Marie-France Daniel

Mathieu Gagnon

Emmanuèle Auriac-Slusarczyk

DIALOGICAL CRITICAL THINKING IN KINDERGARTEN AND ELEMENTARY SCHOOL Studies on the impact of philosophical praxis in pupils

.

Lipman's thesis remains controversial in the community of academic philosophers and researchers, and continues to fuel debate. To shed light on this controversy, it is essential to evaluate pupils' cognitive and epistemological development in order to, first, investigate the philosophical quality of the exchanges and, as a corollary, to illustrate the positive impact of P4C on learning in schools.

To that end, many philosophers and researchers involved in P4C have studied 'evolution' or 'epistemic progress' manifested within pupils' discourse (among others:

we briefly describe the empirical model of the developmental process of CT that emerged from our analyses of pupils' exchanges during P4C praxis. Finally, using this model as an analysis grid, we present the results of research projects we conducted with apprentice philosophers aged five to 12 years. Results show that these pupils did in fact engage in the process of CT, as defined in our model.

Critical thinking

Where does the concept of critical thinking come from? How is it conceptualized in the field of P4C? How is critical thinking studied empirically in the fields of psychology and epistemology? CT seems to have its source in informal or applied logic. At the beginning of the twentieth century, John Dewey, building on the theoretical model of Charles Sanders Peirce, proposed a reflective method in five steps to provide individuals with a tool to evaluate the causes and consequences of their actions (Dewey, 1903(Dewey, /1933)). Dewey stressed the importance of teaching/learning to think, in order to help students to reflect in an autonomous and critical manner. His aim was to improve the quality of the individual and of the democratic experience (1916/1983). The movement of informal or applied logic continued to develop during the 1950s with philosophers such as Max Black and Monroe Beardsley, who attempted to make logic accessible to students. Black did so by associating logic with language, and Beardsley by emphasizing the search for meanings (vs. the search for truth). Then in 1962, Robert Ennis proposed a first definition of the concept of CT based on logical reasoning. Later on, Ennis's definition of CT came to include two thinking modes, logical and creative, and the intellectual predispositions associated with these modes. These two modes of thinking are included in Lipman's P4C programme. In the 1980s, the concept of CT was further developed by philosophers such as Richard-Stanley Peters, John Passmore and Michael Scriven, to name only a few. Their works represent a significant contribution to recognition of the importance of thinking in education. Today, the most quoted definitions of CT are those of [START_REF] Ennis | A concept of critical thinking[END_REF][START_REF] Ennis | Critical thinking and the curriculum[END_REF], [START_REF] Lipman | Critical thinking: What can it be?[END_REF]), John McPeck (1990), Richard [START_REF] Paul | Critical and reflexive thinking: a philosophical perspective[END_REF] and Harvey Siegel (1988) (for more information, see [START_REF] Auriac-Slusarczyk | Tester les prédispositions à l'esprit critique au primaire[END_REF].

Since Lipman is the creator of the P4C programme, with his first collaborator Ann-Margaret Sharp and with Frederick Oscanyan, we succinctly present his definition of CT, inspired by the works of Peirce and Dewey. For Lipman, CT contributes to reinforcing complex thinking, which is mobilized through peer dialogue concerning situations, concepts or principles associated with daily life [START_REF] Lipman | À l'école de la pensée[END_REF]. CT is a tool to counter non-reflective thinking and action. Individuals need critical thinking in order to think well and to evaluate, among all the information received, the most relevant in accordance with the objectives they pursue. At the core of Lipman's definition is the notion of 'good judgment', that is, judgment that takes into account the central elements of a problem and follows the inquiry steps to solve it. In this sense, CT is governed by criteria, is self-correcting and is sensitive to context [START_REF] Lipman | Critical thinking: What can it be?[END_REF][START_REF] Lipman | Thinking in Education[END_REF].

Like Lipman, many proponents of P4C consider CT to be a meaningful tool for the reconstruction of society, but only to the extent that it is embedded in a non-indoctrinating pedagogy such as the community of inquiry [START_REF] Fisherman | Thinking as two -philosophy, critical thinking and community of inquiry[END_REF][START_REF] Matt | Coming to terms on critical thinking[END_REF][START_REF] Tsiplakides | Critical and creative thinking in the English language classroom[END_REF][START_REF] Weinstein | Reason and critical thinking[END_REF]. Some maintain that CT aims at the rational validation of achievements, structures and values based on criteria [START_REF] Weinstein | Reason and critical thinking[END_REF][START_REF] Weinstein | Critical thinking and the psycho-logic of race prejudice[END_REF]. Others define CT as thinking that is imaginative [START_REF] Pritchard | Critical thinking: Problem-solving or problem creating?[END_REF], autonomous and open to differences [START_REF] Matt | Coming to terms on critical thinking[END_REF], and that falls within the scope of inter-relational and dialogical ethics [START_REF] Boe | Critical thinking in kindergarten[END_REF]. CT is thinking that questions [START_REF] Scholl | Students' questions: Developing critical and creative thinkers[END_REF] and searches for meanings [START_REF] Nowell | Education as meaning-making and the development of critical thinking[END_REF] to the extent that the content of discussions among pupils is contextualized based on the pupils' school and family experiences [START_REF] Weinstein | Reason and critical thinking[END_REF]. Proponents of P4C consider that CT, unlike the problem-solving approach with which it is sometimes confused, cannot be evaluated with standardized tests, which are too restrictive [START_REF] Gagnon | La pratique de la philosophie en communauté de recherche et le développement de la pensée critique d'adolescents[END_REF][START_REF] Matt | Coming to terms on critical thinking[END_REF][START_REF] Pritchard | Critical thinking: Problem-solving or problem creating?[END_REF].

Over the last thirty years, researchers from various fields (i.e. health sciences, developmental psychology, epistemology) have been empirically investigating CT in students (students not involved in P4C). Among these, some establish a correlation between the development of CT and the personal epistemological development of students.

A review of the literature indicates that most studies are conducted with adolescents and particularly with young adults attending college and university (for an overview: [START_REF] Hofer | Personal epistemology. The psychology of beliefs about knowledge and knowing[END_REF]. To measure the development of CT, most researchers use interviews or individual tests that are primarily associated with applying the rules of formal logic [START_REF] Kwak | Re-conceptualizing critical thinking for moral education in culturally plural societies[END_REF].

Empirical research

Over the past ten years, we have conducted research projects on CT 1 in 28 groups of pupils aged five to 12 years from several countries (Quebec, Ontario, France, Mexico and Australia). 2 Analyses were carried out on transcripts of pupils' exchanges during P4C sessions. 3

Phase 1. Toward a model of the developmental process of (dialogical) critical thinking

The first research result we obtained was based on the qualitative methodology of Grounded Theory (GT) [START_REF] Charmaz | Grounded theory in the 21st century: Applications for advancing social justice studies[END_REF]. The objective of a GT analysis is to draw out a new understanding of a phenomenon from data collected on the ground. Thus, although we were familiar with the classical definitions associated with CT, we focused on the pupils' manifestations of CT.

A developmental process of dialogical CT emerged from our analyses of pupils' exchanges. In this process, CT is referred to as 'dialogical' CT (DCT) to highlight and clarify that CT mobilized within the P4C community of inquiry emerges from interactions and dialogue among peers. Philosophical dialogue stimulates divergence in points of view, which raises doubts or cognitive conflicts in pupils' minds. The teacher can participate in raising these doubts by asking pupils for counter-examples, different alternatives and so on. These doubts and cognitive conflicts represent the starting point of the reflective and critical process (Dewey, 1903(Dewey, /1933)).

The analyses revealed that DCT is not exclusively oriented towards argumentation and conceptualization (logical thinking). DCT also encompasses reflection based on divergent relationships with the intention of transforming perspectives (creative thinking), evaluating behaviours and social/ethical values to improve societal bases for human life (responsible thinking), and reconsidering one's thoughts and perspectives in order to self-correct one's representations of the world and one's relationship to or engagement in the world (meta-cognitive thinking).

The analyses of the transcripts also revealed that the manifestations of these four thinking modes (logical, creative, responsible, meta-cognitive) included in our interpretative model are articulated along a spectrum from the simpler to the more complex. We used the notion of 'epistemological perspectives' to express the diversity of manifestations inherent in each thinking mode and also to account for the increasingly complex process of DCT in pupils.

Here epistemology refers to a relational epistemology (versus an individual one) that situates the development of thinking within a process of social construction: a process of dynamic thought and language construction with others [START_REF] Thayer-Bacon | Relational '(e)pistemologies[END_REF]. The concept of perspective refers to the groups of pupils' representations of the world (i.e. are their representations of the world centred on the self? Are they open to others' points of view or alternatives? Are they striving towards responsible improvement of society?).

To guide the interpretation of subsequent transcript analyses, six epistemological perspectives that progressively emerged from repeated transcript analyses were named as follows: Egocentricity, Post-Egocentricity, Pre-Relativism, Relativism, Post-Relativism/ Pre-Intersubjectivity and Intersubjectivity (examples to illustrate the epistemological perspectives can be found in [START_REF] Daniel | A developmental model of dialogical critical thinking in groups of pupils aged 4 to 12 years[END_REF][START_REF] Daniel | The incidence of philosophy on discursive and language competencies of pupils aged four years[END_REF]. In the perspective of Egocentricity, pupils' representations of the world are personal and concrete, and statements are simple and generally isolated from peers' ideas (vs. relationships). In Post-Egocentricity there is a slight decentring of representations that refer to the specific experience of the pupils' immediate surroundings (i.e. family members). In Pre-Relativism, points of view underlie the beginning of generalized representations but remain rooted in the concrete environment. In Relativism, pupils' statements are issued from reasoning, based on experience; they may present concrete and/or incomplete justifications; and statements imply convergent relationships between points of view. In Post-Relativism/Pre-Intersubjectivity, representations involve conceptualized statements that introduce divergent relationships; they presuppose the beginnings of a constructive evaluation. In Intersubjectivity, statements are conceptualized and justified with criteria; they are not expressed as closed conclusions but rather as questionings; the relationships they underlie are divergent; and they seek correction and transformation of perspectives toward a Common Good (see [START_REF] Daniel | Dialogical critical thinking with 5 to 12 year-old pupils: A continuous epistemological development[END_REF]. The operational model (see Appendix) became a meaningful tool for evaluating the pupils' developmental process toward DCT.

Phase 2. Some empirical results

The model of the developmental process of DCT was used as a grid to analyse pupils' exchanges during one full school year, and thus to identify which thinking modes and epistemological perspectives are mobilized when pupils philosophize (see [START_REF] Daniel | Modeling the development process of dialogical critical thinking in pupils aged 10 to 12 years[END_REF][START_REF] Daniel | A developmental model of dialogical critical thinking in groups of pupils aged 4 to 12 years[END_REF][START_REF] Lam | Continuing Lipman's and Sharp's pioneering work on philosophy for children: using Harry to foster critical thinking in Hong Kong students[END_REF][START_REF] Daniel | The incidence of philosophy on discursive and language competencies of pupils aged four years[END_REF]. In this chapter, we limit the presentation of results to the epistemological perspectives without differentiating the thinking modes.

For kindergarten pupils, results reveal that in most groups, at the end of the school year, the epistemological perspective of Pre-Relativism predominates, no matter what the subject, and no matter who acted as facilitator.

A comparative study was conducted with two groups of kindergarten pupils (experimental and control groups) who shared the same socio-economic and school conditions. The experimental group of apprentice philosophers tried to answer the question: 'Are we free?', whereas the control group asked themselves: 'What is love?' By grouping results linked to Egocentric (Egocentricity and Post-Egocentricity) and Relativist (Pre-Relativism and Relativism) perspectives, we obtain the following results (see Table 27.1): 35 percent of pupils' interventions in the experimental group were manifested in an Egocentric manner and 65 percent in a Relativist manner, whereas in the control group, 60 percent of interventions were manifested in an Egocentric manner and 40 percent in a Relativist manner.

Is Pre-Relativism a simple or complex perspective for kindergarten pupils? To find out, we compared results from the kindergarten experimental group with a group of fifth graders who had no experience of philosophical praxis, and who were participating for the first time in an exchange on a question related to the same theme as the kindergarten group: 'What does it mean to be free?' Results (see Table 27.2) indicate that a greater percentage of kindergarten pupils' interventions was situated in Egocentricity in comparison to fifth graders (11% vs. 1%); a greater percentage of fifth graders' interventions was situated in Relativism (17% vs. 11%) and in Post-Relativism (1% vs. 0%) compared to those of kindergarten pupils; in both groups of pupils the predominant epistemological perspective was Pre-Relativism (54% and 53%). From this latter data, we might infer that Pre-Relativism is a complex epistemological perspective for kindergarten children.

We compiled results on epistemological mobilization in other groups of elementary school pupils who participated in the research (pupils from different countries who had different discussion themes and whose teachers had different facilitating styles). These results show that after one year of philosophical praxis with P4C, the groups from the first cycle of elementary school were generally situated in a Pre-Relativist epistemology with a tendency toward Relativism, and that the groups from the second cycle of elementary school were generally situated in Relativist epistemology with a tendency toward Post-Relativism. A single group from the second cycle of elementary school that had more than two years of P4C praxis mobilized DCT consistent with Intersubjectivity.

Again, we can ask: is Relativism a complex perspective for pupils at the end of elementary school? Results from two distinct studies, using the developmental process model of DCT as an analysis grid, and conducted among undergraduate university students, indicate that the epistemology of these young adults fluctuates between Egocentricity and Relativism [START_REF] Forges | Formation initiale des enseignants en enseignement de l'ÉPS. Étude des manifestations d'une pensée critique visée[END_REF][START_REF] Lechasseur | Modélisation de la mobilization des savoirs par une pensée critique chez des étudiantes en sciences infirmières lors de stages cliniques[END_REF]. Given these results, Relativism would appear to be a complex epistemological perspective for pupils at the end of elementary school.

Discussion

From a theoretical point of view DCT differs from CT, as the components of DCT were not 'inferred' from existing definitions, but rather 'emerged' from the analyses of philosophical exchanges among pupils. As a result, DCT comprises four thinking modes and six epistemological perspectives. It is essential that these components be included in a definition of CT because they better illustrate the complexity of pupils' thinking and its progress following P4C praxis.

From an empirical point of view, results compiled from the literature show the impact of P4C on the development of logical reasoning in elementary and secondary school students, and they showed the mobilization of CT in adolescents. The results of the research presented in this chapter demonstrate for the first time that P4C has a significant impact on the mobilization of DCT among pupils in kindergarten and elementary school.

Moreover, most of the results compiled on logical reasoning relied on a quantitative paradigm, that is, they issued from standardized tests. The appropriateness of using standardized tests is regularly questioned in the study of CT and P4C. Conversely, the results on DCT issued from observing classroom interactions and analysing transcripts of exchanges among pupils. The qualitative methodology employed not only supports the quantitative results concerning the impact of P4C on the pupils' cognitive development, but also, and mainly, provides indications of the thinking modes and epistemological perspectives that are involved in DCT. In that sense, the qualitative method allows us to widen and increase our comprehension of the processes of mobilization and development of DCT when children engage in philosophical dialogue.

Conclusion

Can P4C contribute to the development of CT in kindergarten and elementary school pupils? Innovative and exploratory results corroborate Lipman's thesis insofar as the results show that apprentice philosophers from kindergarten and elementary school are able to engage in the co-construction of DCT when they are stimulated by an approach such as P4C. These results do not indicate that children mobilize an accomplished DCT, one that is anchored in Intersubjectivity, but the results do indicate that children's representations of the concepts discussed transcend the simple perspectives associated with Egocentricity, being situating within more complex epistemologies.

In addition, the developmental model of DCT provides researchers and teachers with a tool for systematically evaluating pupils' cognitive and epistemological evolution and progress when they are engaged in P4C praxis, and thus for assessing the value of the philosophical quality of their exchanges.

It would be appropriate to conduct further empirical research in other groups of pupils, of different cultural, socio-economic and educational backgrounds to verify/complete the current results. It would be relevant to examine, for example, the components inherent in DCT and their interactions. It would also be useful to examine the influence of educational policy on the developmental process of DCT; to study the impact of the adults' teaching style on pupils cognitive and epistemological development; and to verify whether pupils can mobilize DCT on their own without the intervention of a teacher, and whether DCT is retained as a competency in the long term, when philosophical praxis is no longer present. [START_REF] Daniel | Modeling the development process of dialogical critical thinking in pupils aged 10 to 12 years[END_REF][START_REF] Daniel | A developmental model of dialogical critical thinking in groups of pupils aged 4 to 12 years[END_REF]

Appendix: Operational model of the developmental process of DCT

Table 27 .

 27 1 Comparison between two kindergarten classrooms

		Group with P4C	Group without P4C
	Egocentricity	11%	35%	35%	60%
	Post-Egocentricity	24%		25%
	Pre-Relativism	54%	65%	36%	40%
	Relativism	11%		4%

  The research projects were subsidized by the Social Sciences and Humanities Research Council of Canada. 2 Principal investigator: Marie-France Daniel. Collaborators: Louise Lafortune, Richard Pallascio, Teresa de la Garza, Christina Slade, Laurance Splitter (Project 1). Mathieu Gagnon, Emmanuèle Auriac-Slusarczyk (Project 2). 3 For methodological details, see

	Intersubjectivity	Justification based	Evaluative	Doubt that	Evaluative
			on criteria.	relationship	underlies the		statement
		Mode/	Logical Conceptualization	Creative that provides	Responsible evaluation of		Meta-cognitive that expresses
		Epistemological	based on	a different	categories (rules,		a change in
		perspective	evaluative	meaning and	principles,		perspective
		Egocentricity	Statement based on the perceptual experience of a specific and reasoning. Conceptualization	Statement that gives meaning to a personal and concrete point of transforms the perspective. Transformation Categorization Statement that is related to a personal and specific social/moral values).	following the Retrospective integration of statement criticism. about a personal and Correction
			personal fact.	view.	behaviour		specific task,
					tied to a		point of view,
					social or moral		feeling, etc.
				Notes	belief.
	1	Post-Egocentricity	Statement based	Statement that	Particular/concrete	Retrospective
			on experience	gives meaning	statement tied		statement
			(personal or	to a personal	to a moral		about a
			someone close) +	point of view	or social rule		personal task,
			reasoning.	(but distanced	(learnt). Not		point of view,
				from self).	contextualized.		feeling, etc.
							(distanced
							from self).
		Pre-Relativism	Somewhat	Statement that is	Statement linked		Descriptive
			generalized	new, divergent	to a somewhat		retrospective
			statement that is	or that presents	generalized		of a personal
			not justified, or	different	action in a		task, point of
			has an implicit,	situations/	moral or social		view, feeling,
			circular or false	solutions/	perspective.		etc. (distanced
			justification.	hypotheses			from self).
				(units) in relation
				to a personal idea
				or to someone
				else's idea (peer
				or text).
		Relativism	Incomplete	Relationship that	Statement that		Descriptive/
			or concrete	gives meaning	explains a will		explanatory
			justification	to a peer's	to understand/		retrospective
			(explanation)/	point of view	include others		of another
			reasoning based	(by completing	(from the		person's task,
			on experience.	it or adding a	immediate		thought, etc.
				nuance).	environment).		(immediate
					(contextualized).	environment).
		Post-Relativism/	Justification	Relationship	Statement that		Descriptive/
		Pre-	based on 'good	that presents a	justifies a desire		explanatory
		Intersubjectivity	reasons'/simple	different context	to understand/		retrospective
			reasoning.	that takes	include		of another
				into account	others (distant		person's task,
				the group's	environment)		thought,
				perspective.	(contextualized).		etc. (distant
							environment).


Pascal Zongo

René Dorville

Elisée Gouba

Joseph Ki-Zerbo

Method for identifying spatial reservoirs of malaria infection and control strategies

Keywords: Control, Optimization, Metapopulation, Basic reproductive number, cost-effectiveness analysis

Managing spatial reservoirs of malaria infection plays a crucial role in effective disease control. In this paper, a reservoir of infection refers to one or more interconnected subpopulations that sustain the epidemic at the level of the metapopulation to which applying a (linear) control strategy suffices to eradicate the disease in the whole system. We propose a numerical method to explain the steps for identifying reservoirs of malaria infection within n connected regions with the explicit movement of human population from the previous theoretical results in order to design an efficient computational tool. Furthermore, we determine the minimal percentage (critical vaccination fraction) of susceptible individuals in the reservoirs that should be protected to eliminate malaria. The costs and cost-effectiveness of malaria control interventions were analysed considering two strategies of control. (i) protecting the minimal fraction of susceptible individual; (ii) protecting any fraction greater than the minimal fraction. Cost-effectiveness analysis shows that the less cost and more effective strategy is to vaccinate (or protect) the minimal fraction of susceptible human in the reservoir of infection to halt outbreak. A numerical example provides insight into the efficiency of this approach.

I. INTRODUCTION

Malaria is a mosquito-borne infections disease caused by protozoa of the genus plasmodium (parasite). It is estimated that about 1.5-3 million of people, mostly children, die of malaria every year [29]. Malaria control requires an integrated approach, including prevention and prompt treatment [START_REF] Who | Guidelines for the Treatment of Malaria[END_REF].

The parasites are transmitted indirectly from human to human by the bite of infected female mosquitoes of the genus Anopheles. There is some natural acquired partial immunity to the pathogen in humans developed after many years of repeated infections [START_REF] Aron | Mathematical modeling of immunity to malaria[END_REF], [START_REF] Drakeley | Transmission reducing immunity is inversely related to age in plasmodium falciparum gametocyte carriers[END_REF], [START_REF] Kaslow | Transmission-blocking immunity against malaria and other vector-borne diseases[END_REF], [START_REF] Ponnudurai | Transmission blockade of Plasmodium falciparum: its variability with gametocyte numbers and concentration of antibody[END_REF].

Models have already been proposed to provide an explicit framework for understanding malaria transmission dynamics in human population for over 100 years [START_REF] Chitnis | Bifurcation analysis of a mathematical model for malaria transmission[END_REF], [START_REF] Ducrot | A mathematical model for malaria involving differential susceptibility, exposedness and infectivity of human host[END_REF], [START_REF] Mandal | Mathematical models of malariaa review[END_REF], [START_REF] Ngwa | A mathematical model for endemic malaria with variable human and mosquito populations[END_REF] and references there in. Human movement has rarely been taken into account in models. Recently, it was shown that the role of human movement plays a significant role on disease reemergence and persistence [START_REF] Arino | Quarantine in a multispecies epidemic model with spatial dynamics[END_REF], [START_REF] Brauer | Mathematical Models in Population Biology and Epidemiology[END_REF], [START_REF] Martens | Malaria on the move: human population movement and malaria transmission[END_REF]. There are two standard approaches to study the spatial dynamics of vector borne disease such as malaria: partial differential equations [START_REF] Lou | A reaction-diffusion malaria model with incubation period in the vector population[END_REF], [START_REF] Xiao | Mathematical Biology Transmission dynamics for vector-borne diseases in a patchy environment[END_REF], [START_REF] Xu | A reaction-diffusion model of dengue transmission[END_REF], [START_REF] Yu | A nonlocal spatial model for Lyme disease[END_REF] and meta-population models [START_REF] Arino | A metapopulation model for malaria with transmission-blocking partial immunity in hosts[END_REF], [START_REF] Lee | the role of residence times in two-patch dengue transmission dynamics and optimal strategies[END_REF], [START_REF] Xiao | Mathematical Biology Transmission dynamics for vector-borne diseases in a patchy environment[END_REF]. More precisely, in [START_REF] Arino | A metapopulation model for malaria with transmission-blocking partial immunity in hosts[END_REF], a metapopulation malaria model was proposed using SI and SIRS models for the vectors and hosts, respectively. The mobility of human population has been proved to have a significant impact on the epidemic behavior. For example, the impact of the movement of human population on malaria transmission in different realistic situations: from rural into urban areas and colonization of heretofore unused territories was performed. Using type reproduction numbers approach developed in [START_REF] Heesterbeek | The type-reproduction number T in models for infectious disease control[END_REF], [START_REF] Roberts | A new method to estimate the effort required to control an infectious disease[END_REF], the authors identify the reservoirs of infection and evaluate the effect of control measures. We point out that the reservoirs of infection remain variously and loosely defined in the literature [START_REF] Ghinai | Malaria epidemiology in central Myanmar: identification of a multi-species asymptomatic reservoir of infection[END_REF], [START_REF] Haydon | Identifying reservoirs of infection: a conceptual and practical challenge[END_REF], [START_REF] Shannon | Sub-clinical P. falciparum infections act as year-round reservoir for malaria in the hypoendemic Chittagong Hill Districts of Bangladesh[END_REF], [START_REF] Viana | Assembling evidence for identifying reservoirs of infection[END_REF]. In this paper, a reservoir of infection is a subpopulation to which applying a (linear) control strategy suffices to eradicate the disease in the whole system [START_REF] Arino | A metapopulation model for malaria with transmission-blocking partial immunity in hosts[END_REF], [START_REF] Heesterbeek | The type-reproduction number T in models for infectious disease control[END_REF], [START_REF] Roberts | A new method to estimate the effort required to control an infectious disease[END_REF]. A reservoir may comprise multiple connected subpopulation of human and/or mosquitoes. Thus to reduce or eliminate malaria over time, a control should be applied simultaneously in the different reservoirs of infection, but the minimal fraction of susceptible individuals of each reservoir to be protected is not specified, thus a control like that should be cost a great deal of money. Therefore arise the following question, what minimal fraction of each group of the reservoir should be protected to eliminate malaria? This minimal fraction can be interpreted as the critical vaccination fraction (fraction of population to vaccinate to halt outbreak).

The main purpose of this paper is threefold: (i) to explain the steps for identifying reservoirs of malaria infection from the theoretical results derived in [START_REF] Arino | A metapopulation model for malaria with transmission-blocking partial immunity in hosts[END_REF] in order to design an efficient computational tool; (ii) determine the minimal percentage of susceptible individuals in the reservoir namely, those that should be protected to eliminate the malaria over time in the whole of the region; (iii) analysis the cost and cost-effectiveness when controlling infection within the reservoir from the minimal percentage of susceptible individuals to protect. Cost-effectiveness analysis is very important because it compares the costs and health effects of an intervention to assess the extent to which it can be regarded as providing value for money [START_REF] Alphey | A Model Framework to Estimate Impact and Cost of Genetics-Based Sterile Insect Methods for Dengue Vector Control[END_REF], [START_REF] Rich | An integrated epidemiologicaleconomic analysis of foot and mouth disease : applications to the southern cone of South America[END_REF]. This informs decision-makers who have to determine where to allocate limited healthcare resources.

II. DESCRIPTION OF MODEL FORMULATION

In this section we give a summary of the model as already discussed in [START_REF] Arino | A metapopulation model for malaria with transmission-blocking partial immunity in hosts[END_REF] before to extend it. The space was split into n geographical regions. For each geographical unity i, i = 1, . . . , n, human population was divided into three subclasses: susceptible S H,i (t), infectious I H,i (t) and semiimmune R H,i (t). Total size of the human population H i (t) = S H,i (t) + I H,i (t) + R H,i (t) and the mosquito population into two subclasses: susceptible S V,i (t) and infectious I V,i (t). Humans were assumed move from patch to patch but the movement of mosquitoes was neglected, so humans were assumed do not change their epidemiological status during travel. The model was read as follows:

Total mosquito population

V i (t) = S V,i (t)+I V,i (t).
for each i = 1, . . . , n, dS H,i dt = Λ H,i + β H,i R H,i + ρ H,i I H,i -µ H,i S H,i -Φ H,i S H,i + n j=1 m S ij S H,j - n j=1 m S ji S H,i , dI H,i dt = Φ H,i S H,i -ǫ H,i I H,i + n j=1 m I ij I H,j - n j=1 m I ji I H,i , dR H,i dt = α H,i I H,i -δ H,i R H,i + n j=1 m R ij R H,j - n j=1 m R ji R H,i , dS V,i dt = Λ V,i -µ V,i S V,i -Φ V,i S V,i , dI V,i dt = Φ V,i S V,i -µ V,i I V,i (1)
with initial conditions S H,i (0), S V,i (0) > 0,

I H,i (0), R H,i (0), I V,i (0) ≥ 0, ǫ H,i = α H,i + γ H,i + ρ H,i + µ H,i , δ H,i = β H,i + µ H,i .
In the above formulation, m π ij , π = S, I, R, denote the constant rate of travel of humans from patch j to patch i, for all i, j = 1, . . . , n, i = j. M π = [m π ij ], π = S, I, R, is the travel rate matrices. The matrices M π , π = S, R, was assumed to be irreducible and m π ii = 0, for π = S, I, R and i = 1, . . . , n. In this paper, we use the force of infection derived in [START_REF] Chitnis | Bifurcation analysis of a mathematical model for malaria transmission[END_REF] as follows:

Φ H,i = a V,i a H,i V i a V,i V i + a H,i H i σ ViHi I V,i V i , (2a)
Φ V,i = a V,i a H,i H i a V,i V i + a H,i H i σ HiVi I H,i H i + σ HiVi R H,i H i . (2b)

A. Disease free state

An equilibrium solution of system (1) at which there is no disease in any of the patches is called a disease-free equilibrium. The local stability of this point is governed by the basic reproduction number denoted by R 0 . This latter is the expected number of secondary cases produced by a typical infective individual introduced into a completely susceptible population, in the absence of any control measure [START_REF] Diekmann | On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations[END_REF]. Next generation approach was used to derive R 0 (see [START_REF] Van Den Driessche | Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission[END_REF]). When R 0 < 1 the infection will die out in the long run. But if R 0 > 1 the infection will be able to spread in a population. It was shown that when disease induce mortality is large, a backward bifurcation may occur at R 0 = 1, that to say, reduce R 0 below 1 is not always sufficient to eliminate malaria.

B. Reservoir of infection

In this paper, a subgroup of patches is said to be a reservoir of infection when only targeting a control strategy which linearly reduce the number of susceptible is sufficient to eliminate the malaria in the whole of the patches. Define by J h = {H 1 , H 2 . . . , H n } the set of humans population, it was shown that a control only targeted on human population was possible to eliminate malaria. In this paper we focus on a control type targeted only to the human population. A new next generation operator M J h was defined by

M J h = [R HiHj ] 1≤i,j≤n , (3)
where R HiHj can be interpreted as the expected number of secondary infected humans in patch j that would arise from a single infected human case in patch i, in a situation where all the patches contain a completely susceptible population. Moreover we have

ρ(M J h ) < 1 ⇔ R 0 < 1 (4)
where ρ(A) is the spectral radius of a matrix A (see [START_REF] Arino | A metapopulation model for malaria with transmission-blocking partial immunity in hosts[END_REF] for details).

C. Sufficient condition for a patch to be a reservoir of infection

In [START_REF] Arino | A metapopulation model for malaria with transmission-blocking partial immunity in hosts[END_REF], it was shown that if there exists some patch ℓ in the subset {1, 2, . . . , n} such that R H ℓ H ℓ ≥ 1, then patch ℓ is an infection reservoir. In this case, we need to target simultaneously a control to the whole of susceptible population of the reservoir to eliminate malaria over time.

III. MINIMAL AND EFFICIENT CONTROL

From results obtained in [START_REF] Arino | A metapopulation model for malaria with transmission-blocking partial immunity in hosts[END_REF], arise the following question: What minimal fraction of each group of the reservoir should be protected to eliminate malaria?

Let J res be the set of reservoirs of infection. From section II-C, we have

J res = {ℓ ∈ {1, 2, . . . , n} : R H ℓ H ℓ ≥ 1} (5)
In the sequel, denotes by p the number of patches reservoirs, it is clear that p = cardinal(J res ) so that (np) represents the number of un-reservoirs patches while n is the number of patches. From [START_REF] Heesterbeek | The type-reproduction number T in models for infectious disease control[END_REF], one can define a new next generation matrix as follows:

M Jres = E T J h M J h [I n -(I n -P J h )M J h ] -1 E J h , (6)
where E J h and P J h are, respectively, n × p and n × n projection matrices satisfying (E J h ) jj = (P J h ) jj = 1 for j ∈ J res and (E J h ) jj = (P J h ) jj = 0 otherwise. M Jres is a p × p matrix satisfying ρ(M Jres ) < 1 ⇔ R 0 < 1. In the sequel, we set

M Jres = [M ij ] 1≤i,j≤p . (7)
The entries of M Jres are similar in concept to the entries of M J h .

A. Formulation of the objective function

Let S * H ℓ be the number of susceptible individuals within each patch ℓ at the disease free equilibrium, ℓ = 1, . . . , p. this number is obtained from Theorem [START_REF] Ducrot | A mathematical model for malaria involving differential susceptibility, exposedness and infectivity of human host[END_REF] in [START_REF] Arino | A metapopulation model for malaria with transmission-blocking partial immunity in hosts[END_REF].

Let f ℓ denotes the fraction of these susceptible that should be protected in patch ℓ. Then,

S * H ℓ f ℓ (resp. S * H ℓ (1 -f ℓ )
) represents the number of protected (resp. not protected) individuals at the disease-free state in the patch ℓ, ℓ = 1, . . . , p.

The fraction of susceptible in the whole population is denoted by F = (f 1 , f 2 , . . . , f p ) and those who will not need to be protected is denoted by

F = (1-f 1 , 1-f 2 , . . . , 1-f p ).
From now, one can define a new next generation matrix with variable F denoted by M Jres (F ) :

M Jres (F ) =      (1 -f 1 )M 11 (1 -f 2 )M 12 • • • (1 -f p )M 1p (1 -f 1 )M 21 (1 -f 2 )M 22 • • • (1 -f p )M 2p . . . . . . . . . . . . (1 -f 1 )M p1 (1 -f 2 )M p2 • • • (1 -f p )M pp      Note that when F = 0 then M Jres (F ) = M Jres .
To determine the minimal fractions of susceptible individuals, we set

ρ(M Jres (F )) = 1 ⇔ ρ(M Jres ) = 1 ⇔ R 0 = 1. (8)
Now, one can formulate the objective function to minimized as follows:

min 0≤F ≤1 J (F ) = p ℓ=1 S * H ℓ f ℓ , (9a)
subject to the constraint ρ(M Jres (F )) = 1

Existence of an unique solution of Eqs (9 can be easily derived by a similar argument as in [START_REF] Hill | The critical fraction for heterogeneous epidemic models[END_REF]. A powerful tool for numerical solving of Eqs ( 9) is the method of Lagrange multipliers [START_REF] Lasdon | Optimization theory for large systems[END_REF]. In the sequel we denote by F * = (f * 1 , . . . , f * p ) T the solution of Eq [START_REF] Van Den Driessche | Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission[END_REF].

B. Dynamics of system (1) with control over time

We recall that the minimal control is obtained by solving Eqs. [START_REF] Van Den Driessche | Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission[END_REF]. To test its effect on the dynamic of the system (1), we define a new force of infection, Φ H,i (t), with the value of the minimal control as follows:

Φ H,i (t) = (1 -f * i )Φ H,i (t) if i ∈ J res Φ H,i (t) otherwise. ( 10
)
for all i = 1, . . . , n. In the above equation, Φ H,i (t) represents the initial force of infection defined in Eq. ( 2), J res represents the set of reservoirs of infection defined by Eq (5), the control is only introduced within the reservoir of infection.

C. Analysis of Optimal control

We point out that the minimal control derived in the previous section is not an optimal control. The minimal control does not depend on the time, it is constant over time. An optimal control can be considered when we would like to apply a prevention depending of the time with a given final time T for disease eradication as well as budget constraint, this type of control was already investigated in [START_REF] Zorom | Optimal Control of a Spatio-Temporal Model for Malaria: Synergy Treatment and Prevention[END_REF]. In that case an optimal control problem can be formulated in the set of the identified reservoirs {1, . . . , p} ⊂ {1, . . . , n} with the following objective (cost) functional:

J(f ) = p i=1 T 0 I H,i (t) + R H,i (t) + a i 2 f 2 i (t) dt - p i=1 C i S H,i (T ) (11)
where f = (f 1 , f 2 , . . . , f p ). I H,i and R H,i represent the number of infectious and semi-immune in patch i respectively, p is the number of reservoirs patches, (np) represents the number of unreservoirs patches while n is the number of patches. The term ai 2 f 2 i (t) is the cost of prevention with a i > 0 are the weight factor in the cost of control. C i S H,i (T ) is the fitness of the susceptible at the end of the process as a result of the prevention efforts implements for the patch i = 1, . . . , p. Using a similar argument as in [START_REF] Zorom | Optimal Control of a Spatio-Temporal Model for Malaria: Synergy Treatment and Prevention[END_REF] and under suitable condition one can prove the existence of an optimal control.

D. Methodology for numerical implementation

Here, we present the steps for numerical implementation. Step 3: Compute M J h from in [3, Eq. ( 17)].

Step 4: Compute M Jres from Eq. (6).

Step 5: Solving Eq. ( 9) to find minimal control.

Step 6: Solving Eqs. (1) using the minimal control obtained in step 5 to represent the dynamic of malaria model.

IV. COSTS AND COST-EFFECTIVENESS OF MALARIA CONTROL FOR TWO STRATEGIES

Cost-effectiveness analysis compares the costs and health effects of an intervention to assess the extent to which it can be regarded as providing value for money [START_REF] Alphey | A Model Framework to Estimate Impact and Cost of Genetics-Based Sterile Insect Methods for Dengue Vector Control[END_REF], [START_REF] Rich | An integrated epidemiologicaleconomic analysis of foot and mouth disease : applications to the southern cone of South America[END_REF]. This informs decision-makers who have to determine where to allocate limited healthcare resources. It is performed in this section to assess the effectiveness of a control targeted on the minimal fraction obtained in previous section considering two strategies of control over a reference period T -T I , where T I is the initial time for which the control is introduced and T the final time.

Strategy 1 : Protecting the minimal fraction of susceptible individuals to eliminate malaria in the reservoir of infection over the period T -T I .

Strategy 2 : Protection of any fraction of susceptible individuals greater than the minimal fraction for eradicate malaria at the same over the period T -T I .

To achieve this purpose for our above strategies, we need to compare the differences between the cost and health outcome. This is done by calculating the incremental cost effectiveness ratio (ICER) defined as follows:

ICER = C * -C HE * -HE (12)
where HE * (resp. HE) denotes the health effect of strategy 1 (resp. strategy 2) and C * (resp. C) denotes the present value (at time t = 1) of costs for the whole project of strategy 1 (resp. strategy 2). A similar method is used in [START_REF] Okosun | Impact of optimal control on the treatment of HIV/AIDS and screening of unaware infectives[END_REF].

A. Method to calculate costs C (resp. C * ) for strategy 2 (resp. strategy 1) Let c t ℓ denotes financial cost of protecting one individual for one unit of time t (here one month) in patch ℓ. Because f l S * H l is the number of individuals to be protect in patch ℓ then c t ℓ f ℓ S * H ℓ is the total required financial cost to protect all individuals in patch ℓ = 1, . . . , p. It follows that the total cost allowed to strategy 2 (resp. strategy 1) is given by

C t = p l=1 c t l f l S * H l , and
C * t = p l=1 c t l f * l S * H l t = 1, . . . , T
When cost effectiveness ratios are reported with discounting of future costs and benefits due to the longer-term implementation time, if we denote by r s the social discount rate, we have

C * := T t=1 C * t (1 + r) t-1 and C := T t=1 C t (1 + r) t-1 ,

B. Method to calculate health effect of strategy 1 HE (resp. of strategy 2 HE

* ) HE = 1 T -T I T TI p l=1 S f H l (t)dt and HE * = 1 T -T I T TI p l=1 S f * H l (t)dt,
where S f H l (t) denote the number of susceptible at time t when Φ H,i (t) = Φ H,i and S f * H l (t) denote the number of susceptible at time t when

Φ H,i (t) = (1 -f * i )Φ H,i .

SIMULATION EXPERIMENTS

A. Parameters values

To test the method, we have assumed that the travel rates of humans depend on the distance between cities. We set

M S = M I = M R = 10 -3 n -1 • [|i -j|], 2 ≤ i,

B. Results and Discussion

The computation provides the basic reproductive number value: R 0 = 1.3239. That to say, without control, one has malaria that persists in the human population.

1) Reservoir of infection: Numerical implementation of the method shown in Appendix allowed to identify the set of reservoirs shown on Table II andFigure 2) Dynamics of system [START_REF] Alphey | A Model Framework to Estimate Impact and Cost of Genetics-Based Sterile Insect Methods for Dengue Vector Control[END_REF] with/ or without minimal control: Figure 2 show the endemicity of malaria without control. Indeed, Although others cities are not reservoirs of The comparison between ICER(1) and ICER(2) shows a cost saving of 7,2325 for strategy 1 over strategy 2. The ICER for strategy 1 indicates the strategy 2 is "strongly dominated". Strategy 2 is then more costly and less effective than strategy 1. With this result, we therefore conclude that strategy 1 is the less cost and most effective.

VI. CONCLUSION

In this article, we have extended the model derived in [START_REF] Arino | A metapopulation model for malaria with transmission-blocking partial immunity in hosts[END_REF] to give a numerical method to explain the steps for identifying reservoirs of malaria infection within n connected regions with the explicit movement of population. This method is intended to design an efficient computational tool.

In the one hand, we have shown how to determine the minimal percentage of susceptible individuals in the reservoirs that should be protected to eliminate malaria over time in the whole population. The minimal percentage quantifies the degree of malaria risk in the reservoirs areas. It is regulated by the mobility flux of human population between patches. Thus for the public health decision makers, when datasets on human migration flux, demographic and epidemic are known, this method can aid to estimate the minimal percentage of susceptible individuals to be protected. We have explained that the minimal percentage is not necessarily an optimal control but allows to reduce the cost of intervention. It is constant over time. An optimal control can be considered when we would like to apply a preventative depending of the time with a given final time T for disease eradication as well as a budget constraint. In that case we have shown how to formulate such a problem.

In the second hand, the costs and cost-effectiveness of malaria control interventions is performed considering two strategies of control: (i) protecting the minimal fraction of susceptible individual; (ii) protecting any fraction greater than the minimal fraction. Cost-effectiveness analysis shows that the less cost and more effective strategy is to protect the minimal fraction of susceptible individual in the reservoir of infection. Biologically relevant parameters have been estimated and used to fulfill numerical simulations of the model. These simulations was implemented through an example using Matlab (www.matlab.org). Simulation of the evolution over month of the density of human in infectious class (I H,i ) and semi-immune class (R H,i ), as well as the density of infectious mosquitoes (I V,i ) for 20 cities was shown on Figure 23. Together with the both Figures and the Cost-infectiveness results, we argue that Strategy (ii) is then more costly and less effective than strategy (i). With this result, we therefore conclude that strategy (i) is the less cost and most effective, namely the minimal fraction of population to vaccinate (or protect) to halt outbreak is efficient.

Our study will be useful for spatial vaccination programs in which optimization methods are needed to minimize the costs. Moreover thanks to this study, when the datasets on human migration flux, demographic and epidemic are known, one can estimate the minimal percentage of susceptible individuals to be protected. The method developed in this paper may be allow to the implementation of a software for monitoring spatio-temporal variations in malaria epidemic risk.

Step 1 : 2 :

 12 Compute principal next generator matrix K from [3, Theorem 3]. Step Compute next generator matrix extracted K from [3, Corollary 1].

1 .

 1 Cities number 2,3,5,6,8,10,12,13,15,16 and 20 represent the reservoirs of infection based on the value of the minimal fraction F * := (f * 1 , . . . , f * 20 ). Indeed, one considers that a city ℓ is a reservoir when f * ℓ = 0.

Fig. 1 .

 1 Fig. 1. This figure shows the cities which represent reservoirs of infection when we use the data in Table III and the algorithm presented in this paper. Cities number 2,3,5,6,8,10,12,13,15,16 and 20 represent the reservoirs of infection while cities 1,4,7,9,11,14,17,18,19 are not. Moreover this figure provides the minimal percentage 100F * , where F * := (f * 1 , . . . , f * 20 ) is shown on Table (II)) of human within each reservoir that should be protected to eliminate malaria in the whole of the population (human and mosquitoes).

[

Fig. 2 .

 2 Fig. 2. Simulation of the evolution over month of the density of human in infectious class (I H,i ) and semi-immune class (R H,i ), as well as the density of infectious mosquitoes (I V,i ) for 20 cities when we have no control measures. Initial condition: S H,i (0) = 500, i = 1, . . . , 20, I H,1 (0) = 10, and I H,i (0) = 0, i=2,. . . , 20; R H,i (0) = 0, i = 1, . . . , 20; S V,i (0) = 100, i = 1, . . . , 20; I V,i (0) = 0, i = 1, . . . , 20.

[

Fig. 3 .

 3 Fig. 3. Simulation of the evolution over month of the density of human in infectious class (I H,i ) and semi-immune class (R H,i ), as well as the density of infectious mosquitoes (I V,i ) for 20 cities when the minimal percentage of susceptible human shown in Table II are protected 100 months post-infection. Initial condition: S H,i (0) = 500, i = 1, . . . , 20, I H,1 (0) = 10, and I H,i (0) = 0, i=2,. . . , 20; R H,i (0) = 0, i = 1, . . . , 20; S V,i (0) = 100, i = 1, . . . , 20; I V,i (0) = 0, i = 1, . . . , 20.

  Table III summarizes the model parameters as well as their biological interpretation and Table I summarizes the state variables.

TABLE II THE

 II MINIMAL FRACTION F * AND THE MAXIMAL FRACTION F DEPENDING OF THE PARAMETERS VALUES OF a H AND a V .

	City	1	2	3	4	5	6	7
	a H	1.6	2	2.5	1.5	2.1	2.3	1.2
	a V	0.6	0.65	0.35	0.4	0.3	0.5	0.2
	F *	0	0.6	0.80	0	0.63	0.70	0
	F	0	1	1	0	1	1	0
	City	8	9	10	11	12	13	14
	a H	1.9	1.7	2.4	1.6	2	2.5	1.5
	a V	0.55	0.8	0.45	0.6	0.65	0.35	0.4
	F *	0.48	0	0.70	0	0.51	0.72	0
	F	1	0	1	0	1	1	0
	City	15	16	17	18	19	20
	a H	2.1	2.3	1.2	1.9	1.7	2.4
	a V	0.3	0.5	0.2	0.55	0.8	0.45
	F *	0.56	0.65	0	0	0	0.71
	F	1	1	0	0	0	1

TABLE III BASELINE

 III VALUES FOUND IN THE LITERATURE[START_REF] Arino | A metapopulation model for malaria with transmission-blocking partial immunity in hosts[END_REF],[START_REF] Ducrot | A mathematical model for malaria involving differential susceptibility, exposedness and infectivity of human host[END_REF],[START_REF] Chitnis | Bifurcation analysis of a mathematical model for malaria transmission[END_REF],[START_REF] Zongo | Modélisation mathématique de la dynamique de la transmission du paludisme[END_REF]. WE HAVE ASSUMED THAT THE PARAMETERS ARE IDENTICAL IN ALL THE PATCHES EXCEPTED a H,i THAT VARIES FROM A PATCH TO ANOTHER AND THE RATE OF TRAVEL m π ij THAT DEPENDS ON THE DISTANCE BETWEEN PATCHES.

	Symbol	Description	Values
	Λ H,i :	recruitment into the susceptible class	0.4
	α H,i :	rate of progression from the
		infectious to the semi-immune class	0.0035
	ρ H,i :	rate of recovery from being infectious	0.035
	β H,i :	rate of recovery from being semi-immune	5.5 × 10 -4
	γ H,i :	disease induced death rate	9 × 10 -5
	µ H,i :	natural death rate	5 × 10 -4
	Λ V,i :	recruitment into the susceptible class	500
	µ V,i :	natural death rate	0.04
	σ H i V i :	probability of transmission from an infect-
		ious human to a susceptible mosquito	0.48.
	σ H i V i :	probability of transmission from a semi-
		immune human to a susceptible mosquito	0.048
	σ V i H i :	probability of transmission from an infec-
		tious mosquito to a susceptible human	0.022
	a H,i :	maximum number of mosquito bites
		a human can receive per unit time	Table II
	a V,i :	number of times one mosquito would
		bite humans per unit time	Table II
	M π :	π = S, I, R rate of travel
		of humans from patch j to patch i	Eq. (13)