Patent Application: US-201514720784-A

Abstract:
measurements of individual - level infectivity , susceptibility and baseline infection risk to biological or social contagion are made for large number of entities from their contact relation , sequential infection occurrences and environmental data , using computer implemented mcmc for a bayesian estimation of an integrative latent trait response model . the method is useful for precise and efficient contagion control and prevention .

Description:
an interaction network is defined by a set of multilaterally interacted entities and the contact relation among them . a matrix n represents how the individual entities are linked with each other by physical contact in disease contagion or any form of communication in social contagion . if entity i contacts with entity j , the corresponding matrix element n ij is 1 , otherwise 0 . the elements of the contact matrix can also be decimals or some weighted numbers to reflect the likelihood of contact between any two entities , if the explicit and accurate contact between them is not available . for example , the spatial or temporal proximity of two entities can be used as proxy of their contact likelihood . as shown in the upper part of fig1 , the elements of contact relation for four multilaterally interacted individuals a , b , c , and d , are calculated using the inverse distance between every two individuals . in a contagion , we also observe a sequence of infection occurrences of all the individual entities , i . e ., we know their infection status ( infected or not infected ) and the corresponding time . as shown in the lower part of fig1 , during the observation period , individuals c , a , and d sequentially get infected while individual b has not . assume there are no two infections that occurring at the exactly same time and any two entities are potentially interacted with some contact likelihood . the probability of the susceptible entity i infecting disease l ( or adopting product l ) at occasion t is defined according to the present invention as α i c ζ j x jl ( t - 1 ) n ijlt + x ′ ilt α i − c + ε ijlt , ( 1 ) where ζ j is the infectivity of potential infectious entity j , x jl ( t − 1 ) is the prior infection status ( or adoption behavior ) of j on disease ( or product ) l before occasion t , n ijlt is the contact probability between i and j in term of disease ( or product ) l till occasion t , x ilt =[ x ilt1 , x ilt2 , . . . , x iltk ]′ is i &# 39 ; s confronted environmental factors such as temperature , humidity and wind speed etc . ( or marketing mix such as price , promotion ) and disease ( or product ) attributes at occasion t , α c i is entity i &# 39 ; s susceptibility to contagion and vector α − c i is i &# 39 ; s sensitivity to environmental factors and other attributes . different with aggregate - level models for infectious disease and product diffusion , which typically neglect individual - level dynamics , the present specification accounts for the contingent nature of contagion upon the intrinsic traits , infection status and contact of hosts , as well as individuals &# 39 ; responses to specific environmental and disease ( or product ) attributes , and thus allows for jointly inference of both infectivity and susceptibility of individual entities . we notice that the infectivity and susceptibility are allowed to be continuous values , not only “ yes - or - no ” dichotomy or other discrete classifiers , so that they can be further used for comparison and ranking . assuming the error term ε ijlt to follow logistic distribution or standard normal distribution , the probability of entity i infecting disease ( or adopting product ) l conditional on entity j &# 39 ; s potential infectivity and all the other factors is : where y ijlt is the incident ( or adoption behavior ) of entity i as being potentially infected by entity j , such that y ijlt = 1 denotes an infected status ( or adoption ) of the disease ( or product ) l on occasion t and y ijlt = 0 otherwise . equation ( 2 ) and ( 3 ) indicate the probability that entity i will infect disease ( or adopt product ) l on occasion t based on the effects defined in the equation ( 3 ) with only negligible error . therefore if the value of equation ( 3 ) is very low , the probability of infection or adoption is near to zero . however , sometimes an individual might get infected without contact with other individuals , for instance , via a secondary host such as insects , and thus bears a baseline infection probability greater than zero . similarly , in some situations , an individual may have non - negligible preference for certain products , which contributes to positive baseline adoption likelihood . for such situations , equation ( 2 ) and ( 3 ) can be enhanced to incorporate an individual specific baseline infection risk ( or adoption likelihood ). let β il denote the baseline probability of entity i infecting disease ( or adopting product ) l due to unobservable and non - contagion factors . then corresponding to the logistic or standard normal distribution of error term , the probability of entity i infecting a disease ( or adopting product ) l on occasion t is : p ( y ijlt = 1 | ζ j , α i c , α i − c , x jl ( t − 1 ) , n ijlt , x ilt )= β il +( 1 − β ) λ ( α i c ζ j x jl ( t − 1 ) n ijlt + x ′ ilt α i − c ) ( 4 ) p ( y ijlt = 1 | ζ j , α i c , α i − c , x jl ( t − 1 ) , n ijlt , x ilt )= β il +( 1 − β ) φ ( α i c ζ j x jl ( t − 1 ) n ijlt + x ′ ilt α i − c ) ( 5 ) this enhancement makes the present method more general as we can see that if β il = 0 , i . e ., entity i has zero baseline infection probability for disease ( or product ) l due to unobservable and non - contagion factors , equation ( 4 ) and ( 5 ) reduces to equation ( 2 ) and ( 3 ) respectively . in principle , maximum likelihood estimates of the parameters can be obtained by jointly maximizing the likelihood function assuming conditional independence among the infection of disease or adoption of product across different occasions as well as individuals , which means that an individual &# 39 ; s infection of disease or adoption of product after contacting to different potential infectious individuals across different occasions may be positively correlated but all this correlation can be entirely explained by the infectivity and susceptibility of the individuals and the correlation among environmental covariates . we can see that presently the model is over parameterized and the maximum likelihood estimation is not identifiable . one solution is to impose restrictions to the latent trait parameters , such as σα c i = 1 or to set specific distribution , for instance , ζ j ˜ n ( 0 , 1 ) and to focus on susceptibility parameters by integrating the parameters ζ j out of the joint likelihood function . assuming ε ilt to be logistic distribution we can solve out the unknown susceptibility and sensitivity parameters from the below equations : since this is computationally demanding and cannot be easily done using standard numerical methods , the present invention provides a markov chain monte carlo estimation which is based on the one proposed by johnson and albert ( 1999 ), but extends it to incorporate observable covariates and additional parameters in a setting of multilateral interactions among infectious and susceptible entities . to implement the gibbs sampler , a latent variable w is introduced so that w ijlt ≧ 0 if y ijlt = 1 and w ijlt & lt ; 0 otherwise . under the assumption of standard normal distributed error term , the joint posterior distribution of ( w , α , ζ ) conditional on the observed data is assume a normal prior for the individual parameter ζ j ˜ n ( v , σ 2 ), which means that the individual entities &# 39 ; infectivity is normally distributed among the population . for the susceptibility and sensitivity parameter vector of the susceptible entity i , assume a conjugate multivariable normal prior α i ˜ n k ( μ , σ ) and restrict the susceptibility parameter α i c to be positive which guarantees that entity i who interacts with a entity j with a positive (/ negative ) ζ j should have a higher (/ lower ) probability to purchase . with prior distributions being specified , the fully conditional distributions of w , α , and ζ for bayesian estimation are given as below : to estimate the enhanced model ( equation 5 ), a new latent variable z is included such that p ( z ijlt = 1 | y ijlt = 1 )= φ ( α i c ζ j x jl ( t − 1 ) n ijlt + x ′ ilt α i − c ); ( z ijlt = 0 | y ijlt = 1 )= β il ( 1 − φ ( α i c ζ j x jl ( t − 1 ) n ijlt + x ′ ilt α i − c )) 19 ) p ( z ijlt = 1 | y ijlt = 0 )= 0 ; p ( z ijlt = 0 | y ijlt = 0 )= 1 ( 20 ) and the joint posterior distribution of ( z , w , α , ζ , β ) conditional on the observed data is assuming a conjugate bata prior distribution β il ˜ bata ( c il , d il ), the posterior conditional distributions of β il for bayesian estimation is as below : f ( β il |*)∝ bata ( c il + σ ( t | z ilt = 0 ) y ilt , d il + σ t l ( z ilt = 0 )− σ ( t | z ilt = 0 ) y ilt ) ( 22 ) fig2 shows a flowchart of one embodiment of the present invention , which is a computer implemented markov chain monte carlo process for implementing the bayesian estimation for all the parameters and latent variables . at step 101 , the sequential occurrences , i . e ., values of 0 or 1 , of all the individual entities &# 39 ; disease infections or product adoptions are obtained as input data . at step 102 , the conditions of all the environmental and other factors such as temperature , humidity and wind speed etc ., or marketing mix such as price , promotion , as well as other disease ( or product ) related attributes confronted by the individual entities at various occasions are input . at step 103 , the contact relation matrix between any two individuals that represent their explicit or implicit contact in different occasions are input . an element of the matrix can be 1 or 0 if the contact relation is accurate and explicit , or can be decimals and weighted value to reflect the likelihood of contact . the matrix can have more than two dimensions for multiple diseases , products or behaviors . at step 201 , the obtained data is input into the storage devices for later use . at step 202 , initial values for all the unknown parameters are set arbitrarily or randomly by computer . at step 203 , the conjugate prior distributions are assumed and their hyper - parameters are set arbitrarily , or by field experts . at step 204 , the computer iteratively computes and draws from the full conditional distributions of individuals &# 39 ; infectivity , susceptibility , sensitivity and baseline infection risk , and other latent variables in accordance with formulas given by equation ( 12 ) to equation ( 22 ) stated in this specification . after an early burn - in period of iterations in step 204 , markov chains for each of the unknown parameters are generated and recorded by the computer in step 205 . at step 206 , the convergence of the chains are evaluated by comparison of the between and within variances for markov chains or the r statistics suggested by gelman and rubin ( 2004 ). if the markov chains converge , the values of each individual &# 39 ; s infectivity , susceptibility , sensitivity and baseline infection risk parameters are determined at step 207 . otherwise , the computer return back to step 204 , iteratively compute and draw more from the full conditional distributions . besides this embodiment of the present invention , a maximum likelihood estimation for susceptibility and sensitivity parameters based on logistic distribution assumption can be obtained by solving equations ( 7 )-( 10 ) stated in this specification , which demands more skills in numerical methods . there are many ways that the present invention and its output can be used . for instance , the method can be used to rank and identify the most and least infectious and susceptible individuals for targeted vaccination , quarantine and in - depth genetic and pathological examinations . it can be used to identify key individual consumers for word - of - mouth marketing . it can also be used for identifying and monitoring problematic financial institutions or social riot sources . moreover , there are many ways that this method can be adopted or altered for various purposes . as stated in the summary , this method provides an integrative latent trait framework that allows for various factors to be included into the model to play different roles . for example , more covariates , factors and latent traits can be added into the specifications , and altered forms of them can be used . moreover , instead of using occurrences data , quantity data such as money can be examined , and different prior distributions and different link functions can be used . further , the method can be used in various contexts wherein individual - level asymmetric dynamics occur , and for various types of entities among whom multilateral interactions and relation exist . watts and dodds , “ influentials , networks , and public opinion formation ,” journal of consumer research , 34 ( 4 ), 441 - 458 , 2007 . goldenberg et al , “ the role of hubs in the adoption processes ,” journal of marketing , 73 , 1 - 13 . 2009 keeling and rohani , modeling infectious diseases in humans and animals . nj , usa 2008 . katz , “ the two - step flow of communication : an up - to - date report on a hypothesis ,” public opinion quart , 21 , 61 - 67 1957 . weimann , “ the influentials : back to the concept of opinion leaders ?” public opinion quarterly , 55 ( 2 ), 267 - 279 , 1991 . chaney , “ opinion leaders as a segment for marketing communications ,” marketing intelligence and planning , 19 ( 5 ), 302 - 308 , 2001 . thoburn , “ who are the influencers ,” marketing , 109 ( 29 ), 21 , 2004 . birnbaum , a ., “ some latent trait models and their use in inferring an examinee &# 39 ; 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