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Some modified versions of susceptible-infected-recovered-susceptible (SIRS) model are defined on small-world networks. Latency, incubation and variable susceptibility are separately included. Phase transitions in these models are studied. Then inhomogeneous models are introduced. In some cases, the application of the models to small-world networks is shown to increase the epidemic region.

We show that Spatially Inhomogeneous (SI) and Irrotational dust models admit a six-dimensional algebra of Intrinsic Conformal Vector Fields (ICVFs) $X_{\alpha }$ satisfying $p_a^c{p}_b^d{{ℒ}}_{X_{\alpha }}{p}_{cd}=2\varphi (X_{\alpha })p_{ab}$, where $p_{ab}$ is the associated metric of the two-dimensional distribution ${𝒳}$ normal to the fluid velocity $u^a$ and the radial unit space-like vector field $x^a$. The Intrinsic Conformal (IC) algebra is determined for each of the curvature value $𝜖$ that characterizes the structure of the screen space ${𝒳}$. In addition the conformal flatness of the hypersurfaces $u=0$ indicates the existence of a ten-dimensional algebra of ICVFs of the three-dimensional metric $h_{ab}$. We illustrate this expectation and propose a method to derive them by giving explicitly the seven proper ICVFs of the Lemaître–Tolman–Bondi (LTB) model which represents the simplest subclass within the Szekeres family.

The consistency of the constraint with the evolution equations for spatially inhomogeneous and irrotational silent (SIIS) models of Petrov type I, demands that the former are preserved along the timelike congruence represented by the velocity of the dust fluid, leading to new nontrivial constraints. This fact has been used to conjecture that the resulting models correspond to the spatially homogeneous (SH) models of Bianchi type I, at least for the case where the cosmological constant vanish. By exploiting the full set of the constraint equations as expressed in the 1+3 covariant formalism and using elements from the theory of the spacelike congruences, we provide a direct and simple proof of this conjecture for vacuum and dust fluid models, which shows that the Szekeres family of solutions represents the most general class of SIIS models. The suggested procedure also shows that, the uniqueness of the SIIS of the Petrov type D is not, in general, affected by the presence of a nonzero pressure fluid. Therefore, in order to allow a broader class of Petrov type I solutions apart from the SH models of Bianchi type I, one should consider more general "silent" configurations by relaxing the vanishing of the vorticity and the magnetic part of the Weyl tensor but maintaining their "silence" properties, i.e. the vanishing of the curls of E_ab, H_ab and the pressure p.

We study the volume averaging of inhomogeneous metrics within GR and discuss its shortcomings, such as gauge dependence, singular behavior as a result of caustics, and causality violations. To remedy these shortcomings, we suggest some modifications to this method. As a case study we focus on the inhomogeneous structured FRW model based on a flat LTB metric. The effect of averaging is then studied in terms of an effective backreaction fluid. It is shown that, contrary to the claims in the literature, the backreaction fluid behaves like a dark matter component, instead of dark energy, having a density of the order of 10^-5 times the matter density, and, most importantly, it is gauge-dependent.

The Szekeres inhomogeneous models are used in order to fit supernova combined data sets. We show that with a choice of the spatial curvature function that is guided by current observations, the models fit the supernova data almost as well as the LCDM model without requiring a dark energy component. The Szekeres models were originally derived as an exact solution to Einstein’s equations with a general metric that has no symmetries and are regarded as good candidates to model the true lumpy universe that we observe. The null geodesics in these models are not radial and require numerical integration. The best fit model found is also consistent with the requirement of spatial flatness at the CMB scale. Interestingly, apparent acceleration due to inhomogeneities is also linked to the profound problem of averaging in cosmology. The first results presented here seem to encourage further investigations of apparent acceleration using various inhomogeneous models using full CMB spectra as well as large structure.

Exact Lemaître-Tolman-Bondi solutions are obtained in -Tolman-Bondi solutions are obtained inf(R)-gravity. In particular, we discuss how such solutions can be consistently compared with General Relativity according to post-Newtonian and post-Minkowskian limits. The cosmological application of the results is considered towards the observed apparent accelerated behavior of the cosmic fluid.