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In this paper, an epidemic model with spatial dependence is studied and results regarding its stability and numerical approximation are presented. We consider a generalization of the original Kermack and McKendrick model in which the size of the populations differs in space. The use of local spatial dependence yields a system of partial-differential equations with integral terms. The uniqueness and qualitative properties of the continuous model are analyzed. Furthermore, different spatial and temporal discretizations are employed, and step-size restrictions for the discrete model's positivity, monotonicity preservation, and population conservation are investigated. We provide sufficient conditions under which high-order numerical schemes preserve the stability of the computational process and provide sufficiently accurate numerical approximations. Computational experiments verify the convergence and accuracy of the numerical methods., Comment: 34 pages, 4 figures, 4 tables

We present and analyse numerical methods with operator splitting procedures, applied to an epidemic model which takes into account the space-dependence of the infection. We derive conditions on the time step, under which the numerical methods preserve the non-negativity and monotonicity properties of the exact solution. Our results are illustrated by numerical experiments.

In this paper we present some extensions of the classical SIR model with non-symmetric spatial dependence. SIR-type models usually describe the epidemic in a population, which is split into three categories, namely the susceptibles (S), the infected (I) and the recovered (R). The proposed model yields a system of partial integro-differential equations. Two methods handling the integrals of the equations are presented. We give numerical examples which show that the discrete models preserve the basic qualitative properties of the original biological process, and these results are supported by the oretical theorems.

We have used simple learning rules to study how firing maps containing triangular grids-as found in in vivo experiments-can be developed by Hebbian means in realistic robotic simulations. We started from typical non-local postrhinal neuronal responses. We found that anti-Hebbian weight pruning can develop triangular grids under certain conditions. Experimental evidences and the present study suggest that within this model, whitening is a bottom-up process, whereas weight pruning and possibly the non-linear extension of whitening to bottom-up information maximization are guided by top-down influences that reorganize entorhinal responses. We connect our model to the computational model of the entorhinal-hippocampal region of Lorincz and Buzsaki. In the joined model, the hippocampus is the origin of response reorganization. The joined model may provide insights for memory reorganization guided by hippocampal supervision.

It has been argued that the processing of sensory information in the entorhinal-hippocampal loop involves independent component analysis (ICA) on temporally concatenated inputs. Here, we demonstrate that ICA in a realistic robot simulation on a U-shaped track forms place fields similar to those found in rat experiments in vivo.

It has been suggested that sensory information processing makes use of a factorial code. It has been shown that the major components of the hippocampal-entorhinal loop can be derived by conjecturing that the task of this loop is forming and encoding independent components (ICs), one type of factorial codes. However, continuously changing environment poses additional requirements on the coding that can be (partially) satisfied by extending the analysis to the temporal domain and performing IC analysis on concatenated inputs of time slices. We use computer simulations to decide whether IC analysis on temporal sequences can produce place fields in labyrinths or not.