Mapping Rift Valley Fever Vectors and Prevalence Using Rainfall Variations

VECTOR-BORNE AND ZOONOTIC DISEASES Volume 4, Number 1, 2004 © Mary Ann Liebert, Inc.

Research Paper

D.J. BICOUT and P. SABATIER

ABSTRACT

High activity of the Rift Valley Fever (RVF) virus is related to a tremendous increase of associated mosquito vectors, which follows periods of high rainfall. Indeed, rainfall creates an ecologically humid environment that in- sures the proliferation of breeding sites and the development of RVF vectors. Data collected by Fontenille et al. (1998) from 1991 to 1996 in the Barkedji area in the northern Senegal are employed to discuss and quantify the in- cidence of rainfall upon the abundances of RVF vectors. We have constructed a non-linear mapping of vector abundances versus rainfall variations, and developed a stochastic model and a corresponding algorithm allowing on output the simulation of RVF mosquito vectors as a function of rainfall trajectories in the course of time. This stochastic mapping of vector abundance is subsequently used to assess the prevalence of RVF in a population of susceptible hosts as a consequence of rainfall. Keywords: Arboviruses—Stochastic dynamics—Rainfall— Vectors— RVF prevalence. Vector-Borne Zoonotic Dis. 4, 33–42.

INTRODUCTION severe disease with high mortality and abor- tions in pregnant females. Human infection IFT VALLEY FEVER (RVF) is an arboviral dis- with RVF virus is characterized by the onset of Rease that causes epizootics and associated high fever, hepatitis, encephalitis, haemor- human epidemics throughout Africa (Meegan rhagic fever orocular disease and sometimes and Bailey 1988, Peters and Linthicum 1994). leading to death (Laughlin et al. 1979, Meegan RVF virus belongs to the genus Phlebovirus , et al. 1980, McIntosh et al. 1980). family Bunyaviridae . It is transmitted to verte- RVF outbreaks are commonly correlated brates by infected floodwater mosquitoes Aedes with periods of widespread and heavy rainfall spp. and other Culex spp. Humans can also be (Meegan and Bailey 1988, Wilson et al. 1994, infected via inhalation of aerosols of vireamic Digoutte and Peters 1989, Linthicum et al. blood during slaughtering, or other direct con- 1999). It is assumed that ecologically wetter tact with infected animals. Vertical transmis- conditions increases the number of breeding sion from mosquito to mosquito also occurs in sites for mosquitoes resulting in an increase in some species (Meegan and Bailey 1988, Davies the number of vectors and therefore more in- and Highton 1980, Wilson et al. 1994, Fontenille tense virus transmission and circulation. Rain- et al. 1998). The RVF virus affects domestic an- fall floods the mosquito breeding habitats imals, specially sheep, cattle and goats, causing which contain eggs of the primary vector

Unité Biomathé matiques et Epidémiologie, Ecole Nationale Vété rinaire de Lyon–INRA, Marcy L’Etoile, France.

33 34 BICOUT AND SABATIER species and reservoirs (transovarially infected fluential since maximum vector-host contact Aedes spp.), and these subsequently serve as a occurs during periods when vectors are active habitat for development of secondary vectors and their populations are high. To address the (e.g., Culex spp.). It follows that excessive flood- issue of direct influence of rainfall on RVF virus ing allows for massive hatching and a tremen- circulation we have focused our analysis on m. dous increase of the number of both primary We have used real data collected in Barkedji and secondary vectors, which may results in an area in the northern Senegal during 1991–1996 outbreak (Davies et al. 1985, Zeller et al. 1997, (Fontenille et al. 1998) to construct a framework Woods et al. 2002). Although other weather fac- for a rational study of correlation between rain- tors may be important, this paper focuses on fall and vector abundance in a region where analysing and modelling the relationship be- temperature shows little variation. Our ap- tween rainfall, vector abundance, and the oc- proach is fundamentally stochastic. It consists currence and maintenance of RVF outbreaks. of a statistical analysis for determining firstly In general, the importance of climate condi- the distribution laws describing the data and tions to vector-borne diseases is highlighted in secondly, constructing the correlations be- the basic reproductive number given ( R0) in the tween observable components of the problem. formula below (Macdonald 1957, Anderson In this context, the observable factors are the and May 1991): rainfall and vectors abundances over the course 2 of time. The parameters obtained from the sta- C0 a mbc R0 5 } 5 RvRh 5 } exp(2mtx) (1) tistical analysis are next used to formulate a sto- a am chastic model describing the dynamics of the where a is the vector biting rate, m the number observables and, to develop an algorithm for of vectors per host, b the transmission proba- stochastic simulations. Accordingly, we derive bility from vector to vertebrate, c the transmis- mathematical expressions allowing for the sto- sion probability from vertebrate to vector, a the chastic mapping of vectors abundance versus rate of recovery of the vertebrate from infec- rainfall over the course of time. Numerical re- tiousness, m the vector mortality rate, and tx the sults are compared to collected data and, sim- extrinsic incubation period. The basic repro- ulated stochastic trajectories of vector popula- ductive number is the product of the vectorial tion densities are used to generate the pattern capacity C0 and the duration of the infectious of the prevalence of RVF virus infection and period in the vertebrate hosts. The vectorial ca- immunity in a population of hosts. pacity is defined as the mean number of po- tentially infective contacts made by a mosquito population per infectious host and per unit MATERIALS AND METHODS time (Garrett-Jones 1964) and the R0 yields the average number of secondary host infections Data characterization which one infected host could produce during the duration of the infection among a suscep- The Barkedji area (15°17 9N, 14°179W) is lo- tible host population. The R0 comprises of two cated in the northern Senegal in the Sahelian components: Rv 5 (ac/m)exp(2mtx) the num- Ferlo region. This area was found to sustain a ber of infective contacts for transmission from focus of an enzootic maintenance of the RVF vertebrate to vector and Rh 5 mab/a the num- virus in 1993 (Zeller et al. 1997). Rainfall in this ber of infective contacts for transmission from region averages 250–350 mm annually, and oc- vector to vertebrate. If R0 . 1 each infected host curs in a short rainy season (July to September) will generate on average, a further case and an which fills the temporary ground pools. These epizootic or epidemic can occur (with proba- latter remain the only source of water until to bility 1 2 1/R0). When R0 , 1 the outbreak can- January during the 6th-month dry season (De- not maintain itself and will stutter to extinction. cember through May). After the first rains, the Among all parameters influencing C0 and R0, number of mosquitoes dramatically increases. vector mortality (or survival) rates and vector More than 228,000 mosquitoes from 52 species density (m) are strongly related to environ- were collected in Barkedji and tested for virus mental conditions. Rainfall is particularly in- isolation (Fontenille et al. 1998). Aedes species RVF VECTORS AND PREVALENCE VERSUS RAINFALL 35 represented 28.8% of the mosquitoes collected, among which the Ae. vexans was the most abundant, followed by Ae. ochraceus . Other Aedes species like, for example, Ae. mcintoshi and Ae. dalzieli, were rare but Culex species and sand flies were very abundant during the dry season. Records of monthly rainfall and the distributions of RVF vectors ( Aedes vexans and Culex poicilipes) abundances captured in Barkedji, Senegal, over 6 years from 1991 to 1996 (Fonte- nille et al. 1998) were used for the statistical analyses (Fig. 1). Simple inspection of the data shows that periods of greater mosquito abundances coincide with that of heavy rainfall in amount and the maximum numbers of Culex is FIG. 2. Monthly reduced mean values fi(t) of the rain- attained couple months after that of Aedes. To fall, Aedes and Culex. Filled triangles, circles, and squares are calculated using the field data from Fontenille et al. characterize the intra- and inter-annual varia- (1998), while the solid, dashed, and dot-dashed lines rep- tion of the data, we calculate the between-year resent formula in Eqs. 3 and 5, respectively. monthly means and variances of the rainfall and vector abundance. Let r (t) be the observable vector (in the math- i mum value of kr (t)l given by R 5 128.67 mm, ematical sense) at time t (measured in months) i 1 R 5 4981.48 and R 5 5814.82. If it is assumed with i 5 1,2,3 corresponding to the rainfall, 2 3 that the population of hosts within the study Aedes and Culex abundances, respectively. We area is almost constant during the period of the calculate for each month the average kri(t)l and

study, then the Aedes and Culex densities are standard deviation s2(t) 5 kr2(t)l 2 kr (t)l2 over i i i calculated as m (t) 5 r (t)/N and m (t) 5 all the years under study. As a result, we find 2 2 3 r (t)/N, respectively, where N is a population for all three variables, that s (t) < kr (t)l for all 3 i i scaling constant. t such that 0 # t # T 5 12 months. Figure 2 dis- These findings indicate firstly that as s (t) < plays the reduced mean f 5 kr (t)l/R (sym- i i i i kr (t)l then the inter-annual values of r (t), at bols) as a function of t, where R is the maxi- i i i each month t within a year, can be described by the non-stationary exponential distribution,

P(r1,r2,r3;t)

3 exp{2ri/kri(t)l} 5 P } } for t 5 1,2,…,12 (2) 3i 5 1 kri(t)l 4

Secondly, kri(t)l describes an average of the intra-annual variation of ri(t) and correlations between coordinates. To characterize the intra- annual variation of ri(t), we first focused on the rainfall, i.e., r1(t). Next, assuming that mosquito abundances are only controlled and triggered by rainfall, we express the mosquito abundances as a function of rainfall.

Rainfall. Figure 2 shows that the following FIG. 1. Distributions of Aedes vexans and Culex poicilipes heuristic formula (curve) describes well the reduced abundances captured by monthly rainfall in Barkedji, Senegal (data from Fontenille et al. 1998). Ver- data of the rainfall, tical axis represents the monthly field data divided by 2 2 kr1(t)l v (t 2 t) R1 5 128.67 mm, R2 5 4981.48, and R3 5 5814.82 for the f1(t) 5 } 5 exp 2 } } (3) rainfall, Aedes vexans , and Culex poicilipes , respectively. R1 5 2 6 36 BICOUT AND SABATIER where t is an average of the date (within a year) Equations 5 and 6 indicate that in average the at which the rainfall is maximum and 2 Ï2/v Aedes abundance appears to be synchronized is the rainy season extension. For the data un- at the scale of the month with the rainfall vari- der study, we have: ations but with different rates in the increas- ing and decreasing phases of the rainfall, i.e., v 5 0.921/month and t 5 8 months (4) for 0 , t , t and t , t , T. When the rainfall increases from zero in the course of months, meaning that, on average, the maximum rain- the number of Aedes mosquitos rapidly in- fall occurs in August and the rainy season lasts creases and tends to saturate, while its slowly about 3.1 months. However, we note that this decreases down to zero with the decreasing expression does not fit the rainfall at t 5 2 rainfall. For the Culex, however, Eqs. 5 and 7 months. The form of Eq. 3 is deliberately cho- indicate that their abundance is closely re- sen on purpose, based on a detailed analysis of lated to the rainfall variations but with a de- data that shows no evident correlation between lay of about 2 months. These features are dis- vector abundances and the rainfall at t 5 2 played in Figure 2, which shows a good months. Thus, for simplicity, the rainfall at t 5 agreement between formulas in Eq. 5 and col- 2 months will be neglected in the following lected data. analysis, and Eq. 3 will be considered as the Now, we use these above defined character- best description to the overall data. istics to formulate a simple stochastic model for the dynamics of ri(t), i.e., the simultaneous dy- RVF vector abundances versus rainfall . In the namics of rainfall and RVF vector abundances. absence of more information, we assume in our analysis that the abundance of Aedes and Culex are only related to the rainfall. To establish the Stochastic modeling relationship between the rainfall and RVF vec- Rainfall and RVF vector abundances . As char- tors abundances, we consider the variations of acterized above, the stochastic dynamics of the the reduced mean f2(t) (for Aedes) and f3(t) (for observable ri(t) at time t can be modelled using Culex) as a function of f1(t) (for rainfall) the evolution equations, recorded at the same date t. The following 2 2 heuristic formulas accurately interpolate the r1(t) 5 Z1(t)exp{2v [ f(t) 2 t] /2}(rainfall) dependence of mosquito abundances as a function of rainfall, r2(t) 5 Z2(t)g2[r1(t)/Z1(t)] (Aedes) (8) f2(t) 5 g2[f1(t)] and f3(t) 5 g3[f1(t 2 2)](5) r3(t) 5 Z3(t)g3[r1(t 2 2)/Z1(t 2 2)] (Culex) where the non-linear mapping functions g2(…) where f(t) 5 mod[t,T] with mod[…] being the and g3(…) are given by, modulo operation, v and t given in Eq. 4, the mapping functions g2(…) and g3(…) defined in g2[x(t)] 5 Eq. 5 and Zi is a Poisson random vector of mean and correlation 1 2 exp{0.53x(t) 2 9.13x2(t)} } } } } H(t 2 t) ` 3 1 2 exp{28.6} 4 2 kZi(t)l 5 Ri; kZi(t)Zj(t9)l 5 2Ri dij E ci(t0)dt0 (9) ut2t9u 1 2 exp{1.04x(t)} 1 } } H(t 2 t) (6) in which R are defined in Section Characteri- 3 1 2 exp{1.04} 4 i zation of Data, dij is the Krönecker symbol and, where H(L) is the Heaviside step function defined as H(z) 5 0 for z , 0 and H(z) 5 1 for z . ci(t) 5 nid(nit 2 1) (10) 0, t is given in Eq. 4 and, is the distribution of pausing time between two 1 2 exp{1.4x(t)} environmental events with the mean pausing g3[x(t)] 5 } } (7) 1 2 exp{1.4} time 1/ni and d(…) the delta function. The dy- RVF VECTORS AND PREVALENCE VERSUS RAINFALL 37 namics goes as follows. The initial Zi is ob- m3 5 R3/N associated to Aedes and Culex mos- tained from the exponential distribution, quitoes, respectively, where N is a population scaling constant. Rh2 5 m2ab/a is the component exp{2Zi/Ri} p(Zi) 5 } } (11) of the basic reproduction number for transmis- R i sion from infected Aedes vectors to vertebrate and the time variation of ri(t) is given by the hosts and « the proportion of transovarially in- non-linear equations Eq. 8. Zi(t) remains con- fected Aedes mosquitoes. stant equal to its initial value until it suffers an environmental event of zero duration which updates Zi(t) according to the distribution p(Zi). RESULTS AND DISCUSSION After each event, ri(t) still follows Eq. 8 but with a new Zi(t). For each Zi(t), ni is the corre- Within the framework of stochastic dynam- sponding frequency of events and the pausing ics, the monthly records in Figure 1 represent time between successive environment events is a single trajectory of the stochastic process that a random variable with the distribution ci(t). we have characterized above in the Character- Combination of both p(Zi) and ci(t) are re- ization of Data section and described in the Sto- quired to mimic the stochastic nature of ri(t) chastic Modelling section. The aim does not and the expressions of p(Zi) and ci(t) originate exactly reproduce the time variation of ri(t) from Eq. 2. (which is impossible in the stochastic sense) but it does reproduce the same trends of the ob- Infection dynamics of host population . To illus- served rainfall and mosquito abundance data. trate the relationship between rainfall, through To this end, we have generated ri(t) trajectories the force of infection applied by Aedes and using the algorithm described above with Culex vectors, and the occurrence and spread mean pausing times, 1/ ni 5 1 month (with ni of RVF virus infection within a population of corresponding to the frequency of data records), susceptible hosts, we consider a simple model and seasonal period T 5 12 months. To illus- that incorporates the basic features of interac- trate the mapping method, we compare the tions between the population of vertebrate simulated results with the raw data of rainfall hosts and densities of mosquito vectors. The and mosquito abundance. We can then simu- classic dynamic model for a homogeneous pop- late the trajectories of mosquito abundances to ulation of hosts with constant size reads, predict the pattern of infection and immunity dy a«R r (t) levels in a population of hosts. } 5 2ay 1 }h}2 2 (1 2 y 2 z) dt R 3 2 4 Rainfall (1-«)R02r2(t) R r (t) 1 a } } 1 }03 3 y(1 2 y 2 z), (12) Comparison between the data and simula- 3 R2 R3 4 tions is displayed in Figure 3, where the his- tograms represent the data and dashed lines dz } 5 ay 2 gz, simulations. It is clear that simulations describe dt quite well the general behavior of the rainfall. where y and z are the proportions of infected As expected, larger deviations from the data and immune hosts, respectively, 1/ a the mean are observed for a single trajectory in consis- duration of the host infectiousness, the renewal tency with stochastic processes (simulations rate g of susceptible hosts is the rate of decay of not shown). However, performing few aver- the immune population, r2(t) and r3(t) the num- ages (e.g., 10 averages) over trajectories seems ber of Aedes and Culex vectors, respectively, gen- to provide a closer reproduction of the data. In- erated using the algorithm described above, R2 creasing the number of averages smoothes the and R3 the maximum values of the mean num- curves and ultimately leads to the mean be- ber of Aedes and Culex mosquitoes, respectively. havior, i.e., to r1(t) as given in Eq. 3. Further- R02 and R03 are the basic reproductive numbers more, one can also consider the cumulated as defined in Eq. 1 with densities m2 5 R2/N and rainfall defined as, 38 BICOUT AND SABATIER

Culex in October 93 seems to be not related with the level of rainfall. Thus, the stochastic mapping of mosquito trajectories as generated from the above algorithm and depicted in Figure 5 can be considered as the rainfall contribution to the observed mosquito abundances. The dis- crepancies in the total abundances between simulations and field data indicate both the need of improving the distribution function P(ri;t) and that of including factors other than rainfall in the mapping of mosquito abundances. As an additional test of the stochastic mapping, we have also examined the monthly dis- FIG. 3. Monthly rainfall over 6 years. Histogram repre- tribution of relative abundances, i.e., the ratio sents field data from Fontenille et al. (1998), and the of the number of Aedes or Culex over the total dashed line corresponds to an average over 20 stochastic trajectories. All trajectories are generated using the algo- number of mosquitoes versus months (simula- rithm described in Section Stochastic Modeling with R1 5 tions not shown). We found a very good qual- 128.67 mm, c1(t) 5 d(t 2 1), and T 5 12 months. itative and quantitative agreement between simulations and observations. It is worthwhile to note that the maximum of the relative abun- 1 t dance for Culex appears to occur about 5 cr(t) 5 }E r1(mod[t9,T])dt9 (13) D 0 months late (forward shift) compare to that for where mod[…] is the modulo operation, D 5 1 Aedes. 5 Given that the stochastic mapping of vector month, and T 12 months. Figure 4 shows comparison between data and simulations av- abundance versus rainfall is verified and that eraged over twenty trajectories. Here again, the rainfall variations are known, we believe that agreement between simulations and data is our algorithm could be used to predict vector very good not just in the general trends but also abundance, without the necessity of population in absolute values. sampling. However, we have to keep in mind the assumption made above in mapping the Aedes and Culex abundances mosquito distributions from the rainfall only. Now, as prescribed in Eq. 8, the rainfall trajectories are used to obtain corresponding trajectories for Aedes and Culex abundances. Briefly, Figure 5 shows the comparison between the data (histogram) and simulations (dashed lines) averaged over twenty rainfall trajectories. It appears that simulations follow rather well the general behavior of the distributions of Aedes and Culex in the course of months, and even the locations of maxima of mosquitoes coincide with those of collected data. According to observations by Fontenille et al. (1998), the adult Aedes mosquitoes ap- peared about 4 days after first rainfall and rapidly disappeared about 2 months after flooding of ground pools, while the density of FIG. 4. Monthly cumulated rainfall, cr(t), corresponding other species like Culex increases at the end of to Figure 3. Solid line represents data from Fontenille et rainy season. However, the higher peak of al. (1998), and dashed line corresponds to simulations. RVF VECTORS AND PREVALENCE VERSUS RAINFALL 39

A B

FIG. 5. Monthly distributions of Aedes vexans and Culex poicilipes abundances over 6 years. Vertical axis represents the monthly mosquito abundance divided by R2 5 4981.48 for Aedes vexans and by R3 5 5814.82 for Culex poicilipes . Histogram represents field data from Fontenille et al. (1998), while the dashed lines correspond to simulated data generated using rainfall trajectories in Figure 3.

As mentioned above, additional environmen- lations of Aedes and Culex are R2 and R3, re- tal factors like the intensity of rainfall, the vari- spectively, ation of ground pools, the evapotranspiration, a temperature, and humidity may need to be R 2 1 2 «R 1 1 }} 0 h2 g taken into account for a better description of g } }1}2 ys 5 } zs 5 a mosquito abundances. a 2R 1 1 }} 01 g 2 Host infection prevalence a 2 Figure 6 illustrates the simulated pattern of R0 1 1 1 «Rh2 1 1 }} 2 4R0 monthly prevalence of infected y and immune 3 1 g 24 1 !}§§}§}§}§§ , (14) z proportions of a population of domestic ru- a 2R 1 1 }} minants exposed to the distributions of Aedes 01 g 2 and Culex vectors depicted in Figure 5. Basic reproductive numbers—R02 and Rh2 for Aedes with R0 5 (1-«)R02 1 R03. Likewise, the preva- and R03 for Culex—are chosen close to the crit- lence of immune population of hosts oscillates ical level to see large changes in prevalence. in synchronization with that of infected hosts Four insights are illustrated in this figure. but stays always below its stationary value zs. First, as expected, the onset and spread of the The third insight is that, as vectors of RVF RVF prevalence are highly seasonal following virus, the relative importance between Aedes years where the periodic variations of vector and Culex mosquitoes appears in their relative abundance are correlated with rainfall as dis- abundance and in the timing of their emer- cussed above. Thus, this can be considered as gence and disappearance in the course of rainy an indirect stochastic mapping of RVF preva- season. This is illustrated on Figures 6A and 6B lence using rainfall variations. where R02 , R03 and R02 . R03, respectively, Second, the dynamical scales of y and z are such that Rh2 and R02 1 R03 remain constant. in the ratio of about [ a/g]1/2 , 10. The preva- Without going into details, year-to-year com- lence of infected hosts increases every year parison between panels A and B shows that from zero, attains maximum values greater there are qualitative and quantitative differ- than ys by a factor of about 10, turns over and ences in prevalence of infected hosts in contrast decreases to zero as the rainy season goes on; to the prevalence of immune hosts which is al- ys is the stationary value of y when the popu- most identical for the two panels. The differ- 40 BICOUT AND SABATIER

A B

FIG. 6. Monthly prevalence of infected y and immune z proportions of a population of hosts obtained from inte- gration of Eq. 12 associated with vector abundances ri(t) given in Figure 5 and subjected to initial conditions, and y(0) 5 0 and z(0) 5 0.5. Parameters used in calculations are: 1/ a 5 5 days, 1/g 5 1.64 year for domestic animls, « 5 0.01, Rh2 5 1, and the basic reproductive numbers R02 and R03. Straight horizontal lines represent the steady means ys and zs. (A) (R02,R03) 5 (1.5,2) and ( ys,zs) 5 (0.00663,0.794). ( B) (R02,R03 5 (2,1.5) and ( ys,zs)(0.00663,0.794). ( C) (R02,R03 5 (0,2) and ( ys,zs) 5 (0.00602,0.721). [The value of g used in the illustration represents the observed rate of decay of the number of individuals positive with RVF IgG antibody in a population of domestic ruminants, estimated from Figure 3 of the following reference: Thiongane, Y, Thonnon, J, Zeller, HG, et al. Données récentes de l’é pidémi- ologie de la Fièvre de la Vallée du Rift (F.V.R.) au Séné gal. Dakar-Medical Spécial Congrès, Commun 1996; 1–6.]

ences between Figures 6A and 6B originate have set R02 , 0 in Figure 6C (or equivalently, mainly from the fluctuations in the relative Rn2 , 0 because for example mtx .. 1). This abundance of Aedes and Culex mosquitoes. may correspond to the situation where the The last and fourth insight concerns the role transovarially infected newborn Aedes, issued primary Aedes vectors (which maintain and in- from infected Aedes mosquitoes of the previous troduce viruses into the population of suscep- rainy seasons, are the only ones capable of tible hosts) and the role of the secondary vec- transmitting the virus to susceptible hosts. In tors, mainly played by Culex but also by Aedes such a picture, the Aedes primarily introduce and some other species, (which, with interac- RVF virus into the host population and then tions with hosts, amplify the circulation of Culex species amplify and circulate the virus. viruses [Zeller et al. 1997, Linthicum et al. 1999, As shown in Figure 6C, peaks of prevalence of Woods et al. 2002]). To emphasize the different infected hosts coincide with peaks of Culex roles played by Aedes and Culex mosquitoes, we abundance. Interestingly, the highest peak ob- RVF VECTORS AND PREVALENCE VERSUS RAINFALL 41 served between months 30 and 36 is consistent incorporated in parallel into other types of with the enzootic circulation of RVF virus models related to time-sensitive infection dy- found in Barkedji in 1993 (Zeller et al. 1997). In namics of RVF prevalence in a population of contrast to Figure 6A and Figure 6B where hosts. Such a procedure is also developed for peaks of prevalence for infected hosts between the purpose of predicting infection risks for hu- months 30 and 36 (i.e., in year 1993) are as large man and livestock populations. However, the as the others, the situation in Figure 6C appears method discussed above is being currently im- very much like of outbreak with a single great proved by considering different kind of distri- peak in 1993 embedded in a low noise circula- bution P(ri;t) and by including additional cli- tion of virus for previous and following years. matic factors having significant impact on R0 It appears from this that investigating the rela- like rainfall intensity, temperature range and tionship between the Aedes versus Culex mos- humidity and, when possible, the oscillations quitoes can be very informative in determining of temporary mosquito breeding habitats. the epidemic behavior. What would be important for an outbreak to occur is the conjunction of two things: the initiation of transmission by ACKNOWLEDGMENTS transovarial reservoirs or vertical vectors fol- Thanks go to K. Chalvet-Monfray for critical lowed by the continuation and amplification of reading of the manuscript. The financial sup- transmission by secondary or horizontal vec- port of DGA and CNES in the form of a grant tors. Nevertheless, more information and in- is gratefully acknowledged. We also thank S. vestigations on both vector and host popula- Higgs for his help in making the paper read- tions are needed to figure out which scenarios able. of Figure 6 are likely taking place in this area.

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