Development of Mathematical models predicting the density of vectors: Case of sandflies vectors of leishmaniasis

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1 Development of Mathematical models predicting the density of vectors: Case of sandflies vectors of leishmaniasis HABIBA MEJHED Département Génie Informatique, Ecole Nationale des sciences appliquées Université Cadi Ayyad, BP 575, Boulevard Abdelkarim Khattabi, Guéliz, Marrakech, Morcco SAMIA BOUSSA Laboratoire d Ecologie et Environnement, Faculté des Sciences Semlalia, Université Cadi Ayyad, Marrakech, Morocco & Laboratoire de Parasitologie, Faculté de Pharmacie, Université Louis Pasteur-Strasbourg I, B.P , 74 Route de Rhin, Illkirch, France.France., France. samiaboussaa@yahoo.fr NOUR EL HOUDA MEJHED Département Génie Informatique, Ecole Nationale des sciences appliquées Université Sidi Mohammed Ben Abdallah, Fès, Maroc noramej@hotmail.com Abstract: In order to reduce the transmission of leishmaniasis in Morocco and to control the vectors, we developed a simple mathematical model that can predict the vector density according to weather conditions and taking into account the global climate change. Key Words: Zoonotic cutaneous leishmaniasis, Anthroponotic cutaneous leishmaniasis, Visceral leishmaniasis, Mathematical model, Morocco 1 Introduction Leishmaniasis represents a complex of diseases with an important clinical and epidemiological diversity. Leishmaniasis complex (parasite, vector and reservoir) is highly affected by bioclimatic factors. In Morocco, leishmaniases remain a severe public health problem. Many foci were described in rural area of Ouarzazate [10], Essaouira [7], Azilal [7], Chichaoua [4] and Al Haouz [3, 8] but also in urban area of Taza [5] and Fès [9]. Marrakech is an interesting study site because it lies close to the focus of cutaneous leishmaniasis in the south of Morocco (Ouarzazate, Chichaoua and Al Haouz) and current studies classified the area of Marrakech as being at risk of cutaneous leishmaniasis [1, 2]. As the best choice of a vector-control strategy is dictated by sandfly ecology, we try to simplify this complex of diseases and quantify the climatic factors which can determine the repartition and activity of sandflies vectors in Marrakech city. According to l OMS [6], the activity of sand fly fauna is mainly affected by many climatic factors as temperature, humidity and win but also by season and according to sand fly species. In the aim to understand the interaction between environmental factors and phlebotomine vectors and to design the appropriate disease prevention and control strategies, we present a simple mathematical model predicting the density of vectors according to local climatic conditions and global climatic changes. We used datawarehouse and datamining to study three forms of leishmaniasis: - Zoonotic cutaneous leishmaniasis Vector: Phlebotomus papatasi Reservoir host: Meriones shawi grandis - Visceral leishmaniasis Vector: P. longicusis Reservoir host: dogs - Anthroponotic cutaneous leishmaniasis Vector: P. sergenti Reservoir host: Human ISSN: ISBN:

2 2. Mathematical model for the activity of sandflies vectors of cutaneous leishmaniasis: In the epidemiology, as in other domains we can distinguish in particular, two major types of models: Deterministic models: the initial conditions that determine clearly the future of the system, these models often studying the phenomena at the macroscopic level. In the epidemiological for example, we modelize the evolution of the status of the populations. The stochastic models: They are based on the probability and studying more the phenomena at the microscopic level. In the epidemiology, the modelling focuses on the individual. To fight against the spread of the cutanius or visceral leishmaniasis, we chose to define a mathematical model that combines deterministic SIR model [11], [12] with the existence of the vectors that transmit leishmaniasis. Our model suggests that is not necessary to eliminate completely the vectors but the epidemic could be stopped if the vector population were reduced. 2.1 Model of Anthroponotic cutaneous leishmaniasis We schematize the model of anthroponotic cutaneous leishmaniasis as showing. M : sandfly dansity per humans r1 : rate of human recovery r2: probability of infectious sandfly become not infectious (to rid of parasite) μ1 : rate of mortality of humans μ2 : rate of mortality of sandflies According to model of Ross, we get the following equations: To simplify our equation, we put : With: Hence : V : population of sandflies H : population of humans For leishmaniasis, we have : Set: r=r1 et u=u2, the previous equation is of the form: Set : a : number of biting per sandfly at time t b2 : rate of infectious biting which can give infection for human b1 : rate of biting per not infectious sandfly which can give infection for vector ISSN: ISBN:

3 with : To stop leishmaniasis transmission, we look for : The system becomes : Hence : The presentation of the system as a matrix gives the following equation: 2.2. Model of zoonotic cutaneous leishmaniasis and visceral leishmaniasis: We have the equilibrium system if the infectious were eliminated, therefore: So : Set : The result is : with : a : number of biting per sandfly at time t b3 : rate of biting which can give infection for reservoir b2 : rate of infectious biting which can give infection for human b1 : rate of biting per not infectious sandfly which can give infection for vector V : sandfly population H : human population R : reservoir population (Canines or Rodent) M=V/H: sandfly dansity per humans MR=V/R : sandfly density per reservoir (Canines or Rodent) ISSN: ISBN:

r1 : rate of human recovery r2: probability of infectious sandfly become not infectious (to rid of parasite) r3 : probability of reservoir become not infectious μ1 : rate of mortality of humans μ2 :

4 r1 : rate of human recovery r2: probability of infectious sandfly become not infectious (to rid of parasite) r3 : probability of reservoir become not infectious μ1 : rate of mortality of humans μ2 : rate of mortality of sandflies μ3 : rate of mortality of reservoir Following the same reasoning, we get the mathematical model: We get the following results : We apply the same method for get the factor R0 following: The vector seasonality can be explain by the variation of factor a (number of biting per sandfly) at time t. According to factor R0, we have : if : leishmaniasis disappear if leishmaniasis propagate 3.2. Visceral leishmaniasis (P. longicusis) With the following data: 3. Simulation of factor R0 (t) by Scilab software: In this chapter, we present the results of numeric simulation by Scilab software ( 3.1 Zoonotic cutaneous leishmaniasis (P. papatasi) With the following data: ISSN: ISBN:

This site was selected considering the presence, in this site, of all the sand fly species inventoried in the urban area of Marrakech [2].

5 We get the following results: We get the following results : 3.3. Anthroponotic cutaneous leishmaniasis (P. sergenti) With the following data: Our investigations were conducted in Akioud, an urban district in Marrakech city (31 36 N, 8 02 W, 471m a.s.l.) during the 1-year-study (from October to September). This site was selected considering the presence, in this site, of all the sand fly species inventoried in the urban area of Marrakech [2]. According to the correlation between the weekly density of the three vectors (P. papatasi, P. sergenti and P. longicuspis) and the factor R0 we can prevent risk of leishmaniasis in this area. - For P. papatasi population, R0 factor is superior to 01 during two periods of year: November and May-June-July, which correspond to the periods of risk of zoonotic ISSN: ISBN:

6 cutanous leishmaniais caused by L. major in this area. - For P. sergenti population, R0 is superior to 1 during the period July August and inferior to 1 in the rest of the year. So, this period correspond to phase of risk of anthroponotic cutanous leishmaniais caused by L. tropica in this area. - For P. longicuspis, the results have showed two periods of risk of visceral leishmaniasis in this area: October and May- June. threshold conditions for infection persistence.acta Tropica 92, (2004) 4 Conclusion The purpose of this work is the modeling of leishmaniasis in its various forms, as well as the exploitation of actual data collected to assist in the decision on the issue of the fight against leishmaniasis in Morocco. Relying on the principles of epidemiology modeling, we developed a mathematical model for each type of leishmaniasis, these models can be used to analyze the spread of leishmaniasis in any areas, particularly in Marrakech city. Indeed, it is spatio-temporal models witch permit to follow and control the evolution of leishmaniasis in terms of time and the region and to find the appropriate threshold of the population of sandflies for stopping the propagation of the leishmaniasis. References: [1] Boussaa et al Acta Trop., 95, [2] Boussaa et al Ann. Trop. Med. Parasitol. 101: [3] Boussaa et al Ann. Trop. Med. Parasitol. 103, [4]Guernaoui et al J. Med. Entomol., 42, [5]Guessous-Idrissi et al Trans. R. Soc. Trop. Med. Hyg., 91, [6] OMS Lutte contre les leishmanioses. Série de Rapports Techniques. [7] Pratlong et al Ann. Parasitol. Hum. Comp., 66, [8] Ramaoui et al Parasitology research. [9] Rhajaoui et al Trans. R. Soc. Trop. Med. Hyg., 98, [10] Rioux et al Leishmania. Taxonomie et Phylogenèse. ed. Rioux, J. A. pp [11] Chaves, L.F., Hernandez, M.J.: Mathematical modelling of American Cutaneous Leishmaniasis: incidental hosts and ISSN: ISBN:

Original Article. Epidemiological Aspects of Cutaneous Leishmaniasis in Yazd Province within

Original Article. Epidemiological Aspects of Cutaneous Leishmaniasis in Yazd Province within Journal of Community Health Research. 2016;5(2): 131-139. Original Article Epidemiological Aspects of Cutaneous Leishmaniasis in Yazd Province within 2004-2013 1. 2. 3. 4. 5. 6. Hadis Barati 1, Mohammad