Arnaud Castelltort

Anne Laurent

Fuzzy Historical Graph Pattern Matching A NoSQL Graph Database Approach for Fraud Ring Resolution

Keywords: NoSQL Graph Databases, Time Information, Fraud Rings

Graphs have been studied and used for many years as they allow to represent in an efficient manner real data such as biological or social data. Graph databases have recently emerged within the NoSQL framework and are implemented in systems like Neo4J, OrientDB, etc. Recent works have shown that the management of history is crucial in such systems. In this paper, we show how such historical graph databases can be queried in order to retrieve fraud rings, also known as fraud cycles. Frauds are indeed often based on sophisticated chains of successive transactions (money, communications, etc.). We thus claim that the indirect link between fraudsters can be retrieved by considering historical NoSQL graph databases. We study how the model of historical NoSQL databases can be extended for better address this goal and we propose the associated queries that have been tested on a synthetical database.

Introduction

Graphs are known to be efficient for representing data in many applications, from linguistics to chemistry and social networks. For instance, such a graph allows to draw in a very intuitive manner the relationships among people and between these people and the organizations they belong to.

Graphs are recognized to play an important role within the pattern recognition field [START_REF] Conte | Thirty Years Of Graph Matching In Pattern Recognition[END_REF], thus being a key technology for retrieving relevant information, as for fraud detection or in social/biological interactions. Techniques and algorithms can be distinguished depending on the fact that they are meant to mine relevant patterns or to retrieve patterns.

If considering frauds (bank and insurance), frauds represent billions of dollars lost by companies every year. For instance more than £52 billions have been lost in UK in 2013 [START_REF]Annual fraud indicator[END_REF]. Fraud can be organised or can be individual. It can impact individuals or organizations (e.g., banks) as in first-party and third-party fraud. Graphs can help for retrieving frauds through the modelization of fraud rings which are hidden within the graph of interactions [START_REF] Sadowski | Fraud detection: Discovering connections with graph databases[END_REF]. A fraud ring is a set of connections between actors. It can be found in many fraud frameworks.

Although graphs have been studied since the very beginning of computer science in the so-called graph theory field [START_REF] Bondy | Graph Theory With Applications[END_REF], their integration within database management systems is more recent. Some of the first systems have been proposed with the emergence of ontologies and RDF triplets queried through SPARQL [26]. More recently, NoSQL databases have proposed efficient engines devoted to graph databases: GraphDB, Neo4J,. . . [START_REF] Board | Technology radar[END_REF] compares some of these engines and points the advantages of the Neo4J system, which is the one we consider.

In this paper, we propose a framework for defining fuzzy historical pattern matching from NoSQL graph databases. For this purpose, we first recall the basic concepts of NoSQL graph databases, historical queries and graph pattern matching in Section 2. We then detail the problem we address in Section 3, before presenting our contribution and its resolution using the NoSQL Neo4j graph database in Section 4. Section 5 sums up this paper and presents the future work we would like to address.

Related Work

Graphs and NoSQL Graph Databases

Graphs have been studied for a long time by mathematicians and computer scientists. A graph can be directed or not. It is defined as follows.

Definition 1 (Graph).

A graph G is given by a pair (V, E) where V stands for a set of vertices and E stands for a set of edges with E ⊆ (V × V ).

Definition 2 (Directed Graph).

A directed graph G is given by a pair (V, E) where V stands for a set of vertices and E stands for a set of edges with E ⊆ {V × V }. That is E is a subset of all ordered permutations of V element pairs. When used in real world applications, graphs need to be provided with the capacity to label nodes and relations, thus leading to the so-called labeled graphs, or property graphs as shown in Fig. 1 and defined bellow : Definition 3 (Labeled Oriented Graph). A labeled oriented graph G, also known as oriented property graph, is given by a quadruplet (V, E, α, β) where V stands for a set of vertices and E stands for a set of edges with E ⊆ {V × V }, α stands for the set of attributes defined over the nodes, and β the set of attributes defined over the relations.

NoSQL graph databases [START_REF] Robinson | Graph Databases[END_REF] are based on these concepts, attributes and values over the attributes being stored thanks to the (key, value) paradigm which is very common in NoSQL databases.

Fig. 2 shows a graph and its structure in (key, values) pairs. Studies have shown that these technologies present good performances, much better than classical relational databases for representing and querying such large graph databases. Their exist several NoSQL graph database engines (OrientDB, Neo4J,HyperGraphDB, etc.) [START_REF] Angles | Survey of graph database models[END_REF]. Neo4J is recognised as being one of the top ones regarding performance [START_REF] Board | Technology radar[END_REF]. It has been recently extended for managing the successive versions of the graph database in the Mnemosyne system described below.

2.2 Mnemosyne: an innovative historical data management system [START_REF] Castelltort | Representing history in graph-oriented nosql databases: A versioning system[END_REF] aims at proposing an innovative model of history management in NoSQL graph databases. This contribution is based on three key concepts:

-Use the graph to manage historical data; -Data historization must be totaly decoupled from the representation of the data in the graph datasource; -Use generic graph traversals regardless of the graph datasource.

One of the main specificity of the Mnemosyne system is that it uses two graphs :

-DataGraph: the current graph in use -VersionGraph: a VersionGraph storing the history of different versions of a data graph. The VersionGraph model does not depend on the DataGraph.

Fig. 3 shows an example of a DataGraph and a VersionGraph. In this system, every node and relation in the graph database is tracked in the VersionGraph with a node called T raceElement. From this node, all the modifications (insert, update, create) are kept in a list of RevisionElements that can be queried generically for elements (nodes or relations) of the graph source.

Fig. 4 illustrates this proposal. In this example, three nodes (a,b,c) and three relations (r1,r2,r3) from Fig. 5 are traced.

When several modifications have occured, the versiongraph becomes more complex and several revisions appear. These revisions can be traversed with different points of view, depending on the fact that the user needs to trace nodes or elements, as illustrated by Fig. 6.

Pattern Matching and Querying

Graph pattern matching is a very difficult algorithmic problem that has attracted many works that cannot be all recalled here. Graph pattern matching is distinguished from graph mining where frequent subgraphs are searched for [START_REF] Yan | gspan: Graph-based substructure pattern mining[END_REF][START_REF] Kuramochi | Frequent subgraph discovery[END_REF]. Historical queries have been studied since the beginning of databases, in relational databases [START_REF] Chawathe | Representing and querying changes in semistructured data[END_REF], datawarehouses [START_REF] Lee | An efficient method for maintaining data cubes incrementally[END_REF] and Web data [START_REF] Auer | A versioning and evolution framework for rdf knowledge bases[END_REF][START_REF] Papavasileiou | High-level change detection in rdf(s) kbs[END_REF]. However, very few works have proposed to represent and query history in graph-oriented NoSQL databases [START_REF] Khurana | Efficient snapshot retrieval over historical graph data[END_REF].

Many works on temporal queries are based on the Allen interval algebra [START_REF] Andrei Krokhin | Reasoning about temporal relations: The tractable subalgebras of allen's interval algebra[END_REF] and some predicates are defined on A and B which hold or not depending on some tests over the boundaries a -, a + , b -, b + (for instance bef ore(A 1 , B 1 ) does not hold as M arch2013 ≮ F ebruary2013). Temporal SQL has been extensively studied, and has been fuzzified [START_REF] Schockaert | Fuzzifying allen's temporal interval relations[END_REF] as for instance to describe that the event B occurs long before event A, or that event A occurs before or approximately at the same date as the event B.

For all the above-mentioned topics, graphs can be queried through languages. Some of them have been studied in the literature. Some languages have been proposed by scientists and some other ones have been issued by the editors.

[28] proposes a survey of all the languages defined over the last 25 years, where subgraph matching appears to be one of the most powerful and necessary query, including approximate matching.

[3] proposes a propositional dynamic logic that extends several graph database languages.

[13] introduces GraphQL, a graph algebra capable of taking into account nodes, relations, and attributes on both nodes and relations. In this language, the authors define the so-called graph pattern as a pair P = (M, F) where M is a graph motif and F is a predicate on the attributes of the motif. The algebra contains several operations, including selection, cartesian product, join, composition and demonstrate that their algebra is contained into Datalog, thus allowing for rewriting their queries in Datalog.

We show below how these works can be used for fuzzy pattern matching in the framework of fraud ring detection and what their limits are.

Problem Statement

Preliminary statements

Fraud detection is often considered as a subtopic of anomaly detection. Anomaly detection has been widely studied [START_REF] Chandola | Anomaly detection: A survey[END_REF]. Fraud data are often intrinsically graphs. However, when focusing on graphs, few works have been done [START_REF] Noble | Graph-based anomaly detection[END_REF]. In such problems, fraudsters try to stay hidden by acting in groups so that no action is then a direct one. Intermediate layers are meant to hide the association of fraudsters. It should be noted that if the size of the group is too small then the fraud can more easily be discovered, but if the size is too big, then there are more probabilities that some problem occurs, should it be caused by coincidence or caused by a weak link in the circle of persons. The size of the group being organized for the fraud ranges from 2 to several.

In our work, we focus on fraud rings [START_REF] Jedrzejek | Graph mining for detection of a large class of financial crimes[END_REF] defined as follows.

Definition 4 (Fraud Ring). Given a graph G, a fraud ring (also known as fraud cycle) can be defined as a subgraph F ⊆ G where there exist at least two nodes n 1 , n 2 ∈ F that are indirectly connected, betraying illegal links between the nodes n 1 and n 2 .

Many problems are based on fraud rings, from corruption, insurance and bank fraud, to shell companies, etc. Even in managing human resources, it can be the case that a few people make benefit their brotherhood from unfair positions. Many works and methods have addressed this problem but fraudsters have built innovative manners that are not easily discovered by existing tools.

For instance, for insurance fraud, some people are claiming millions of dollars after declaring fake accidents, fake passengers and fake witnesses. Fig. 8 shows how this appears on a graph.

Fraud rings can be retrieved by connected analysis when for instance one people acts once as a driver and then twice as a witness or passenger in another car accident, as shown by [START_REF] Moody | Advanced techniques for detecting complex fraud schemes in large datasets[END_REF] and illustrated by Fig. 9.

Use Case

We focus on the bank fraud detection as depicted in [START_REF] Bastani | Bank Fraud Detection[END_REF]. In such frauds, a circle of persons share some legal documents and create accounts. The credit lines and accounts are used, and gradually merged with unsecured lines. As depicted in [START_REF] Bastani | Bank Fraud Detection[END_REF]: "One day the ring busts out, coordinating their activity, maxing out all of their credit lines, and disappearing. Sometimes fraudsters will go a step further and bring all of their balances to zero using fake checks immediately before the prior step, doubling the damage".

For this purpose, link analysis and discrete data analysis has been proposed and applied, for instance using the Neo4j tools or in [START_REF] Leontjeva | Fraud detection: Methods of analysis for hypergraph data[END_REF][START_REF] Horvth | Cyclic pattern kernels for predictive graph mining[END_REF]. For instance it can help to retrieve the "account holders who share more than one piece of legitimate contact information" which is very easy when data are represented using graph structures.

Problem

In this paper, we focus on the exploitation of the relations, of the history and of the labels contained in the graph for retrieving fraud rings. We claim that:

history is a key feature for discovering such rings. Time is indeed important and fraudsters are playing with delays in order to gradually root their actions. However, time is not always represented in the systems used to store the information, which prevents organizations from efficiently wiping out fraud and corruption. fuzziness is required as such patterns are approximate.

many algorithms have been designed, but their implementation on real world problems and engines is not always tractable. We thus consider the use of adapted tools with the capabilities to express such queries in declarative languages.

The next section introduces various means to address these points.

Resolution and Experiments

As shown above, fraud detection relies on the discovery of graph patterns in the successive states of the data. This section relies on a classical example of bank fraud. The dataset used in this section is defined in Fig. 10. Green circled nodes represent fraudsters and red circled nodes represent the connection points between these persons.

In this section, we first introduce a standard pattern matching resolution to detect fraud rings in a graph. This kind of resolution works well when all the data are available (which means that no valuable information for resolution have been erased or replaced). To handle this problem, there are two main ways:

either by embedding the history within the operational model. This creates difficulties as the patterns for fraud detection must then be written in an adhoc manner, depending on the specific model either by considering a generic model for describing the history that is compatible with pattern matching. This allows to write generic pattern matching queries for retrieving the fraud rings.

This latter case is explored by presenting how the Mnemosyne model can help.

Finaly, to go further, we show that fuzziness can help to address approximate pattern matching.

Pattern Matching Resolution

Classic Resolution The challenge is to determine if there is a ring in the dataset, and if so, what is its size and what is the finacial risk it represents. A classical solving resolution looks like Listing 1.1: That is to say, finding all the account holders that have at least one piece of information in common (line 1 to 5) and then calculate the ring size (line 3) and the induced financial risk (lines 8 to 12).

The result of the execution of Listing 1.1 declarative request is shown in Fig. 11.

Resolution Generalisation

In fact, resolution of this kind of problems can be generalized as answering this question: "Do some people share some information?". Expressing the query for answering such a question can be more genericaly answered with queries like: We can then apply this algorithm in the same maner to other graphs that are not in the same business area.

Historical Pattern Matching Resolution

In this section, historical queries rely on the Mnemosyne model presented in Section 2.

Fraud Ring: Temporal Cheating It should be noted that expressing pattern matching defined in Section 4.1 can be done on existing insurance/bank data as such applications are meant to store and trace the history. However, this is not always the case. Fig. 12 shows how, if history was not managed in such systems, the fraudsters can cheat to stay hidden by creating, updating and deleting some information over time.

In this case, the above algorithms will not detect any fraud ring as they cannot manage history. To help organizations to find frauds, it is thus necessary to record historical tracks.

Mnemosyne Extension

The Mnemosyne system presented above offers an efficient manner to trace the history within graph data. If a node or a relation is impacted by an operation, should it be an update, delete or insert operation, then this is recorded in an other graph called VersionGraph which handle the history of each node or relation.

It should be noted that the VersionGraph manages the history in the same manner whatever the business area should be. The Mnemosyne system represents the history in the same way whatever the heterogeneity of the DataGraph (in terms of types as well as structures of data). This is due to the generic nature offered by the system which does not care about the semantics of data and which provides a strong dissociation between the data (in the sense of information) and the data structure.

However, in this model, the representation does not materialize the links between the impacts on the neighborhood. For instance, if a relation is added, then the incoming and outgoing nodes are changed in the graph. All these operations will thus be traced in the Mnemosyne VersionGraph, but no link is traced between the elements (to manage performance optimization). We thus propose to materialize these links in order to speed up the process. If a relation is updated, then the Mnemosyne will materialize (by a relationship in the VersionGraph) the impacted nodes and relations. Fig. 13 presents an example of such a situation, the blue relationships representing the materialized relationships which are added.

Resolution Resolving such patterns in NoSQL graph databases requires to define original queries. Such queries are built to navigate in the graph and historical graph (Mnemosyne VersionGraph) in order to retrieve the fraud rings. The efficiency of the system relies on the NoSQL engine.

Listing 1.2 shows how to query all the links from the history in order to retrieve the potential fraud ring. As shown on this Cypher query, all the elements impacted in the graph can easily be retrieved by exploiting the materialized links of the history from the extension we propose. However, we claim that the patterns to be found cannot

Fuzzy Historical Pattern Matching Resolution

In many cases, the parameters of the patterns cannot be defined in a strict manner. For instance, the minimum and maximum size of the fraud ring are fuzzy parameters. In the same manner, the time delays between the fraudsters' actions are fuzzy.

We thus propose to use fuzzy clauses in the pattern matching queries. Temporal fuzzy clauses can be defined either by considering the fuzzy Allen intervals or by considering user defined clauses. Fig. 15 displays a membership function defining fuzzy weeks that is exploited in Listing 1.5. Rings can be characterized in the same way as the length of the paths forming the relations and the ring ranges within fuzzy bounds, as shown on Listing 1.4.

Conclusion and Further Work

In this paper, we address the problem of fraud detection by considering fraud rings in graph databases. The use of NoSQL graph databases is a key element of our work. We especially claim that representing the successive versions of the graph data allows to better retrieve the chains of successive transactions that represent a fraud. For this purpose, we consider the Mnemosyne system that is extended here for materializing temporal relations between objects. This allows us to directly apply pattern matching on the graph for retreiving the fraud rings. Fuzziness allows us to take better account of the fraud characteristics. The patterns may indeed differ from one case to another and considering crisp values when scanning the data may prevent from retrieving the relevant patterns.

Our further works will especially address the comparison of various strategies for materializing the history links in the Mnemosyne system. We also aim at testing our methods on various datasets, for instance for financial crime detection and counter-terrorism.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Labeled Graph

Fig. 3 :

 3 Fig. 3: DataGraph (left) and VersionGraph (right)

Fig. 4 :Fig. 5 :Fig. 6 :

 456 Fig. 4: Version Graph Model

Fig. 7 :

 7 Fig. 7: Pattern Matching

  describing the temporal relations between two intervals A = [a -, a + ] and B = [b -, b + ].In this framework, A and B are two intervals of dates (for instance, A 1 = [January2013, M arch2013] and B 1 = [F ebruary2013, June2013])

Fig. 8 :

 8 Fig. 8: Fraud Ring -fake car accidents, passengers and witnesses

Fig. 10 :

 10 Fig. 10: Fraud dataset

Listing 1 . 1 :

 11 Classical pattern resolution 1 MATCH ( a c c o u n t H o l d e r : A c c o u n t H o l d e r ) -[ ] -> ( c o n t a c t I n f o r m at i o n ) 2 WITH c o n t a c t I n f o r m a t i o n , 3 c o u n t ( a c c o u n t H o l d e r ) A S R i n g S i z e 4 MATCH ( c o n t a c t I n f o r m a t i o n )< -[ ] -( a c c o u n t H o l d e r ) , 5 ( a c c o u n t H o l d e r ) -[ r : H A S _ C R E D I T C A R D | H A S _ U N S E C U R E D L O A N ] -> ( ← u n s e c ur e d A c c o u n t ) 6 WITH c o l l e c t ( D I S T I N C T a c c o u n t H o l d e r . U n i q u e I d ) A S A c c o u n t H o l d e r s ← , 7 c o n t a c t I n f o r m a t i o n , R i n g S i z e , 8 S U M (CASE t y p e ( r ) 9 WHEN ' H A S _ C R E D I T C A R D ' THEN u n s e c u r e d A c c o u n t . L i m i t 10 WHEN ' H A S _ U N S E C U R E D L O A N ' THEN u n s e c u r e d A c c o u n t . B a l a n c e 11 E L S E 0 12 E N D ) a s F i n a n c i a l R i s k 13 WHERE R i n g S i z e > 1 14 RETURN A c c o u n t H o l d e r s A S F r a u d R i n g , 15 l a b e l s ( c o n t a c t I n f o r m a t i o n ) A S C o n t a c t T y p e , 16 R i n g S i z e , 17 r o u n d ( F i n a n c i a l R i s k ) a s F i n a n c i a l R i s k 18 ORDER BY F i n a n c i a l R i s k DESC

Fig. 11 :

 11 Fig. 11: Classical pattern resolution results

Fig. 12 :

 12 Fig. 12: Example of temporal cheating applied to current dataset

Listing 1 . 2 :

 12 Exploiting Historical Revisions1 match ( T E A H 1 : T r a c e E l e m e n t ) , ( T E A H 2 : T r a c e E l e m e n t )2 w i t h T E A H 1 , T E A H 2 3 match p=( T E A H 1 ) -[ * 1 . . 7 ] -( T E A H 2 ) 4where a l l ( r e l I N r e l a t i o n s h i p s ( p ) where t y p e ( r e l ) <> ' R E V I S I O N ' ) 5 r e t u r n T E A H 1 , T E A H 2 , e x t r a c t ( r e l I N r e l a t i o n s h i p s ( p ) | t y p e ( r e l ) )

Fig. 13 :

 13 Fig. 13: Materializing Elements impacted by an operation in the VersionGraph

Listing 1 . 3 :

 13 Fig. 14: Mnemosing Ring: Exploiting Historical Revisions

MATCH (A : A C C O U N T H O L D E R ) -[ * ] -> ( B : A C C O U N T H O L D E R ) -[ * ] -> ( C : A C C O U N T H O L D E R )

WITH c o u n t ( a c c o u n t H o l d e r ) A S R i n g S i z e

RETURN A , B , C , R i n g S i z e


Leila Khouaja
email: leilakhouaja85@gmail.com

Ben -Slimane
email: slimane.benmiled@pasteur.rns.tn

Miled

Hassan Hbid
email: hbid@ucam.ac.ma

Modélisation en dynamique des populations Impacts des changements climatiques sur les populations de tiques

Keywords: Epidémiology, McKendrick-Von Foerster equation, Partial differential equation(PDE), Transport equation MOTS-CLÉS : Epidemiologie, Equation de

Epidemiology had an important development these last years allowing the resolution of a large number of problems and had good prediction on disease evolution. However, the transmission of several vector-borne diseases is closely connected to environmental protagonists, specially in the parasite-host interaction. Moreover, understanding the disease transmission is related to studying the ecology of all protagonists. These two levels of complexity(epidemiology and ecology) cannot be separated and have to be studied as a whole in a systematic way. Our goal is to understand the interaction of climate change on the evolution of a disease when the vector has ecological niche that depends on physiological state of development. We are particularly interested in tick vector diseases which are serious health problem affecting humans as well as domestic animals in many parts of the world. These infections are transmitted through a bite of an infected tick, and it appears that most of these infections are widely present in some wildlife species.

RÉSUMÉ. L'épidémiologie a connu un développement important ces dernières années. Cette discipline a permis une meilleure compréhension de l'évolution de maladies. Cependant, plusieurs maladies à transmission vectorielle sont étroitement liées aux protagonistes environnementaux. Ce constat est particulièrement vrai dans le contexte des interactions du parasite avec son hôte. De plus, comprendre la transmission de maladie est lié à l'étude de l'écologie de tous les protagonistes. Notre objectif est de comprendre l'influence du changement climatique sur l'évolution des maladies lorsque la niche écologique du vecteur dépend de l'état de développement physiologique de son hôte. Nous sommes particulièrement intéressés par les maladies vectorielles à tiques qui constituent un grave problème de santé touchant l'être humain et les animaux domestiques dans de nombreuses régions du monde. Ces infections sont généralement transmises par la piqûre d'une tique infectée et il apparaît que la plupart de ces infections sont largement présentées dans certaines espèces fauniques.

Introduction

Tick-borne diseases (theileriosis, rickettsiosis, Lyme disease, Ehrlichiosis, relapsing fever, TBE(tick-borne encephalitis)) are serious health problem affecting humans as well as domestic animals in many parts of the world. These infections are generally transmitted through a bite of an infected tick, and it appears that most of these infections are widely present in some wildlife species; hence, an understanding of tick population dynamics and its interaction with hosts is essential to understand and control such diseases [START_REF] Hudson | Trophic interactions and population growth rates: describing patterns and identifying mechanisms[END_REF]. For example, the vector of tropical theileriosis in North Africa, the tick Hyalomma detritum, has seasonal activity, while Hyalomma anatolicum is active throughout the year in several parts of Africa and Asia leading to animals being challenged with infection all over the year, this provides a solid immunity during the year contrasting with a very high infection leading to possible endemic stability.

The object of the present work is to develop a tick-borne biology model specific to Hyalomma detritum species in Tunisia. The model will be fitted to field data that have been previously gathered from several Tunisian farms [START_REF] Bouattour | Cattle infestation by Hyalomma ticks and prevalence of Theileria in H. detritum species in Tunisia[END_REF].

Our ultimate goal in this paper is to construct models in order to study :

-Epidemiology: The effect of climate change on the evolution of tick-borne diseases particularly Theileriosis.

-Ecological question: What is the most important fact of tick life cycle regulation: Seasonality vs food.

-Control result: The effect of different control actions on tick population. In order to achieve our goal, we need to solve the two following steps:

1) Modeling the tick life cycle, taking account of temperature fluctuation and seasonality: In this part our objective is to model tick life cycle in order to study the effect of temperature and seasonality on density of the ticks. The model used here will be a partial differentiable equation. The model will be tested using the data from [START_REF] Randolph | Tick ecology: processes and patterns behind the epidemiological risk posed by ixodid ticks as vectors[END_REF] that have been previously gathered from several Tunisian farms [START_REF] Bouattour | Cattle infestation by Hyalomma ticks and prevalence of Theileria in H. detritum species in Tunisia[END_REF]. This model will be the foundation of the late epidemiological model.

2) Integration the preceding model of tick life cycle into an epidemiological model: -Tick: SI model and host: SIR model. Our work is organized as follows: in the next section, we describe the biology of tick population and present the epidemiological interactions between ticks and their hosts. In section three, we describe both the tick life cycle and its mathematical models; introduce the model which represents the host-parasite epidemiological interaction. Section four is devoted to the conclusion and recommendation.

Biological Model

Several field observations on tick biology show a huge polymorphism in their biology (prolificity, mortality, phenology). This polymorphism is enhanced during the parasitic stages of the tick (during feeding stages) because of the interaction between the tick and the host (immunity of the host, surface of exposure, biology of the host). This degree of interaction is again more complicated when the tick-borne infections are considered. Describing this biology of the tick is possible by monitoring infested animals and questing instars and presenting the observations as descriptive results. Nevertheless, understanding and predicting the mechanisms leading to a determined phenology is quite impossible. Moreover, the prediction of the impact of different control actions is difficult. Modeling represents a powerful tool offering the opportunity to counter account these difficulties. It is possible to model in silico both tick dynamic and the impact of different control options before implementing them, offering then a dramatic decrease of the control costs.

Mathematical modeling represents a powerful tool offering the opportunity to avoid these difficulties. Indeed, it is possible to model in silico both tick-host and epidemiological interactions in order to investigate and understand climate change on disease evolution. Moreover, modeling offer tools to test impact of different control options before implementing them, offering then a dramatic decrease of the control costs.

Effect of vector life cycle on disease transmission

The tick life cycle includes three post-embryonic developmental stages: larva, nymph and adult. Each stage can be subdivided in turn according to the activity phases: 'questing', in which the unfed tick seeks a host and 'feeding', in which the attached tick feeds, becomes engorged and drops off. After dropping off their hosts, the cattle, ticks go through a period of development, after which they emerge as questing ticks at the next stage (or eggs hatch, if the feeding ticks are adult females). The transition from one stage to an other depends closely on the successful questing period that depends on host density. Moreover, the physiological development depends on temperature fluctuations . These two phenomenons are strictly connected to climate change. Indeed, on one hand, it is evident that temperature fluctuation depends on the climate change and on the other hand cattle populations are strictly connected to the agricultural habit which depends on environment.

A variety of approaches have been used to model the tick population with various degrees of complexity. Models often describe in a discrete way the various stages of tick development from egg-larvae-nymph-adult, whether the ticks are attached to hosts, and if disease is part of the model, whether the ticks themselves are infected [START_REF] Rosà | Thresholds for disease persistence in models for tick-borne infections including non-viraemic transmission, extended feeding and tick aggregation[END_REF], [START_REF] Rosà | Effects of tick population dynamics and host den-sities on the persistence of tick-borne infections[END_REF].

Therefore, we propose in this paper two kind of models. The first model is a system of ordinary differential equations with delay where physiological structure is described in a discrete form. This time delay cannot be ignored because the development of the ticks between stages takes time. Moreover, the time delay depends on the weather and climate situation. For this first model our aim is to model tick life cycle in order to study the effect of temperature and seasonality on ticks density. For the second model, the transition from one physiological stage to an other is considered as a continuous process. In this case, we propose to build a PDE model where tick population density satisfies the McKendrick-Von Foerster model with or without blood meal as a limiting factor. All models constructed will be tested using the data from [START_REF] Randolph | Tick ecology: processes and patterns behind the epidemiological risk posed by ixodid ticks as vectors[END_REF], data that have been previously gathered from several Tunisian farms [START_REF] Bouattour | Cattle infestation by Hyalomma ticks and prevalence of Theileria in H. detritum species in Tunisia[END_REF] and several data from laboratory colonies. These models will be the foundation of the previous epidemiological model.

Host -Tick epidemiological interaction

The infection transmission is incorporated into models by adding more states to record the infected status of the ticks and hosts. Typically a mass action law assumption is adopted by the rate of new infections which is directly proportional to the product of susceptible hosts and infectious ticks. However, if larval and nymphs bites are statistically independent, then such clustering would tend to reduce tick and host infection prevalence. In this sense, as positive co-variance of larval and nymphal bites would tend to increase infection prevalence, as larval bites would be clustered on the host individuals most likely to be infected and infective. An alternative approach to explicitly modeling the host and tick populations was provided by [4] who instead consider the life cycle of the Theileria parva parasite as it progresses through the vertebrate and tick hosts and estimates the time in days (from infection) of disease characteristics in cattle considering challenges from different numbers of infective ticks. It is often assumed that infected ticks behave in the same way as uninfected ones with the mortality of ticks being independent of their infection status. Although as has been seen a pathogen may have a negative impact on the tick in the same manner as a host. Generally models do not consider non-systemic infection (see above) although in a study by [START_REF] Rosà | Effects of tick population dynamics and host den-sities on the persistence of tick-borne infections[END_REF] this possibility was introduced. However trans-ovarian infection is usually excluded due to the lack of evidence for this in the literature. Reservoir decay or host turnover might enhance positive feedback of infection transmission, for example an increasing prevalence of infected nymphs would increase the frequency at which hosts are re-infected, keeping hosts in a state of high specific infective with a greater probability of infecting the next generation of ticks.

Infection is a one-way through the tick vectors, larvae/nymphae can transmit (transstadially) to the hosts of the adults they become, and adults can transmit (trans-ovally) to hosts of the larvae/nymphae that they become.There are two basic frameworks: those who treat the tick density as a parameter, and those who include the processes determining the density of ticks.

Implementation of models

Tick life cycle models

A structured population model is a summary of rules specifying how the number and distribution of individuals within a population changes over time [START_REF] Tuljapurkar | Structured-population models in marine, terrestrial, and freshwater systems[END_REF]. Most structured population models fall into one of three categories: matrix models, ordinary differential equation (ODE) models, and partial differential equation (PDE) models. In this classification, model type is determined by whether time is discrete (matrix) or continuous (ODE, PDE), and whether the individual-level state is treated as a discrete (matrix, ODE) or a continuous (PDE) variable.

Matrix projection models are popular, because they have relatively simple structure and provide useful information. The eigenvalues and eigenvectors of the projection matrix provide estimates of the population growth rate, the stable age or stage distribution, reproductive value, and the sensitivities of population growth rate to changes in life history parameters [START_REF] Caswell | Matrix Population Models: construction, analysis, and interpretation[END_REF]. However, whenever a matrix projection or ODE model is applied to population characterized by a continuous state variable (e.g., age, mass, or physiological stage), individuals must be divided into a discrete set of classes.

In partial differential equation models, the individual-level state variables are continuous, and individuals are not lumped into categories. Like the matrix models, PDE models can incorporate a variety of biological situations, including density dependence, and stage-or age-structured populations [START_REF] Metz | The dynamics of physiologically structured populations[END_REF], that's why the basic model structure is the same in all cases. Furthermore, tick population dynamics can be expressed by the McKendrick-Von Foerster equation which is based on partial differential equations (PDE).

As we announced previously, our first objective is to construct a physiological stage dependent PDE model for the tick population dynamics. This model will be in order to fit to field data from Tunisia that have been previously gathered from several Tunisian farms and several data from laboratory colonies.

As a second step we will investigate the effects of climate on geographic range and seasonality of the tick and compare our results with the ones in [START_REF] Ogden | A dynamic population model to investigate effects of climate on geographic range and seasonality of the tick ixodes scapularis[END_REF].

Parameter definition

We denote by s the tick physiological parameter and t the time parameter and suppose that host populations are fixed at given densities H. To understand the relationship between our PDE model and classical ODE model: we use a physiological parameters s and interstadal development rate, g and let us define s max egg , s max larvae , s max nymph ,s max adult , the maximum length in the eggs, larvae, nymph and adult class.

To properly model the tick population the rates of tick mortality, reproduction rate (egg-laying) K, and interstadal development rate, g, must be obtained, while to prevent the tick population exponentially increasing issues regarding density dependence should be addressed.

Mathematical Models

We describe here the mathematical models that we propose to study.

Model 1

The functional equation considered in this model is derived from a physiologicalstructured model for a population divided into several stages in which individuals change their stage when a certain magnitude reaches a predetermined threshold value. This means that the physiological parameter s of passing from one stage to the next is time-dependent, giving rise to a moving boundary. More details can be found in [START_REF] Arino | A Mathematical model of population of fish in the larval stage: density-dependence effects[END_REF].

To illustrate the ideas underlying the model, consider a population divided into two stages, larvae (l) and adults (L), each one being structured by the age in the stage.

Denote by l(s, t) the density of larvae, n(s, t) the density of nymph and a(s, t) the density of adult at time t and in physiological state s. Capitals, L, N and A, denote the total population of larvae, nymph and adult respectively at time t.

Transition from larvae to nymph stage

Let us describe the passage through the larvae stage. We assume that the larvae turn adult when some variable reaches a prescribed value. For example, in [START_REF] Arino | A Mathematical model of population of fish in the larval stage: density-dependence effects[END_REF] the passage to (n) is described in terms of a blood meal which can be measured by weight function of larvae w l (s, t) representing the quantity of blood eaten until time t by an individual until reaches stage s. Larvae turn nymph when the food index reaches a prescribed value Q l > 0. We also assume that there is a finite maximum age s max larvae > 0 for individuals in the larval stage: individuals which have not acquired the amount Q of food past s max larvae > 0 will die or never reach the nymph stage.

In the model considered in [START_REF] Arino | A Mathematical model of population of fish in the larval stage: density-dependence effects[END_REF], the weight function of larvae depends on the total population of larvae, so that:

w l (s, t) := t t-a K l L(σ) + J l dσ; L(t) := s max larvae 0 l(s, t) ds (1)
which means that the quantity of food available is shared in equal parts by all the individuals occupying the same space at time t. K l > 0 is the quantity of food entering the species habitat per unit of volume and per unit of time, which for simplicity is considered to be constant. The constant J > 0 represents the food (converted into a number of individuals) taken per unit of volume by consumers other than larvae.

Then, the age of passage to the (l) stage, denoted by s * l (t), is defined by the threshold condition:

w l (s, t) = Q (2)
so that

s * l (t) = s(t)
solution to (2), if it exists and satisfies 0 ≤ s(t) ≤ s max larvae , s max larva otherwise.

Bearing in mind the above considerations, the density of larvae l(s, t) satisfies the following model:

               ∂l ∂t (s, t) + ∂ ∂s (g l (s, t)l(s, t)) = -µ l (s)l(s, t), 0 < s < s * l (t), t > 0, l(s, t) = 0, s * l (t) ≤ s ≤ s max larvae , t > 0, l(s, 0) = 0, 0 ≤ s ≤ s max larvae , l(0, t) = B(t), t > 0,
where µ l (s) is the age-dependent mortality rate of larvae, the initial condition expresses the fact that at time t = 0 no individuals are in the (l) stage and B(t) is the recruitment of larvae at time t. We will assume that µ l is a nonnegative continuous function on [0, s max larva ) such that 1) l(s, t) is the density of larvae that have absorbed a quantity of blood s at time t.

2) n(s, t) is the density of nymphs that have absorbed a quantity of blood s at time t.

3) a(s, t) is the density of adults that have absorbed a quantity of blood s at time t. The tick population dynamic is given by the following system which is composed by three PDE:

Equation of larvae:

         ∂l(s, t) ∂t + ∂ ∂s (g l (s, t)l(s, t)) = -µ l (s)l(s, t), l(0, t) = λ s max l 0 B(σ)a(σ, t)dσ, l(s, 0) = ψ l (s). (3)
Equation of nymphs

         ∂n(s, t) ∂t + ∂ ∂s (g n (s, t)n(s, t)) = -µ n (s)n(s, t), n(0, t) = δ t t-sn l(s * n (σ), σ)dσ, n(s, 0) = ψ n (s), (4)
with sn = sup

[0,t] s * n (t).
Dynamic equation of adults:

         ∂a(s, t) ∂t + ∂ ∂s (g a (s, t)a(s, t)) = -µ a (s)a(s, t), a(0, t) = γ t t-sa n(s * a (σ), σ)dσ, a(s, 0) = ψ a (s), (5)
with sa = sup

[0,t]
s * a (t).

-The functions g l , g n and g a are the growth of blood's quantity which have absorbed by ticks at stages l, n and a respectively.

Model 2

The tick population density varies satisfying the following model for all t ∈ [0, T] and s ∈ [s min , s max ] given by

             ∂n(s, t) ∂t + ∂ ∂s (g(s, t)n(s, t)) = -µ(n(s, t))n(s, t), n(s min , t) = smax smin K(n(s, t))n(s, t)ds, n(s, 0) = n 0 (s),
with s min and s max the min and the max physiological stage of the tick life cycle and T the maximum study time.

We are going to work on a set of differentiable C 1 periodic functions on t, and C 1 non-negative functions on s such that n(s max , t) = 0 for all times t.

On this basis, since we are dealing with C 1 non-negative functions on s, we may consider that a(s) and b(s), functions appearing in the somatic growth rate g(s, t) as nonnegative functions of s so that g(T ) ≤ a(s) for all s ∈ [s min , s max ] and t ∈ [0, T]. Moreover, we may suppose that a(s) is a bounded function i.e there exists a constant

C > 0 such that a(s) ≤ C, ∀s ∈ [s min , s max ]. Thus, g(T ) ≤ C, ∀s ∈ [s min , s max ], ∀t ∈ [0, T].
Also let N be the maximum tick population density so that n(s, t) ≤ N , ∀s ∈ [s min , s max ], ∀t ∈ [0, T].

Model 3

The following model aims to analyze the impact of climate change on life cycle tick and especially on hibernation period. Let x 1 (t) be the density of larvae at timet, x 2 (t) the density of nymphs after hibernation at time t and x 3 (t) the density of adults at time t.

We suppose that T (t) the temperature at time t, ρ(T ) the fertility rate of adult females at time t and µ(t, T ) the mortality rate. Thus, the system equations are written in the following way:

       dx1(t) dt = t t-τ ρ(T (σ))x 3 (σ)dσ -µ 1 (t, T (t))x 1 (t) -f 1 (t, T, H, x 1 ), dx2(t) dt = λ 1 α 1 (t, T, H, x 1 (t -r(t, T (t))))x 1 (t -r(t, T (t))) -µ 2 (t, T (t))x 2 (t) -f 2 (t, T, H, x 2 (t)), dx3(t) dt = λ 2 α 2 (t, T, H, x 2 (t))x 2 (t) -µ 3 (t, T (t))x 3 (t).
Impacts des changements climatiques sur les populations de tiques -23

With -f 1 (t, T, H, x 1 ) = α 1 (t, T, H, x 1 (t))x 1 (t). -f 2 (t, T, H, x 1 ) = α 2 (t, T, H, x 2 (t))x 2 (t).

Physiological SIS for ticks and SIR ODE for host Model

We consider that the number of tick is governed by the equation of section

n s T (t) = 1 -n i T (t) (9)

Conclusion and Recommendation

In this paper, we present the various stages of tick population dynamics which is composed by three partial differential equations.

Our aim in the future is to adapt the numerical method developed in subsection 3.2 in order to solve the correlated EDO-EDP equations. This method will be used to test impact of climate change on transmission of Tick disease, for that the model will be fitted to field data that have been previously gathered from several Tunisian farms [START_REF] Bouattour | Cattle infestation by Hyalomma ticks and prevalence of Theileria in H. detritum species in Tunisia[END_REF] and several data from laboratory colonies (Darghouth, unpublished data) and with data from Tunisian National Institute of Meteorology. Furthermore, our goal is to develop new numerical methods in order to approximate solutions of the previous type of equations in subsection 3.3.2.

  [START_REF] Hope-Cawdery | Mathematical models for predicting the prevalence of liver fluke disease and its control from biological and meteorological data[END_REF].2 and that the tick are subdivided in two class Susceptible and Infected. Let n s T (t) and n i T (t) be respectively the frequencies of susceptible and infected tick parasites, i.e., is total population of tick parasites, a min and a max are the min and the max physiological age of the tick life cycle.And that the host population number is constant, let n s H (t), n i H (t) and n r H (t) be respectively the frequencies of susceptible, infected and removed host population, i.e. Consider the following assumptions:-Let φ(n) be the factor representing the influence of ticks on the host. Thus the model representing the host-parasite epidemiological interaction is given by the following SIR-SIS model:

		n s T (t) + n i T (t) = P (t).
	But
	P (t) =
	n s H (t) + n i H (t) + n r H (t) = 1.
	dn s H (t) dt	= -K 1 n s H (t)n i H (t)	(6)
	dn i H (t) dt	= K 1 n s H (t)n i H (t) -K 2 n i T (t)n i H (t)	(7)
	dn r H (t) dt	= K 2 n i T (t)n i H (t)	(8)
	dn i T (t) dt	= K 1 n s H (t)n i H (t) -φ(n)K 2 n i T (t)n i H (t) + n s T (t)

amax amin (l(a, t) + n(a, t) + a(a, t))da, ∀a ∈ [a min , a max ],

where P (t)

-ARIMA -Volume 22 -2016

-ARIMA -Volume 22 -2016

-ARIMA -Volume 22 -2016 ARIMA Journal