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Mathematical analysis of a weather-driven model for the population ecology of mosquitoes

†. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona, USA

‡. Present address: Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Kingdom of Saudi Arabia

Received: , Accepted: , Published:

A new deterministic model for the population biology of immature and mature mosquitoes is designed and used to assess the impact of temperature and rainfall on the abundance of mosquitoes in a community. The trivial equilibrium of the model is globally-asymptotically stable when the associated *vectorial reproduction* number $({\mathcal R}_0)$ is less than unity. In the absence of density-dependence mortality in the larval stage, the autonomous version of the model has a unique and globally-asymptotically stable non-trivial equilibrium whenever $1 < {\mathcal R}_0 < {\mathcal R}_0^C$ (this equilibrium bifurcates into a limit cycle, *via* a Hopf bifurcation at ${\mathcal R}_0={\mathcal R}_0^C$). Numerical simulations of the weather-driven model, using temperature and rainfall data from three cities in Sub-Saharan Africa (Kwazulu Natal, South Africa; Lagos, Nigeria; and Nairobi, Kenya), show peak mosquito abundance occurring in the cities when the mean monthly temperature and rainfall values lie in the ranges $[22 -25]^{0}$C, $[98 -121]$ mm; $[24 -27]^{0}$C, $[113 -255]$ mm and $[20.5 -21.5]^{0}$C, $[70 -120]$ mm, respectively (thus, mosquito control efforts should be intensified in these cities during the periods when the respective suitable weather ranges are recorded).

# References

[1] A. Abdelrazec,A. B. Gumel, Mathematical assessment of the role of temperature and rainfall on mosquito population dynamics, Journal of Mathematical Biology, 74 (2017): 1351-1395.

[2] F. B. Agusto,A. B. Gumel,P. E. Parham, Qualitative assessment of the role of temperature variations on malaria transmission dynamics, Journal of Biological Systems, 23 (2015): 597-630.

[3] N. Ali,K. Marjan,A. Kausar, Study on mosquitoes of Swat Ranizai sub division of Malakand, Pakistan Journal of Zoology, 45 (2013): 503-510.

[4] *Anopheles* Mosquitoes, Centers for Disease Control and Prevention,
http://www.cdc.gov/malaria/about/biology/mosquitoes/. Accessed: May, 2016.

[5] N. Bacaër, Periodic matrix population models: Growth rate, basic reproduction number and entropy, Bulletin of Mathematical Biology, 71 (2009): 1781-1792.

[6] N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population, Bulletin of Mathematical Biology, 69 (2007): 1067-1091.

[7] N. Bacaër,S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, Journal of Mathematical Biology, 53 (2006): 421-436.

[8] N. Bacaër,R. Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor, Mathematical Biosciences, 210 (2007): 647-658.

[9] N. Bacaër,X. Abdurahman, Resonance of the epidemic threshold in a periodic environment, Journal of Mathematical Biology, 57 (2008): 649-673.

[10] N. Bacaër,H. Ait Dads el, Genealogy with seasonality, the basic reproduction number, and the influenza pandemic, Journal of Mathematical Biology, 62 (2011): 741-762.

[11] M. Belda,E. Holtanová,T. Halenka,J. Kalvová, Climate classification revisited: From Köppen to Trewartha, Climate Research, 59 (2014): 1-13.

[12] K. Berkelhamer,T. J. Bradley, Mosquito larval development in container habitats: The role of rotting *Scirpus californicus*, Journal of the American Mosquito Control Association, 5 (1989): 258-260.

[13] B. Gates,
*Gatesnotes: Mosquito Week* The Deadliest Animal in the World, https://www.gatesnotes.com/Health/Most-Lethal-Animal-Mosquito-Week. Accessed: May, 2016.

[14] S. M. Blower,H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model, as an example, International Statistical Review, 2 (1994): 229-243.

[15] P. Cailly,A. Tranc,T. Balenghiene,C. Totyg,P. Ezannoa, A climate-driven abundance model to assess mosquito control strategies, Ecological Modelling, 227 (2012): 7-17.

[16] J. Cariboni,D. Gatelli,R. Liska,A. Saltelli, A. The role of sensitivity analysis in ecological modeling, Ecological Modeling, 203 (2007): 167-182.

[17] J. Carr, null, Applications of Centre Manifold Theory, , Springer-Verlag, New York, 1981.

[18] C. Castillo-Chavez,B. Song, Dynamical models of tuberculosis and their applications, Mathematical Bioscience Engineering, 1 (2004): 361-404.

[19] N. Chitnis,J. M. Cushing,J. M. Hyman, Bifurcation analysis of a mathematical model for malaria transmission, SIAM Journal on Applied Mathematics, 67 (2006): 24-45.

[20] S. Chow,C. Li,D. Wang, null, Normal Forms and Bifurcation of Planar Vector Fields, , Cambridge University Press, Cambridge, 1994.

[21] J. Couret,E. Dotson,M. Q. Benedict, Temperature, Larval diet, and density effects on development rate and survival of *Aedes aegypti* (Diptera: Culicidae), PLoS One, 9 (2014).

[22] J. M. O. Depinay,C. M. Mbogo,G. Killeen,B. Knols,J. Beier, A simulation model of African Anopheles ecology and population dynamics for the analysis of malaria transmission, Malaria Journal, 3 (2004): p29.

[23] O. Diekmann,J. Heesterbeek,J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990): 365-382.

[24] F. Dufois, Assessing inter-annual and seasonal variability Least square fitting with Matlab: Application to SSTs in the vicinity of Cape Town, http://www.eamnet.eu/cms/sites/eamnet.eu/files/Least_square_fitting_with_Matlab-Francois_Dufois.pdf. Accessed: October, 2016.

[25] Durban Monthly Climate Average, South Africa, http://www.worldweatheronline.com/Durban-weather-averages/Kwazulu-Natal/ZA.aspx. Accessed: May 2016.

[26] J. Dushoff,W. Huang,C. Castillo-Chavez, Backward bifurcations and catastrophe in simple models of fatal diseases, Journal of Mathematical Biology, 36 (1998): 227-248.

[27] T. G. George, Positive Definite Matrices and Sylvester's Criterion, The American Mathematical Monthly, 98 (1991): 44-46.

[28] H. M. Giles,D. A. Warrel, null, Bruce-Chwatt's Essential Malariology, 3rd edition, Heinemann Medical Books, Portsmouth, NH, 1993.

[29] J. E. Gimnig,M. Ombok,S. Otieno,M. G. Kaufman,J. M. Vulule,E. D. Walker, Density-dependent development of *Anopheles gambiae* (Diptera: Culicidae) larvae in artificial habitats, Journal of Medical Entomology, 39 (2002): 162-172.

[30] R. E. Harbach, Mosquito Taxonomic Inventory, (2011). http://mosquito-taxonomic-inventory.info/simpletaxonomy/term/6045. Accessed: May, 2016.

[31] D. Hershkowitz, Recent directions in matrix stability, Linear Algebra and its Applications, 171 (1992): 161-186.

[32] W. M. Hirsch,H. Hanisch,J. P. Gabriel, Differential equation models for some parasitic infections: Methods for the study of asymptotic behavior, Communications on Pure and Applied Mathematics, 38 (1985): 733-753.

[33] S. S. Imbahale,K. P. Paaijmans,W. R. Mukabana,R. van Lammeren,A. K. Githeko,W. Takken, A longitudinal study on Anopheles mosquito larval abundance in distinct geographical and environmental settings in western Kenya, Malaria Journal, 10 (2011).

[34] K. C. Kain,J. S. Keystone, Malaria in travelers, Infectious Disease Clinics, 12 (1998): 267-284.

[35] V. Kothandaraman, Air-water temperature relationship in Illinois River, Water Resources Bulletin, 8 (1972): 38-45.

[36] Lagos Monthly Climate Average, Nigeria, http://www.worldweatheronline.com/lagos-weather-averages/lagos/ng.aspx. Accessed: May 2016.

[37] V. Lakshmikantham,S. Leela, null, Differential and Integral Inequalities: Theory and Applications, , Academic Press, New York-London, 1969.

[38] V. Laperriere,K. Brugger,F. Rubel, Simulation of the seasonal cycles of bird, equine and human West Nile virus cases, Preventive Veterinary Medicine, 88 (2011): 99-110.

[39] J. P. LaSalle,
*The Stability of Dynamical Systems* Regional Conference Series in Applied Mathematics. SIAM Philadephia. 1976.

[40] Y. Lou,X.-Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM Journal on Applied Mathematics, 70 (2010): 2023-2044.

[41] A. M. Lutambi,M. A. Penny,T. Smith,N. Chitnis, Mathematical modelling of mosquito dispersal in a heterogeneous environment, Journal of Mathematical Biosciences, 241 (2013): 198-216.

[42] P. Magal,X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM Journal on Mathematical Analysis, 37 (2005): 251-275.

[43] Malaria Atlas Project: Mosquito Malaria Vectors, http://www.map.ox.ac.uk/explore/mosquito-malaria-vectors/, Accessed: May: 2016.

[44] S. Marino,I. B. Hogue,C. J. Ray,D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, Journal of Theoretical Biology, 254 (2008): 178-196.

[45] M. D. Mckay,R. J. Beckman,W. J. Conover, Comparison of 3 methods for selecting values of input variables in the analysis of output from a computer code, Technometrics, 21 (1979): 239-245.

[46] R. G. McLeod,J. F. Brewster,A. B. Gumel,D. A. Slonowsky, Sensitivity and uncertainty analyses for a SARS model with time-varying inputs and outputs, Mathematical Biosciences and Engineering, 3 (2006): 527-544.

[47] E. A. Mordecai, Optimal temperature for malaria transmission is dramatically lower than previously predicted, Ecology Letters, 16 (2013): 22-30.

[48] Mosquito Life Cycle. American Mosquito Control Association, http://www.mosquito.org/life-cycle, Accessed: May, 2016.

[49] Mosquitoes of Michigan -Their Biology and Control, Michigan Mosquito Control Organization, 2013. http://www.mimosq.org/mosquitobiology/mosquitobiology.htm. Accessed: May: 2015.

[50] Nairobi Monthly Climate Average, Kenya, http://www.worldweatheronline.com/nairobi-weather-averages/nairobi-area/ke.aspx. Accessed: May 2016.

[51] G. A. Ngwa, On the population dynamics of the malaria vector, Bulletin of Mathematical Biology, 68 (2006): 2161-2189.

[52] G. A. Ngwa,A. M. Niger,A. B. Gumel, Mathematical assessment of the role of non-linear birth and maturation delay in the population dynamics of the malaria vector, Applied Mathematics and Computation, 217 (2010): 3286-3313.

[53] A. M. Niger,A. B. Gumel, Mathematical analysis of the role of repeated exposure on malaria transmission dynamics, Differential Equations and Dynamical Systems, 16 (2008): 251-287.

[54] T. E. Nkya,I. Akhouayri,W. Kisinza,J. P. David, Impact of environment on mosquito response to pyrethroid insecticides: Facts evidences and prospects, Insect Biochemistry and Molecular Biology, 43 (2013): 407-416.

[55] K. O. Okuneye,A. B. Gumel, Analysis of a temperature-and rainfall-dependent model for malaria transmission Dynamics, Mathematical Biosciences, 287 (2017): 72-92.

[56] H. J. Overgaard,Y. Tsude,W. Suwonkerd,M. Takagi, Characteristics of *Anopheles minimus* (Diptera: Culicidae) larval habitats in northern Thailand, Environmental Entomology, 31 (2002): 134-141.

[57] K. P. Paaijmans,S. S. Imbahale,M. B. Thomas,W. Takken, Relevant microclimate for determining the development rate of malaria mosquitoes and possible implications of climate change, Malaria Journal, 9 (2010): p196.

[58] K. P. Paaijmans,M. O. Wandago,A. K. Githeko,W. Takken, Unexpected high losses of *Anopheles gambiae* larvae due to rainfall, PLOS One, 2 (2007).

[59] P. E. Parham,E. Michael, Modeling the effects of weather and climate change on malaria transmission, Environmental Health Perspectives, 118 (2010): 620-626.

[60] P. E. Parham,D. Pople,C. Christiansen-Jucht,S. Lindsay,W. Hinsley,E. Michael, Modeling the role of environmental variables on the population dynamics of the malaria vector Anopheles *gambiae sensu* stricto, Malaria Journal, 11 (2012): p271.

[61] P. C. Park, A new proof of Hermite's stability criterion and a generalization of Orlando's formula, International Journal of Control, 26 (2012): 197-206.

[62] J. M. Pilgrim, X. Fang and H. G. Stefan,
*Correlations of Minnesota Stream Water Temperatures with Air Temperatures* Project Report 382, prepared for National Agricultural Water Quality Laboratory Agricultural Research Service U. S. Department of Agriculture Durant, Oklahoma, 1995.

[63] T. Porphyre,D. J. Bicout,P. Sabatier, Modelling the abundance of mosquito vectors versus flooding dynamics, Ecological Modelling, 183 (2005): 173-181.

[64] E. B. Preud'homme and H. G. Stefan,
*Relationship Between Water Temperatures and Air Temperatures for Central U. S. Streams* Project Report No. 333, prepared for Environmental Research Laboratory U. S. Environmental Protection Agency Duluth, Minnesota, 1992.

[65] F. Rubel,K. Brugger,M. Hantel,S. Chvala-Mannsberger,T. Bakonyi,H. Weissenbock,N. Nowotny, Explaining Usutu virus dynamics in Austria: Model development and calibration, Preventive Veterinary Medicine, 85 (2008): 166-186.

[66] M. A. Safi,M. Imran,A. B. Gumel, Threshold dynamics of a non-autonomous SEIRS model with quarantine and isolation, Theory in Biosciences, 131 (2012): 19-30.

[67] J. Shaman,J. Day, Reproductive phase locking of mosquito populations in response to rainfall frequency, Plos One, 2 (2007): p331.

[68] O. Sharomi,C. N. Podder,A. B. Gumel,E. H. Elbasha,J. Watmough, Role of incidence function in vaccine-induced backward bifurcation in some HIV models, Mathematical Biosciences, 210 (2007): 436-463.

[69] H. L. Smith, null, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, , American Mathematical Society, 1995.

[70] H. L. Smith,P. Waltman, Perturbation of a globally stable steady state, American Mathematical Society, 127 (1999): 447-453.

[71] H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, Journal of Mathematical Biology, 30 (1992): 755-763.

[72] H. R. Thieme, Persistence under relaxed point dissipativity (with application to an endemic model), SIAM Journal on Mathematical Analysis, 24 (1993): 407-435.

[73] P. Van den Driessche,J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002): 29-48.

[74] E. Van Handel, Nutrient accumulation in three mosquitoes during larval development and its effect on young adults, Journal of the American Mosquito Control Association, 4 (1988): 374-376.

[75] W. Wang,X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, Journal of Dynamics and Differential Equations, 20 (2008): 699-717.

[76] World Health Organization, A global brief on vector-borne diseases, 2014.

[77] World Health Organization, World health report. Executive summary, Insect-borne diseases, 1996.

[78] World Health Organization, WHO global health days, http://www.who.int/campaigns/world-health-day/2014/vector-borne-diseases/en/. Accessed: June, 2016.

[79] P. Wu,G. Lay,R. Guo,Y. Lin,C. Lung,J. Su, Higher temperature and urbanization affect the spatial patterns of dengue fever transmission in subtropical Taiwan, Science of The Total Environment, 407 (2009): 2224-2233.

[80] F. Zhang,X.-Q. Zhao, A periodic epidemic model in a patchy environment, Journal of Mathematical Analysis and Applications, 325 (2007): 496-516.

[81] Z. Zhang, T. W. Ding, T. Huang and Z. Dong,
*Qualitative Theory of Differential Equations* American Mathematical, 2006.

[82] X.-Q. Zhao, null, Dynamical Systems in Population Biology, , Springer, New York, 2003.

[83] X.-Q. Zhao, Permanence implies the existence of interior periodic solutions for FDEs, International Journal of Qualitative Theory of Differential Equations and Applications, 2 (2008): 125-137.

[84] X.-Q. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications, Canadian Applied Mathematics Quarterly, 3 (1995): 473-495.

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