A new stage-structured model for the population dynamics of the mosquito (a major vector for numerous vector-borne diseases), which takes the form of a deterministic system of non-autonomous nonlinear differential equations, is designed and used to study the effect of variability in temperature and rainfall on mosquito abundance in a community. Two functional forms of eggs oviposition rate, namely the Verhulst-Pearl logistic and Maynard-Smith-Slatkin functions, are used. Rigorous analysis of the autonomous version of the model shows that, for any of the oviposition functions considered, the trivial equilibrium of the model is locally-and globally-asymptotically stable if a certain vectorial threshold quantity is less than unity. Conditions for the existence and global asymptotic stability of the non-trivial equilibrium solutions of the model are also derived. The model is shown to undergo a Hopf bifurcation under certain conditions (and that increased density-dependent competition in larval mortality reduces the likelihood of such bifurcation). The analyses reveal that the Maynard-Smith-Slatkin oviposition function sustains more oscillations than the Verhulst-Pearl logistic function (hence, it is more suited, from ecological viewpoint, for modeling the egg oviposition process). The non-autonomous model is shown to have a globally-asymptotically stable trivial periodic solution, for each of the oviposition functions, when the associated reproduction threshold is less than unity. Furthermore, this model, in the absence of density-dependent mortality rate for larvae, has a unique and globally-asymptotically stable periodic solution under certain conditions. Numerical simulations of the non-autonomous model, using mosquito surveillance and weather data from the Peel region of Ontario, Canada, show a peak mosquito abundance for temperature and rainfall values in the range [20−25] • C and [15-35] mm, respectively. These ranges are recorded in the Peel region between July and August (hence, this study suggests that anti-mosquito control effects should be intensified during this period).
There are more than 300 avian species that can transmit West Nile virus (WNv). In general, the corvid and non-corvid families of birds have different responses to the virus, with corvids suffering a higher disease-induced mortality rate. By taking both corvids and non-corvids as the primary reservoir hosts and mosquitoes as vectors; we formulate and study a system of ordinary differential equations to model a single season of the transmission dynamics of WNv in the mosquito-bird cycle. We calculate the basic reproduction number and analyze the existence and stability of the equilibria. The existence of a backward bifurcation gives a further sub-threshold condition beyond the basic reproduction number for the spread of the virus. We also discuss the role of corvids and non-corvids in spreading the virus. We conclude that knowledge of the relative abundance of corvid bird species and other mammals assist us in accurate estimation of the epidemic of WNv.
We prove convergence of the Adomian decomposition method for an abstract initial-value problem using the method of majorants from the Cauchy-Kowalevskaya theorem for differential equations with analytic vector fields. Convergence rates of the Adomian method are investigated in the context of the nonlinear Schrödinger equation.
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