Disease Spreading in Time-Evolving Networked Communities

Human communities are organized in complex webs of contacts that may be represented by a graph or network. In this graph, vertices identify individuals and edges establish the existence of some type of relations between them. In real communities, the possible edges may be active or not for variable periods of time. These so-called temporal networks typically result from an endogenous social dynamics, usually coupled to the process under study taking place in the community. For instance, disease spreading may be affected by local information that makes individuals aware of the health status of their social contacts, allowing them to reconsider maintaining or not their social contacts. Here we investigate the impact of such a dynamical network structure on disease dynamics, where infection occurs along the edges of the network. To this end, we define an endogenous network dynamics coupled with disease spreading. We show that the effective infectiousness of a disease taking place along the edges of this temporal network depends on the population size, the number of infected individuals in the population and the capacity of healthy individuals to sever contacts with the infected, ultimately dictated by availability of information regarding each individual’s health status. Importantly, we also show how dynamical networks strongly decrease the average time required to eradicate a disease.

Introduction
Understanding disease spreading and evolution involves overcoming a multitude of complex, multi-scale challenges of mathematical and biological nature [1,2]. Traditionally, the contact process between an infected individual and the susceptible ones was assumed to affect equally any susceptible in a population (mean-field approximation, well-mixed population approximation) or, alternatively, all those susceptible living in the physical neighborhood of the infected individual (spatial transmission). During recent years, however, it has become clear that disease spreading [2][3][4][5] transcends geography: the contact process is no longer restricted to the immediate geographical neighbors, but exhibits the stereotypical small-world phenomenon [6][7][8][9], as testified by recent global pandemics (together with the impressive amount of research that has been carried out to investigate them) or, equally revealing, the dynamics associated with the spreading of computer viruses [5,[10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Recent advances in the science of networks [3,4,19,24,25] also provided compelling evidence of the role that the networks of contacts between individuals or computers play in the dynamics of infectious diseases [4,7]. In the majority of cases in which complex networks of disease spreading have been considered [9], they were taken to be a single, static entity. However, contact networks are intrinsically temporal entities and, in general, one expects the contact process to proceed along the lines of several networks simultaneously [11, 13-16, 18, 23, 24, 26-36]. In fact, modern societies have developed rapid means of information dissemination, both at local and at centralized levels, which one naturally expects to alter individuals' response to vaccination policies, their behavior with respect to other individuals and their perception of likelihood and risk of infection [37]. In some cases one may even witness the adoption of centralized measures, such as travel restrictions [38,39] or the imposition of quarantine spanning parts of the population [40], which may induce abrupt dynamical features onto the structure of the contact networks. In other cases, social media can play a determinant role in defining the contact network, providing crucial information on the dynamical patterns of disease spreading [41]. Furthermore, the knowledge an individual has (based on local and/or social media information) about the health status of acquaintances, partners, relatives, etc., combined with individual preventive strategies [42][43][44][45][46][47][48][49][50] (such as condoms, vaccination, the use of face masks or prophylactic drugs, avoidance of visiting specific web-pages, staying away from public places, etc.), also leads to changes in the structure and shape of the contact networks that naturally acquire a temporal dimension that one should not overlook.
Naturally, the temporal dimension and multitude of contact networks involved in the process of disease spreading render this problem intractable from an analytic standpoint. Recently, sophisticated computational platforms have been developed to deal with disease prevention and forecast [5,10,11,18,27,[29][30][31][32][33][34][35][36][51][52][53][54][55]. The computational complexity of these models reflects the intrinsic complexity of the problem at stake, and their success relies on careful calibration and validation procedures requiring biological and socio-geographic knowledge of the process at stake.
Our goal here, instead, will be to answer the following question: What is the impact of a temporal contact network structure in the overall dynamics of disease progression? Does one expect that it will lead to a rigid shift of the critical parameters driving disease evolution, as one witnesses whenever one includes spatial transmission patterns? Or even to an evanescence of their values whenever one models the contact network as a (static and infinite) scale-free network, such that the variance of the network degree distribution becomes arbitrarily large? Or will the temporal nature of the contact network lead to new dynamical features? And, if so, which features will emerge from the inclusion of this temporal dimension?
To answer this question computationally constitutes, in general, a formidable challenge. We shall attempt to address the problem analytically, and to this end some simplifications will be required. However, the simplifications we shall introduce become plausible taking into consideration recent results (i) in the evolutionary dynamics of social dilemmas of cooperation, (ii) in the dynamics of peer-influence, and even (iii) in the investigation of how individual behavior determines and is determined by the global, population wide behavior. All these recent studies point out to the fact that the impact of temporal networks in the population dynamics stems mostly from the temporal part itself, and not so much from the detailed shape and structure of the network [56][57][58][59][60][61][62][63]. Indeed, we now know that (i) different models of adaptive network dynamics lead to similar qualitative features regarding their impact in what concerns the evolution of cooperation [56][57][58][59][60][61][62][63], (ii) the degree of peer-influence is robust to the structural patterns associated with the underlying social networks [62], and (iii) the impact of temporal networks in connecting individual to collective behavior in the evolution of cooperation is very robust and related to a problem of N-body coordination [61,63]. Altogether, these features justify that we model the temporal nature of the contact network in terms of a simple, adaptive network, the dynamics of which can be approximately described in terms a coupled system of ODEs. This "adaptive-linking" dynamics, as it was coined [28,[57][58][59], leads to network snapshot structures that do not replicate what one observes in real-life, in the same sense that the small-world model of Watts and Strogatz does not lead to the heterogeneous and diverse patterns observed in data snapshots of social networks. Notwithstanding, the active-linking dynamics allows us to include, analytically, the temporal dimension into the problem of disease dynamics. The results [28], as we elaborate in Sects. 3 and 4, prove rewarding, showing that the temporal dimension of a contact network leads to a shift of the critical parameters (defined below) which is no longer rigid but, instead, becomes dependent on the frequency of infected individuals in the population. This, we believe, constitutes a very strong message with a profound impact whenever one tries to incorporate the temporal dimension into computational models of disease forecast.
This chapter is organized as follows. In the following Sect. 2, we introduce the standard disease models we shall employ, as well as the details of the temporal contact network model. Section 3 is devoted to present and discuss the results, and Sect. 4 contains a summary of the main conclusions of this work.

Models and Methods
In this section, we introduce the disease models we shall employ which, although well-known and widely studied already, are here introduced in the context of stochastic dynamics in finite populations, a formulation that has received less attention than the standard continuous model formulation in terms of coupled Ordinary Differential Equations (ODEs). Furthermore, we introduce and discuss in detail the temporal contact network model.

Disease Spreading Models in Finite Populations
Here we introduce three standard models of disease transmission that we shall employ throughout the manuscript, using this section at profit to introduce also the appropriate notation associated with stochastic dynamics of finite populations and the Markov chain techniques that we shall also employ in the remainder of this chapter. We shall start by discussing the models in the context of well-mixed populations, which will serve as a reference scenario for the disease dynamics, leaving for the next section the coupling of these disease models with the temporal network model described below. We investigate the popular Susceptible-Infected-Susceptible (SIS) model [2,4], the Susceptible-Infected (SI) model [2] used to study, e.g., AIDS [2,64], and the Susceptible-Infected-Recovered (SIR) model [2,65], more appropriate to model, for instance, single season flu outbreaks [2] or computer virus spreading [7]. It is also worth pointing out that variations of these models have been used to successfully model virus dynamics and the interplay between virus dynamics and the response of the immune system [66].

The SIS Model
In the SIS model individuals can be in one of two epidemiological states: Infected (I) or Susceptible (S). Each disease is characterized by a recovery rate (•) and an infection rate (oe). In an infinite, well-mixed population, the fraction of infected individuals (x) changes in time according to the following differential equation where y D 1 x is the fraction of susceptible individuals and hki the average number of contacts of each individual [4]. There are two possible equilibria (P x D 0): x D 0 and x D 1 R 1 0 , where R 0 D hki/ı denotes the basic reproductive ratio. The value of R 0 determines the stability of these two equilibria: x D 1 R 1 0 is stable when R 0 > 1 and unstable when R 0 < 1. Let us now move to finite populations, and consider the well-mixed case where the population size is fixed and equal to N. We define a discrete stochastic Markov process describing the disease dynamics associated with the SIS model. Each configuration of the population, which is defined by the number of infected individuals i, corresponds to one state of the Markov chain. Time evolves in discrete steps and two types of events may occur which change the composition of the population: infection events and recovery events. This means that, similar to computer simulations of the SIS model on networked populations, at most one infection or recovery event will take place in each (discrete) time step. Thus, the dynamics can be represented as a Markov chain M with NC1 states [67,68] -as many as the number of possible configurations -illustrated in the following Fig. 13.1.
In a finite, well-mixed population, the number i of infected will decrease at a rate given by where 0 denotes the recovery time scale, i N the probability that a randomly selected individual is infected and ı the probability that this individual recovers. Adopting 0 as a reference, we assume that the higher the average number of contacts hki, the smaller the time scale INF at which infection update events occur ( INF D 0 /hki) [4]. Consequently, the number of infected will also increase at a rate given by Equations (13.1) and (13.2) define the transitions between different states. This way, we obtain the following transition matrix for M: where each element p kj of P represents the probability of moving from state k to state j during one time step. The state without any infected individual (iD0) is an absorbing state of M. In other words, the disease always dies out and will never re-appear, once this happens.

Average Times to Absorption
At this level of approximation, it is possible to derive an analytical expression for the average time t i it takes to reach the single absorbing state of the SIS Markov chain (i.e., the average time to absorption) starting from a configuration in which there are i infected individuals. Denoting by P i (t) the probability that the disease disappears at time t when starting with i infected individuals at time 0, we may write [69] Using the properties of P i (t) we obtain the following recurrence relation for t i whereas for t N we may write Defining the auxiliary variables i D l , a little algebra allows us to write, for t 1 such that t i can be written as a function of t 1 as follows The intrinsic stochasticity of the model, resulting from the finiteness of the population, makes the disease disappear from the population after a certain amount of time. As such, the population size plays an important role in the average time to absorption associated with a certain disease, a feature we shall return to below.

Quasi-Stationary Distributions in Finite Populations
Equations (13.1) and (13.2) define the Markov chain M just characterized. The fraction of time the population spends in each state is given by the stationary distribution of M, which is defined as the eigenvector associated with eigenvalue 1 of the transition matrix of M [67,68]. The fact that in the SIS model the state without infected (iD0) is an absorbing state of the Markov chain, implies that the standard stationary distribution will be completely dominated by this absorbing state, which precludes one to gather information on the relative importance of other configurations. This makes the so-called quasi-stationary distribution of M [70] the quantity of interest. This quantity allows us to estimate the relative prevalence of the population in configurations other than the absorbing state, by computing the stationary distribution of the Markov chain obtained from M by excluding the absorbing state iD0 [70]. It provides information on the fraction of time the population spends in each state, assuming the disease does not go extinct.

The Infinite, Well-Mixed Populations as a Limiting Case
The Markov process M defined before provides a finite population analogue of the well-known mean-field equations written at the beginning of Sect. 2.1.1. Indeed, in the limit of large populations, 0 G.i/ D T C .i/ T .i/ provides the rate of change of infected individuals. For large N, replacing i N by x and N i N by y, the gradients of infection which characterize the rate at which the number of infected are changing in the population, are given by Again, we obtain two roots: 0 G(i) D 0 for i D 0 and i r 0 D N .N 1/ ı hki . Moreover, i r 0 becomes the finite population equivalent of an interior equilibrium for R 0 Á ı hki N N 1 > 1 (note that, for large N we have that N N 1 1). The disease will most likely expand whenever i < i r 0 , the opposite happening otherwise.

The SI Model
The SI model is mathematically equivalent to the SIS model with ı D 0, and has been employed to study for instance the dynamics of AIDS. The Markov Chain representing the disease dynamics is therefore defined by transition matrix Eq. (13.3), with T i D 0 for all i. The remaining transition probabilities T C i (0 < i < N) are exactly the same as for the SIS model. Since all T i equal zero, the Markov Chain has two absorbing states: the canonical one without any infected (iD0) and the one without any susceptible (iDN). The disease will expand monotonically as soon as one individual in the population gets infected, ultimately leading to a fully infected population. The average amount of time after which this happens, which we refer to as the average infection time, constitutes the main quantity of interest. This quantity can be calculated analytically [28]: The average number of time steps needed to reach 100% infection, starting from i infected individuals is given by (13.9)

The SIR Model
With SIR one models diseases in which individuals acquire immunity after recovering from infection. We distinguish three epidemiological states to model the dynamics of such diseases: susceptible (S), infected (I) and recovered (R), indicating those who have become immune to further infection. The SIR model in infinite, well-mixed populations is defined by a recovery rate ı and an infection rate . The fraction of infected individuals x changes in time according to the following differential equation where y denotes the fraction of susceptible individuals, which in turn changes according to P y D hki xy: (13.11) Finally, the fraction of individuals z in the recovered class changes according to P z D ı x: (13.12) To address the SIR model in finite, well-mixed populations, we proceed in a way similar to what we have done so far with SIS and SI models. The Markov Chain describing the disease dynamics becomes slightly more complicated and has states (i, r), where i is the number of infected individuals in the population and r the number of recovered (and immune) individuals (i C r Ä N). A schematic representation of the Markov Chain is given in Fig. 13.2.
Note that the states (0, r), with 0 Ä r Ä N, are absorbing states. Each of these states corresponds to the number of individuals that are (or have become) immune at the time the disease goes extinct.
Consider a population of size N with average degree hki. The number of infected will increase with a rate where 0 denotes the recovery time scale. As before, the gradient of infection G(i), such that 0 G.i/ D T C .i/ T .i/, measures the likelihood for the disease to either expand or shrink in a given state, and is given by Note that we recover Eq. (13.10) in the limit N ! 1. For a fixed number of recovered individuals r 0 , we have that 0 G(i, r 0 ) D 0 for i D 0 and for i r 0 D N becomes the finite population analogue of an interior equilibrium. Furthermore, one can show that the partial derivative @G.i;r/ @i has at most one single root in (0, 1), possibly located at i r 0 D i r 0 2 Ä i r 0 . Hence, G(i, r 0 ) reaches a local maximum at i r 0 (given that at that point @ 2 G.i;r/ @i 2ˇi r 0 D 2hki N.N 1/ < 0). The number of infected will therefore most likely increase for i < i r 0 (assuming r 0 immune individuals), and most likely decrease otherwise.
The gradient of infection also determines the probability to end up in each of the different absorbing states of the Markov chain. These probabilities can be calculated analytically [28]. To this end, let us use y a i;r to denote the probability that the population ends up in the absorbing state with a recovered individuals, starting from a state with i infected and r recovered. We obtain the following recurrence relationship for y a i;r y a i;r D T .i; r/ y a i 1;rC1 C T C .i; r/ y a iC1;r C 1 T .i; r/ T C .i; r/ y a i;r ; (13.16) which reduces to The following boundary conditions (13.18) allow us to compute y a i;r for every a, i and r.

Network Model
Our network model explicitly considers a finite and constant population of N individuals. Its temporal contact structure allows, however, for a variable number of overall links between individuals, which in turn will depend on the incidence of disease in the population. This way, infection proceeds along the links of a contact network whose structure may change based on each individual's health status and the availability of information regarding the health status of others. We shall assume the existence of some form of local information about the health status of social contacts. Information is local, in the sense that individual behavior will rely on the nature of their links in the contact network. Moreover, this will influence the way in which individuals may be more or less effective in avoiding contact with those infected while remaining in touch with the healthy. Suppose all individuals seek to establish links at the same rate c. For simplicity, we assume that new links are established and removed randomly, a feature which usually does not always apply in real cases, where the limited social horizon of individuals or the nature of their social ties may constrain part of their neighborhood structure (see below). Let us further assume that links may be broken off at different rates, based on the nature of the links and the information available about the individuals they connect: Let us denote these rates by b pq for links of type pq (p , q 2 fS, I, Rg. We assume that links are bidirectional, which means that we have links of pq types SI, SR, and IR. Let L pq denote the number of links of type pq and L M pq the maximum possible number of links of that type, given the number of individuals of type S, I and R in the population. This allows us to write down (at a mean-field level) a system of ODEs [57,58] for the time evolution of the number of links of pq-type (L pq ) [57,58] which depends on the number of individuals in states p and q (L M pp D p .p 1/ =2 and L M pq D pq for p ¤ q) and thereby couples the network dynamics to the disease dynamics. In the steady state of the linking dynamics ( P L pq D 0), the number of links of each type is given by L pq D ' pq L M pq , with ' pq D c/(c C b pq ) the fractions of active pq-links, compared to the maximum possible number of links L M pq , for a given number of S, I and R. In the absence of disease only SS links exist, and hence SS determines the average connectivity of the network under disease free conditions, which one can use to characterize the type of the population under study. In the presence of I individuals, to the extent that S individuals manage to avoid contact with I, they succeed in escaping infection. Thus, to the extent that individuals are capable of reshaping the contact network based on available information of the health status of other individuals, disease progression will be inhibited. In the extreme limit of perfect information and individual capacity to immediately break up contacts with infected, we are isolating all infected, and as such containing disease progression. Our goal here, however, is to understand how and in which way local information, leading to a temporal reshaping of the network structure, affects overall disease dynamics.

Computer Simulations
We investigate the validity of the approximations made to derive analytical results as well as their robustness by means of computer simulations. All individual-based simulations start from a complete network of size ND100. Disease spreading and network evolution proceed together under asynchronous updating. Disease update events take place with probability (1 C ) 1 , where D NET / DIS . We define DIS as the time-scale of disease progression, whereas NET is the time scale of network change. The parameter D NET / DIS provides the relative time scale in terms of which we may interpolate between the limits when network adaptation is much slower than disease progression ( ! 0) and the opposite limit when network adaptation is much faster than disease progression ( ! 1). Since D NET / DIS is the only relevant parameter, we can make, without loss of generality, DIS D 1. For network update events, we randomly draw two nodes from the population. If connected, then the link disappears with probability given by the respective b pq . Otherwise, a new link appears with probability c. When a disease update event occurs, a recovery event takes place with probability (1 C hki) 1 , an infection event otherwise. In both cases, an individual j is drawn randomly from the population. If j is infected and a recovery event has been selected then j will become susceptible (or recovered, model dependent) with probability •. If j is susceptible and an infection event occurs, then j will get infected with probability oe if a randomly chosen neighbor of j is infected. The quasi-stationary distributions are computed (in the case of the SIS model) as the fraction of time the population spends in each configuration (i.e., number of infected individuals) during 10 9 disease event updates (10 7 generations; under asynchronous updating, one generation corresponds to N update events, where N is the population size; this means that in one generation, every individual has one chance, on average, to update her epidemic state). The average number of infected hIi and the mean average degree of the network hki observed during these 10 7 generations are kept for further plotting. We have checked that the results reported are independent of the initial number of infected in the network. Finally, for the SIR and SI models, the disease progression in time, shown in the following sections, is calculated from 10 4 independent simulations, each simulation starting with 1 infected individual. The reported results correspond to the average amount of time at which i individuals become infected.

Results and Discussion
In this section we start by (i) showing that a quickly adapting community induces profound changes in the dynamics of disease spreading, irrespective of the underlying epidemic model; then, (ii) we resort to computer simulations to study the robustness of these results for intermediate time-scales of network adaptation; finally, (iii) we profit from the framework introduced above to analyze the impact of information on average time for absorption and disease progression in adaptive networks.

Disease Spreading in a Quickly Adaptive Network Structure
Empirically, it is well-known that often individuals prevent infection by avoiding contact with infected once they know the state of their contacts or are aware of the potential risks of such infection [31,33,[42][43][44][45][46][47][48][49][50]: such is the case of many sexually transmitted diseases [42,[71][72][73], for example, and, more recently, the voluntary use of face masks and the associated campaigns adopted by local authorities in response to the SARS outbreak [40,[43][44][45] or even the choice of contacting or not other individuals based on information on their health status gathered from social media [41,74,75]. In the present study, individual decision is based on available local information about the health state of one's contacts. Thus, we can study analytically the limit in which the network dynamics -resulting from adaptation to the flow of local information -is much faster than disease dynamics, as in this case, one may separate the time scales between network adaptation and contact (disease) dynamics: The network has time to reach a steady state before the next contact takes place. Consequently, the probability of having an infected neighbor is modified by a neighborhood structure which will change in time depending on the impact of the disease in the population and the overall rates of severing links with infected. Let us start with the SIR model. The amount of information available translates into differences mostly between the break-up rates of links that may involve a potential risk for further infection (b SI , b IR , b II ), and those that do not (b SS , b SR , b RR ). Therefore, we consider one particular rate b I for links involving infected individuals (b I Á b SI D b IR D b II ), and another one, b H , for links connecting healthy . In general, one expects b I to be maximal when each individual has perfect information about the state of her neighbors and to be (minimal and) equal to b H when no information is available, turning the ratio between these two rates into a quantitative measure of the efficiency with which links to infected are severed compared to other links. Note that we reduce the model to two break-up rates in order to facilitate the discussion of the results. Numerical simulations show that the general principles and conclusions remain valid when all break-up rates are incorporated explicitly. It is worth noticing that three out of these six rates are of particular importance for the overall disease dynamics: b SS , b SR and b SI . These three rates, combined with the rate c of creating new links, define the fraction of active SS, SR and SI links, and subsequent correlations between individuals [76], and therefore determine the probability for a susceptible to become infected (see Models and Methods). This probability will increase when considering higher values of c (assuming b I > b H ). In other words, when individuals create new links more often, therefore increasing the likelihood of establishing connections to infected individuals (when present), they need to be better informed about the health state of their contacts in order to escape infection. In the fast linking limit, the other three break-up rates (b II , b IR and b RR ) will also influence disease progression since they contribute to changing the average degree of the network.
When the time scale for network update ( NET ) is much smaller than the one for disease spreading ( DIS ), we can proceed analytically using at profit the separation of times scales. In practice, this means that the network has time to reach a steady state before the next disease event takes place. Consequently, the probability of having an infected neighbor is modified by a neighborhood structure which will change in time depending on the impact of the disease in the population and the overall rates of severing links with infected individuals. For a given configuration (i,r) of the population, the stationary state of the network is characterized by the parameters ' SS , ' SI and ' SR . Consequently, the number of infected increases at a rate [28] where we made 0 D 1. The effect of the network dynamics becomes apparent in the third factor, which represents the probability that a randomly selected neighbor of a susceptible is infected. In addition, Eq. (13.14) remains valid, as the linking dynamics does not affect the rate at which the number of infected decreases. It is noteworthy that we can write Eq. (13.19) in the form which is formally equivalent to Eq. (13.13) and shows that disease spreading in a temporal adaptive network is equivalent to that in a well-mixed population with (i) a frequency dependent average degree hki and (ii) a transmission probability that is rescaled compared to the original according to Note that this expression remains valid for both SIR, SIS (r D 0) and SI (ı D 0, r D 0) models. Since the lifetime of a link depends on its type, the average degree hki of the network depends on the number of infected in the population, and hence becomes frequency (and time) dependent, as hki depends on the number of infected (through L M pq ) and changes in time. Note that Á scales linearly with the frequency of infected in the population, decreasing as the number of infected increases (assuming SS ı SI > 1); moreover, it depends implicitly (via the ratio SS ı SI ) on the amount of information available.
It is important to stress the distinction between the description of the disease dynamics at the local level (in the vicinity of an infected individual) and that at the population wide level. Strictly speaking, a dynamical network does not change the disease dynamics at the local level, meaning that infected individuals pass the disease to their neighbors with probability intrinsic to the disease itself. At the population level, on the other hand, disease progression proceeds as if the infectiousness of the disease effectively changes, as a result of the network dynamics. Consequently, analyzing a temporal network scenario at a population level can be achieved via a renormalization of the transmission probability, keeping the (mathematically more attractive) well-mixed scenario. In this sense, from a well-mixed perspective, dynamical networks contribute to changing the effective infectiousness of the disease, which becomes frequency and information dependent. Note further that this information dependence is a consequence of using a single temporal network for spreading the disease and information. Interestingly, adaptive networks have been shown to have a similar impact in social dilemmas [63]. From a global, population-wide perspective, it is as if the social dilemma at stake differs from the one every individual actually plays.
As in Sect. 2, one can define a gradient of infection G, which measures the tendency of the disease to either expand or shrink in a population with given configuration (defined by the number of individuals in each of the states S, I and R). To do so, we study the partial derivative @G.i;r/ @i at i D 0 This quantity exceeds zero whenever Note that taking r D 0 yields the basic reproductive ratio R A 0 for both SIR and SIS: On the other hand, whenever R A 0 < 1, eradication of the disease is favored in the SIS model (G(i)<0), irrespective of the fraction of infected, indicating how the presence of information (b H < b I ) changes the basic reproductive ratio.
In Fig. 13.3 we illustrate the role of information in the SIS model by plotting G for different values of b I (assuming b H < b I ) and a fixed transmission probability . The corresponding quasi-stationary distributions are shown in the right panel and clearly reflect the sign of G. Whenever G(i) is positive (negative), the dynamics will act to increase (decrease), on average, the number of infected. Figure 13  population and, once again, allows us to identify when disease expansion will be favored or not. Figure 13.4 gives a complete picture of the gradient of infection, using the appropriate simplex structure in which all points satisfy the relation iCrCsDN. The dashed line indicates the boundary G(i, r) D 0 in case individuals do not have any information about the health status of their contacts, i.e., links that involve infected individuals disappear at the same rate as those that do not (b I D b H ). Disease expansion is more likely than disease contraction (G(i, r) > 0) when the population is in a configuration above the line, and less likely otherwise. Similarly, the solid line indicates the boundary G(i, r) D 0 when individuals share information about their health status, and use it to avoid contact with infected. Once again, the availability of information modifies the disease dynamics, inhibiting disease progression for a broad range of configurations.

Analysis of Intermediate Time-Scales Through Computer Simulations
Up to now we have assumed that the network dynamics proceeds much faster than disease spreading (the limit ! 0). This may not always be the case, and hence it is important to assess the domain of validity of this limit. In the following, we use computer simulations to verify to which extent these results, obtained analytically via time scale separation, remain valid for intermediate values of the relative timescale for the linking dynamics. We start with a complete network of size N, in which initially one individual is infected, the rest being susceptible. As stated before, disease spreading and network evolution proceed simultaneously under asynchronous updating. Network update events take place with probability (1 C ) 1 , whereas a disease model (SI, SIS or SIR) state update event occurs otherwise. For each value of , we run 10 4 simulations. For the SI model, the quantity of interest to calculate is the average number of generations after which the population becomes completely infected. These values are depicted in Fig. 13.5. The lower dashed line indicates the analytical prediction of the infection time in the limit ! 1 (the limit when networks remain static), which we already recover in the simulations for > 10 2 . When is smaller than 10 2 , the average infection time significantly increases, and already reaches the analytical prediction for the limit ! 0 (indicated by the upper dashed line) when < 1. Hence, the validity of the time scale separation does again extend well beyond the limits one might expect.
For the SIR model, we let the simulations run until the disease goes extinct, and computed the average final fraction of individuals that have been affected by is given by Eqs. (13.17) and (13.18). One observes that linking dynamics does not affect disease dynamics for > 10. Once drops below ten, a significantly smaller fraction of individuals is affected by the disease. This fraction reaches the analytical prediction for ! 0 as soon as < 0.1. Hence, and again, results obtained via separation of time scales remain valid for a wide range of intermediate time scales. We finally investigate the role of intermediate time scales in the SIS model. We performed computer simulations in the conditions discussed already, and computed several quantities that we plot in Fig. 13.7. Figure 13.7 shows the average hIi of the quasi-stationary distributions obtained via computer simulations (circles) as a function of the relative time scale of network update. Whenever ! 1, we can characterize the disease dynamics analytically, assuming a well-mixed population (complete graph), whereas for ! 0 we recover the analytical results obtained in the fast linking limit. At intermediate time scales, Fig. 13.7 shows that as long as is smaller than ten, network dynamics contributes to inhibit disease spreading by effectively increasing the critical infection rate. Overall, the validity of the time scale separation extends well beyond the limits one might anticipate based solely on the time separation ansatz. As long as the time scale for network update is smaller than the one for disease spreading ( < 1), the analytical prediction for the limit ! 0, indicated by the lower dashed line in Fig. 13.7, remains valid. The analytical result in the extreme opposite limit ( ! 1), indicated by the upper dashed line in Fig. 13.7, holds as long as > 10 5 . Moreover, it is noteworthy that the network dynamics influences the disease dynamics both by reducing the frequency of interactions between susceptible and infected, and by reducing the average degree of the network. These complementary effects are disentangled in intermediate regimes, in which the network dynamics is too slow to warrant sustained protection of susceptible individuals from contacts with infected, despite managing to reduce the average degree (not shown). In fact, for > 10 the disease dynamics is mostly controlled by the average degree, as shown by the solid lines in Fig. 13.7. Here, the average stationary distribution was determined by replacing, in the analytic expression for static networks, hki by the time-dependent average connectivity hki computed numerically. This, in turn, results from the frequency dependence of hki. When b I > b H , the network will reshape into a configuration with smaller hki as soon as the disease expansion occurs. For < 1, hki reflects the lifetime of SS links, as there are hardly any infected in the population. For 10 0 < < 10 3 , the network dynamics proceeds fast enough to reduce hki, but too slowly to reach its full potential in hindering disease progression. Given the higher fraction of infected, and the fact that SI and II links have a shorter lifetime than SS links, the average degree drops when increasing from 1 to 10 3 . Any further increase in leads to a higher average degree, as the network approaches its static limit.

Average Time to Absorption in Adaptive Networks
Contrary to the deterministic SIS model, the stochastic nature of disease spreading in finite populations ensures that the disease disappears after some time. However, this result is of little relevance given the times required to reach the absorbing state (except, possibly, in very small communities). Indeed, the characteristic time scale of the dynamics plays a determinant role in the overall epidemiological process and constitutes a central issue in disease spreading. Figure 13.8 shows the average time to absorption t 1 in adaptive networks for different levels of information, illustrating the spectacular effect brought about by the network dynamics on t 1 . While on networks without information (b I D b H ) t 1 rapidly increases with the rate of infection oe, adding information moves the fraction of infected individuals rapidly to the absorbing state, and, therefore, to the disappearance of the disease.
Moreover, the size of the population can have a profound effect on t 1 . With increasing population size, the population spends most of the time in the vicinity of the state associated with the interior root of G(i). For large populations, this acts to reduce the intrinsic stochasticity of the dynamics, dictating a very slow extinction of the disease, as shown in Fig. 13.9.
When recovery from the disease is impossible, a situation captured by the SI model, the population will never become disease-free again once it acquires at least one infected individual. The time to reach absorbing state in which all individuals are infected, again depends on the presence of information. When information prevails, susceptible individuals manage to resist infection for a long time, thereby delaying the rapid progression of the disease, as shown in the inset of Fig. 13.10. Naturally, the average number of generations needed to reach a fully infected population increases with the availability of information, as illustrated in the main panel of Fig. 13.10.

Conclusions
Making use of three standard models of epidemics involving a finite population in which infection takes place along the links of a temporal graph, the nodes of which are occupied by individuals, we have shown analytically that the bias introduced into the graph dynamics resulting from the availability of information about the health status of others in the population induces fundamental changes in the overall dynamics of disease progression.
The network dynamics employed here differs from those used in most other studies [29,[32][33][34][35][36][51][52][53][54][55]. We argue, however, that the differences obtained stem mostly from the temporal aspect of the network, and not so much from the detailed dynamics that is implemented. Importantly, temporal network dynamics leads to additional changes in R 0 compared to those already obtained when moving from the well-mixed assumption to static networks [77]. An important ingredient of our model, however, is that the average degree of the network results from the selforganization of the network structure, and co-evolves with the disease dynamics. A population suffering from high disease prevalence where individuals avoid contact in order to escape infection will therefore exhibit a lower average degree than a population with hardly any infected individuals. Such a frequency-dependent average degree further prevents that containment of infected individuals would result in the formation of cliques of susceptible individuals, which are extremely vulnerable to future infection, as reported before [36,51,54].
The description of disease spreading as a stochastic contact process embedded in a Markov chain constitutes a second important ingredient of the present model. This approach allows for a direct comparison between analytical predictions and individual-based computer simulations, and for a detailed analysis of finite-size effects and convergence times, whose exponential growth will signal possible bistable disease scenarios. In such a framework, we were able to show that temporal adaptive networks in which individuals may be informed about the health status of others lead to a disease whose effective infectiousness depends on the overall number of infected in the population. In other words, disease propagation on temporal adaptive networks can be seen as mathematically equivalent to disease spreading on a well-mixed population, but with a rescaled effective infectiousness. In accord with the intuition advanced in the introduction, as long as individuals react promptly and consistently to accurate available information on whether their acquaintances are infected or not, network dynamics effectively weakens the disease burden the population suffers. Last but not least, if recovery from the disease is possible, the time for disease eradication drastically reduces whenever individuals have access to accurate information about the health state of their acquaintances and use it to avoid contact with those infected. If recovery or immunity is impossible, the average time needed for a disease to spread increases significantly when such information is being used. In both cases, our model clearly shows how availability of information hinders disease progression (by means of quick action on infected, e.g., their containment via link removal), which constitutes a crucial factor to control the development of global pandemics.
Finally, it is also worth mentioning that knowledge about the health state of others may not always be accurate or available in time. This is for instance the case for diseases where recently infected individuals remain asymptomatic for a substantial period. The longer the incubation period associated with the disease, the less successful individuals will be in escaping infection, which in our model translates into a lower effective rate of breaking SI links, with the above mentioned consequences. Moreover, different (social) networks through which awareness of the health status of others proceeds may lead to different rates of information spread. One may take these features into account by modeling explicitly the spread of information through a coupled dynamics between disease expansion and individuals' awareness of the disease [31,33].
Creation and destruction of links may for instance not always occur randomly, as we assumed here, but in a way that is biased by a variety of factors such as social and genetic distance, geographical proximity, family ties, etc. The resulting contact network may therefore become organized in a specific way, promoting the formation of particular structures, such as networks characterized by long-tailed degree distributions or with strong topological correlations among nodes [3,[78][79][80] which, in turn, may influence the disease dynamics. The impact of combining such effects, resulting from specific disease scenarios, with those reported here will depend on the prevalence of such additional effects when compared to linkrewiring dynamics. A small fraction of non-random links, or of ties which cannot be broken, will likely induce small modifications on the average connectivity of the contact network, which can be incorporated in our analytic expressions without compromising their validity regarding population wide dynamics. On the other hand, when the contact network is highly heterogeneous (e.g., exhibiting pervasive long-tail degree distributions), non-random events may have very distinct effects, from being almost irrelevant (and hence can be ignored) to inducing hierarchical cascades of infection [81], in which case our results will not apply.


extfieldsofstudy:
Computer Science
Mathematics
Engineering

text:
Learning to Fuse Music Genres with Generative Adversarial Dual Learning

FusionGAN is a novel genre fusion framework for music generation that integrates the strengths of generative adversarial networks and dual learning. In particular, the proposed method offers a dual learning extension that can effectively integrate the styles of the given domains. To efficiently quantify the difference among diverse domains and avoid the vanishing gradient issue, FusionGAN provides a Wasserstein based metric to approximate the distance between the target domain and the existing domains. Adopting the Wasserstein distance, a new domain is created by combining the patterns of the existing domains using adversarial learning. Experimental results on public music datasets demonstrated that our approach could effectively merge two genres.

I. INTRODUCTION
Computational creativity is a lively research area that focuses on understanding and facilitating human creativity through the implementation of software programs [2]. With the rapid advances in data-driven algorithms such as deep learning [7], the exploration into computational creativity via machine learning approaches has become increasingly popular. Examples showing the potential of deep learning for creative approaches are the artistic style transfer on images [4] and videos [12]. Music, with its complex hierarchical and sequential structure and its inherent emotional and aesthetic subjectivity, is an intriguing research subject at the core of human creativity. While there has been some work on generative models for music, research that investigates the capabilities of deep learning for creative applications such as style transfer is limited. This paper aims to fill this space by exploring the idea of style fusion in music with generative adversarial dual learning.
In the field of unsupervised generative learning, generative adversarial networks (GAN) [6] have recently gained considerable attention. It is important to note, however, that GANs are designed to learn from a single domain and cannot discover cross-domain knowledge. Inspired by dual learning in machine translation, several recent publications propose methods to pair unlabeled data points in different domains by optimizing bi-directional reconstruction errors [14], [8], [16]. Although these methods are designed to address the challenge of multimodal learning, they do not address the problem of domain fusion.
The goal of this paper is to provide a solution for fusing two or more groups of sequential patterns using unsupervised methods. Specifically, we focus on the problem of generating music with a fused music genre. To combine genres using generative adversarial learning, we propose a framework for integrating multiple GANs. Unlike previous work, our study targets the creation of an unknown domain by mixing two given domains; a multi-way GAN-based model is proposed to absorb existing patterns and yield a new mixture. With the utilization of the Wasserstein measure [1], we pose a distance constraint on the new domain that automatically balances its relation to the given domains. As a result, our model is capable of generating a mixed pattern after convergence. The main contributions of this paper are: • A novel framework for unsupervised music fusion: The dual learning scheme is extended to involve multiple domains by leveraging mutual regularization. In this way, the proposed framework enables the new domain to keep equal similarities with the given domains and produce a mixture of existing sequential patterns. • A sequence fusion method using GANs: To apply unsupervised fusion learning, we extend the generative adversarial model so that multiple generators and discriminators can be updated by one another. • Formulation of an objective function indicating fusion progress: An objective function based on Wasserstein distance is designed to integrate the information from multiple domains and represent learning progress.
II. RELATED WORK Music Genre Fusion: Fusion is mostly known as the sub-genre of jazz that emerged in the late '60s, and that combines several musical styles such as funk, rock, and blues with the jazz harmony and improvisation [5]. The term can be generalized, however, to any combinations of music genres. The creative combination of two or more genres obviously requires not only intimate knowledge of the stylistic characteristics of the involved genres but also extensive compositional experience and skill to combine genres in a meaningful and satisfying way. Recently, Engel et al. [3] proposed a new approach of generating new musical sounds through interpolating the latent space learned by WaveNet [11] autoencoders. Additionally, Gatys et al. demonstrated the Deep Neural Networks' (DNNs) capabilities of learning the artistic styles by blending the style of the training materials to the testing images using ConvNet [4]. Both examples show the great potential of using DNNs for merging two abstract concepts without the need for domain knowledge and explicitly defined rules.
Cross-domain GAN: GAN [6], which has quickly risen to one of the most popular generative approaches, learns patterns without requiring adversarial examples or labels. In CGAN [10], the generator G has a conditional parameter c and will learn the conditional distribution of the data.
However, those conditions need to be provided manually, somewhat similar to supervised learning. CoGAN [9] is proposed to learn a joint distribution with only samples drawn from the marginal distributions. DiscoGAN [8] solves the cross-domain pairing by minimizing bi-directional reconstruction loss, while DualGAN [14] takes advantage of the dual learning paradigm in machine translation. CycleGAN [16] presents a method whose mapping functions are cycleconsistent and proposes a cycle consistency loss function to further reduce the space of possible mapping functions.
Different from the previous studies, our work raises a new problem, i.e., merging two sequential patterns into one. We propose a GAN-based framework that merges two domains without manually tuning the distance between the given domains and the target domain.

III. LEARNING TO BLEND MUSIC WITH FUSIONGAN A. Problem Definition
This paper aims to characterize music fusion across different genres in an unsupervised manner. Based on the data X A and X B from existing music domains D A , D B respectively, our goal is to create a new domain D F which resembles both D A and D B . In the first phase, the new domain D F is expected to learn from D A , D B and keep an equal distance from both. Specifically, the generator G F of D F obtains feedback from the discriminators D A , D B of existing domains D A , D B , while the discriminator D F collects data from G F and D A , D B for iterative updating. The G F from the new domain minimizes the distance from the domains D A , D B , respectively, and keeps the same distance from D A and D B .

B. FusionGAN
First, the framework initializes individual GAN models for existing domains. Maximum likelihood estimation using Long Short Term Memory (LSTM) is applied to initialize the sequence generators G A (G B ). Then D A (D B ) are trained given the domain data and sequence sampled from G A (G B ). Then GAN is employed to iteratively enhance the G A (G B ) and D A (D B ). After pre-training, we build a GAN model for the new domain using the feedback from the models constructed in the pre-training procedure. First, a new domain D F is initialized randomly and trained by both D A and D B . Following the dual learning strategy, D A is enhanced by D B and D F in the second phase. Similarly, D B is improved by D A and D F . Such learning proceeds until convergence. The framework overview is shown in Figure 1. In FusionGAN, we employ the Wasserstein-1 distance for all discriminators because it is sensible when the learning distributions are supported by low dimensional manifolds [1]. To avoid the intractable infimum in Wasserstein-1 distance, we resort to its Kantorovich-Rubinstein duality [13]: For the vanilla GAN, the goal is to find the optimal configuration of the parameters φ of discriminator (f = D). When the discriminator is optimized, the maximized Wasserstein distance can be used as reward in the policy gradient process of the generator. Similarly, we define a three-way Wasserstein distance as the objective function: W (P XA , P XB , P θ ) = L to measure the integrated distance between P θ and P XA , P XB . This measure should consider the relationship between every pair of domains. The framework consists of three pairs of generators and discriminators, i.e., and two input data, i.e., X A , X B . For each discriminator, there are five possible inputs, i.e., X A , X B , G A , G B , G F . Collecting all possible inputs w.r.t. the three discriminators, the objective function to maximize is defined as: where the first five terms are for D A , the second five terms are for D B , and the last five terms are for D F . This loss is used to update all the generators and discriminators. The following two subsections, III-C and III-D, will present the update rules using Eq. 1.

C. D F Update of FusionGAN
Removing the terms that are unrelated to D F in Eq. 1 (zero derivative w.r.t. D F and G F ), we have the objective function for updating D F and G F : First, L F is calculated w.r.t. G F so as to update G F . Considering differentiating ∇ θ L F , the optimal G F is approximated as its convergence. The proof is provided below to show that this derivative is principled under the optimality assumption. Theorem III.1 (FusionGAN Optimality Theorem). Let P XA and P XB be any distribution, Let P θ be the distribution of G Fθ (Z) with Z, a random variable with density p and G Fθ a function satisfying E z∼p [L(θ, z)] < +∞, where L is Lipschitz constants. All functions are well-defined. Then there is a solution D F : X → R to the problem: and we have: Proof of Theorem III.1: Let us define By the Kantorovich-Rubinstein duality, there is an f ∈ F that satisfies: is continuous and differentiable for any D F ∈ X * (θ) when both terms are well-defined. Let D F ∈ X * (θ), which we knows exists since X * (θ)is non-empty for all θ. Then: since D F is 1-Lipschitz. Furthermore, G Fθ (z) is locally Lipschitz as a function of (θ, z). Therefore, G Fθ (z) is locally Lipschitz on (θ, z) with constants L(θ, z). By Radamachers Theorem, D F (G Fθ (z)) has to be differentiable almost everywhere for (θ, z) jointly. Rewriting this, the set By Fubinis Theorem, this implies that for almost every θ the section A θ = {z : (θ, z) ∈ A} has measure 0. Let's fix a θ 0 such that the measure of A θ0 is null. For this θ 0 we have ∇ θ D F (G Fθ (z))|θ 0 is well-defined for almost any z, and since p(z) has a density, it is defined p(z) almost everywhere. Given the condition E z∼p [L(θ, z)] < +∞, we have so E z∼p(z) [∇ θ f (g θ (z))] is well-defined for almost every θ 0 .
To keep the notation simple, we leave it implicit in the following equation that the E subjects to z ∼ p(z). : where i ∈ {A, B, F }. By differentiability, the term inside the integral converges p(z) to 0 as θ → θ 0 . Furthermore, and since E z∼p(z) [6L(θ 0 , z)] < +∞, we get by dominated convergence that Eq. 5 converges to 0 as θ → θ 0 . So the result of Eq. 5 equals to: Therefore, the update rules for the parameters θ of G F is Eq. 4. The physical meaning is that all three discriminators can offer feedback to generator G F as shown in Fig. 2. The left cyan rectangle indicates D A , the middle pink one is D F , and the right yellow one D B . Similarly, the gradient w.r.t. the parameters φ of D F is: As shown in Fig. 3, the update for D F actually implicitly balances itself between D A and D B . However, this update does not cover the degree of blending which may lead to an unbalanced mixture. Specifically, a good learning process for the discriminator D F should consider two factors: (1) minimizing the distance with D A , D B and D F simultaneously and (2) maintaining a equal distance from D A and D B so as to satisfy both domains. To satisfy (2) and push the discriminator D F to have a well-proportioned blending, an additional constraint is employed to further balance the ratio of existing domains, i.e., D A , D B :

D. D A and D B Update of FusionGAN
After updating D F , FusionGAN will improve D A and D B . Since D A and D B are symmetric in the framework, we only discuss D A as shown in Fig. 9. Keeping related terms in Eq. 1, we get the loss function for D A : Taking the derivative of L A w.r.t. D A , we have There are four negative terms and one positive term in Eq. 11 w.r.t. D A . The training implicitly increases the positive terms and decreases the value of the negative terms. However, the inequalities in the two groups are not controlled. To further control the distance among the three domains, we propose an inequality constraint for D A : Then combining Eq. 12 into loss function Eq. 11, it can be rewritten as:

E. Algorithm Description
The Algorithm 1 1 initializes the parameters of the discriminators and generators (line 1). TextCNN [15] and LSTM are employed to build the discriminators and generators respectively. Given X A , the generator is updated by maximum likelihood estimation (line 3). From lines 4 to 5, D A receives the sequences generated by G A as negative data, and real data as positive samples. Between lines 6 and 16, a GAN procedure is applied to optimize D A and G A . D A returns regression scores that are treated as rewards for the policy gradient of G A . Similarly, D A improves itself by the real world data X A and the data generated by G A . Our method proceeds to update D B and G B by using the same process from lines 3 to 16. D F repeats the same pre-training procedure from 3 to 16 with D F , G F , X F = X A + X B .

A. Quantitative Study
Due to the aesthetic nature of the task, a quantitative evaluation can only give a first impression of the power of the generative system. Here, we investigate the distance between pitch and note duration distributions, knowingly discarding all sequential information of the melodies.
The ideal output should keep equally close to the given domains and minimize this distance. Two distributions of properties are selected for the study.
1) Duration Distribution (DD): Notes are categorized by their durations. 2) Normalized Pitch Distribution (NPD): All tokens are categorized into C/C#/D/D#/E/F/F#/G/G#/A/A#/B. To quantify the similarity between distributions, two symmetric metrics, the Euclidean distance (EUD) and the Wasserstein-1 metric (Earth Mover distance (EM)), are employed to calculate the distance between the distributions of generated sequences and those of domains X A and X B . Intuitively, the distance between X A and X B , between X A , D F , and between X B , D F should satisfy the triangle inequality, and the optimal configuration should minimize the inequality, i.e., Figure 6 (top) shows the note duration distributions (DD) and Table I (top) gives the corresponding distance measures. The note duration distribution of jazz and folk have shapes that resemble the power law distribution. The shape of RM is very different from jazz or folk. Visually, RL's duration distribution does not match that of jazz and folk because the portion of notes at the length of 0 and 1 (the peak) were relatively higher than that of jazz or folk. The other baselines (GAN, MLE, MC) appear similar to the given domains because they conform to the power law distributions. Fusion is roughly at the same level of MC. Note that, given our evaluation metrics, MC has the best performance in this experiment since it directly samples notes from the true distribution. This disregard of sequential information, however, results in unnatural rhythmic qualities of the music clips generated by MC. Figure 6 (bottom) shows the pitch distributions (NPD) and Table I (bottom) gives the corresponding distance measures. RL's pitch distribution is -due to overfitting-different from that of jazz and folk, which shows its low capacity in modeling pitch distribution. The single peak at D# as generated by GAN does not match either the jazz or folk distributions. The distributions of the other baselines are difficult to categorize as a good or bad match. Since MC is directly sampled from real distributions, its configurations should be exactly the same as that of the optimal fusion. MC shows two valleys at C# and F#, and two peaks at C and G, which should be expected. RM satisfies one of them (F# is a valley), while RM meets only one standard (C is the peak). GAN fails in all the characteristic pitch classes, and both MLE and Fusion meet all of them. Quantitatively, the ratio of RM in NPD is the highest which means that RM is the worst in modeling pitch. GAN and RL improve RM by around 10%. As one of the two distributions that look most similar to the original distributions, MLE outperforms RM, RL, and GAN. Our method Fusion has the second lowest ratio after MC. However, as mentioned above, the MC does not consider the sequential consistence, resulting in bad evaluations in the listening test described in Sect. IV-B.

B. User Study via Listening Test
To evaluate the subjective quality of the generated sequences, a user study was conducted via Amazon Mechanical Turk. The listening test was divided into three steps: 1) Qualification test: "Listen to the music, and choose one genre below that most matches the music". This is to test the evaluators' qualification level. The provided choices are (A) jazz, (B) folk, (C) neither. 2) Fusion recognition: "Listen to the attached music, and then choose the choice that most matches the music" Possible answers are (A) pure jazz, (B) pure folk, (C) mixture of jazz and folk, (D) neither. Each of the pieces of music generated by our proposed method and the baselines are assigned to one such question. 3) Musicality: "Listen to the attached music in 2nd step, who do you think its composer", the allowed answers are (A) expert, (B) newbie, (C) robot. Similarly, each generated sample is associated with one such question. We randomly drew from the training data and prepared 800 sets for the first question. In the first step, 66.5% of candidates chose the correct answer, which is an acceptable rate given the inherent difficulties of genre identification for average listeners. The statistics resulting from the valid answers are shown in Table II. Each of the methods generated 500 samples, which were randomly selected for the second and third questions.
As shown in Table II, the best system based on the percentage of mixture is MLE. A closer look at MLE, however, reveals the unbalanced distribution between jazz and folk, which implies its bias towards the jazz genre. On the other hand, RM is the best system based on the balance between jazz and folk, but the high percentage of neither suggests the confusion of the listeners. It is clear that a single criterion is insufficient for determining the best system. Therefore, we design a metric to summarize the results and represent the fusion level (FL) of the evaluated systems with the following equation: where C i indicates the count of i. C jazz − C f olk means the unbalanced error, and C neither can be treated as another type of error. A higher FL value implies better fusion.
Overall, Fusion shows a balanced performance in both F usionrecognition and M usicality tests, which demonstrates the effectiveness of our proposed method. V. CONCLUSION In this paper, we proposed a three-way GAN-based learning framework to integrate multiple domains. A Wasserstein distance based metric is introduced to indicate the blending progress. The evaluation investigated objective metrics looking at the pitch and note duration distributions of the generated data. A user study explicitly illustrates the validity of the proposed method as compared to the baselines, as the melodies generated by our model were preferred by the majority of users. While the listening test results are encouraging, future work will benefit from removing key dependence of the training data, a careful look into perceptually meaningful evaluation metrics that take into account the sequential nature of music, and a careful listening test design, including human-composed melodies for comparison.


extfieldsofstudy:
Computer Science
Engineering

text:
Compression Fractures Detection on CT

The presence of a vertebral compression fracture is highly indicative of osteoporosis and represents the single most robust predictor for development of a second osteoporotic fracture in the spine or elsewhere. Less than one third of vertebral compression fractures are diagnosed clinically. We present an automated method for detecting spine compression fractures in Computed Tomography (CT) scans. The algorithm is composed of three processes. First, the spinal column is segmented and sagittal patches are extracted. The patches are then binary classified using a Convolutional Neural Network (CNN). Finally a Recurrent Neural Network (RNN) is utilized to predict whether a vertebral fracture is present in the series of patches.

Vertebral Fracture
The annual incidence of osteoporotic fractures in the United States surpasses the cumulative incidence of breast cancer, heart attack and stroke. The most common osteoporotic fractures are those which result in compression of a vertebral body [1]. In addition to being a direct source of morbidity, vertebral compression fractures (VCFs) indicate substantially elevated risk of future osteoporotic hip fractures, which represent the most consequential sequelae of osteoporosis.
Within a year of suffering osteoporotic hip fracture, 30% of previously independent indiviuals will lose the ability to walk without assistance [2], 25% will become totally dependent or require nursing home facilities [3,4] and 20% will not survive [5].
Prophylactic treatment of at-risk osteoporotic individuals has been shown to reduce the rate of future hip fracture by 40-70% [6,7,8]. Yet, despite the burden and potential to prevent hip fractures in an ageing population, osteoporosis screening remains profoundly underutilized: less than 20% of people older than 65 years undergo bone mineral density screening via Dual Energy X-ray Absorptiometry (DXA) evaluation, [9] as recommended by the National Osteoporosis Foundation.
VCF detection on CT examination is as predictive for future osteoporotic hip fractures as a DXA diagnosis of osteoporosis: detection of a VCF confers a relative risk ratio of 2.3 for experiencing a hip fracture and a 4.4 fold risk of incurring other additional insufficiency fractures [8,10,11,12]. Yet only 13%-16% of retrospectively confirmed VCFs are actually reported at the time of computed tomographic (CT) interpretation [13,14,15]. The expertise required to diagnose VCF's is far less than that required to make the vast majority of routine radiologic observations on CT imaging. In Carberry et al.s study of VCF's, examining over 2000 CT examinations, the retrospective detection was accomplished by a medical student with one hour of dedicated compression fracture detection training [13]. The reason why VCF's are routinely missed is more likely due to the fact that they are considered incidental findings relative to the primary clinical indication which prompted the CT study.
Here we describe a method to increase identification of at-risk individuals via automatic opportunistic detection of VCFs on CT imaging.

Convolutional Neural Networks
Convolutional Neural Networks (CNN's) have proven immense utility when applied to Computer Vision tasks such as detection, segmentation and classification. Among CNN's advantages is the ability to learn a hierarchical representation from the input images and extract relevant features that generalize well across a large volume of data. This stands in contrast to conventional methods reliant on handcrafted feature design. Among the most successful CNN architectures for 2D image inputs are AlexNet, VGG, ResNet and Inception [16,17,18,19].

Recurrent Neural Networks
Recurrent Neural Networks (RNN's) are a key Deep Learning tool used to model sequences and time series. RNN's are often combined with CNN's, using the CNN as a feature extractor and the RNN to model the sequence. RNN's are commonly utilized in tasks such as video classification [20], language modeling [21], and speech recognition [22], among others.
The algorithm proposed here comprises three main steps, once provided a CT of the chest and/or abdomen. First, a segmentation process extracts sagittal patches along the vertebral column. These patches are then binary classified using a CNN. Finally, an RNN is run on the resulting vector of probabilities. The RNN output is a prediction of the probability for the presence of a compression fracture within the input CT scan.

RELATED WORK
Previous work on spine segmentation and VCF detection focused primarily on achieving an accurate segmentation of each vertebra. Yao et al [23] defined a novel method for segmentation of vertebrae, extracting the vertebral cortical circumference and mapping it into 2D for fracture detection. Ghosh et al [24] describe a VCF detection method based upon a 2D sagittal reconstruction, with relatively unstable results. Kelm et al [25] describe vertebral segmentation method using a disc-centered approach based on a probabilistic model which requires the full vertebral column to be included in the scan. Yao et al [26] accomplishes the segmentation on axial slices assessing the vertebral body as compared to an axial vertebral model. Discs are detected by low similarity to the model. This method was later used for feature-extraction based classification of vertebral fractures of osteoporotic or neoplastic etiology [27].

DATA
We assembled an initial dataset from 3701 individuals over the age of 50 years who had undergone CT scans of Chest and/or Abdomen for any clinical indication. Two expert radiologists reviewed each CT and assessed them for the presence of VCF's as defined by Genant criteria for vertebral compression. A third radiologist (EE) served to mediate in instances of non-consensus. Of the 3701 CT studies, 2681 (72%) were designated as negative for the presence of VCF and 1020 (28%) were tagged as VCF positive, including a bounding box annotation indicating the fracture position.
The positive class was comprised of 61% women of an average age of 73 years (std. 12.4), and 39% men of average age 66.8 years (std. 16.8). The negative class was comprised of 47% women of average age 56.7 years (std. 17.4) and 53% men of average age 56.1 years (std. 17.9). These results are not surprising: the prevalence of VCF's is known to be substantially higher in women and elderly individuals. We determined however that such pronounced inter-group differences in demographic characteristics could introduce biases with unintended results. Training a classifier on this dataset might, for example, have resulted in an algorithm which has learned to distinguish between the spines of elderly women and younger men.
Considering the above, a more demographically balanced subset of CT studies was derived including 1673 CT studies, of which 849 were VCF negative and 824 VCF positive. The VCF negative and VCF positive groups were filtered to balance age and sex. The negative class contains 43% men of average age of 64.7 years (std. 15.9) and 57% women of average age years 69.4 (std. 11.8). The positive class contains 42% men of average age 65.7 years (std. 16.0) and 58% women of average age 70.4 years (std. 11.9). All algorithmic training was performed upon this more balanced data set.

METHOD
To automatically detect VCF's from a provided CT Chest or Abdomen, we constructed an algorithmic pipeline of three stages: first, we segment the portion of the spine included in the provided CT and extract sagittal patches. We then classify the patches using a CNN and employ a RNN on the resulting vector of probabilities. The training phase was performed on the patches extracted using the algorithm below.

Spine Segmentation and Patch Extraction
The spine segmentation algorithmic goal is to enable deterministic sagittal section patches extracted along the vertebral column. These patches are used as input samples for training and inference. The use of the patches will be further explained in subsection 4.3.
Several spine segmentation approaches have been described. Ghosh et al [24] achieve spinal segmentation based upon the mid vertebral body in sagittal projection -a technique which has limited precision when applied to individuals with spinal scoliosis. To overcome limitations of variant spinal curvatures, we included a preprocessing step in generating a "virtual sagittal section" which first identifies and aligns the spinal cord in a straight cranial-caudal projection and then adjusts vertebral body location accordingly. We preferred this more relaxed segmentation approach for the task of VCF detection relative to several more accurate segmentation techniques for vertebrae [23,25] or discs [24,26].
The virtual sagittal section creation is based on localization of the spinal cord in axial view, followed by tracking the back (posterior) of the individual along the spinal cord in coronal projection and bone intensity Hounsfield Unit (HU) threshold. The virtual sagittal section is used for segmentation of the vertebral column and extraction of sagittal patches of vertebrae to be learned and classified. The process of the algorithm is described in Figure 3.  This method was employed on the dataset described in section 3 and was used to extract both positive and negative samples.

Patch-based CNN
We applied the present algorithm to the CT data set to extract patches along the vertebral column. Figure 7 demonstrates the output of the segmentation algorithm which is an input to the CNN training phase. The patches were re-scaled to 32x32. We found this size ideal -there were no significant improvements in our experiments for larger patch sizes such as 64x64 and 128x128. 15% of all CT studies were sequestered as an exclusive validation set -patches derived from any individual CT study reside in either the training or validation set but not both. The validation set held a balanced number of samples between the two classes.
In contrast to other computer vision tasks where various data augmentation techniques are used in the training process, we opted not to apply any augmentation tools which might misrepresent the anatomic structure of the vertebral column. Instead applied only light rotations of -18 to 18 degrees, as the resulted transformation can naturally match the curvature and variability of the spine.

Patch sequence classification
RNN's have been a dominant tool in handling sequences for many tasks in fields such as video classification [20], language modeling [21]and speech recognition [22]. A popular approach is to use LSTM based RNN's [28] which are more robust to long sequences as they don't suffer the vanishing and exploding gradient during training, which is a disadvantage for normal RNN's.
Given a CT scan, we extract the sagittal vertebral patches and apply the CNN described to obtain a vector of probabilities. Figure 7 illustrates a subset of a resulting vector of probabilities which represents a CT scan of a patient. The input vector size is not fixed; this is due to variance in physical attributes such as the height and the curvature of the spine. In addition, the vector of probabilities obtained is not a regular vector of features, but rather represents a sequence of predictions for fractures along the spine, therefore justifying an RNN based approach. Specifically, we employed a single layered LSTM with 128 cells, which were connected to a 2D output trained via the cross entropy loss. Figure 8 illustrates the RNN model used.

RESULTS
The CNN was trained for 15 epochs and reached 92.9% accuracy over the validation set. Visualizing training samples such in Figure 9, demonstrates that the trained CNN learned to focus on VCF's. We then applied the trained CNN to the entire training and validation data sets to generate vectors for the RNN based classifier. The RNN was trained on the training data set and evaluated upon the validation vectors, resulting in 89.1% accuracy, 83.9% sensitivity and 93.8 specificity.

CONCLUSIONS
VCF's are clinically important findings which are readily identifiable but routinely overlooked by radiologists. Only 13%-16% of retrospectively confirmed VCFs are actually reported at the time of CT study interpretation. This is likely due to the fact that they are considered incidental findings relative to the primary reason for which a given CT study was performed. In this work we present a robust algorithm to detect VCF's in CT scans of the Chest and/or Abdomen. First, a deterministic segmentation algorithm extracts patches along the patients vertebral column. Then these patches are classified using a CNN followed by RNN classifier on the resulting vector of probabilities. To the best of our knowledge this work is the first to address the problem of VCF detection using deep learning methodologies. This application may serve to increase the diagnostic sensitivity for VCF's in routinely acquired CT's, potentially triggering preventative measures to reduce the rate of future hip fracture.