Virial theorem in scalar tensor fourth order gravity

In this paper, we study, in the Newtonian limit, the virial theorem in the context of a scalar tensor fourth order gravity. In particular, we show, that for a isolated galaxy in viral equilibrium, a specific class of scalar tensor fourth order gravity, i.e. $$f(R,\phi )+\omega (\phi )\,\phi _{;\alpha }\,\phi ^{;\alpha }$$ f ( R , ϕ ) + ω ( ϕ ) ϕ ; α ϕ ; α in not suitable to explain the large fraction of dark matter necessary to have the flatness of the galaxies rotation curves experimentally observed.

A mixed SIR-SIS model to contain a virus spreading through networks with two degrees

Due to the fact that the “nodes” and “links” of real networks are heterogeneous, to model computer viruses prevalence throughout the Internet, we borrow the idea of the reduced scale free network which was introduced recently. The purpose of this paper is to extend the previous deterministic two subchains of Susceptible-Infected-Susceptible (SIS) model into a mixed Susceptible-Infected-Recovered and Susceptible-Infected-Susceptible (SIR–SIS) model to contain the computer virus spreading over networks with two degrees. Moreover, we develop its stochastic counterpart. Due to the high protection and security taken for hubs class, we suggest to treat it by using SIR epidemic model rather than the SIS one. The analytical study reveals that the proposed model admits a stable viral equilibrium. Thus, it is shown numerically that the mean dynamic behavior of the stochastic model is in agreement with the deterministic one. Unlike the infection densities [Formula: see text] and [Formula: see text] which both tend to a viral equilibrium for both approaches as in the previous study, [Formula: see text] tends to the virus-free equilibrium. Furthermore, since a proportion of infectives are recovered, the global infection density [Formula: see text] is minimized. Therefore, the permanent presence of viruses in the network due to the lower-degree nodes class. Many suggestions are put forward for containing viruses propagation and minimizing their damages.

Global Stability of a Computer Virus Propagation Model with Two Kinds of Generic Nonlinear Probabilities

Vaccination is one of the most effective measures for suppressing the spread of computer virus, and the bilinear incidence rate assumption for the majority of previous models, which is a good first approximation of the general incidence rate, is in disagreement with the reality. In this paper, a new dynamical model with two kinds of generic nonlinear probabilities (incidence rate and vaccination probability) is established. An exhaustive mathematical analysis of this model shows that (a) there are two equilibria, virus-free equilibrium and viral equilibrium, and (b) the virus-free (or viral) equilibrium is globally asymptotically stable when the basic reproduction number is less (or greater) than unity. The analysis of the basic reproduction number is also included. Additionally, some numerical examples are given to illustrate the main results, from which it can be seen that the generic nonlinear vaccination is helpful to strengthen computer security.

Propagation Behavior of Virus Codes in the Situation That Infected Computers Are Connected to the Internet with Positive Probability

All the known models describing the propagation of virus codes were based on the assumption that a computer is uninfected at the time it is being connected to the Internet. In reality, however, it is much likely that infected computers are connected to the Internet. This paper is intended to investigate the propagation behavior of virus programs provided infected computers are connected to the Internet with positive probability. For that purpose, a new model characterizing the spread of computer virus is proposed. Theoretical analysis of this model indicates that (1) there is a unique (viral) equilibrium, and (2) this equilibrium is globally asymptotically stable. Further study shows that, by taking active measures, the percentage of infected computers can be made below an acceptable threshold value.