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Global behavior of a multi-group SEIR epidemic model with spatial diffusion in a heterogeneous environment

Liu Pengyan, Li Hong-Xu

International Journal of Applied Mathematics and Computer Science

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2022

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Vol. 32, no. 2

271--283

EN

In this paper, we propose a multi-group SEIR epidemic model with spatial diffusion, where the model parameters are spatially heterogeneous. The positivity and ultimate boundedness of the solution, as well as the existence of a global attractor of the associated solution semiflow, are established. The definition of the basic reproduction number is given by utilizing the next generation operator approach, whereby threshold-type results on the global dynamics in terms of this number are established. That is, when the basic reproduction number is less than one, the disease-free steady state is globally asymptotically stable, while if it is greater than one, uniform persistence of this model is proved. Finally, the feasibility of the main theoretical results is shown with the aid of numerical examples for a model with two groups.

2

Przestrzenne zróżnicowanie stanu i rozwoju bazy hotelowej w Polsce według województw w okresie 2002–2017

Wojdacki Krzysztof

Biuletyn Komitetu Przestrzennego Zagospodarowania Kraju PAN

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2018

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No 272

77--93

EN

The main purpose of the author was to determine the spatial diversity of hotel base development in the regional (voivodship) system. Assuming that many factors determine the level of hotel base development in a given area, the author has used a multidimensional approach, or more precisely – a multidimensional comparative analysis in its dynamic variant. Following the adopted algorithm, a potential set of diagnostic variables was identified on the basis of secondary information sources (date of CSO local data bank), then using factor analysis to reduce it to the final set of quasi-variables (four main factors). Based on them, a synthetic measure of the development of the hotel base of particular voivodships was constructed. This in turn allowed to determine the voivodship hierarchical structure due to particular main factors and synthetic variable, with its changes in the years 2002–2017. The voivodship were also divided into homogeneous groups, according to the value of the synthetic measure of development, as well as the direction and intensity of its temporal changes.

3

The enigma of metropolis: its spatial diversity and methods of diagnosis

Bonenberg W.

Czasopismo Techniczne. Architektura

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2015

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Y. 112, iss. 8-A

33--38

EN

The type and scale of internal diversification are the main characteristics of metropolitan areas. There are some agglomerations where areas differ only slightly and some where territorial disparities are significant. The question is whether this internal diversification is a factor which stimulates spatial and economic development? In other words, which strategy should be chosen in terms of planning solutions: the egalitarian strategy targeted at equalisation or the one focused on using the competitive advantage of diversification? These questions are related to the problem of diagnosing the diversity of metropolitan areas.

PL

Typ i skala wewnętrznej zróżnicowania to główne cechy charakterystyczne terenów metropolitalnych. Istnieją aglomeracje, których obszary różnią się tylko nieznacznie i takie, gdzie zachodzą znaczące dysproporcje terytorialne. Powstaje pytanie, czy wspomniane wewnętrzne zróżnicowanie jest czynnikiem stymulującym rozwój przestrzenny i gospodarczy? Innymi słowy, którą strategię należy dobrać w kategoriach rozwiązań planistycznych: ukierunkowaną na wyrównywanie strategię egalitarną, czy tę skoncentrowaną na wykorzystywaniu konkurencyjnej przewagi dywersyfikacji? Powyższe pytania są związane z problemem diagnozowania różnorodności obszarów metropolitalnych.

4

Spatial Heterogeneity and Local Oscillation Phase Drifts Individual-Based Simulations of a Prey-Predator System

Waniewski J., Jędruch W.

International Journal of Applied Mathematics and Computer Science

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2000

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Vol. 10, no 1

175-192

EN

Individual-based simulations of a simple prey-predator system of Lotka-Volterra type were carried out on a tessellation of identical squares with discrete time steps. The particles representing individuals moved freely along (roughly) straight lines with constant (on the average) velocity, and changed their movement during a collision with another particle. Individuals were of two types: preys (with free exponential population growth) and predators (with exponential population decrease in the absence of a prey, they attack with probability one and are characterized by zero handling and gestationtimes). Therefore the system might be also interpreted as a chemical reactionin a gas. For this simple system, a spontaneous generation of complex spatio-temporal pattern was observed with wavy spatial patterns and tendency for preys to form clusters surrounded by predators if the population density was high. The oscillations of the system were investigated at different spatial scales, and the phase lag between the oscillations in different local observation windows was demonstrated. The parameters of the classical Lotka-Volterra equations were estimated and the impact of the migration and the oscillation phase drift on the parameter values was discussed.