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We study the local and global existence and uniqueness of strict solution and the local well-posedness for a class of abstract "explicit" neutral integro-differential equations with state-dependent delay. In the last section some examples motivated by explicit neutral ordinary integro-differential equations arising in the theory of population dynamics are presented. [ABSTRACT FROM AUTHOR]

We represent the optimal control functions by neural networks and solve optimal control problems by deep learning techniques. Adjoint sensitivity analysis is applied to train the neural networks embedded in differential equations. This method can not only be applied in classic epidemic control problems, but also in epidemic forecasting, discovering unknown mechanisms, and the ideas behind can give new insights to traditional mathematical epidemiological problems. [ABSTRACT FROM AUTHOR]

An immuno-epidemiological model of pathogen transmission is developed. This model incorporates two main features: (i) the epidemiological model includes within-host pathogen dynamics for an infectious disease, (ii) the susceptible individuals to the infection experience a series of exposures via the pathogen before becoming infectious. It is shown that this model leads naturally to a system of differential delay equations of the threshold type and that these equations can be transformed, in a biologically natural way, to differential equations with state-dependent delay. An interesting dynamical behavior of the model is the bistability phenomena, when the basic reproductive ratio $$R_{0}$$ is less than unity, which raises many new challenges to effective infection control. [ABSTRACT FROM AUTHOR]

From February to May, 2013, 132 human avian influenza H7N9 cases were identified in China resulting in 37 deaths. We developed a novel, simple and effective compartmental modeling framework for transmissions among (wild and domestic) birds as well as from birds to human, to infer important epidemiological quantifiers, such as basic reproduction number for bird epidemic, bird-to-human infection rate and turning points of the epidemics, for the epidemic via human H7N9 case onset data and to acquire useful information regarding the bird-to-human transmission dynamics. Estimated basic reproduction number for infections among birds is 4.10 and the mean daily number of human infections per infected bird is 3.16*10−5 [3.08*10−5, 3.23*10−5]. The turning point of 2013 H7N9 epidemic is pinpointed at April 16 for bird infections and at April 9 for bird-to-human transmissions. Our result reveals very low level of bird-to-human infections, thus indicating minimal risk of widespread bird-to-human infections of H7N9 virus during the outbreak. Moreover, the turning point of the human epidemic, pinpointed at shortly after the implementation of full-scale control and intervention measures initiated in early April, further highlights the impact of timely actions on ending the outbreak. This is the first study where both the bird and human components of an avian influenza epidemic can be quantified using only the human case data. [ABSTRACT FROM AUTHOR]

Abstract: We develop a global Hopf bifurcation theory for a system of functional differential equations with state-dependent delay. The theory is based on an application of the homotopy invariance of -equivariant degree using the formal linearization of the system at a stationary state. Our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local Hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state-dependent delay. [Copyright &y& Elsevier]

We consider the following system of delay differential equations \[ \dot{x}_i(t)=-x_i(t)-\alpha [S(x_{i-1}(t-\tau)) + S(x_{i+1}(t-\tau))], \] where i (mod 3), and S(x) is the hyperbolic tangent function. The system models the evolution of a network of three identical neurons with delayed feedback. We provide a full classification of all equilibria and their stability in the (α,τ)-parameter space. Such a classification is essential for the description of spatio-temporal patterns of the model system and for the applications to dynamic memory storage and retrieval. [ABSTRACT FROM AUTHOR]

Abstract: We propose a non-local PDE model for the evolution of a single species population that involves delayed feedback, where the delay such as the maturation time in the delayed birth rate, is selective and the selection depends on the status of the system. This delay selection, in contrast with the usual state-dependent delay widely used in ordinary delay differential equation, ensures the Lipschitz continuity of the nonlinear functional in the classical phase space. We also develop the local theory, and the existence and upper semi-continuity of the global attractor with respect to parameters. [Copyright &y& Elsevier]

Abstract: We consider a class of hyperbolic–parabolic equation with time delay and a non-local feedback due to the maturation for the adult population density of a single species population, and we show that the wave profile is described by a hybrid system that consists of an integral transformation and an ordinary differential equation. We show the existence and uniqueness of a travelling wavefront of the hyperbolic–parabolic system in the so-called bistable case, by considering the same problem for a properly parametrized parabolic system, and then by considering the continuous dependence of the wave speed on the parameter involved. [Copyright &y& Elsevier]

This paper is concerned with the existence, smoothness and attractivity of invariant manifolds for evolutionary processes on general Banach spaces when the nonlinear perturbation has a small global Lipschitz constant and locally Ck-smooth near the trivial solution. Such a nonlinear perturbation arises in many applications through the usual cut-off procedure, but the requirement in the existing literature that the nonlinear perturbation is globally Ck-smooth and has a globally small Lipschitz constant is hardly met in those systems for which the phase space does not allow a smooth cut-off function. Our general results are illustrated by and applied to partial functional differential equations for which the phase space C([−r,0],X) (where r>0 and X being a Banach space) has no smooth inner product structure and for which the validity of variation-of-constants formula is still an interesting open problem. [Copyright &y& Elsevier]

In this paper, we establish the existence and describe the global structure of traveling waves for a class of lattice delay differential equations describing cellular neural networks with distributed delayed signal transmission. We describe the transition of wave profiles from monotonicity, damped oscillation, periodicity, unboundedness and back to monotonicity as the wave speed is varied. We also describe an interval of the wave speed where the structure of the wave solution is unknown since the corresponding profile equation involves distributed argument of both advanced and retarded types, and we present some preliminary numerical simulation to illustrate the complexity. [ABSTRACT FROM AUTHOR]

We study the existence and uniqueness of C 1 + α -strict solutions for a general class of abstract differential equations with state dependent delay. We also study the local well-posedness of this type of problems on subspaces of C 1 + α ( [ − p , 0 ] ; X ) . Some examples involving partial differential equations are presented. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we derive a population model for the growth of a single species on a two-dimensional strip with Neumann and Robin boundary conditions. We show that the dynamics of the mature population is governed by a reaction–diffusion equation with delayed global interaction. Using the theory of asymptotic speed of spread and monotone traveling waves for monotone semiflows, we obtain the spreading speed , the non-existence of traveling waves with wave speed , and the existence of monotone traveling waves connecting the two equilibria for . [Copyright &y& Elsevier]

Abstract: Entire solutions for monostable reaction–diffusion equations with nonlocal delay in one-dimensional spatial domain are considered. A comparison argument is employed to prove the existence of entire solutions which behave as two traveling wave solutions coming from both directions. Some new entire solutions are also constructed by mixing traveling wave solutions with heteroclinic orbits of the spatially averaged ordinary differential equations, and the existence of such a heteroclinic orbit is established using the monotone dynamical systems theory. Key techniques include the characterization of the asymptotic behaviors of solutions as in term of appropriate subsolutions and supersolutions. Two models of reaction–diffusion equations with nonlocal delay arising from mathematical biology are given to illustrate main results. [Copyright &y& Elsevier]

Abstract: We consider a network of two coupled neurons with delayed feedback. We show that the connection topology of the network plays a fundamental role in classifying the rich dynamics and bifurcation phenomena. Regarding eigenvalues of the connection matrix as bifurcation parameters, we obtain codimension 1 bifurcations (including a fold bifurcation and a Hopf bifurcation) and codimension 2 bifurcations (including fold–Hopf bifurcations and Hopf–Hopf bifurcations). We also give concrete formulae for the normal form coefficients derived via the center manifold reduction that give detailed information about the bifurcation and stability of various bifurcated solutions. In particular, we obtain stable or unstable equilibria, periodic solutions, quasi-periodic solutions, and sphere-like surfaces of solutions. We also show how to evaluate critical normal form coefficients from the original system of delay-differential equations without computing the corresponding center manifolds. [Copyright &y& Elsevier]

Abstract: The Vector Integration to Endpoint (VITE) circuit describes a real-time neural network model simulating behavioral and neurobiological properties of planned arm and hand movements by the interaction of two populations of neurons. We analyze the speed-accuracy trade-off generated by this circuit, generalized to include delayed feedback. With delay, two important new properties of the circuit emerge: a breakdown of Fitts'' law when the movement time is small relative to the delay; and a positive Fitts'' law Y-intercept. This breakdown of Fitts'' law for tasks with small Index of Difficulty has been previously observed experimentally, and we suggest it may be attributed at least in part to delay effects in the nervous system elaborated by the model. Additionally, this gives a theoretical explanation for why positive Fitts'' law Y-intercept should occur, and that it is related to the delay within the movement circuit. [Copyright &y& Elsevier]

Abstract: We consider a delayed network of two neurons with both self-feedback and interaction described by an all-or-none threshold function. The model describes a combination of analog and digital signal processing in the network and takes the form of a system of delay differential equations with discontinuous nonlinearity. We show that the dynamics of the network can be understood in terms of the iteration of a one-dimensional map, and we obtain simple criteria for the convergence of solutions, the existence, multiplicity and attractivity of periodic solutions. [Copyright &y& Elsevier]