Iggidr A, Sallet G (2014) Aedes, Wolbachia and Dengue. Technical report
[43] Kot M (2001) Elements of mathematical ecology. Cambridge University Press, Cambridge · Zbl 1060.92058
[44] Kroese DP, Taimre T, Botev ZI (2011) Handbook of Monte Carlo methods, vol 706. Wiley series in probability and statistics. Wiley, Hoboken · Zbl 1213.65001
[45] Lawson AB (2006) Statistical methods in spatial epidemiology, 2nd edn. Wiley series in probability and statistics. Wiley, Hoboken · Zbl 1096.62118
[46] Lenhart S, Workman JT (2007) Optimal control applied to biological models. Chapman & Hall/CRC, Boca Raton · Zbl 1291.92010
[47] Liles, JN, Effects of mating or association of the sexes on longevity in aedes aegypti (l.), Mosquito News, 25, 434-439, (1965)
[48] Lindström, J; Kokko, H, Sexual reproduction and population dynamics: the role of polygyny and demographic sex differences, Proc R Soc Lond B Biol Sci, 265, 483-488, (1998)
[49] Manore, CA; Hickmann, KS; Xu, S; Wearing, HJ; Hyman, JM, Comparing dengue and chikungunya emergence and endemic transmission in A. aegypti and A. albopictus, J Theor Biol, 356, 174-191, (2014) · Zbl 1412.92292
[50] McGraw, EA; O’Neill, SL, Beyond insecticides: new thinking on an ancient problem, Nat Rev Microbiol, 11, 181-193, (2013)
[51] McMeniman, C; etal., Stable introduction of a life-shortening wolbachia infection into the mosquito aedes aegypti, Science, 323, 141-144, (2009)
[52] McMeniman, CJ; O’Neill, SL, A virulent wolbachia infection decreases the viability of the dengue vector aedes aegypti during periods of embryonic quiescence, PLoS Negl Trop Dis, 4, e748, (2010)
[53] Moreira, LA; Iturbe-Ormaetxe, I; Jeffery, JA; Lu, G; Pyke, AT; Hedges, LM; Rocha, BC; Hall-Mendelin, S; Day, A; Riegler, M; etal., A wolbachia symbiont in aedes aegypti limits infection with dengue, chikungunya, and plasmodium, Cell, 139, 1268-1278, (2009)
[54] Moulay, D; Aziz-Alaoui, MA; Kwon, HD, Optimal control of chikungunya disease: larvae reduction, treatment and prevention, Math Biosci Eng, 9, 369-392, (2012) · Zbl 1260.92068
[55] Ndii, MZ; Hickson, R; Allingham, D; Mercer, G, Modelling the transmission dynamics of dengue in the presence of wolbachia, Math Biosci, 262, 157-166, (2015) · Zbl 1315.92083
[56] Nguyen, T; Nguyen, H; Nguyen, T; Vu, S; Tran, N; Le, T; Vien, Q; Bui, T; Le, H; Kutcher, S; Hurst, T; Duong, T; Jeffery, J; Darbro, J; Kay, H; Iturbe-Ormaetxe, I; Popovici, J; Montgomery, B; Turley, A; Zigterman, F; Cook, H; Cook, P; Johnson, P; Ryan, P; Paton, C; Ritchie, S; Simmons, C; O’Neill, S; Hoffmann, A, Field evaluation of the establishment potential of wmelpop wolbachia in Australia and Vietnam for dengue control, Parasit Vectors, 8, 1, (2015)
[57] Okosun, KO; Ouifki, R; Marcus, N, Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity, Biosystems, 106, 136-145, (2011)
[58] Okosun, KO; Rachid, O; Marcus, N, Optimal control strategies and cost-effectiveness analysis of a malaria model, BioSystems, 111, 83-101, (2013)
[59] Patterson, MA; Rao, AV, GPOPS-II: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming, ACM Trans Math Softw (TOMS), 41, 1, (2014) · Zbl 1369.65201
[60] Popovici, J; Moreira, LA; Poinsignon, A; Iturbe-Ormaetxe, I; McNaughton, D; O’Neill, SL, Assessing key safety concerns of a wolbachia-based strategy to control dengue transmission by aedes mosquitoes, Memórias do Instituto Oswaldo Cruz, 105, 957-964, (2010)
[61] Ritchie, SA; Montgomery, BL; Hoffmann, AA, Novel estimates of aedes aegypti (diptera: culicidae) population size and adult survival based on wolbachia releases, J Med Entomol, 50, 624-631, (2013)
[62] Ritchie, SA; Townsend, M; Paton, CJ; Callahan, AG; Hoffmann, AA, Application of wmelpop wolbachia strain to crash local populations of aedes aegypti, PLoS Negl Trop Dis, 9, e0003930, (2015)
[63] Roberts SM, Shipman JS (1972) Two-point boundary value problems: shooting methods. In: Modern analytic and computational methods in science and mathematics, vol 31. Elsevier, New York
[64] Rockwood LL (2015) Introduction to population ecology, 2nd edn. Wiley, Hoboken
[65] Ross, PA; Endersby, NM; Yeap, HL; Hoffmann, AA, Larval competition extends developmental time and decreases adult size of wmelpop wolbachia-infected aedes aegypti, Am J Trop Med Hyg, 91, 198-205, (2014)
[66] Ruang-Areerate, T; Kittayapong, P, Wolbachia transinfection in aedes aegypti: a potential gene driver of dengue vectors, Proc Natl Acad Sci, 103, 12534-12539, (2006)
[67] Schraiber, JG; Kaczmarczyk, AN; Kwok, R; Park, M; Silverstein, R; Rutaganira, FU; Aggarwal, T; Schwemmer, MA; Hom, CL; Grosberg, RK; etal., Constraints on the use of lifespan-shortening wolbachia to control dengue fever, J Theor Biol, 297, 26-32, (2012) · Zbl 1336.92085
[68] Sepúlveda, LS; Vasilieva, O, Optimal control approach to dengue reduction and prevention in cali, Colombia, Math Methods Appl Sci, 39, 5475-5496, (2016) · Zbl 1353.92100
[69] Sepúlveda-Salcedo, LS; Vasilieva, O; Martínez-Romero, HJ; Arias-Castro, JH, Ross-Macdonald: un modelo para la dinámica del dengue en cali, Colombia. Revista de Salud Pública, 17, 749-761, (2015)
[70] Sinkins, SP, wolbachia and cytoplasmic incompatibility in mosquitoes, Insect Biochem Mol Biol, 34, 723-729, (2004)
[71] Sinkins, SP, wolbachia and arbovirus inhibition in mosquitoes, Future Microbiol, 8, 1249-1256, (2013)
[72] Soares-Pinheiro, V; Dasso-Pinheiro, W; Trindade-Bezerra, J; Tadei, W, Eggs viability of aedes aegypti linnaeus (diptera, culicidae) under different environmental and storage conditions in Manaus, amazonas, Brazil. Braz J Biol, 77, 396-401, (2017)
[73] Styer, LM; Minnick, SL; Sun, AK; Scott, TW, Mortality and reproductive dynamics of aedes aegypti (diptera: culicidae) fed human blood, Vector Borne Zoonotic Dis, 7, 86-98, (2007)
[74] Telschow, A; Flor, M; Kobayashi, Y; Hammerstein, P; Werren, JH, wolbachia-induced unidirectional cytoplasmic incompatibility and speciation: mainland-island model, PLoS ONE, 2, e701, (2007)
[75] Thieme H (2003) Mathematics in population biology. Mathematical biology series. Princeton University Press, Princeton · Zbl 1054.92042
[76] Turelli, M, Cytoplasmic incompatibility in populations with overlapping generations, Evolution, 64, 232-241, (2010)
[77] Turelli, M; Hoffmann, AA, Rapid spread of an inherited incompatibility factor in California drosophila, Nature, 353, 440-442, (1991)
[78] Walker, T; Johnson, P; Moreira, L; Iturbe-Ormaetxe, I; Frentiu, F; McMeniman, C; Leong, Y; Dong, Y; Axford, J; Kriesner, P; Lloyd, A; Ritchie, S; O’Neill, S; Hoffmann, A, The wmel wolbachia strain blocks dengue and invades caged aedes aegypti populations, Nature, 476, 450-453, (2011)
[79] Williams, CR; Johnson, P; Ball, T; Ritchie, S, Productivity and population density estimates of the dengue vector mosquito aedes aegypti (stegomyia aegypti) in Australia, Med Vet Entomol, 27, 313-322, (2013)
[80] Woolfit, M; Iturbe-Ormaetxe, I; Brownlie, JC; Walker, T; Riegler, M; Seleznev, A; Popovici, J; Rancès, E; Wee, BA; Pavlides, J; etal., Genomic evolution of the pathogenic wolbachia strain, wmelpop, Genome Biol Evol, 5, 2189-2204, (2013)
[81] Xi, Z; Khoo, CC; Dobson, SL, wolbachia establishment and invasion in an aedes aegypti laboratory population, Science, 310, 326-328, (2005)
[82] Yamauchi, A; Telschow, A; Kobayashi, Y, Evolution of cytoplasmic sex ratio distorters: effect of paternal transmission, J Theoret Biol, 266, 79-87, (2010) · Zbl 1407.92086
[83] Yeap, HL; Mee, P; Walker, T; Weeks, AR; O’Neill, SL; Johnson, P; Ritchie, SA; Richardson, KM; Doig, C; Endersby, NM; Hoffmann, AA, Dynamics of the “popcorn” wolbachia infection in outbred aedes aegypti informs prospects for mosquito vector control, Genetics, 187, 583-595, (2011)
[84] Yeap, HL; Axford, JK; Popovici, J; Endersby, NM; Iturbe-Ormaetxe, I; Ritchie, SA; Hoffmann, AA, Assessing quality of life-shortening wolbachia-infected aedes aegypti mosquitoes in the field based on capture rates and morphometric assessments, Parasites Vectors, 7, 1-13, (2014)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.