Qualitative analysis for a Wolbachia infection model with diffusion

We consider a reaction-diffusion model which describes the spatial Wolbachia spread dynamics for a mixed population of infected and uninfected mosquitoes. By using linearization method, comparison principle and Leray-Schauder degree theory, we investigate the influence of diffusion on the Wolbachia infection dynamics. After identifying the system parameter regions in which diffusion alters the local stability of constant steady-states, we find sufficient conditions under which the system possesses inhomogeneous steady-states. Surprisingly, our mathematical analysis, with the help of numerical simulations, indicates that diffusion is able to lower the threshold value of the infection frequency over which Wolbachia can invade the whole population.

[1]  A. Turing.  The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

[2]  C. Calisher.  Persistent Emergence of Dengue , 2005, Emerging infectious diseases.

[3]  S. Ritchie,et al.  The wMel Wolbachia strain blocks dengue and invades caged Aedes aegypti populations , 2011, Nature.

[4]  Ernst Caspari,et al.  ON THE EVOLUTIONARY IMPORTANCE OF CYTOPLASMIC STERILITY IN MOSQUITOES , 1959 .

[5]  Eva Harris,et al.  Global spread and persistence of dengue. , 2008, Annual review of microbiology.

[6]  Zhiyong Xi,et al.  Generation of a novel Wolbachia infection in Aedes albopictus (Asian tiger mosquito) via embryonic microinjection. , 2005, Insect biochemistry and molecular biology.

[7]  Jifa Jiang,et al.  Saddle-point behavior for monotone semiflows and reaction–diffusion models☆ , 2004 .

[8]  Louis Nirenberg,et al.  Topics in Nonlinear Functional Analysis , 2001 .

[9]  Paul H. Rabinowitz,et al.  Some global results for nonlinear eigenvalue problems , 1971 .

[10]  Wei-Ming Ni,et al.  Turing patterns in the Lengyel-Epstein system for the CIMA reaction , 2005 .

[11]  H. Weinberger,et al.  Maximum principles in differential equations , 1967 .

[12]  M. Turelli,et al.  Factors affecting the distribution of cytoplasmic incompatibility in Drosophila simulans. , 1990, Genetics.

[13]  S. Ritchie,et al.  Successful establishment of Wolbachia in Aedes populations to suppress dengue transmission , 2011, Nature.

[14]  Bo Zheng,et al.  Modeling Wolbachia Spread in Mosquitoes Through Delay Differential Equations , 2014, SIAM J. Appl. Math..

[15]  Thomas Walker,et al.  Wolbachia and the biological control of mosquito‐borne disease , 2011, EMBO reports.

[16]  Bodil N. Cass,et al.  Stable Introduction of a Life-Shortening Wolbachia Infection into the Mosquito Aedes aegypti , 2009, Science.

[17]  Diffusion-induced extinction of a superior species in a competition system , 1998 .

[18]  Kazuo Kishimoto,et al.  The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains , 1985 .

[19]  A. James,et al.  Microbe‐induced cytoplasmic incompatibility as a mechanism for introducing transgenes into arthropod populations , 1999, Insect molecular biology.

[20]  Jianshe Yu,et al.  Wolbachia infection dynamics by reaction-diffusion equations , 2015 .

[21]  Jianshe Yu,et al.  Wolbachia spread dynamics in stochastic environments. , 2015, Theoretical population biology.

[22]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[23]  Ary A. Hoffmann,et al.  Cytoplasmic incompatibility in insects , 1997 .

[24]  M J Keeling,et al.  The invasion and coexistence of competing Wolbachia strains , 2003, Heredity.

[25]  Zhiyong Xi,et al.  The Endosymbiotic Bacterium Wolbachia Induces Resistance to Dengue Virus in Aedes aegypti , 2010, PLoS pathogens.

[26]  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 .

[27]  R. G. Casten,et al.  Stability Properties of Solutions to Systems of Reaction-Diffusion Equations , 1977 .

[28]  Mingxin Wang,et al.  Non‐Constant Positive Steady States of a Predator‐Prey System with Non‐Monotonic Functional Response and Diffusion , 2004 .

[29]  Zhiyong Xi,et al.  Wolbachia Establishment and Invasion in an Aedes aegypti Laboratory Population , 2005, Science.