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Journal of Biological Systems, Vol. 16, No. 4 (2008) 565–577 c World Scientiﬁc Publishing Company

ASSESSING THE SUITABILITY OF STERILE INSECT TECHNIQUE APPLIED TO AEDES AEGYPTI

CLAUDIA PIO FERREIRA IBB — Departamento de Bioestat´ıstica — UNESP Caixa Postal 510, CEP: 18618-000, Botucatu, SP, Brasil [email protected] HYUN MO YANG IMECC — Departamento de Matem´ atica Aplicada Caixa Postal 6065, CEP: 13083-859, Campinas, SP, Brasil [email protected] LOURDES ESTEVA Departamento de Matem´ aticas, Faculdad de Ciencias UNAM 04510, Mexico, D.F., Mexico [email protected] Received 29 September 2007 Revised 31 July 2008 The eﬃcacy of biological control of Aedes aegypti mosquitoes using Sterile Insect Technique (SIT) is analyzed. This approach has shown to be very eﬃcient on agricultural plagues and has become an alternative control strategy to the usual technique of insecticide application, which promotes resistance against chemical controls and is harmful to other species that live in the same mosquito habitat. By using a discrete cellular automata approach we have shown that in the case of Aedes aegypti, the spatially heterogeneous distribution of oviposition containers and the mosquito behavior, especially with respect to mating, make the application of STI diﬃcult or impracticable. Keywords: Cellular Automata; Biological Control; Aedes aegypti; STI Technique.

1. Introduction The use of genetic control of pest insects population initiated in the 1930s–59s with the works of Knipling, Bushland, Muller, and Serebrovsky.1–4 Since then, the developed techniques have been based on the introduction of genetic mutations in the native population during mating. In general, it aims at suppressing or displacing the native population in favor of one species that is resistant to the agent causing the disease. The Sterile Insect Technique (SIT) is based on the release of a large number of sterile males over the area aﬀected by plague. These sterile males compete with the native ones to mate with wild females, which mate with them and produce nonviable eggs. This technique is species-speciﬁc, environmentally non-polluting 565

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and consists in three stages: mass rearing, sexual sterilization usually with gamma radiation and release of the sterile males over the target area. In general, the viability of the biological control depends on several factors, such as: complete sterilization of the insects; massive creation in laboratory must be easy and relatively cheap; elimination of the females during the production process; sterile males have to compete successfully with the native ones for the females; mating number of females must also be small; the radiated insects should not cause damage or transmit pathogenic agents to man, animals or plants and dispersion in the ﬁeld must be easy. Moreover, since the cost of the technique is proportional to the size of wild populations, it should be combined with the usual techniques of insect control, or used at the time of the year when the pest population is at its lowest. Also, the choice of the target area size is an important factor due to the possibility of recolonization from adjacent infested untreated areas. Therefore, it is essential to know the biology, ecology and behavior of the target species.5 In 1955 the eradication of the screwworm ﬂy, the second most important plague caused by arthropods, was carried out for the ﬁrst time on the Caribbean island of Curacao.6 The entire island was treated weekly with about 800 sterile ﬂies/mile2 and eradication was reached after seven weeks. Since then, this technique has been studied and applied in the control of insect pests, mainly agricultural. Because vector control is one of the few proved ways to reduce transmission of many vectorborne diseases, there is an increasing eﬀort to use this technique on mosquito populations.7 In particular, mathematical models can be a useful tool to create and test different scenarios concerning biological problems. Diﬀerent, continuous or discrete approaches can be done and the choice between them depends on the speciﬁc problem. For example, there has recently been a growing interest in spatial issues in ecology, such as how locality of competition and the range of dispersal play a role in ecosystem dynamics. These models are usually investigated via simulation or else they are simpliﬁed by essentially throwing away the whole detailed spatial structure, resulting in the so-called metapopulations or mean-ﬁeld models. However, when the local interactions are important for the dynamics of the process, the cellular automata models (CA) are a very useful and simple tool. In summary, a CA is a mathematical model with discrete time, space and variables, composed of lots of identical, simple components arrangement on a lattice. These components interacted with each other and change states according to local rules that specify how the automaton develops in time. The emergence of a complex behavior from simple local rules is a characteristic of the CA models.8 First introduced by John von Neumann in the early 1950s to act as simple models of biological self-reproductions, CA has been used for model plant growth, propagation of infection disease, social dynamics, forest ﬁres and also as a discrete version of partial diﬀerential equations in one or more variables.9

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In this work, a cellular automata model to analyze the biological control eﬀorts used on the mosquito Aedes aegypti was developed. This approach captures essential features of mosquito Aedes, such as spatial heterogeneity, spreading and colonization.10 The main objective of this paper is to compare the eﬃciency of biological control when we consider two situations: homogeneous and heterogeneous distributions of mosquito populations associated with the spatial distribution of oviposition containers. Also, the control eﬃcacy related to the time interval of the SIT technique application as well as the proportion between irradiated and wild males have been analyzed. This paper is organized as follows: Sec. 2 presents the cellular automata model used to describe the population dynamics of native and irradiated mosquitoes; Sec. 3 shows the results related to biological control eﬃciency in a homogeneous and heterogeneous spatial distribution of mosquito population; ﬁnally, Sec. 4 presents the conclusion.

2. The Cellular Automata Model The major diﬃculty associated with the mosquito Aedes control is associated with the oviposition containers. The doctrine of container homogeneity related to size and spatial distribution results in non-prioritized mosquito controls and a basic lack of understanding of the causes of the problem.11 In general, the wild male mosquito emerges before the female and the meeting between them, which results in fertile female mosquito, takes place in the neighborhood of the containers. This mosquito behavior as well as the spatial distribution of oviposition containers are easily reproduced using the cellular automata approach. Also, colonization, persistence and distribution of the mosquito population in controlled regions can be reproduced using this formalism.10 To model the space localization of the containers together with the mosquitoes behavior, we developed a stochastic two-dimensional (L × L) cellular automata. With each site we associated a ﬁve-state automaton that represents the non-fertile female (not mating yet), fertile female (can produce viable eggs), wild male, sterile male and empty state. The empty sites represent females that had mated with sterile males and cannot produce viable eggs, therefore do not contribute to the overall dynamics.12 One time step corresponds to the parallel update of the entire lattice and will correspond to one day. Between time steps, diﬀusion is implemented using Margolus neighborhood.8 Brieﬂy, the Margolus neighborhood is a cell-interconnection scheme constructed following the steps described below: 1. The array of cells is partitioned into a collection of ﬁnite, disjoint and uniformly arranged pieces called blocks. 2. A block rule is given that looks at the contents of a block and updates the whole block (rather than a single cell as in a ordinary CA). The same rule is applied

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to every block. Blocks do not overlap and no information is exchanged between adjacent blocks. 3. The partition is changed from one step to the next, so as to have some overlap between blocks used at one step and those used at the next one. Therefore, the temporal evolution of mosquito populations is given by the following local rules applied in a Moore neighborhood, which comprises the eight cells surrounding a central cell on a two-dimensional square lattice: 1. Sites occupied by non-fertile female (I) can become empty with probability µI + γ1 β1 , because of mortality or mating with sterile males; or can become occupied by fertile females with probability γ2 β2 , where γ1 and γ2 are the numbers of, respectively, sterile and wild males in the neighborhood; β1 and β2 are meeting probabilities; 2. Sites occupied by female mosquitoes (F), wild (M) or sterile males (R) can become empty with probability µ (mortality); 3. Empty sites (V) can become occupied with probability γ by new mosquitoes that emerge from the oviposition containers (aquatic emergence rate). The proportion between female and male emergence is given by the parameter r. Emergence occurs in the neighborhood of the oviposition containers, deﬁned by the ratio D with a center in the position of the oviposition container; 4. Mosquito oviposition occurs inside oviposition containers and is proportional to Cγ3 φ, where C, γ3 and φ are, respectively, the carrying capacity of the container, the number of fertile females in the oviposition neighborhood and the oviposition rate; and 5. There are N oviposition containers distributed on the lattice, and in each of them a fraction of the aquatic population dies with probability µA times the number of aquatic individuals. The transition rules among the automaton states deﬁned above were constructed based on the compartmental model described in Esteva.12 To compare the control eﬃcacy in homogeneous and heterogeneous spatial distribution of mosquito populations, all results are obtained using the two distinct lattices shown in Fig. 1.

3. Results and Discussion Oviposition containers and the wild population are distributed on the lattice at the beginning of the simulation (in our examples 20% of the lattice is occupied by native female and male mosquitoes distributed randomly). All simulations were performed taking into account a square toroidal lattice with a linear dimension L = 250. Meanwhile, the oviposition containers were always distributed inside a square region smaller that the previous one (L = 190), in such a way that the same results are obtained for an open square lattice.10

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(b)

Fig. 1. Two diﬀerent spatial distributions of oviposition containers used in simulations: (a) homogeneous and (b) heterogeneous. In the both cases, the number of recipients is N = 324 and the lattice linear size L = 250.

Initially, the simulation is carried on until the native population becomes stationery, which is deﬁned as in Caswell and Etter,13 i.e., when the last-squares regression line through the values of mosquito population for the last 100 time steps had a slope of less than 0.001, the simulation was considered close enough to equilibrium and then the suppression process is started by distributing randomly the sterile male mosquitoes in the empty sites of the lattice during a time interval, ∆, in a proportion of sterile mosquitoes for each native male mosquito. The results are shown for D = 15, C = 30 and 2 diﬀusion steps. The other parameters are: φ = 0.1, µA = 0.015 (66 days), µI = 0.03 (33 days), µ = 0.04 (25 days), γ = 0.1 (10 days). Mean values were obtained from 50 simulations. We assume that r = 0.5 and β1 = β2 = 0.1077, therefore, the maximum probability of oviposition is 9 × 0.1077 ∼ 1 if all cells in a Moore neighborhood of the oviposition containers are occupied, and that sterile and wild males are equally ﬁt. For a detailed analysis of the parameters space see Ferreira.10 We remark that for this set of parameters the basic oﬀspring of natural mosquitoes (average number of viable female descendants that an adult female mosquito produces during its fertile period10 ) is bigger than one, and therefore the continuous model given in Esteva12 establishes the persistence of the mosquito population, regardless of the carrying capacity (related to the number and size of the containers) as long as it is diﬀerent from zero. Figure 2 shows the number of non-fertile female population (I) in the steady state when we consider random and uniform distributions of the oviposition containers. We notice that for 110 < N < 250, persistence of mosquitoes is obtained

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3000

2000 I

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0

0

100

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300

400

500

N

Fig. 2. Steady state of the non-fertile female population, I, as a function of the number of oviposition containers, N, without biological control. The diﬀerent curves are related to uniform (•) and random (◦) distribution of oviposition containers.

only for the random distribution of the containers; for 250 < N < 350 persistence is obtained for uniform and random distribution, but the number of mosquitoes is much smaller for the uniform distribution. Finally, for N > 350 the results obtained with the two distributions are practically the same. Therefore, for the cellular automata model, there exists a critical point depending on the environmental carrying capacity, N × C, where N is the number of containers and C, the size of each container, below which population becomes extinct. This critical point depends on the spatial distribution of the containers (Fig. 1), mosquito diﬀusion speed, and the size of the neighborhood containers. Then, for a ﬁxed C, there is a threshold number Nc such that if N < Nc the mosquito population becomes extinct. For a uniform distribution we obtained Nc ≈ 250, whereas for a random distribution we obtained Nc ≈ 110. Therefore, a random distribution facilitates dispersion and colonization of mosquitoes, since the threshold number of containers for persistence, Nc is smaller than the one for uniform distribution. Furthermore, for Nc large the cellular automata model reproduces the qualitative behavior obtained by the continuous model.12 In order to understand why a random distribution of oviposition containers promotes mosquito persistence, in Fig. 3 we display the average distance, Φ, and the minimum distance, η, between successive oviposition containers. We can see that Φ is almost the same for both distributions, but for all values of N the random distribution generates, on average, space distributions with a lower minimum distance between the containers, facilitating colonization and persistence of mosquito populations. In fact, the threshold obtained for both spatial distributions of oviposition containers (Fig. 2) are related to the percolation properties. For N > Nc we have a cluster of female mosquitoes extending from one side of the lattice to the other,

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Φ

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η

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0 0

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110

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330 N

440

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Fig. 3. Distance average, Φ, and the minor distance, η, between oviposition containers as a function of the number of oviposition containers, N. The diﬀerent curves are related to uniform (•) and random (◦) distribution of oviposition containers.

whereas for N < Nc no such cluster exists. However, determining critical behavior from steady-state simulations is often very diﬃcult due to large ﬂuctuations, critical slowing down, ﬁnite-size eﬀects and diﬃculties in exactly locating the critical point. Therefore, these phase transitions will be studied in a future paper using time-dependent simulations.14 Figure 4 shows the time evolution of the mosquito population [fertile female (F), wild male (M) and sterile male (R)] for a uniform distribution of N = 326 oviposition containers. The release of the sterile mosquitoes occurs at t ≈ 1000

F, M, R

3000

1500

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Fig. 4. Time evolution of fertile female, F (hatch line), wild males, M (dot line), and sterile males, R (continuous line), for a uniform distribution of oviposition containers.

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during a period of time ∆ = 20 days, and it is proportional to the number of native mosquitoes = I/M = 7. The native population is reduced during the time interval ∆, and then increases when control application is discontinued. The native population after the SIT control is smaller than the population at the beginning of the control, and its ﬁnal size depends on the value. This fact is associated with a local extinction of the population in regions of the lattice that were colonized before control, (bear in mind that the initial mosquito population was distributed randomly on the lattice, and new colonizations started from oviposition containers). In general, the number of positive containers which have immature forms of mosquito is lower than the number of containers distributed on the lattice. For a random distribution of containers, the recovery dynamics after control is faster and therefore local extinction is less probable (results not shown). To measure the eﬃciency of the SIT technique, we deﬁne J, the accumulated T index control parameter, as J = (1 − Ac /A0 ) × 100, where A• = 0 F (t)dt is the the accumulated number of female mosquitoes with (Ac ) and without (A0 ) controlling mechanisms applied on the mosquito population. Observe that Ac − A0 measures the accumulated reduction in the number of female mosquitoes by controlling mechanism in the range from 0 to T, therefore, J is the percentage of reduction of female mosquitoes over the interval [0, T ].15 A similar index was used to measure vaccine eﬃcacy on malaria control programs.16 Figure 5 shows control eﬃcacy measurement as a function of the time interval when sterile mosquitoes are released, ∆, for a uniform distribution of N = 326 oviposition containers. The proportion of sterile and wild male mosquito is ﬁxed in = 7. According to the above deﬁnition of J, note that it is not allowed to reach the value of 100%, even when female mosquitoes are eliminated. For instance, when

100

80

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J 40

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0

10

20

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Fig. 5. Control eﬃcacy, J, measures as a function of sterile mosquito time release, ∆, for a uniform distribution of oviposition containers. The proportion of sterile and wild male mosquito is ﬁxed on = 7.

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60 J

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ε

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Fig. 6. Control eﬃcacy, J, measures as a function of the proportion of sterile and wild male mosquito, , for a uniform distribution of oviposition containers. Sterile mosquito time release is ﬁxed on ∆ = 20.

the release of sterile mosquitoes occurs during a period of time larger than 40 days, or ∆ ≥ 40, the mosquito population is extinguished, but the control index assumes J ≈ 74%. The corresponding elimination index for a random containers distribution is attained when ∆ ≥ 400, but mosquito extinction is not observed. Figure 6 shows control eﬃcacy measurement as a function of the proportion between sterile and wild male mosquitoes, , for a uniform distribution of oviposition containers. The period of time that sterile mosquitoes are released is ﬁxed in ∆ = 20, and all other parameters have the values deﬁned previously. At = 15 the mosquito population is extinguished. Notwithstanding that, for a random distribution of oviposition containers and = 30, the eﬃcacy observed was below 5%, and mosquito extinction is never observed. Comparing Figs. 5 and 6 with respect to tangent slopes in each point, we notice that for SIT control, increasing the ratio between wild and native males was more eﬀective than increasing the time for releasing sterile mosquitoes. Furthermore, the two graphs increase monotonically until they reach a constant value near 80%, which represents mosquito extinction. In order to increase control eﬃcacy for a random distribution of oviposition containers, J was measured for a set of parameters above the critical value of the carrying capacity given by N = 300. Also, the proportion of sterile and wild males and the time for releasing sterile males were changed for these new simulations. In this case, mosquito control is easier to obtain than for the previous one described above. Figure 7 shows control measures eﬃcacy as a function of sterile mosquito time release, ∆, for a random distribution of oviposition containers. The proportion between sterile and wild male mosquitoes is ﬁxed on = 7. At ∆ = 200, the index J is almost 74%, but extinction is not observed yet.

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∆

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Fig. 7. Control eﬃcacy, J, measures as a function of sterile mosquito time release, ∆, for a random distribution of oviposition containers. The proportion of sterile and wild male mosquitoes is ﬁxed on = 7.

100

80

60 J

40

20

0 0

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10

ε

15

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25

Fig. 8. Control eﬃcacy, J, measures as a function of the proportion of sterile and wild male mosquito, , for a random distribution of oviposition containers. The sterile mosquito time release is ﬁxed on ∆ = 100.

Figure 8 shows control eﬃcacy measurement as a function of the proportion of sterile and wild male mosquitoes, , for a random distribution of oviposition containers (N = 300). The sterile mosquito time release is ﬁxed on ∆ = 100. At = 25 the index J is almost 62%, but extinction has not been observed yet. The continuous compartmental model described in Esteva12 shows that in the absence of sterile insects the condition for the existence of natural insects is R > 1, where the parameter R is the basic oﬀspring number. However, when sterile insects

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are released, the threshold values are situated at R = R∗ (> 1), indicating that one of the conditions for the existence of natural insects is furnished by increasing R. This is due to the decrease in mating fertilized females with sterile male insects, resulting in diminishing net production of oﬀspring. This value depends on the parameter S that measures the number of mated but not fertilized female insects in comparison with the fertilized ones, which depends on the proportion of sterile insects that are sprayed in adequate places and on the lost ﬁtness associated with the sterilization process. If S is suﬃciently high, the next generation of wild insects will be smaller than the present one since a proportion of eggs will not hatch. Therefore, if sterile male insects are sprayed for a long period of time, natural insect populations will be driven to zero. In particular, the trivial solution (mosquito extinction) is always stable and if R > 1 and S > S c the non-trivial equilibrium coexist with the trivial one. In the region of coexistence, numerical simulations indicate that SIT would have a low chance of success. The results shown in Figs. 5, 6, 7 and 8 for the cellular automata model agree with those provided by the continuous compartmental model.12 The two models show that control eﬃcacy is reduced by the non-homogeneous distribution of breeding sites but control eﬃcacy is overestimated by continuous compartmental modeling. In this case, the increase of S can be oﬀset by increasing the number of sterile or applying alternative controls (like larvacide and insecticide) together with SIT to decrease the number of viable female mosquitoes. For the automata cellular model, extinction is not observed for a spatially heterogeneous distribution of mosquitoes, even in the best situation. 4. Conclusion The automata cellular model proposed here was able to reproduce aspects of Aedes mosquito behavior, such as spatial heterogeneity, infestation and persistence. As in other CA models, such a success shows the importance of local interactions for the mosquito population dynamics. The eﬃcacy of SIT control was measured for two distributions of oviposition containers: homogeneous and heterogeneous, which reproduce diﬀerent spatial distributions of mosquitoes. The results show that even in the best situation, when sterile males have the same ﬁtness as the wild mosquito and the number of mosquito is not large, spatially heterogeneous distribution of mosquitoes makes the use of SIT technique diﬃcult or impracticable. In fact, this result is correlated with the observation that non-homogeneous spatial distribution on breeding sites facilitates colonization and persistence of Aedes aegypti. This result is not accessed by the compartmental model described in Esteva.12 The model proposed here predicts that for a spatially uniform distribution of breeding sites, and consequently quasi-homogeneous distribution of female mosquitoes, the SIT technique is suitable to eliminate wild mosquito populations. This result can be borrowed to explain the success in controlling agricultural plagues or screwworm ﬂy by SIT technique: reproduction does not depend on the spatial

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distribution of breeding sites and, naturally, the target area is also well delimited. This technique is also suitable for other mosquitoes which have dispersion, sexual behavior and adult habitat preferences diﬀerent from aedes mosquito, such as Anopheles, the malaria vector. Furthermore, as cited before, the threshold obtained for the two spatial distribution of oviposition containers (Fig. 2) are related to the percolation properties. Therefore, a great amount of interesting theoretical work can be done, such as cluster analysis and determination of the transition universality class.17

Acknowledgments The work of C.P.F. is supported by Funda¸c˜ao de Amparo a Pesquisa do Estado de S˜ ao Paulo (FAPESP), Project No. 05/51671-6 (fellowship awarded by FAPESP). H.M.Y. is support in part by Conselho Nacional de Desenvolvimento Cient´ıﬁco e Tecnol´ogico (CNPq) and FAPESP, Project No. 04/07075-7. The third author of this paper, Lourdes Esteva was supported by grant IN108607-03 of PAPIIT UNAM.

References 1. Knipling EF, Sterile-male method of population control, Science 130(3380):415–420, 1959. 2. Bushland RC, Hopkings DE, Sterilization of screwworm ﬂies with X-rays and gamma rays, J Econ Entomol 55(5):725–731, 1951. 3. Muller HJ, Radiation damage to the genetic material. Part II. Eﬀects manifested mainly in the exposed individuals, Am Sci 38:399–425, 1950. 4. Serebrovskii AS, On the possibility of a new method for the control of insect pests, Zollogicheskii Zhurnal 19:618–630, 1940. 5. Coleman PG, Alphey L, Editorial: genetic control of vector population: an imminent prospect, Trop Med Int Health 9(4):433–437, 2004. 6. Baumhover AH, A personal account of developing the sterile insect technique to eradicate the screwworm from curacao, Florida, and the Southeastern United States, Florida Entomol 85(4):666–673, 2002. 7. Benedict MQ, Robinson AS, The ﬁrst release of transgenic mosquitoes: an argument for the sterile insect technique, Trend Parasitol 19(8):349–355, 2003. 8. Toﬀoli T, Margolus N, Cellular Automata Machines, The MIT Press, Cambridge, Massachusetts, London, England, 1987. 9. Ilachinski A, Cellular Automata, World Scientiﬁc Publishing Co. Pte. Ltd, 2001. 10. Ferreira CP, Pulino P, Yang HM, Takahashi LC, Controlling dispersal dynamics of Aedes aegypti, Math Popul Stud 13(4):215–236, 2006. 11. Kay B, Nam V, New strategies against Aedes aegypti in Vietnam, Lancet 356:613–617, 2005. 12. Esteva L, Yang YM, Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique, Math Biosci 198:132–147, 2005. 13. Caswell H, Etter RJ, Ecological interactions in patchy environments: from patch occupancy models to cellular automata, in Powell T, Levin SA, Steele J (eds.), Patch Dynamics, Springer-Verlag, New York, pp. 93–109, 1993.

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14. Ferreira CP, Fontanari JF, Nonequilibrium phase transitions in a model for the origin of life, Phys Rev E 65(2):art. no. 021902 Part 1 Feb., 2002. 15. Yang HM, Ferreira CP, Assessing the eﬀects of vector control on dengue transmission, Appl Math Comput 198:401–413, 2008. 16. Dietz K, Raddatz G, Molineaux L, Mathematical model of the ﬁrst wave of Plasmodium falciparum asexual parasitemia in non-immune and vaccinated individuals, Am J Trop Med Hyg 75(Suppl 2):46–55, 2006. 17. Stauﬀer D, Ahorony A, Introduction to Percolation Theory, Taylor & Francis Ltd, London, 1994.

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Journal of Biological Systems, Vol. 16, No. 4 (2008) 565–577 c World Scientiﬁc Publishing Company

ASSESSING THE SUITABILITY OF STERILE INSECT TECHNIQUE APPLIED TO AEDES AEGYPTI

CLAUDIA PIO FERREIRA IBB — Departamento de Bioestat´ıstica — UNESP Caixa Postal 510, CEP: 18618-000, Botucatu, SP, Brasil [email protected] HYUN MO YANG IMECC — Departamento de Matem´ atica Aplicada Caixa Postal 6065, CEP: 13083-859, Campinas, SP, Brasil [email protected] LOURDES ESTEVA Departamento de Matem´ aticas, Faculdad de Ciencias UNAM 04510, Mexico, D.F., Mexico [email protected] Received 29 September 2007 Revised 31 July 2008 The eﬃcacy of biological control of Aedes aegypti mosquitoes using Sterile Insect Technique (SIT) is analyzed. This approach has shown to be very eﬃcient on agricultural plagues and has become an alternative control strategy to the usual technique of insecticide application, which promotes resistance against chemical controls and is harmful to other species that live in the same mosquito habitat. By using a discrete cellular automata approach we have shown that in the case of Aedes aegypti, the spatially heterogeneous distribution of oviposition containers and the mosquito behavior, especially with respect to mating, make the application of STI diﬃcult or impracticable. Keywords: Cellular Automata; Biological Control; Aedes aegypti; STI Technique.

1. Introduction The use of genetic control of pest insects population initiated in the 1930s–59s with the works of Knipling, Bushland, Muller, and Serebrovsky.1–4 Since then, the developed techniques have been based on the introduction of genetic mutations in the native population during mating. In general, it aims at suppressing or displacing the native population in favor of one species that is resistant to the agent causing the disease. The Sterile Insect Technique (SIT) is based on the release of a large number of sterile males over the area aﬀected by plague. These sterile males compete with the native ones to mate with wild females, which mate with them and produce nonviable eggs. This technique is species-speciﬁc, environmentally non-polluting 565

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and consists in three stages: mass rearing, sexual sterilization usually with gamma radiation and release of the sterile males over the target area. In general, the viability of the biological control depends on several factors, such as: complete sterilization of the insects; massive creation in laboratory must be easy and relatively cheap; elimination of the females during the production process; sterile males have to compete successfully with the native ones for the females; mating number of females must also be small; the radiated insects should not cause damage or transmit pathogenic agents to man, animals or plants and dispersion in the ﬁeld must be easy. Moreover, since the cost of the technique is proportional to the size of wild populations, it should be combined with the usual techniques of insect control, or used at the time of the year when the pest population is at its lowest. Also, the choice of the target area size is an important factor due to the possibility of recolonization from adjacent infested untreated areas. Therefore, it is essential to know the biology, ecology and behavior of the target species.5 In 1955 the eradication of the screwworm ﬂy, the second most important plague caused by arthropods, was carried out for the ﬁrst time on the Caribbean island of Curacao.6 The entire island was treated weekly with about 800 sterile ﬂies/mile2 and eradication was reached after seven weeks. Since then, this technique has been studied and applied in the control of insect pests, mainly agricultural. Because vector control is one of the few proved ways to reduce transmission of many vectorborne diseases, there is an increasing eﬀort to use this technique on mosquito populations.7 In particular, mathematical models can be a useful tool to create and test different scenarios concerning biological problems. Diﬀerent, continuous or discrete approaches can be done and the choice between them depends on the speciﬁc problem. For example, there has recently been a growing interest in spatial issues in ecology, such as how locality of competition and the range of dispersal play a role in ecosystem dynamics. These models are usually investigated via simulation or else they are simpliﬁed by essentially throwing away the whole detailed spatial structure, resulting in the so-called metapopulations or mean-ﬁeld models. However, when the local interactions are important for the dynamics of the process, the cellular automata models (CA) are a very useful and simple tool. In summary, a CA is a mathematical model with discrete time, space and variables, composed of lots of identical, simple components arrangement on a lattice. These components interacted with each other and change states according to local rules that specify how the automaton develops in time. The emergence of a complex behavior from simple local rules is a characteristic of the CA models.8 First introduced by John von Neumann in the early 1950s to act as simple models of biological self-reproductions, CA has been used for model plant growth, propagation of infection disease, social dynamics, forest ﬁres and also as a discrete version of partial diﬀerential equations in one or more variables.9

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In this work, a cellular automata model to analyze the biological control eﬀorts used on the mosquito Aedes aegypti was developed. This approach captures essential features of mosquito Aedes, such as spatial heterogeneity, spreading and colonization.10 The main objective of this paper is to compare the eﬃciency of biological control when we consider two situations: homogeneous and heterogeneous distributions of mosquito populations associated with the spatial distribution of oviposition containers. Also, the control eﬃcacy related to the time interval of the SIT technique application as well as the proportion between irradiated and wild males have been analyzed. This paper is organized as follows: Sec. 2 presents the cellular automata model used to describe the population dynamics of native and irradiated mosquitoes; Sec. 3 shows the results related to biological control eﬃciency in a homogeneous and heterogeneous spatial distribution of mosquito population; ﬁnally, Sec. 4 presents the conclusion.

2. The Cellular Automata Model The major diﬃculty associated with the mosquito Aedes control is associated with the oviposition containers. The doctrine of container homogeneity related to size and spatial distribution results in non-prioritized mosquito controls and a basic lack of understanding of the causes of the problem.11 In general, the wild male mosquito emerges before the female and the meeting between them, which results in fertile female mosquito, takes place in the neighborhood of the containers. This mosquito behavior as well as the spatial distribution of oviposition containers are easily reproduced using the cellular automata approach. Also, colonization, persistence and distribution of the mosquito population in controlled regions can be reproduced using this formalism.10 To model the space localization of the containers together with the mosquitoes behavior, we developed a stochastic two-dimensional (L × L) cellular automata. With each site we associated a ﬁve-state automaton that represents the non-fertile female (not mating yet), fertile female (can produce viable eggs), wild male, sterile male and empty state. The empty sites represent females that had mated with sterile males and cannot produce viable eggs, therefore do not contribute to the overall dynamics.12 One time step corresponds to the parallel update of the entire lattice and will correspond to one day. Between time steps, diﬀusion is implemented using Margolus neighborhood.8 Brieﬂy, the Margolus neighborhood is a cell-interconnection scheme constructed following the steps described below: 1. The array of cells is partitioned into a collection of ﬁnite, disjoint and uniformly arranged pieces called blocks. 2. A block rule is given that looks at the contents of a block and updates the whole block (rather than a single cell as in a ordinary CA). The same rule is applied

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to every block. Blocks do not overlap and no information is exchanged between adjacent blocks. 3. The partition is changed from one step to the next, so as to have some overlap between blocks used at one step and those used at the next one. Therefore, the temporal evolution of mosquito populations is given by the following local rules applied in a Moore neighborhood, which comprises the eight cells surrounding a central cell on a two-dimensional square lattice: 1. Sites occupied by non-fertile female (I) can become empty with probability µI + γ1 β1 , because of mortality or mating with sterile males; or can become occupied by fertile females with probability γ2 β2 , where γ1 and γ2 are the numbers of, respectively, sterile and wild males in the neighborhood; β1 and β2 are meeting probabilities; 2. Sites occupied by female mosquitoes (F), wild (M) or sterile males (R) can become empty with probability µ (mortality); 3. Empty sites (V) can become occupied with probability γ by new mosquitoes that emerge from the oviposition containers (aquatic emergence rate). The proportion between female and male emergence is given by the parameter r. Emergence occurs in the neighborhood of the oviposition containers, deﬁned by the ratio D with a center in the position of the oviposition container; 4. Mosquito oviposition occurs inside oviposition containers and is proportional to Cγ3 φ, where C, γ3 and φ are, respectively, the carrying capacity of the container, the number of fertile females in the oviposition neighborhood and the oviposition rate; and 5. There are N oviposition containers distributed on the lattice, and in each of them a fraction of the aquatic population dies with probability µA times the number of aquatic individuals. The transition rules among the automaton states deﬁned above were constructed based on the compartmental model described in Esteva.12 To compare the control eﬃcacy in homogeneous and heterogeneous spatial distribution of mosquito populations, all results are obtained using the two distinct lattices shown in Fig. 1.

3. Results and Discussion Oviposition containers and the wild population are distributed on the lattice at the beginning of the simulation (in our examples 20% of the lattice is occupied by native female and male mosquitoes distributed randomly). All simulations were performed taking into account a square toroidal lattice with a linear dimension L = 250. Meanwhile, the oviposition containers were always distributed inside a square region smaller that the previous one (L = 190), in such a way that the same results are obtained for an open square lattice.10

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(b)

Fig. 1. Two diﬀerent spatial distributions of oviposition containers used in simulations: (a) homogeneous and (b) heterogeneous. In the both cases, the number of recipients is N = 324 and the lattice linear size L = 250.

Initially, the simulation is carried on until the native population becomes stationery, which is deﬁned as in Caswell and Etter,13 i.e., when the last-squares regression line through the values of mosquito population for the last 100 time steps had a slope of less than 0.001, the simulation was considered close enough to equilibrium and then the suppression process is started by distributing randomly the sterile male mosquitoes in the empty sites of the lattice during a time interval, ∆, in a proportion of sterile mosquitoes for each native male mosquito. The results are shown for D = 15, C = 30 and 2 diﬀusion steps. The other parameters are: φ = 0.1, µA = 0.015 (66 days), µI = 0.03 (33 days), µ = 0.04 (25 days), γ = 0.1 (10 days). Mean values were obtained from 50 simulations. We assume that r = 0.5 and β1 = β2 = 0.1077, therefore, the maximum probability of oviposition is 9 × 0.1077 ∼ 1 if all cells in a Moore neighborhood of the oviposition containers are occupied, and that sterile and wild males are equally ﬁt. For a detailed analysis of the parameters space see Ferreira.10 We remark that for this set of parameters the basic oﬀspring of natural mosquitoes (average number of viable female descendants that an adult female mosquito produces during its fertile period10 ) is bigger than one, and therefore the continuous model given in Esteva12 establishes the persistence of the mosquito population, regardless of the carrying capacity (related to the number and size of the containers) as long as it is diﬀerent from zero. Figure 2 shows the number of non-fertile female population (I) in the steady state when we consider random and uniform distributions of the oviposition containers. We notice that for 110 < N < 250, persistence of mosquitoes is obtained

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3000

2000 I

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0

0

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N

Fig. 2. Steady state of the non-fertile female population, I, as a function of the number of oviposition containers, N, without biological control. The diﬀerent curves are related to uniform (•) and random (◦) distribution of oviposition containers.

only for the random distribution of the containers; for 250 < N < 350 persistence is obtained for uniform and random distribution, but the number of mosquitoes is much smaller for the uniform distribution. Finally, for N > 350 the results obtained with the two distributions are practically the same. Therefore, for the cellular automata model, there exists a critical point depending on the environmental carrying capacity, N × C, where N is the number of containers and C, the size of each container, below which population becomes extinct. This critical point depends on the spatial distribution of the containers (Fig. 1), mosquito diﬀusion speed, and the size of the neighborhood containers. Then, for a ﬁxed C, there is a threshold number Nc such that if N < Nc the mosquito population becomes extinct. For a uniform distribution we obtained Nc ≈ 250, whereas for a random distribution we obtained Nc ≈ 110. Therefore, a random distribution facilitates dispersion and colonization of mosquitoes, since the threshold number of containers for persistence, Nc is smaller than the one for uniform distribution. Furthermore, for Nc large the cellular automata model reproduces the qualitative behavior obtained by the continuous model.12 In order to understand why a random distribution of oviposition containers promotes mosquito persistence, in Fig. 3 we display the average distance, Φ, and the minimum distance, η, between successive oviposition containers. We can see that Φ is almost the same for both distributions, but for all values of N the random distribution generates, on average, space distributions with a lower minimum distance between the containers, facilitating colonization and persistence of mosquito populations. In fact, the threshold obtained for both spatial distributions of oviposition containers (Fig. 2) are related to the percolation properties. For N > Nc we have a cluster of female mosquitoes extending from one side of the lattice to the other,

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Φ

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330 N

440

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Fig. 3. Distance average, Φ, and the minor distance, η, between oviposition containers as a function of the number of oviposition containers, N. The diﬀerent curves are related to uniform (•) and random (◦) distribution of oviposition containers.

whereas for N < Nc no such cluster exists. However, determining critical behavior from steady-state simulations is often very diﬃcult due to large ﬂuctuations, critical slowing down, ﬁnite-size eﬀects and diﬃculties in exactly locating the critical point. Therefore, these phase transitions will be studied in a future paper using time-dependent simulations.14 Figure 4 shows the time evolution of the mosquito population [fertile female (F), wild male (M) and sterile male (R)] for a uniform distribution of N = 326 oviposition containers. The release of the sterile mosquitoes occurs at t ≈ 1000

F, M, R

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Fig. 4. Time evolution of fertile female, F (hatch line), wild males, M (dot line), and sterile males, R (continuous line), for a uniform distribution of oviposition containers.

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during a period of time ∆ = 20 days, and it is proportional to the number of native mosquitoes = I/M = 7. The native population is reduced during the time interval ∆, and then increases when control application is discontinued. The native population after the SIT control is smaller than the population at the beginning of the control, and its ﬁnal size depends on the value. This fact is associated with a local extinction of the population in regions of the lattice that were colonized before control, (bear in mind that the initial mosquito population was distributed randomly on the lattice, and new colonizations started from oviposition containers). In general, the number of positive containers which have immature forms of mosquito is lower than the number of containers distributed on the lattice. For a random distribution of containers, the recovery dynamics after control is faster and therefore local extinction is less probable (results not shown). To measure the eﬃciency of the SIT technique, we deﬁne J, the accumulated T index control parameter, as J = (1 − Ac /A0 ) × 100, where A• = 0 F (t)dt is the the accumulated number of female mosquitoes with (Ac ) and without (A0 ) controlling mechanisms applied on the mosquito population. Observe that Ac − A0 measures the accumulated reduction in the number of female mosquitoes by controlling mechanism in the range from 0 to T, therefore, J is the percentage of reduction of female mosquitoes over the interval [0, T ].15 A similar index was used to measure vaccine eﬃcacy on malaria control programs.16 Figure 5 shows control eﬃcacy measurement as a function of the time interval when sterile mosquitoes are released, ∆, for a uniform distribution of N = 326 oviposition containers. The proportion of sterile and wild male mosquito is ﬁxed in = 7. According to the above deﬁnition of J, note that it is not allowed to reach the value of 100%, even when female mosquitoes are eliminated. For instance, when

100

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J 40

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Fig. 5. Control eﬃcacy, J, measures as a function of sterile mosquito time release, ∆, for a uniform distribution of oviposition containers. The proportion of sterile and wild male mosquito is ﬁxed on = 7.

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60 J

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ε

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Fig. 6. Control eﬃcacy, J, measures as a function of the proportion of sterile and wild male mosquito, , for a uniform distribution of oviposition containers. Sterile mosquito time release is ﬁxed on ∆ = 20.

the release of sterile mosquitoes occurs during a period of time larger than 40 days, or ∆ ≥ 40, the mosquito population is extinguished, but the control index assumes J ≈ 74%. The corresponding elimination index for a random containers distribution is attained when ∆ ≥ 400, but mosquito extinction is not observed. Figure 6 shows control eﬃcacy measurement as a function of the proportion between sterile and wild male mosquitoes, , for a uniform distribution of oviposition containers. The period of time that sterile mosquitoes are released is ﬁxed in ∆ = 20, and all other parameters have the values deﬁned previously. At = 15 the mosquito population is extinguished. Notwithstanding that, for a random distribution of oviposition containers and = 30, the eﬃcacy observed was below 5%, and mosquito extinction is never observed. Comparing Figs. 5 and 6 with respect to tangent slopes in each point, we notice that for SIT control, increasing the ratio between wild and native males was more eﬀective than increasing the time for releasing sterile mosquitoes. Furthermore, the two graphs increase monotonically until they reach a constant value near 80%, which represents mosquito extinction. In order to increase control eﬃcacy for a random distribution of oviposition containers, J was measured for a set of parameters above the critical value of the carrying capacity given by N = 300. Also, the proportion of sterile and wild males and the time for releasing sterile males were changed for these new simulations. In this case, mosquito control is easier to obtain than for the previous one described above. Figure 7 shows control measures eﬃcacy as a function of sterile mosquito time release, ∆, for a random distribution of oviposition containers. The proportion between sterile and wild male mosquitoes is ﬁxed on = 7. At ∆ = 200, the index J is almost 74%, but extinction is not observed yet.

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Fig. 7. Control eﬃcacy, J, measures as a function of sterile mosquito time release, ∆, for a random distribution of oviposition containers. The proportion of sterile and wild male mosquitoes is ﬁxed on = 7.

100

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60 J

40

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0 0

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10

ε

15

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25

Fig. 8. Control eﬃcacy, J, measures as a function of the proportion of sterile and wild male mosquito, , for a random distribution of oviposition containers. The sterile mosquito time release is ﬁxed on ∆ = 100.

Figure 8 shows control eﬃcacy measurement as a function of the proportion of sterile and wild male mosquitoes, , for a random distribution of oviposition containers (N = 300). The sterile mosquito time release is ﬁxed on ∆ = 100. At = 25 the index J is almost 62%, but extinction has not been observed yet. The continuous compartmental model described in Esteva12 shows that in the absence of sterile insects the condition for the existence of natural insects is R > 1, where the parameter R is the basic oﬀspring number. However, when sterile insects

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are released, the threshold values are situated at R = R∗ (> 1), indicating that one of the conditions for the existence of natural insects is furnished by increasing R. This is due to the decrease in mating fertilized females with sterile male insects, resulting in diminishing net production of oﬀspring. This value depends on the parameter S that measures the number of mated but not fertilized female insects in comparison with the fertilized ones, which depends on the proportion of sterile insects that are sprayed in adequate places and on the lost ﬁtness associated with the sterilization process. If S is suﬃciently high, the next generation of wild insects will be smaller than the present one since a proportion of eggs will not hatch. Therefore, if sterile male insects are sprayed for a long period of time, natural insect populations will be driven to zero. In particular, the trivial solution (mosquito extinction) is always stable and if R > 1 and S > S c the non-trivial equilibrium coexist with the trivial one. In the region of coexistence, numerical simulations indicate that SIT would have a low chance of success. The results shown in Figs. 5, 6, 7 and 8 for the cellular automata model agree with those provided by the continuous compartmental model.12 The two models show that control eﬃcacy is reduced by the non-homogeneous distribution of breeding sites but control eﬃcacy is overestimated by continuous compartmental modeling. In this case, the increase of S can be oﬀset by increasing the number of sterile or applying alternative controls (like larvacide and insecticide) together with SIT to decrease the number of viable female mosquitoes. For the automata cellular model, extinction is not observed for a spatially heterogeneous distribution of mosquitoes, even in the best situation. 4. Conclusion The automata cellular model proposed here was able to reproduce aspects of Aedes mosquito behavior, such as spatial heterogeneity, infestation and persistence. As in other CA models, such a success shows the importance of local interactions for the mosquito population dynamics. The eﬃcacy of SIT control was measured for two distributions of oviposition containers: homogeneous and heterogeneous, which reproduce diﬀerent spatial distributions of mosquitoes. The results show that even in the best situation, when sterile males have the same ﬁtness as the wild mosquito and the number of mosquito is not large, spatially heterogeneous distribution of mosquitoes makes the use of SIT technique diﬃcult or impracticable. In fact, this result is correlated with the observation that non-homogeneous spatial distribution on breeding sites facilitates colonization and persistence of Aedes aegypti. This result is not accessed by the compartmental model described in Esteva.12 The model proposed here predicts that for a spatially uniform distribution of breeding sites, and consequently quasi-homogeneous distribution of female mosquitoes, the SIT technique is suitable to eliminate wild mosquito populations. This result can be borrowed to explain the success in controlling agricultural plagues or screwworm ﬂy by SIT technique: reproduction does not depend on the spatial

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distribution of breeding sites and, naturally, the target area is also well delimited. This technique is also suitable for other mosquitoes which have dispersion, sexual behavior and adult habitat preferences diﬀerent from aedes mosquito, such as Anopheles, the malaria vector. Furthermore, as cited before, the threshold obtained for the two spatial distribution of oviposition containers (Fig. 2) are related to the percolation properties. Therefore, a great amount of interesting theoretical work can be done, such as cluster analysis and determination of the transition universality class.17

Acknowledgments The work of C.P.F. is supported by Funda¸c˜ao de Amparo a Pesquisa do Estado de S˜ ao Paulo (FAPESP), Project No. 05/51671-6 (fellowship awarded by FAPESP). H.M.Y. is support in part by Conselho Nacional de Desenvolvimento Cient´ıﬁco e Tecnol´ogico (CNPq) and FAPESP, Project No. 04/07075-7. The third author of this paper, Lourdes Esteva was supported by grant IN108607-03 of PAPIIT UNAM.

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14. Ferreira CP, Fontanari JF, Nonequilibrium phase transitions in a model for the origin of life, Phys Rev E 65(2):art. no. 021902 Part 1 Feb., 2002. 15. Yang HM, Ferreira CP, Assessing the eﬀects of vector control on dengue transmission, Appl Math Comput 198:401–413, 2008. 16. Dietz K, Raddatz G, Molineaux L, Mathematical model of the ﬁrst wave of Plasmodium falciparum asexual parasitemia in non-immune and vaccinated individuals, Am J Trop Med Hyg 75(Suppl 2):46–55, 2006. 17. Stauﬀer D, Ahorony A, Introduction to Percolation Theory, Taylor & Francis Ltd, London, 1994.