Algebraic form and analysis of SIR epidemic dynamics over probabilistic dynamic networks

The outbreak of corona virus disease 2019 has profoundly affected people’s way of life.It is increasingly necessary to investigate epidemics over social networks.This paper studies susceptible-infected-removed(SIR)epidemics via the semi-tensor product.First,a formal susceptible-infected-removed epidemic dynamic model over probabilistic dynamic networks(SIRED-PDN)is given.Based on an evolutionary rule,the algebraic form for the dynamics of individual states and network topologies is given,respectively.Second,the SIRED-PDN can be described by a probabilistic mix-valued logical network.After providing an algorithm,all possible final spreading equilibria can be obtained for any given initial epidemic state and network topology by seeking attractors of the network.And the shortest time for all possible initial epidemic state and network topology profiles to evolve to the final spreading equilibria can be obtained by seeking the transient time of the network.Finally,an illustrative example is given to show the effectiveness of our model.

Limiting stand density and basal area projection models for even-aged Tecomella undulata plantations in a hot arid region of India

This paper presents equations for estimating limiting stand density for Z undulata plantations grown in hot desert areas of Raj asthan State in India. Five different stand level basal area projection models, belonging to the path invariant algebraic difference form of a non-linear growth function, were also tested and compared. These models can be used to predict future basal area as a function of stand variables like dominant height and stem number per hectare and are necessary for reviewing different silvicultural treatment options. Data from 22 sample plots were used for modelling. An all possible growth intervals data structure was used. Both, qualitative and quantitative criteria were used to compare alternative models. The Akaike＇s information criteria differ- ence statistic was used to analyze the predictive ability of the models. Results show that the model proposed by Hui and Gadow performed best and hence this model is recommended for use in predicting basal area development in 12 undulata plantations in the study area. The data used were not from thinned stands, and hence the models may be less accurate when used for predictions when natural mortality is very significant.

Simplification of Shapley value for cooperative games via minimum carrier

Shapley value is one of the most fundamental concepts in cooperative games.This paper investigates the calculation of the Shapley value for cooperative games and establishes a new formula via carrier.Firstly,a necessary and sufficient condition is presented for the verification of carrier,based on which an algorithm is worked out to find the unique minimum carrier.Secondly,by virtue of the properties of minimum carrier,it is proved that the profit allocated to dummy players(players which do not belong to the minimum carrier)is zero,and the profit allocated to players in minimum carrier is only determined by the minimum carrier.Then,a new formula of the Shapley value is presented,which greatly reduces the computational complexity of the original formula,and shows that the Shapley value only depends on the minimum carrier.Finally,based on the semi-tensor product(STP)of matrices,the obtained new formula is converted into an equivalent algebraic form,which makes the new formula convenient for calculation via MATLAB.

Zero Product Determined Jordan Algebras, II

The Albert algebra and the Jordan algebra of a nondegenerate symmetric bilinear form are shown to be zero product determined.