A metapopulation model for the population dynamics of anopheles mosquito

A reactiondiffusion model for the dynamics of anopheles mosquito in a network is designed.A fixed-point theorem is used to establish the existence, uniqueness of a nontrivial equilibrium.A threshold parameter is computed, sensitivity analysis of parameters and variables are given.The monotone operator theory is used to prove the GAS of trivial and positive equilibriums.Assessment of mosquito dispersal, patch heterogeneity on the dynamics of mosquito is given. A more robust assessment of malaria control will come from a better understanding of the distribution and connectivity of breeding and blood feeding sites. Spatial heterogeneity of mosquito resources, such as hosts and breeding sites, affects mosquito dispersal behavior. This paper analyzes and simulates the spreading of anopheles mosquito on a complex metapopulation, that is, networks of populations connected by migratory flows whose configurations are described in terms of connectivity distribution of nodes (patches) and the conditional probabilities of connections between nodes. We examine the impacts of vector dispersal on the persistence and extinction of a mosquito population in both homogeneous and heterogeneous landscapes. For uncorrelated networks in a homogeneous landscape, we derive an explicit formula of the basic offspring number R0(m). Using the theory of monotone operators, we obtain sufficient conditions for the global asymptotic stability of equilibria. Precisely, the value 1 of the basic offspring number is a forward bifurcation for the dynamics of anopheles mosquito, with the trivial (mosquito-free) equilibrium point being globally asymptotically stable (GAS) when R0(m)1, and one stable nontrivial (mosquito-persistent) equilibrium point being born with well determined basins of attraction when R0(m)>1. Theoretical results are numerically supported and the impact of the migration of mosquitoes are discussed through global sensitivity analysis and numerical simulations.

[1]  Thomas Smith,et al.  Mathematical modelling of mosquito dispersal in a heterogeneous environment. , 2013, Mathematical biosciences.

[2]  Jinde Cao,et al.  Effect of time delay on pattern dynamics in a spatial epidemic model , 2014, Physica A: Statistical Mechanics and its Applications.

[3]  Alessandro Vespignani,et al.  Epidemic spreading in scale-free networks. , 2000, Physical review letters.

[4]  Yves Dumont,et al.  Impact of environmental factors on mosquito dispersal in the prospect of sterile insect technique control , 2013, Comput. Math. Appl..

[5]  M. Newman,et al.  Random graphs with arbitrary degree distributions and their applications. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Jinde Cao,et al.  Global stability of an epidemic model with carrier state in heterogeneous networks , 2015 .

[7]  Lourdes Esteva,et al.  Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique. , 2005, Mathematical biosciences.

[8]  Jinde Cao,et al.  Global dynamics of a network epidemic model for waterborne diseases spread , 2014, Appl. Math. Comput..

[9]  R Meiswinkel,et al.  Mosquitoes and Culicoides biting midges: vector range and the influence of climate change. , 2015, Revue scientifique et technique.

[10]  Hal L. Smith,et al.  Monotone Dynamical Systems: An Introduction To The Theory Of Competitive And Cooperative Systems (Mathematical Surveys And Monographs) By Hal L. Smith , 1995 .

[11]  R. Pastor-Satorras,et al.  Epidemic spreading in correlated complex networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  A. Reynolds,et al.  Free-Flight Odor Tracking in Drosophila Is Consistent with an Optimal Intermittent Scale-Free Search , 2007, PloS one.

[13]  J. Watmough,et al.  Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. , 2002, Mathematical biosciences.

[14]  G. Yan,et al.  Influence of host and larval habitat distribution on the abundance of African malaria vectors in western Kenya. , 2002, The American journal of tropical medicine and hygiene.

[15]  S. Wanji,et al.  Spatial distribution, environmental and physicochemical characterization of Anopheles breeding sites in the Mount Cameroon region. , 2009, Journal of Vector Borne Diseases.

[16]  D. Kirschner,et al.  A methodology for performing global uncertainty and sensitivity analysis in systems biology. , 2008, Journal of theoretical biology.

[17]  Herbert W. Hethcote,et al.  Stability of the endemic equilibrium in epidemic models with subpopulations , 1985 .

[18]  B. Knols,et al.  Odors influence mosquito behaviour. , 1997 .

[19]  G. Yan,et al.  A network population model of the dynamics and control of African malaria vectors. , 2010, Transactions of the Royal Society of Tropical Medicine and Hygiene.

[20]  Jinde Cao,et al.  Revisiting node-based SIR models in complex networks with degree correlations , 2015 .

[21]  A L Lloyd,et al.  Spatial heterogeneity in epidemic models. , 1996, Journal of theoretical biology.

[22]  Alessandro Vespignani,et al.  Invasion threshold in heterogeneous metapopulation networks. , 2007, Physical review letters.

[23]  J. Hyman,et al.  Determining Important Parameters in the Spread of Malaria Through the Sensitivity Analysis of a Mathematical Model , 2008, Bulletin of mathematical biology.

[24]  Roumen Anguelov,et al.  Mathematical modeling of sterile insect technology for control of anopheles mosquito , 2011, Comput. Math. Appl..

[25]  Alessandro Vespignani,et al.  Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: theory and simulations. , 2007, Journal of theoretical biology.

[26]  P. Auger,et al.  The Ross-Macdonald model in a patchy environment. , 2008, Mathematical biosciences.

[27]  NAKUL CHITNIS,et al.  Bifurcation Analysis of a Mathematical Model for Malaria Transmission , 2006, SIAM J. Appl. Math..

[28]  Joan Saldaña.  Modelling the Spread of Infectious Diseases in Complex Metapopulations , 2010 .

[29]  M. Takagi,et al.  Heterogeneity in the host preference of Japanese encephalitis vectors in Chiang Mai, northern Thailand. , 2000, Transactions of the Royal Society of Tropical Medicine and Hygiene.

[30]  J. Lubuma,et al.  Global dynamics of a vaccination model for infectious diseases with asymptomatic carriers. , 2016, Mathematical biosciences and engineering : MBE.