M.E. Parrott

Diffusion epidemic models with incubation and crisscross dynamics

A diffusion age-structured epidemic model is analyzed. The model describes an epidemic in a host-vector two-population system. Each population is diffusing in a spatial region. Each population is divided into susceptible, incubating, and infectious subclasses. The incubating and infectious subclasses in each population are determined by a structure variable corresponding to age since infection. The model consists of a system of nonlinear partial differential equations with crisscross dynamics. The existence, uniqueness, and asymptotic behavior of solutions are analyzed.

The weak solution of a functional differential equation in a general Banach space

Abstract The existence of a unique, weak, Lipschitz continuous solution of the abstract functional differential equation u ′( t ) + A ( t ) u ( t ) = G ( t , u t ), t ϵ [0, T ] (FDE) u ( t ) = φ ( t ), t ϵ [− r ,0] is obtained by a fixed point argument. Here X is a general Banach space, A ( t ): D ( A ( t )) ⊂ X → X is m -accretive, t ϵ [0, T ], and G is Lipschitz continuous in both variables. Stability properties of this weak solution are given, and a first order hyperbolic PDE to which the theory can be applied is considered.

Convergence of the Kato approximants for evolution equations involving functional perturbations

The existence of a unique strong solution of the nonlinear abstract functional differential equation u′(t) + A(t)u(t) = F(t,ut), u0 = φeC1(¦−r,0¦,X),te¦0, T¦, (E) is established. X is a Banach space with uniformly convex dual space and, for tϵ ¦0, T¦, A(t) is m-accretive and satisfies a time dependence condition suitable for applications to partial differential equations. The function F satisfies a Lipschitz condition. The novelty of the paper is that the solution u(t) of (E) is shown to be the uniform limit (as n → ∞) of the sequence un(t), where the functions un(t) are continuously differentiate solutions of approximating equations involving the Yosida approximants. Thus, a straightforward approximation scheme is now available for such equations, in parallel with the approach involving the use of nonlinear evolution operator theory.

Global Dynamics of Singularly Perturbed Hodgkin-Huxley Equations

In their well-known work, Hodgkin and Huxley considered the following model for nerve impulse transmission across an axon:

Periodicity in diffusive age-structured SEIR models

\frac{{\partial V}}{{\partial t}} - \frac{{{\partial ^2}V}}{{\partial {x^2}}} = {g_{Na}}{m^3}h\left( {{V_{Na}} - V} \right) + {g_K}{n^4}\left( {{V_K} - V} \right) + {g_L}\left( {{V_L} - V} \right)

Almost periodic solutions of evolution equations

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A diffusive system with age dependency modeling FIV

\frac{{\partial m}}{{\partial t}} = \left( {{m_\infty } - m} \right)/{T_m}