Inter-sector beamforming with MMSE receiver in the downlink of TDD cellular systems The use of beamforming is effective for users in limited power environments. However, when it is applied to the downlink of a cellular system with universal frequency reuse, users near the sector boundary may experience significant interference from more than one sector. The use of a minimum mean square error (MMSE)-type receiver may not sufficiently cancel out the interference unless a sufficient number of receive antennas are used. In this paper, we consider the use of inter-sector beamforming that cooperates with a neighboring sector in the same cell to mitigate this interference problem in time-division duplex (TDD) environments. The proposed scheme can avoid interference from an adjacent sector in the same cell, while enhancing the transmit array gain by using the TDD reciprocity. The performance of the proposed scheme is analyzed in terms of the output signal-to-interference-plus-noise power ratio (SINR) and the output capacity when applied to an MMSE-type receiver. The beamforming mode can be analytically switched between the inter-sector and the single-sector mode based on the long-term channel information. Finally, the effectiveness of the proposed scheme is verified by computer simulation. Demand for higher throughput has motivated advanced wireless systems such as the third generation partnership long-term evolution (3GPP LTE) and mobile worldwide interoperability for microwave access (m-WiMAX) to employ multi-cell configuration with universal frequency reuse [1] , [2] . It has been reported that the use of beamforming is effective in the downlink when users are power limited (i.e., at medium to low signalto-noise power ratio (SNR)) [3] , [4] . In a time-division duplex (TDD) wireless system, base station (BS) can estimate channel state information (CSI) of the uplink using reference signals such as the channel sounding signal [5] and can utilize it for the transmit beamforming by means of channel response reciprocity [5] , [6] . However, the use of beamforming with universal frequency reuse may cause users near the cell boundary to experience serious inter-cell interference [1] , [2] . In particular, when the cell is divided into a number of sectors, users near the sector boundary may experience weak received signal strength (RSS) from the serving sector due to multi-sector antenna pattern as well as strong interference from adjacent sectors [6] , [7] . Recently, advanced wireless systems consider the use of multiple receive antennas in the mobile station (MS) [2] , [6] , [8] , enabling the use of a minimum mean square error (MMSE)-type receiver to suppress interference, while reducing the fading effect [9] . However, when the number of strong interferers is larger than the number of receive antennas minus one, the output signal-to interference-plus-noise power ratio (SINR) of the MMSE receiver significantly deteriorates [10] , [11] .. To improve the performance of users near the sector/cell boundary, the use of BS coordination has recently been considered, where the BSs share the CSI and information streams to minimize the interference effect [12] , [13] . However, this BS cooperation may incur so-called signaling overhead increasing in exponentially proportional to the number of coordinating BSs and suffer from performance degradation due to the signaling delay. A null beamformer has been recently introduced, where the interference to other co-channel cells is removed by exchanging the instantaneous inter-cell CSI [14] . However, it considers the use of a single receive antenna, neglecting the interference cancelation at the receiver. In practice, advanced wireless systems consider the use of real-time cooperation among sectors in the same cell, where the information between these sectors can be exchanged in real time without signaling overhead [8] , [15] .. On the other hand, the use of softer handover, macro diversity handover, and fast sector selection (FSS) with muting can be used to enhance the performance of users near the sector boundary without the exchange of inter-cell information [8] , [16] . In the softer handover, both the serving sector and neighboring sector transmit the same data signal with different sector-specific scrambling codes, and the user softly combines the two signals before decoding. Note that the simultaneous transmission of same data signal may cause mutual interference. In the macrodiversity handover, the MS can demodulate signal without interfering each other, while obtaining radio frequency diversity. The performance depends on the code rate and the propagation delay between the sectors since it is the same as the cyclic shift transmit diversity [17] . In the FSS with muting, only one sector in better condition transmits data signal with doubled transmit power, while the other sector, called muted sector, does not transmit data signal to avoid mutual interference. However, these schemes do not consider the use of TDD reciprocity.. In this paper, to improve the performance of users near the sector boundary, we propose an inter-sector beamforming scheme that cooperates with an adjacent sector in the same cell. The proposed scheme can avoid interference from the adjacent sector in the same cell, while enhancing the transmit array gain using the TDD reciprocity. Moreover, it does not require additional signaling overhead compared to conventional beamform-ing (or single-sector beamforming) scheme. It is assumed that the MS transmits the reference signal using a single transmit antenna, as being considered in the IEEE 802.16e [2] , [6] and IEEE 802.20 systems [18] . This implies that the BS should be able to generate a beamforming weight using partial knowledge of the CSI obtained through a single transmit antenna of the MS. The average output SINR and capacity of the MMSE receiver with the use of inter-sector beamforming are analyzed in the presence of correlation between the transmit antennas. According to the channel condition, the proposed scheme employs either single-sector or inter-sector beamforming to maximize the average capacity.. Following Introduction, Section II describes the system model in consideration. Section III presents the proposed intersector beamforming. The performance of the proposed scheme is analyzed in terms of the output SINR and the output capacity with combined the use of an MMSE-type receiver in Section IV. The performance of the proposed scheme is verified by computer simulation in Section V. Finally, conclusions are given in Section VI. We have considered the use of inter-sector beamforming for the service of users near the sector boundary in TDD based cellular systems. The performance of the inter-sector beamforming has been analyzed in terms of the average output SINR and the capacity in correlated channel environments. The intersector beamforming mode can analytically be switched to the single-sector mode based on the long-term CSI. The simulation results show that the inter-sector beamforming is very effective for users near the sector boundary, outperforming conventional schemes such as the softer handover, macro diversity handover, FSS with muting, and null beamforming.

O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B I O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B I O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B I O O O O O O O O O O O O O O O O B


Example-driven meta-model development The intensive use of models in model-driven engineering (MDE) raises the need to develop meta-models with different aims, such as the construction of textual and visual modelling languages and the specification of source and target ends of model-to-model transformations. While domain experts have the knowledge about the concepts of the domain, they usually lack the skills to build meta-models. Moreover, meta-models typically need to be tailored according to their future usage and specific implementation platform, which demands knowledge available only to engineers with great expertise in specific MDE platforms. These issues hinder a wider adoption of MDE both by domain experts and software engineers. In order to alleviate this situation, we propose an interactive, iterative approach to meta-model construction, enabling the specification of example model fragments by domain experts, with the possibility of using informal drawing tools like Dia or yED. These fragments can be annotated with hints about the intention or needs for certain elements. A meta-model is then automatically induced, which can be refactored in an interactive way, and then compiled into an implementation meta-model using profiles and patterns for different platforms and purposes. Our approach includes the use of a virtual assistant, which provides Model-driven engineering (MDE) makes heavy use of models during the software development process. Models are usually built using domain-specific modelling languages (DSMLs) which are themselves specified through a metamodel. A DSML should contain useful, appropriate primitives, and abstractions for a particular application domain. Hence, the input from domain experts and their active involvement in the meta-model development process are essential to obtain effective, useful DSMLs [34, 37, 38, 43, 53] .. The usual process of meta-model construction requires first building (a part of) the meta-model which only then can be used to build instance models. Even though software engineers are used to this process, it may be counter-intuitive and difficult for non-meta-modelling experts, who may prefer drafting example models first and then abstract those into classes and relations in a meta-model. As Oscar Nierstrasz put it, "… in the real world, there are only objects. Classes exist only in our minds" [45] . In this way, domain experts and final users of MDE tools are used to work with models reflecting concrete situations of their domain of expertise, but not with meta-models. Asking them to build a meta-model before drafting example models is often too demanding if they are not MDE experts. In general, an early exploratory phase of model construction, to understand the main concepts of the language and document the language requirements, is recommended for DSML engineering [12, 37] .. While MDE experts are used to work with specialized meta-modelling tools-like those provided by Eclipse EMF [52] -this is seldom the case for domain experts. These latter may find easier, more intuitive and flexible using sketching and drawing tools in the style of PowerPoint or Visio to build models and examples, than using, e.g. the EMF's tree-based editor. Moreover, once an initial version of a meta-model is built, it needs to be validated in collaboration with the domain experts. While MDE experts are used to inspect meta-models, for domain experts, a validation based on examples (again, built using sketching tools) would be more adequate, as they may lack the required expertise in conceptual modelling to fully understand a meta-model.. Another issue that makes meta-model construction cumbersome is the fact that meta-models frequently need to be fine-tuned depending on their intended use: designing a textual modelling language (e.g. with Xtext 1 ), a graphical language (e.g. with GMF [33] or Eugenia [39] ), or the source or target of a transformation. As illustrated in Fig. 1 , the particular meta-model usage may impact on its design, for instance to decide whether a connection should be implemented as a reference (e.g. for simple graphical visualization), as an intermediate class (e.g. for a more complex visualization, or to enable iterating on all connection instances), as a bidirectional association (e.g. to allow back navigation if it is used in a transformation), or as an intermediate class with composition (e.g. to enable scoping). The use of a specific technological platform, like EMF [52] , has also an impact on how meta-models are actually implemented, e.g. regarding the use of composition, the need to have available a root class, and the use of references. As a consequence, the implementation meta-model for a particular platform may differ from the conceptual meta-model as elicited by domain experts. 1 http://www.eclipse.org/Xtext/.. Specialized technical knowledge is required for this implementation task, hardly ever found in domain experts, which additionally has a steep learning curve.. In order to alleviate this situation, this paper presents a novel way to define meta-models and modelling environments. Its ultimate goal is to facilitate the creation of DSMLs by domain experts without proficiency in meta-modelling and MDE platforms and technologies. For this purpose, we propose an iterative process for meta-model induction in which model fragments are given either sketched by domain experts using drawing tools like Dia 2 or yED, 3 or using a compact textual notation suitable for engineers (not necessarily meta-modelling experts). In both cases, they can annotate the intention of the different modelling elements. From these fragments, a meta-model is automatically induced, which can be refactored if needed. The system provides suggestions of possible improvements based on well-known refactorings [30] , quality issues for conceptual schemas [2] , metamodel design patterns [10] , and anti-patterns [29] . Once an initial version of the meta-model is obtained, it can be validated against new model examples, and the system reports any problematic model element as well as the missing meta-model elements needed to accept the model examples. Finally, the resulting meta-model is compiled into a given technology (e.g. EMF or MetaDepth [20] ), optimized for a particular purpose (e.g. visual or textual language, transformation) and a particular tool (e.g. Xtext or GMF).. This paper is an extended version of the preliminary work presented in [16] . In particular, in this extended version, we provide a detailed account of additional design and domain annotations, some of which are translated into OCL constraints in the synthesized meta-model. We have also extended our tools with additional annotations and refactorings, support for an additional sketching tool and more sophisticated importing capabilities, a virtual assistant module that reports refactoring opportunities and automates their application, and a validation mode to automate testing of meta-models. The paper has also been enlarged with more extensive explanations, a more complete and challenging example, the presentation of the design of our solution, and more comprehensive related work. While in this work, we concentrate on presenting the different concepts, design decisions and tool support of our approach, an empirical evaluation with our industrial partners is left for a future contribution. Paper organization Section 2 overviews the working scheme of our proposal. Its main steps are detailed in the following sections: specification of fragments (Sect. 3), meta-model induction and refactoring (Sect. 4), example-based metamodel validation (Sect. 5), and compilation of the induced meta-model for different purposes and platforms (Sect. 6). Next, Sect. 7 presents tool support. Finally, Sect. 8 compares with related research, and Sect. 9 ends with the conclusions and lines of future work. In this paper, we have presented a novel approach to the development of meta-models to make MDE more accessi-ble to non-experts. For this purpose, we have proposed a bottom-up approach where a meta-model is induced from model fragments, which may be specified using informal sketching tools such as Dia or yED. Sketches are transformed into a specialized textual notation that can be directly used by advanced users. Model fragments can be annotated to guide the automatic induction of the meta-model and document the intention of certain elements. The process is iterative, as fragments are added incrementally, causing updates in the meta-model, which can be refactored in the process based on the recommendations provided by a virtual assistant. To involve the domain experts in the meta-model validation process, we follow an example-based approach to the testing of meta-models. Finally, the meta-model is compiled for specific platforms and usage purposes.. Even though we allow the specification of negative fragments, these are not currently used to induce the meta-model, which is left for future work. We would like to perform an empirical evaluation of the approach with our industrial partners. Among the various aspects to be evaluated, the suitability of the library of annotations we provide is of particular interest. We also foresee the possibility of starting the process of DSML construction using a set of informal annotations, which can be formalized later. We also plan to improve the tool support. One direction is to enhance collaboration by building a web application where domain experts can sketch fragments that are automatically integrated in the environment for their refinement by an engineer. Another goal is to automatically build a visual modelling environment out of the sketched fragments. It would also be interesting to provide automated support for DSML requirements expressed in natural language, in addition to sketches. The integration of different implementation meta-models compiled from the same neutral meta-model, e.g. to support different syntaxes for a DSML, is also future work. We plan to extend our metamodel validator to allow the testing of more complex properties (e.g. meta-model invariants) using a dedicated high-level language and provide our tool with extension mechanisms for defining new annotations, recommendations, and refactorings. There are some works dealing with the inference of metamodels from models. For instance, the MARS system [36] enables the recovery of meta-models from repositories of models using grammar inference. The objective is being able to use a set of models after migrating or losing their meta-model. Actually, the induction process can be seen as a form of structural learning [40] . In contrast, our purpose is enabling the interactive construction of meta-models, also by domain experts.. There are a few works using test-driven development (TDD) to build meta-models iteratively. For instance, in [15] , the authors attach test cases to the meta-classes in a metamodel. Test cases are executable models written in PHP and perform some kind of transformation like code generation. If a test case shows that a meta-model is inadequate, this must be manually modified. Similarly, in [47] , the authors combine specifications and tests to guide the construction of Eiffel meta-models. Specifications are given as Eiffel contracts, whereas tests are written using the acceptance test framework for Eiffel. Another example is [49] , which supports the specification of positive and negative example models from which test models for meta-model testing are generated. In our case, the meta-model is automatically induced from model fragments, and there is a greater level of interactivity. Moreover, meta-models are updated through refactorings, which simplifies their evolution and the propagation of changes to model fragments (i.e. side effects). A catalogue of meta-model refactorings, although not directly related to TDD of meta-models, is available in [44] . We provide support for many of them. Moreover, to promote the collaboration of the domain experts in the meta-model validation process, we support an example-based approach to the testing of metamodels. In the future, we plan to extend this approach to allow, e.g. the testing of meta-model invariants, using some dedicated language.. Techniques to build MDE artefacts "by example" have emerged in the last years, but it is still novel for meta-models. In the position paper [12] , the authors identify some challenges to define DSMLs by demonstration. They discuss the usefulness to bridge informal drawing tools with modelling environments, as the former are the working tools of domain experts. They also recognize the difficulty for experts to manually build meta-models and suggest an iterative process. Recently, the authors have realized their ideas in a framework where domain experts can provide model examples using a concrete syntax, from which a meta-model describing their abstract syntax is inferred [11] . While their proposal is similar to ours, we also stress that meta-models may be different depending on the target platform and usage. Hence, we support the automated induction of a neutral meta-model, its refactoring likely due to automatic recommendations made by a virtual assistant, its validation in collaboration with the domain experts, and different compilations into implementation meta-models guided through annotations and selection of configurations.. Aligned to our proposal, [4] advocates the use of examples for eliciting, modelling, verifying, and validating complex business knowledge in software development projects. The authors suggest that the use of examples is a promising way to improve the quality of structural models and foster the active participation of end-users in the development process. We apply this principle to the construction of meta-models. The work in [4] was further developed in [3] , considering positive and negative examples, and proposing a process that combines abstraction inference and automated example derivation. The process is supported by Clafer, a modelling tool making uniform types and instances. While our process supports abstraction inference (meta-model induction), we do not support negative examples or example derivation yet, but we will consider them in future work.. In software engineering, informal modelling and sketching are normally used in early phases of requirements elicitation [19] . Actually, it is regular practice when gathering user interface requirements in the form of mockups [51] , and many mockup tools exist nowadays. In a more general setting, the work in [56] proposes a sketching tool independent of the notation which is able to transform informal sketches into more formalized models, like UML models. Our proposal can be seen as a means to leverage from DSML requirements in the form of informal sketches which are used for the construction of a meta-model. Additionally, software requirements are often specified using natural language. Several heuristics methods exist, like the noun-phrase [55] , to systematically derive a conceptual model from requirements in natural language. Recently, some tools have emerged for the automated derivation (up to some degree) of class models [22] . We plan to include support for DSML requirements in natural language in future work. Conversely, some authors propose the conversion of UML/OCL conceptual schemas into natural language [9] . In our setting, this idea would be useful as a means to validate the obtained meta-model with domain experts and also as a means to facilitate the comprehension of the constraints attached to model fragments.. Several works recommend integrating end-users in the meta-model construction process as a means to improve the quality of the resulting meta-model. For instance, in [34] , the authors propose a collaborative approach named Collaboro to meta-model construction which involves both domain and technical experts. Their approach is supported by a DSL to represent the collaborations among stakeholders, namely change proposals, solution proposals, and comments. In [35] , we integrated Collaboro and our tool so that domain experts can collaborate by providing examples, from which change proposals are derived.. Another line of related work concerns the expressiveness of model fragments. While one could simply use object diagrams, in [41] , the authors extend object diagrams with modalities to declare positive and negative model fragments and invariants (i.e. fragments that should occur in every valid diagram). Their goal is to check the consistency of a set of object diagrams and for that purpose they use Alloy. In our case, the goal is to automatically induce a meta-model. While we consider negative fragments, they are not yet taken into account by the induction algorithm, but only for validation.. In our approach, newly introduced fragments may raise conflicts if the fragments contain contradictory information. Some application domains where the resolution of conflicts has been extensively studied are model merging [42] , change propagation in software systems [28] , and distributed development [14] . It is up to future work to identify how the conflicts that may arise when evolving a metamodel relate to these previous works. On the other hand, our meta-model refactorings and subsequent propagation to the model fragments can be seen as a simplified scenario of meta-model/model evolution [13] .. A way to simplify and make the development of metamodels systematic is through design patterns. In [10] , some design patterns for meta-models are proposed, while in [50] , the requirements for meta-models are represented as use case diagrams and the meta-models are evolved by applying patterns. Related to our recommender system, in [2] , the authors present a unifying approach to define quality issues on conceptual schemas. These issues include syntactical ones (like invalid cardinalities), naming conventions [1] (like upper camel case for the name of classes), best practices (like redundant generalizations), and basic quality issues (like concrete classes with abstract children). The authors provide a systematic way to define such issues, including their detection and resolution. In [26] , an extensible framework for model recommenders is presented, where different strategies can be plugged in. We will also aim for an extension mechanism enabling the addition of new recommendations.. Our domain annotations serve as a constraint library, in the style of the predicates found in categorical sketches [24, 25] . Categorical sketches 8 are used to formalize diagrammatic notations, like the UML. In this approach, predicates are defined over an arity shape, which can be a graph or a family of graphs. Thus, we believe that categorical sketches could serve as a formal basis for our work.. Research in sketch recognition and understanding is also complementary to our approach [18] , as handwritten, more informal sketches could be turned into fragments to feed our system. In this line, works aiming at recognizing symbols and relationships in handwritten sketches relying on user interaction [54] fit well in our approach, since the user could define the legend (i.e. the types in the meta-model) at the same time that the system is being trained.

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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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Bayesian

evolutionary

hypergraph

learning

for

predicting

cancer

clinical

outcomes

Predicting

the

clinical

outcomes

of

cancer

patients

is

a

challenging

task

in

biomedicine.

A

personalized

and

refined

therapy

based

on

predicting

prognostic

outcomes

of

cancer

patients

has

been

actively

sought

in

the

past

decade.

Accurate

prognostic

prediction

requires

higher-order

representations

of

complex

dependencies

among

genetic

factors.

However,

identifying

the

co-regulatory

roles

and

functional

effects

of

genetic

interactions

on

cancer

prognosis

is

hindered

by

the

complexity

of

the

interactions.

Here

we

propose

a

prognostic

prediction

model

based

on

evolutionary

learning

that

identifies

higher-order

prognostic

biomarkers

of

cancer

clinical

outcomes.

The

proposed

model

represents

the

interactions

of

prognostic

genes

as

a

combinatorial

space.

It

adopts

a

flexible

hypergraph

structure

composed

of

a

large

population

of

hyperedges

that

encode

higher-order

relationships

among

many

genetic

factors.

The

hyperedge

population

is

optimized

by

an

evolutionary

learning

method

based

on

sequential

Bayesian

sampling.

The

proposed

learning

approach

effectively

balances

performance

and

parsimony

of

the

model

using

information-theoretic

dependency

and

complexity-theoretic

regularization

priors.

Using

MAQC-II

project

data,

we

demonstrate

that

our

model

can

handle

high-dimensional

data

more

effectively

than

state-of-the-art

classification

models.

We

also

identify

potential

gene

interactions

characterizing

prognosis

and

recurrence

risk

in

cancer.

Prognostic

prediction

is

an

important

task

in

clinical

medicine.

It

is

crucial

to

estimate

the

clinical

outcomes

of

patients

and

the

potential

effects

of

treatments.

A

refined

treatment

based

on

likely

clinical

outcomes

is

especially

necessary

in

oncology

due

to

the

variation

of

cancer

progression

in

patients.

By

accurately

estimating

the

clinical

response

to

treatments,

clinicians

can

personalize

and

hence

provide

an

improved

therapy

for

a

patient..

Gene

expression

profiling

has

been

widely

used

to

identify

tumor

heterogeneity,

and

it

has

led

to

the

discovery

of

molecular

signatures

of

potential

prognostic

and

therapeutic

interest

As

such,

it

is

recognized

as

a

powerful

source

for

improving

prognostic

assessment

and

treatment

selection

in

cancer

medicine.

Moreover,

cancer

prognosis

is

associated

with

combinatorial

and

modular

regulation

by

multiple

genetic

factors.

Thus,

for

more

precise

prediction

of

cancer

clinical

outcomes,

the

higher-order

relationships

among

genetic

factors

must

be

induced

from

gene

expression

profiles.

However,

the

complexity

of

gene

interactions

renders

this

task

extremely

challenging..

Predictive

methods,

which

classify

patient

outcomes

on

a

genome-wide

scale

from

high-throughput

biological

data,

have

flourished

in

recent

decades.

Many

studies

have

adopted

computational

approaches,

such

as

machine

learning-based

models

and

statistical

methods

to

predict

prognosis

from

cancer

genomic

data.

However,

few

of

the

existing

approaches

address

the

higher-order

interactions

between

genes

involved

in

cancer

prognosis..

Predicting

outcomes

from

higher-order

gene

relationships

requires

searching

an

exponential

search

space

consisting

of

tens

of

thousands

of

genes.

It

is

infeasible

to

exhaustively

explore

such

a

huge

combinatorial

feature

space

using

a

gradient

method,

and

thus

various

feature

selection

methods

are

frequently

used

Typically,

these

approaches

reduce

the

problem

space

by

individually

evaluating

each

gene

under

the

assumption

of

the

independence

between

features.

However,

such

restrictions

may

not

Elsevier

Inc.

All

rights

reserved..

capture

the

important

genes

involved

in

higher-order

relationships

underlying

pathological

processes..

Evolutionary

computation

is

one

of

the

most

useful

techniques

for

solving

real-world

complex

problems.

This

approach

is

conceptually

simple

and

widely

applicable,

and

it

outperforms

classical

optimization

procedures.

Moreover,

it

can

be

easily

hybridized

with

various

existing

methods

and

data

structures.

Evolutionary

methods

include

genetic

algorithm

(GA)

genetic

programing

(GP)

particle

swarm

optimization

(PSO)

learning

classifier

system

(LCS)

multi-objective

optimization

evolutionary

algorithm

(MOEA)

and

estimation

of

distribution

algorithm

(EDA)

These

methods

have

been

successfully

applied

to

problems

that

are

unsolvable

by

deterministic

methods.

Numerous

biological

problems

can

be

formulated

as

optimization

issues,

for

which

evolutionary

approaches

are

eminently

suitable

Also,

especially

in

medicine,

evolutionary

computation

has

expanded

rapidly

in

recent

years,

and

is

currently

adopted

in

diagnosis

and

prognosis

Despite

various

proposals

and

several

successes,

however,

it

remains

difficult

to

evolve

complex

higher-order

systems

involving

tens

of

thousands

of

features..

Here

we

introduce

a

new

population-based

model

that

uses

an

evolutionary

learning

method

to

predict

clinical

outcomes

of

cancer

patients

(Fig.

The

model

handles

complex

genetic

interactions

by

means

of

a

flexible

hypergraph

structure

comprising

a

large

population

of

hyperedges,

encoding

the

multi-variable

combinations

corresponding

to

all

potential

genes

or

markers.

Each

hyperedge

is

weighted

by

its

discriminative

ability

to

predict

prognostic

outcomes.

Thus,

each

hyperedge

potentially

behaves

as

a

prognostic

module

influencing

the

cancer

clinical

outcomes..

The

model

learning

involves

the

search

of

a

high-dimensional

space

characterizing

the

higher-order

relationships

between

factors.

To

learn

the

model

from

a

dataset

comprising

several

tens

of

thousands

of

genetic

variables,

we

use

an

evolutionary

method

based

on

sequential

Bayesian

sampling

scheme.

The

proposed

Bayesian

evolutionary

algorithm

is

designed

upon

a

standard

evolutionary

computation

framework.

The

repeat

of

variation,

evaluation,

and

selection

is

addressed

as

a

sequential

Bayesian

sampling

process,

where

the

posterior

distribution

is

recursively

calculated

from

the

prior

distribution

by

estimating

the

likelihood

from

fitness

measurements.

Using

this

Bayesian

formulation

of

evolutionary

computation,

the

model

can

determine

the

problem-specific

bias

as

a

guideline

for

efficient

search

of

a

huge

combinatorial

feature

space.

This

study

adopts

an

information

theoretic

co-regulatory

measure

i.e.,

mutual

information,

and

the

model

complexity

for

the

prior

distribution.

The

information

theoretic

measure

enhances

the

efficiency

of

the

evolutionary

search,

while

the

complexity

retains

a

compact

model

size

by

controlling

the

parsimony..

The

proposed

model

is

evaluated

on

MAQC-II

breast

cancer

and

multiple

myeloma

gene

expression

data

The

proposed

model

demonstrates

high

classification

performance

for

predicting

prognosis

in

patients,

and

can

identify

higher-order

prognostic

biomarkers

of

cancer

clinical

outcomes.

Moreover,

our

model

directly

identifies

potential

modules

of

informative

genes

that

characterize

prognosis

and

recurrence

risk

in

cancer..

This

paper

is

organized

as

follows.

Section

details

the

proposed

hypergraph-based

model

and

the

Bayesian

evolutionary

algorithm

for

learning

the

model.

In

Section

the

experimental

results

are

described

and

discussed.

Discussion

and

conclusions

are

presented

in

Sections

and

We

proposed

hypergraph

classifiers

based

on

evolutionary

learning

to

predict

cancer

prognoses

from

complex

genetic

interactions,

using

two

archived

datasets.

The

learning

method

evolves

a

population-based

representation

of

hypergraphs

based

on

a

sequential

Bayesian

sampling.

The

Bayesian

evolutionary

hypergraph

model

accommodates

formal

management

of

model

complexity

by

defining

priors

on

a

huge

combinatorial

search

space

comprising

tens

of

thousands

of

genes.

Specifically,

we

controlled

the

evolutionary

search

process

using

two

types

of

prior

distributions.

One

is

defined

in

terms

of

the

mutual

information

between

each

genetic

variable

and

the

class

label

and

it

guides

the

compositional

variation

of

the

variables

in

a

hyperedge.

The

other

is

related

to

the

model

size,

modulating

the

degree

of

a

hyperedge

and

the

number

of

hyperedges

in

the

model..

Cancer

prognosis

is

typically

influenced

by

the

combinatorial

regulation

of

multiple

genetic

factors.

By

analyzing

gene

relationships

at

higher-order

levels,

we

can

better

predict

clinical

outcomes

in

cancer

patients.

We

have

demonstrated

that

higherorder

interactions

discriminate

prognosis

more

precisely

than

pair-wise

analysis

of

single

gene

relationships.

From

this

viewpoint,

we

predicted

that

potential

prognostic

gene

modules

could

be

identified

from

higher-order

gene

interactions..

The

performance

of

the

proposed

method

was

validated

on

MAQC-II

data.

The

classification

performance

of

the

hypergraph

classifiers

was

superior

to

those

of

other

models

including

SVM,

naïve

Bayes

classifier,

decision

tree,

random

forest,

AdaBoost

and

LCS,

especially

in

terms

of

MCC

and

AUC.

Furthermore,

the

interpretable

structures

of

hypergraph

classifiers

proved

useful

for

analyzing

complex

biological

phenomena.

That

is,

the

proposed

model

presents

as

an

alternative

method

for

solving

a

variety

of

biomedical

problems.

Such

contributions

will

greatly

assist

toward

developing

a

personalized

and

refined

therapy.

The

proposed

hypergraph

classifier

predicts

cancer

clinical

outcomes

through

evolutionary

learning

from

genomic

data.

Human

cancer

clinical

outcomes

are

typically

induced

by

the

interactions

of

multiple

genomic

components.

Although

the

analysis

of

multiple

gene

interactions

should

be

important

to

predict

cancer

prognosis

and

to

understand

complex

cancer

phenomena,

however,

it

is

computationally

infeasible

to

explore

all

the

combinatorial

interactions

in

a

higher-order

level..

Herein

we

proposed

a

population-based

model

that

uses

an

evolutionary

learning

method

to

predict

cancer

prognosis.

The

results

of

the

proposed

learning

method

formulated

to

a

sequential

Bayesian

sampling

showed

that

our

model

efficiently

searches

to

a

huge

problem

space

and

provides

competitive

performance

with

other

state-of-the-art

classification

models.

As

shown

in

the

result

of

classification

performance

including

accuracy,

MCC,

and

AUC,

the

performance

of

the

proposed

model

was

competitive

or

superior

to

those

of

existing

models.

In

particular,

the

MCC

score

of

our

model

was

higher

than

that

of

SVMs

for

multiple

myeloma

data

as

with

a

significant

adjusted

p-value,

while

the

MCC

of

LCSs

was

zero

for

both

breast

cancer

and

myeloma

datasets.

This

result

indicates

that

the

proposed

hypergraph

classifiers

are

robust

to

imbalanced

data,

thus

more

precisely

predicting

clinical

outcomes

in

cancer

patients

than

the

existing

models.

We

also

compared

the

performance

of

the

proposed

model

against

two

variants

of

hypergraph

classifiers

and

HCs

without

using

MI

as

prior).

We

observe

that

higher-order

relationships

are

more

important

for

accurately

predicting

cancer

prognosis

than

pair-wise

relationships.

Moreover,

when

hyperedges

were

generated

based

on

mutual

information,

the

MCC

was

improved

for

both

datasets,

indicating

that

the

proposed

method

allowed

the

model

to

efficiently

search

a

huge

combinatorial

feature

space

by

introducing

problem-specific

knowledge

to

the

prior

in

the

evolutionary

learning

process.

Moreover,

the

change

of

the

distribution

of

the

hyperedge

degrees

in

Fig.

notes

that

the

proposed

learning

method

finds

the

best

composition

and

the

number

of

hyperedges

that

increases

the

model

performance

while

keeping

the

model

complexity.

Furthermore,

we

observe

that

higher-order

information

is

useful

for

enhancing

the

prediction

accuracy

in

cancer

prognosis.

Furthermore,

our

method

can

find

significant

genes

and

gene

modules

for

predicting

cancer

clinical

outcomes..

Diagnosing

models

have

been

constantly

proposed

to

predict

cancer

recurrence.

The

recent

proposed

hybrid

model

for

predicting

breast

cancer

prognosis

is

similar

to

our

model

in

view

of

using

evolutionary

computation

technique

for

learning

model.

This

model

is

a

hybrid

neural

network

classifier

combining

with

genetic

algorithms

as

optimization

technique

in

learning

the

weights

of

neural

networks

for

breast

cancer

prognosis

prediction

using

breast

cancer

imaging

data

from

UCI

machine

learning

repository.

In

contrast

to

Belciug

and

Gorunescu'

study,

we

used

gene

expression

profiles

consisting

of

tens

of

thousands

genomic

factors

for

predicting

cancer

prognosis..

However,

the

proposed

model

has

some

limitations

despite

many

advantages

including

classification

performance,

readability,

and

the

discovery

of

important

features.

One

is

the

model

has

relatively

heavy

time

complexity

which

is

O(MN),

where

M

and

N

denote

the

number

of

hyperedges

and

data

instances,

respectively.

Although

our

method

conducts

simple

operation

without

any

Appearance

number

denotes

the

number

of

hypergraph

classifiers

for

which

the

d(v)

of

a

specific

gene

is

larger

than

the

threshold

among

the

learned

models,

and

thus

its

maximum

value

is

Value

in

parenthesis

denotes

MI

rank

of

a

gene

and

it

is

the

rank

of

the

MI

value

between

each

gene

and

the

class

label.

The

genes

not

belonging

to

top

MI

rank

are

listed.

Table

An

example

of

gene

modules

(hyperedges)

from

the

learned

model

in

the

breast

cancer

and

multiple

myeloma.

TTK

TTK

CENPA

HNRNPM

DGKQ

CENPA

Multiple

myeloma

HGFAC

-EDA

The

degree

of

hyperedges,

d(v)

is

range

from

to

in

this

study.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Identifying conserved gene clusters in the presence of homology families The study of conserved gene clusters is important for understanding the forces behind genome organization and evolution, as well as the function of individual genes or gene groups. In this paper, we present a new model and algorithm for identifying conserved gene clusters from pairwise genome comparison. This generalizes a recent model called "gene teams". A gene team is a set of genes that appear homologously in two or more species, possibly in a different order yet with the distance of adjacent genes in the team for each chromosome always no more than a certain threshold. We remove the constraint in the original model that each gene must have a unique occurrence in each chromosome, and thus allow the analysis on complex prokaryotic or eukaryotic genomes with extensive paralogs. Our algorithm analyzes a pair of chromosomes in O(mn) time and uses O(m + n) space, where m and n are the number of genes in the respective chromosomes. We demonstrate the utility of our methods by studying two bacterial genomes, E. coli K-12 and B. subtilis. Many of the teams identified by our algorithm correlate with documented E. coli operons, while several others match predicted operons, previously suggested by computational techniques. Our implementation and data are publicly available at A fundamental question in genomics involves the relationship between a gene's position in a chromosome and its function. The current evidence suggests that genes are not randomly distributed in the chromosomes and that genes which are physically close to each other, even if not adjacent, tend to represent groups of genes with a functional relationship. In prokaryotic genomes, functionally-related genes are often organized into "operons," clusters of genes that are regulated as one transcriptional unit. If a set of genes remain as relative neighbors in multiple species, it may be that evolutionary constraints maintain this arrangement and thus these genes are more likely to represent a functionally-significant group. Such a gene group is often termed a conserved gene cluster in the literature. In this paper, we consider two distinct issues in comparative genomics. Our primary result is the development of an algorithm for identifying conserved gene clusters. A secondary issue is the subsequent evaluation of the significance of identified clusters.. In a recent series of papers, Béal et al. (Béal et al., 2004; Bergeron et al., 2002 ) introduce a concept denoted as a gene team. A gene team is a set of genes that appear homologously in two or more species, possibly in a different order yet with the distance between adjacent genes in the team for each chromosome always no more than a certain threshold. They provide an O(n log 2 n)-time, O(n)-space algorithm which identifies the gene teams common to a pair of chromosomes comprised of n common genes. However, their model includes a restriction that each gene has at most one occurrence in each chromosome. This severely limits the impact of the model, as many genomes contain a large number of multi-gene families known as homologs, which descended from a common ancestor via a combination of duplication and speciation events. This is especially so in higher organisms. In this paper, we extend the notion of "gene teams" to allow more general consideration of such homologs. We refer to these generalizations as "homology teams," for reasons explained in Section 3. We provide a divide-and-conquer algorithm to find such homology teams for pairwise chromosome comparison. The algorithm has O(mn) running time and O(m + n) space requirement, where m and n are the number of genes in the two chromosome for which there exists at least one homologous gene in the opposite chromosome.. When a gene cluster has been identified, a subsequent issue is to evaluate its possible biological significance. It has been shown that gene clusters whose order are conserved have a strong tendency to encode proteins that have physical interaction (Dandekar et al., 1998) . Recent studies also indicate that in some eukaryotic genomes, neighboring genes tend to have similar expression patterns (Spellman and Rubin, 2002) . The functional coupling of genes in conserved clusters has been demonstrated computationally in studies involving multiple prokaryotic genomes (Fujibuchi et al., 2000; Overbeek et al., 1999) . However, the evaluation of a cluster's significance must likely be based upon greater knowledge than solely the combinatorial structure of the two compared genomes. A gene cluster may be conserved simply because the two genomes being compared have not sufficiently diverged from their common ancestor. A non-functional association might presumably be broken only after continued shuffling of the genomes. Conversely, it may be that a set of genes were not located close to each other in an ancestor genome, yet come together during evolution due to coincidence. In this case, an identified gene cluster should not truly be qualified as conserved.. Though the problem of evaluating biological significance of clusters is beyond our primary scope, we wish to demonstrate the practical use of our algorithmic techniques in identifying potential conserved gene clusters. We describe preliminary experiments based upon the comparison of two bacterial genomes, E. coli K-12 and B. subtilis. Because this pair is well studied, we can compare our identified teams to known or predicted operons as benchmarks. Furthermore, these two genomes are sufficiently divergent to assume that genes would no longer be in proximity to each other, if not for the forces of selection. This criteria has been used in a previous study by Fujibuchi et al. (Fujibuchi et al., 2000) , who used phylogenetic distance of 20%, measured by the percent difference in small rRNA sequences, as a cutoff value for any significant conserved clusters. In Appendix A we develop a statistical test to assess the possibility that an identified gene cluster is due to chance.. The remainder of the paper is structured as follows. We discuss several closely related research efforts in Section 2. Our own concept of a homology team is formalized in Section 3. We present the algorithm for identifying homology teams of two chromosomes in Section 4, together with an analysis of running time and space usage. In Section 5, we demonstrate the practical use of this algorithm by performing analysis on genomic data, successfully reconstructing known and predicted operons from pairwise genome comparison. The identification of conserved gene clusters is important to many biological problems, such as the study of transcriptional regulation, genome evolution, etc. We developed a general mathematical model and algorithm for the problem of pairwise conserved gene cluster identification. We used these techniques to perform analysis on two bacterial genomes, E. coli K-12 and B. subtilis, and correctly reconstructed many known or predicted operons.. We would like to point out several avenues for further research. Our algorithm was presented for pairwise genome comparison. The algorithm of Béal et al., with one-to-one homologous relationships, was efficiently generalizable for the simultaneous comparison of more than two chromosomes. Our algorithm does not appear to have such an efficient generalization. The straightforward generalization of our algorithm has time complexity O(n 1 n 2 . . . n k ), where n j is the number of genes of the j th genome.. Secondly, there would be great use for a statistical test to distinguish between coincidental homology teams and those with true biological relevance. Such a test was recently developed by Durand and Sankoff (Durand and Sankoff, 2003) , however using a differing combinatorial model for conserved gene clusters. In Appendix A, we discuss a preliminary model we have developed in the context of homology teams, however more work needs to be done.. Thirdly, for teams that are identified but not known to be a transcription unit, we ask if the transcriptions are indeed coregulated. This can be tested experimentally or computationally by studying the gene expression profiles. In case the hypothesis of coregulation is disproved, one natural question is then what causes the conservation of gene clustering in multiple species.. Finally, the systematic recognition of conserved gene clusters can be used to study the largescale genome reorganization during evolution. The number and distribution of clusters with respect to size or other criteria can be very helpful information to address this question. Sankoff, 2003) . We assume a pessimistic scenario in our analysis to avoid overestimation of the significance of clusters. Specifically, we examine a region of one genome and select a k-gene cluster from this region (called reference region in (Durand and Sankoff, 2003) ). We consider the probability that this set of genes will also occur as a cluster in the second genome, if the order of the second genome is uniformly random. Suppose we have k given genes in a genome of size n, and the genes of this genome are given but their order is random. The k genes are considered to witness a team if the distance of any neighboring two genes of this set is less than a given threshold. In our following analysis, the distance is measured by the number of genes inserted between these two (as opposed to the number of base pairs). Let this distance threshold be d. Hence d = 1 means a single insertion is allowed.. To begin, we consider a special case where there is only one-to-one homologous relationship, as was the case in the original gene team model. We consider the probability that a set of k distinct given genes form a team, with d insertions allowed between any two adjacent genes of this set, in the genome of n distinct genes. There are clearly n! ways of permuting the genome. To count the number of ways of creating a genome with the specified k genes as a cluster, we divide this into three steps. First, we generate a k-cluster. There are k! ways of permuting the k genes. In addition, we need to consider the different ways of adding spaces to the cluster because insertion is allowed. Let the number of spaces within this cluster be i, then 0 ≤ i ≤ (k − 1)d, and the number of ways of generating a k-cluster with i spaces be s(k, d, i). The problem of finding s(k, d, i) is equivalent to the following well-known problem in combinatorics: find the number of integer solutions of the equation:. This problem can be solved by finding the coefficient of x i of the polynomial (1 + x + x 2 + x 3 . . . + x d ) k−1 . The details can be found in standard combinatorics texts, for instance, see (Tucker, 2002) . Since there is no simple closed formula for s(k, d, i), we will assume this function has already been solved. Second, we place this cluster as a whole to the n slots in the genome. The size of the cluster is k + i, hence there are equivalently n − (k + i) + 1 slots in the genome to place this cluster. Last, we place all the rest of genes on the rest of slots. There are (n − k)! ways to do so. In summary, the probability of forming a k-cluster in a genome of size n is:. Next, we consider the general homologous relationship, as is the case in our generalized geneteam model. Suppose there are also m families in the genome, which we will denote as M = {f 1 , f 2 , . . . , f m }. Each gene must belong to one family. Let φ j be the number of member genes that the family f j has. We also assume that the k given genes belong to distinct families to simplify the mathematical analysis. This is justified in most cases since the paralogous genes seldom neighbor with each another. The cluster is now defined differently. If the underlying families of the given k genes form a cluster in the genome, then we will consider these k genes as a cluster. Let the index of the family that the j-th gene belongs to be i j . The only difference between the current problem and the previous one is that we allow more ways of generating a cluster. In other words, for the j-th gene, we have now φ i j possible selections from the family f i j , instead of one. Once we have selected k genes from k families, the previous procedure can be applied without change to the remaining steps. Since there are φ i 1 · φ i 2 . . . φ i k ways of selecting k genes, the new probability is now:. We try to give an approximation of this probability in practice. Generally, the cluster size would be less than 10, and we also suppose that a reasonably small δ is chosen, say 1 to 3. The number of genes in a genome, n, tends to be one to several thousand, which is much larger than k, d and i, with 0 ≤ i ≤ (k − 1)d. Therefore the number of ways of placing a k-cluster with i spaces in n slots is roughly independent of i. The total number of ways of adding spaces to k genes is (d + 1) k−1 since there are (d + 1) choices of adding spaces between any two adjacent genes (0-space, 1-space, up to d-space). So the numerator of P (n, k, d) in Equation 1 is approximately (n − k + 1) · (d + 1) k−1 . Then we have:. For E. coli genome, n ≈ 2000 (note that we only consider the genes that have orthologs in both genomes). Our choice of δ = 1900bp in Section 5, is roughly equivalent to d = 1. Since the majority of COGs have one to three paralogs in E. coli, we can estimate the product term in Equation 2 as 2 k . Then by Equation 2, we have: k = 2, Q(2) ≈ 8 · 10 −3 ; k = 3, Q(3) ≈ 4.8 · 10 −5 ; k = 4, Q(4) ≈ 3.8 · 10 −7 . It can been seen that the identified teams of size greater or equal to 3 should be statistically significant. However, the probability given by Equation 2 depends on the content of a specific cluster. Since the number of genes of families can vary greatly, the actual probabilities can differ significantly from these estimates. In our analysis, we use the estimate of P (n, k, d) for all clusters, but calculate the probability for each individual cluster. Various computational methods have been proposed to identify conserved gene clusters by comparison of multiple genomes, both experimentally and formally. Models of a conserved gene cluster vary across several axes: whether the genes in a cluster are required to be immediate neighbors on a chromosome; whether gene order is allowed to vary among chromosomes; whether paralogs are allowed on a single chromosome, and thus a many-to-many homologous relationship among chromosomes; whether comparison is pairwise between two genomes, or generally between three or more genomes.. If disallowing paralogs and requiring immediate neighboring relationships, each chromosome can be modelled as a permutation over the set of genes. The problem of locating gene clusters is then related to a previously studied combinatorial problem, that of finding common intervals in two or more permutations. Uno and Yagiura present an O(n + K)-time, O(n)-space algorithm to find all K common intervals between two permutations of n items, where K ≤ n 2 (Uno and Tagiura, 2000) . Heber and Stoye generalize this algorithm to an O(cn + K)-time, O(cn)-space algorithm to find all K common intervals between c ≥ 2 permutations (Heber and Stoye, 2001) . Recently, Didier extends this approach to include paralogs by considering a more general sequence definition than a strict permutation, providing an O(n 2 log n)-time and linear-space algorithm to enumerate all such intervals (Didier, 2003) . Schmidt and Stoye show how to solve this problem in O(n 2 )-time and linear-space (Schmidt and Stoye, 2004) . Our algorithm achieves these results as a simple special case where no gap is allowed within a homology team.. A drawback of each of the above approaches is that a single misplaced gene disqualifies the commonality between two otherwise similar intervals. To remedy this drawback, gene clusters can be defined so long as the gaps between adjacent genes in the cluster are less than a given threshold. This is the approach taken by Overbeek et al. (Overbeek et al., 1999) , by Béal et al. (Béal et al., 2004) , as well as by us. We note that the work of Overbeek et al. considers only pairwise comparison of chromosomes, as does ours. Béal et al. do observe that their algorithm can be generalized to find common teams among a set of c chromosomes, resulting in an O(cn log 2 n)-time, O(cn)-space algorithm. However, the insistence on one-to-one homologous relationships severely limits the utility of that generalization. For pairwise comparison, a common technique to ensure one-to-one homologous relationships on experimental data is to rely on the bidirectional best hit method, as in (Luc et al., 2003; Overbeek et al., 1999) . The generalization of this criteria to three or more chromosomes likely precludes the inclusion of many likely homologs. As a proof of concept of our model and algorithm, we have performed experiments demonstrating the use of our techniques in identifying conserved gene clusters. We describe the experiments in this section, exploring several practical issues in the application of our techniques, such as the initial classification of genes into homology families, the effect of the parameter δ on the results, and the statistical significance of the results. Yet, we feel that our experiments provide results of biological significance on their own accord.. Specifically, we perform analysis on two prokaryotic genomes: E. coli K12 and B. subtilis. Because these genomes are well-studied, many of the genes and their presumed homologs have been previously identified and annotated. Furthermore, over 50% of the genes in each genome are identified from homology families common to both genomes, with more than two-thirds of those displaying one-to-many or many-to-many homologous relationships. Thus, these are genomes that could not be effectively analyzed by the previous gene team model, which was restricted to one-to-one orthologous relationships. This highlights the importance of considering more general orthologous relationships when identifying conserved gene clusters.. The assessment of our identified teams is based upon a comparison with known E. coli operons. While operons and conserved gene clusters are two different concepts, they are closely-related in prokaryotic genomes. First, one of the known evolutionary forces behind the conserved gene clusters is the need of coordinated transcription, which is essentially how operon is defined. Therefore, a conserved gene cluster is likely to suggest an operon in the context of a single genome. Second, if an operon is conserved during evolution because of the functional constraint, it will become a conserved gene cluster in the context of pairwise genome comparison. Therefore, the known operons serve as reference data in our experiment, as we would expect to find them correlated with our identified homology teams (and indeed, do).

O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O


Flexible hybrid WDM/TDM PON architectures using wavelength selective switches Abstract: For building an optical access network, we propose some new hybrid WDM/TDM passive optical network (PON) architectures that use wavelength selective switches (WSSs) at the remote node to improve flexibility, data security and power budget. Through simulations we demonstrate that the switching capabilities of a WSS can provide additional gains in terms of wavelength usage by a better statistical multiplexing. Several WSS-based hybrid WDM/TDM PON variants are proposed and assessed. These architectures are also compared with the more commonly used hybrid WDM/TDM PONs consisting of power splitters and/or arrayed wavelength gratings (AWGs), in terms of cost and power budget. Time division multiplexing (TDM) passive optical networks (PONs), like Ethernet PON (EPON) and Gigabit PON (GPON), are now widely accepted as optical access network solutions to distribute reasonably high bandwidths to the customers through an optical fiber network infrastructure [1] [2] [3] . However, due to the advent of bandwidth extensive services like high-definition video or interactive gaming, wavelength division multiplexing (WDM) PONs are introduced to increase the capacity for each individual user. As a pure WDM PON tends to waste bandwidth due to a lack of flexibility, hybrid WDM/TDM PON architectures are proposed in the literature with different ranges of flexibility [4] [5] . A hybrid WDM/TDM PON combines the WDM domain with a TDM PON and can deliver both the benefits of an increased capacity delivered by WDM and the inherent capacity sharing and flexibility of a TDM PON. Note that we refer to 'flexibility' as the architectural capability to offer the same network (PON) performance with a smaller number of used wavelengths. In this paper we have proposed different variants of partially flexible architectures using WSS WSS-based hybrid WDM/TDM PON solutions, however, still need further research in a laboratory environment before they can be considered for commercial use. Moreover, the flexibility advantages, like network extensibility, energy efficiency and network migration, should be studied in more detail to quantify their importance in terms of improved network performance as well as reduced operational costs.

O B I B I O O O O O O O O B I O O O O O B I O O O B I O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B I B I O O O O O O O O O B O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Design of compensated multiport waveguide junctions considering mechanization effects A new tool for the rigorous and efficient design of compensated multiport waveguide junctions, considering the mechanization effects due to low-cost manufacture techniques, is presented. Several new designs for a great variety of key components, present in modern microwave and millimetre-wave equipment, are proposed taking into account the introduction of rounded corners in the rectangular waveguide access ports. The new implemented tool permits to control and compensate for the potential degradation of the wide-band performance of such components prior to their fabrication, thus achieving an optimal design. All the presented results have been successfully validated by comparing the obtained simulated data with the results provided by a commercial software based on the finite-element method. Wide-band multiport rectangular waveguide junctions are a topic of high interest in the field of modern microwave and millimetre-wave equipment, and they have been intensively investigated in the recent years [1] , [2] , [3] . These components are widely used in many telecommunication devices, such as beam forming networks, power dividers or manifold multiplexers, when multiple channels, low insertion losses, and a broadband operation performance are required.. Traditionally, the usable bandwidth of these multiport junctions has been optimized by compensating the structure using metallic insets, such as square prisms, pyramid tuning stubs, and cylindrical posts [4] , [5] , [6] , [7] .. Moreover, it is usual to resort to low-cost waveguide device fabrication techniques, such as computer-controlled milling, electroforming or die-casting, in order to manufacture these optimized components. A common drawback of these previous techniques is the introduction of rounded corners in the crosssection of the waveguide access ports of the cited junctions, thus deteriorating the expected electrical response of the designed device and, therefore, limiting its desired wide-band performance [2] . Nevertheless, if computer-aided design (CAD) tools could accurately consider these negative manufacture effects in the analysis stage, it would be possible to compensate them and achieve optimum electrical designs of such components easily implementable via low-cost fabrication techniques. As a consequence, there is an ever-increasing interest in compensated multiport junctions CAD tools enabling the accurate design of these components, prior to their final hardware implementation using low-cost manufacture procedures.. In this letter, a novel full-wave CAD tool is proposed for the accurate and efficient design of compensated multiport waveguide junctions, taking into account the mechanization effects introduced by the low-cost production methods aforementioned. On the one hand, the proposed CAD tool permits the microwave designer to predict the electrical performance of the final manufactured device as a function of the radius of the considered rounded corners. On the other hand, it also enables to compensate for the degradation introduced in the frequency response of the device by redesigning the junction (i.e. modifying the position and/or the dimensions of the compensating stub) in order to achieve an optimum broadband operation.. In order to fully validate the presented CAD tool, new designs for a great variety of key components present in many communication satellite subsystems are provided. In these devices, the effect of the rounded corners is considered, and partial-height cylindrical metallic posts are used as compensating elements.. The first examples consist of new designs concerning compensated H-plane Tjunctions (see Fig. 1 ) and right-angled bends. A more complex five-port device, specifically a compensated turnstile junction, considering these mechanization effects is also reported. It is important to emphasize that this work provides, for the first time to the authors' knowledge, a rigorous combination of 2-D and 3-D Boundary IntegralResonant Mode Expansion (BI-RME) techniques [2] , [8] for the efficient analysis and design of compensated multiport waveguide junctions considering mechanization effects. To this respect, note that the novelty of this work lies on the fact that previous works of the authors did not considered the presence of rounded corners in the analyzed multiport junctions (i.e. [3] , [6] , [9] ) or did not deal with the design of compensated multiport junctions using the 3-D BI-RME technique (i.e. [2] ). As a consequence, the work developed in this letter, not only constitutes a significative extension of previous contributions, but also provides a novel and very efficient CAD tool aiming at improving the relative bandwidth of a great variety of compensated waveguide junctions considering mechanization effects. A new CAD tool for an accurate and efficient design of compensated multiport waveguide junctions considering the mechanization effects associated to low-cost production techniques has been proposed. It has been shown that, although the wide-band performance of the device can be reduced when high values of the radius of the rounded corners are considered, it is possible to compensate for this degradation by properly re-designing the compensating element, thus achieving an optimum design of these components.

O O O B I I O B I O O O O O O O O O O O B I I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Optimal control for SIRC epidemic outbreak publicly funded repositories, such as the WHO COVID database with rights for unrestricted research re-use and analyses in any form or by any means with acknowledgement of the original source. These permissions are granted for free by Elsevier for as long as the COVID-19 resource centre remains active. c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 1 0 ( 2 0 1 3 ) 333-342 Optimization System analysis a b s t r a c tIn this paper the mathematical SIRC epidemic model is considered. It efficiently describes diseases in which a cross immune class (C) is present, along with the susceptible (S), the infected (I) and the removed (R) ones. Controlling epidemic diseases corresponds to the introduction of vaccination, quarantine and treatment strategies; generally only one of these actions is considered. In this paper the possibility of optimal controls both over the susceptible and the infected subjects is assumed, taking into account also limitations of resources.A suitable cost index is introduced and via the Pontryagin's Minimum Principle the optimal control strategy is determined and the existence of the optimal solution is assessed.Numerical results are developed analyzing the effects of different control strategies. (N. Stasio). protected one (R). Therefore the obtained SIRC model takes into account temporary partial immunity and may well describe, for example, influenza A.The study of mathematical models for epidemics allows the evaluation of the effects of different control strategies: the scheduling of a proper vaccination campaign [3, 4] , the screening and educational campaigns [5], isolation policy [6], the resource allocation [5, 7] . An encouraging approach lies in the framework of the optimal control theory; interesting reviews can be found in [8] . Analytical results on the optimal control problems for epidemics were extensively described in [9] , where models with control by vaccination, quarantine, screening or health campaigns were studied with a rather general choice for the interaction function.Different kind of infectious disease calls for different models and strategies. In [10] the severe acute respiratory syndrome (SARS) disease was considered to find the strategy of tracing and quarantining contacts of identified cases; 0169-2607/$ -see front matter Mathematical models have become important instruments in the analysis and control of infectious diseases. Several models corresponding to epidemic with different characteristics have been proposed and discussed in literature. The fundamental assumption in epidemic models is that the population can be divided into distinct groups; the most common are: the susceptible (S) that are the subjects that may catch the disease; the infected (I) that are the subjects that are already infected and can spread the disease to susceptible individuals; the removed (R) that are the subject that are immune for life. Therefore these models are referred to as SIR models.. Other descriptions may include the presence of subjects in the quarantine state (Q) and in this situation SIRQ models are analyzed [1] . Recently in [2] the class of cross-immune individuals (C) in the population has been introduced: it is an intermediate state between the fully susceptible state (S) and the fully it was shown that this can be very successful in reducing transmission. In [11] it was proved that a maximum quarantine/isolation measures for the exposed and the infectious class would reduce the SARS epidemics; a bang-bang switching control was obtained. In the last decade great attention has been devoted to the modeling and control of the human immunodeficiency virus (HIV) infection; in [12] an output feedback scheme based in CD4 T-cell and viral load measurements has been introduced. For the control in Dengue epidemics optimal and sub-optimal strategies were considered in [13] ; the effects of educational campaigns organized to motivate the population to break the reproduction cycle of the mosquitoes by avoiding the accumulation of still water in open-air recipients were modeled and the assumed cost index reflected a compromise between actual financial spending and population health. Optimal and suboptimal solutions were introduced also in [14] with reference to quarantine and isolation control in SARS epidemics, demonstrating that the maximal applications of the control strategies in the early stage of the epidemic were of very significant impact in both the considered control cases.. In this paper the SIRC model is considered aiming at the control of the epidemic spreading, taking into account the limitations of the resources. Among the cause of death related with influenza, pneumonia, heart and circulatory are the most significant; moreover most of these deaths are in individuals in the age group over 65 years whereas younger children are at significant risk of influenza hospitalization. These considerations lead to recommendation of vaccination for subjects above 64 years (59 in some countries) and for young children (above 6 months). Characteristic of influenza is its seasonality; the peaks of the influenza are related with the weather condition, and social patterns, for example schools holidays.. In this paper, in the framework of multiple optimal controls [15] [16] [17] , the controls both in the susceptible and in the infected class are introduced and the existence of the optimal solution is assessed in the SIRC epidemic model. Note that a similar approach was presented in [18] with reference to vector-borne diseases and not considering the cross-immune class.. The paper is organized as follows; in Section 2 the SIRC model is described and an optimal control strategy is introduced; the proposed control action is analyzed and existence results are assessed. In Section 3 numerical results are presented and discussed. In this paper the SIRC model of influenza is considered aiming at the control of the epidemic spreading; controlling an epidemic disease corresponds, for example, to an ad hoc introduction of vaccine or quarantine or medicine treatment strategy. All the possible strategies should face with the limitations of the resource. In this paper the controls both in the susceptible and in the infected classes are introduced, taking into account the constraints in the controls. A suitable cost index is proposed and existence result is assessed. The aim is to minimize the susceptible and infected subjects with as less resources as possible; the analytical expressions of the optimal controls are provided and their numerical implementation is discussed. Numerical results are presented comparing the effects of dif-

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