×

Modelling Wolbachia infection in a sex-structured mosquito population carrying West Nile virus. (English) Zbl 1387.92082

Summary: Wolbachia is possibly the most studied reproductive parasite of arthropod species. It appears to be a promising candidate for biocontrol of some mosquito borne diseases. We begin by developing a sex-structured model for a Wolbachia infected mosquito population. Our model incorporates the key effects of Wolbachia infection including cytoplasmic incompatibility and male killing. We also allow the possibility of reduced reproductive output, incomplete maternal transmission, and different mortality rates for uninfected/infected male/female individuals. We study the existence and local stability of equilibria, including the biologically relevant and interesting boundary equilibria. For some biologically relevant parameter regimes there may be multiple coexistence steady states including, very importantly, a coexistence steady state in which Wolbachia infected individuals dominate. We also extend the model to incorporate West Nile virus (WNv) dynamics, using an SEI modelling approach. Recent evidence suggests that a particular strain of Wolbachia infection significantly reduces WNv replication in Aedes aegypti. We model this via increased time spent in the WNv-exposed compartment for Wolbachia infected female mosquitoes. A basic reproduction number \(R_0\) is computed for the WNv infection. Our results suggest that, if the mosquito population consists mainly of Wolbachia infected individuals, WNv eradication is likely if WNv replication in Wolbachia infected individuals is sufficiently reduced.

MSC:

92D30 Epidemiology
34D20 Stability of solutions to ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations

References:

[1] Bergsman LD, Hyman JM, Manore C (2016) A mathematical model for the spread of West Nile virus in migratory and resident birds. Math Biosci Eng 13:401-424 · Zbl 1343.92461
[2] Blagrove MSC et al (2012) Wolbachia strain wMel induces cytoplasmic incompatibility and blocks dengue transmission in Aedes albopictus. PNAS 109:255-260 · doi:10.1073/pnas.1112021108
[3] Blayneh KW, Gumel AB, Lenhart S, Clayton T (2010) Backward bifurcation and optimal control in transmission dynamics of West Nile virus. Bull Math Biol 72:1006-1028 · Zbl 1191.92024 · doi:10.1007/s11538-009-9480-0
[4] Bowman C, Gumel AB, van den Driessche P, Wu J, Zhu H (2005) A mathematical model for assessing control strategies against West Nile virus. Bull Math Biol 67:1107-1133 · Zbl 1334.92392 · doi:10.1016/j.bulm.2005.01.002
[5] Caspari E, Watson GS (1959) On the evolutionary importance of cytoplasmic sterility in mosquitoes. Evolution 13:568-570 · doi:10.1111/j.1558-5646.1959.tb03045.x
[6] Dodson BL, Hughes GL, Paul O, Matacchiero AC, Kramer LD, Rasgon JL (2014) Wolbachia enhances West Nile Virus (WNV) infection in the mosquito Culex tarsalis. PLoS Negl Trop Dis 8(7):e2965 · doi:10.1371/journal.pntd.0002965
[7] Dufourd C, Dumont Y (2012) Modeling and simulations of mosquito dispersal. The case of Aedes albopictus. BIOMATH 1:1209262, 7 · Zbl 1368.92141 · doi:10.11145/j.biomath.2012.09.262
[8] Engelstädter J, Telschow A (2009) Cytoplasmic incompatibility and host population structure. Heredity 103:196-207 · doi:10.1038/hdy.2009.53
[9] Engelstädter J, Telschow A, Hammerstein P (2004) Infection dynamics of different Wolbachia-types within one host population. J Theor Biol 231:345-355 · Zbl 1447.92420 · doi:10.1016/j.jtbi.2004.06.029
[10] Farkas JZ, Hinow P (2010) Structured and unstructured continuous models for Wolbachia infections. Bull Math Biol 72:2067-2088 · Zbl 1201.92044 · doi:10.1007/s11538-010-9528-1
[11] Gourley SA, Liu R, Wu J (2006/07) Some vector borne diseases with structured host populations: extinction and spatial spread. SIAM J Appl Math 67:408-433 · Zbl 1123.35083
[12] Hadeler KP (2012) Pair formation. J Math Biol 64:613-645 · Zbl 1252.92047 · doi:10.1007/s00285-011-0454-0
[13] Hadeler KP, Waldstätter R, Wörz-Busekros A (1988) Models for pair formation in bisexual populations. J Math Biol 26:635-649 · Zbl 0714.92018 · doi:10.1007/BF00276145
[14] Hancock PA, Sinkins SP, Godfray HCJ (2011a) Population dynamic models of the spread of Wolbachia. Am Nat 177:323-333 · doi:10.1086/658121
[15] Hancock PA, Sinkins SP, Godfray HCJ (2011b) Strategies for introducing Wolbachia to reduce transmission of mosquito-borne diseases. PLoS Negl Trop Dis 5:e1024 · doi:10.1371/journal.pntd.0001024
[16] Hoffmann, AA; Turelli, M.; Hoffmann, AA (ed.); O’Neill, SL (ed.); Werren, JH (ed.), Cytoplasmic incompatibility in insects, 42-80 (1997), Oxford
[17] Hoffmann AA et al (2011) Successful establishment of Wolbachia in Aedes populations to suppress dengue transmission. Nature 476:454-457 · doi:10.1038/nature10356
[18] Hussain M et al (2013) Effect of Wolbachia on replication of West Nile virus in a mosquito cell line and adult mosquitoes. J Virol 87:851-858 · doi:10.1128/JVI.01837-12
[19] Hughes H, Britton NF (2013) Modelling the use of Wolbachia to control dengue fever transmission. Bull Math Biol 75:796-818 · Zbl 1273.92034 · doi:10.1007/s11538-013-9835-4
[20] Keeling MJ, Jiggins FM, Read JM (2003) The invasion and coexistence of competing Wolbachia strains. Heredity 91:382-388 · doi:10.1038/sj.hdy.6800343
[21] Kendall DG (1949) Stochastic processes and population growth. J R Stat Soc Ser B 11:230-264 · Zbl 0038.08803
[22] Keyfitz N (1972) The mathematics of sex and marriage. In: Proceedings of the sixth Berkeley symposium on mathematical statistics and probability, vol IV: biology and health. Univ. California Press, pp 89-108
[23] Lewis M, Renclawowicz J, van den Driessche P (2006) Traveling waves and spread rates for a West Nile virus model. Bull Math Biol 68:3-23 · Zbl 1334.92414 · doi:10.1007/s11538-005-9018-z
[24] Li J (2011) Modelling of transgenic mosquitoes and impact on malaria transmission. J Biol Dyn 5:474-494 · Zbl 1225.92033 · doi:10.1080/17513758.2010.523122
[25] Li J, Yuan Z (2015) Modelling releases of sterile mosquitoes with different strategies. J Biol Dyn 9:1-14 · Zbl 1448.92220 · doi:10.1080/17513758.2014.977971
[26] McMeniman CJ et al (2009) Stable introduction of a life-shortening Wolbachia infection into the mosquito Aedes aegypti. Science 323:141-144 · doi:10.1126/science.1165326
[27] O’Neill SL, Hoffmann AA, Werren JH (eds) (1997) Influential passengers. Oxford University Press, Oxford
[28] Rasgon JL, Scott TW (2004) Impact of population age structure on Wolbachia transgene driver efficacy: ecologically complex factors and release of genetically modified mosquitoes. Insect Biochem Mol Biol 34:707-713 · doi:10.1016/j.ibmb.2004.03.023
[29] Smith HL (1995) Monotone dynamical systems. An introduction to the theory of competitive and cooperative systems. Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence · Zbl 0821.34003
[30] Stouthamer, R.; Hoffmann, AA (ed.); O’Neill, SL (ed.); Werren, H. (ed.), Wolbachia induced parthenogenesis, 102-124 (1997), Oxford
[31] Telschow A, Hammerstein P, Werren JH (2005a) The effect of Wolbachia versus genetic incompatibilities on reinforcement and speciation. Evolution 59:1607-1619 · doi:10.1111/j.0014-3820.2005.tb01812.x
[32] Telschow A, Yamamura N, Werren JH (2005b) Bidirectional cytoplasmic incompatibility and the stable coexistence of two Wolbachia strains in parapatric host populations. J Theor Biol 235:265-274 · Zbl 1445.92210 · doi:10.1016/j.jtbi.2005.01.008
[33] Turelli M (1994) Evolution of incompatibility-inducing microbes and their hosts. Evolution 48:1500-1513 · doi:10.1111/j.1558-5646.1994.tb02192.x
[34] Vautrin E et al (2007) Evolution and invasion dynamics of multiple infections with Wolbachia investigated using matrix based models. J Theor Biol 245:197-209 · Zbl 1451.92314 · doi:10.1016/j.jtbi.2006.09.035
[35] Walker T et al (2011) The wMel Wolbachia strain blocks dengue and invades caged Aedes aegypti populations. Nature 476:450-453 · doi:10.1038/nature10355
[36] Werren JH (1997) Biology of Wolbachia. Annu Rev Entomol 42:587-609 · doi:10.1146/annurev.ento.42.1.587
[37] Wonham MJ, Lewis MA (2008) A comparative analysis of models for West Nile Virus. Mathematical epidemiology. Lecture Notes in Math., 1945. Springer, Berlin, pp 365-390 · Zbl 1206.92017
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.