The entomological parameters, the parameters that characterize the stages of a insect life, depend upon the conditions of the insect, as well on environmental factors. The dependence on air temperature is known (e.g., Otero et al., 2006) [23] , as well as the threshold amount of water in the breeding sites, for the maintenance of the mosquito life cycle. After the hatching (eclosion) of the larva, that is follows a period of replenishment of the reservoir from rain, the water quantity and the food available in the site must be enough for the survival of the larva. In general, 40 to 60 larva of Aedes aegypti need a water volume on the reservoir of above 10 mm for survival and posterior development of the aquatic cycle as pupa (Christophers, 1960) [12] . Other factors are also important, such as illumination, related to the circadian cycle, and the infection itself, related to the activity level of the mosquito. Experimental results from Lima-Câmara et al. (2011) [24] show a growth in the movement rate of infected female Aedes aegypti in relation to non-infected ones.

There is also which consider that for the increase and maintenance of the epidemic cycles of Dengue, the environmental thermodynamic variables as the enthalpy, entropy and degree-day index, all must accumulate large values, as precondition or concomitant occurrence.

Based on the laws that govern thermodynamics, it is considered that any living organism constitutes a system of chemical reactions and physio-chemical processes in a continuous non-equilibrium state. The maintenance of this state is reached through the consumption of energy, withdraw from the environment. Together with cellular work there dissipation of part of this energy. This energy dissipated as heat is used by animals in the most diverse systemic processes involving the organism, from metabolism to functional activities. Hence, through the variation of enthalpy, entropy and degree day, in a closed system, it is possible to relate the influence of the functions of state in the functional activities of a living being using the dissipation o energy.

The word entropy was first used by Clausius, in 1865, to determine the quantity of dissipative energy, through the transformation oh heat in work. Entropy is a function of state associated with a system in thermodynamic equilibrium. It explains that an isolated physic system evolves to a state of thermodynamic equilibrium when maximum entropy is reached. This maximum entropy corresponds to a systemic disorder, which in turn, is related to a completely random distribution of objects in a statistically homogeneous form. So the order is related to heterogeneity, corresponding to a bigger diversity and smaller systemic redundancy (Atlan, 1992) [25] .

According to Murphy and O’Neill (1997) [26] , Schrödinger postulates that the thermodynamics applicable to biologic systems is in a non-equilibrium state, where a organism survives in its highly organized state withdrawing high quality energy from the external mean and processing it to produce, in itself, a more organized state.

Life is a system away from equilibrium, that in turn keeps its local organization level in expense of a bigger global entropy budget. One of the ways to relate temperature to biologic development is using the thermal units system or degree day, according to Brunini et al. (1976) [27] .

The employ of Degree-Day Index (DG) permits to define a linear relation between increases in air tempera- ture and the ratio of metabolic development, which in despite the simplification hypothesis, allows to determine the basal temperature or even the insect phase duration.

In this work, we propose three thermodynamic variables to stablish the relationships between the environment conditions and the population dynamic. Really, under simplification hypothesis (i.e., associated to barotropy) these variables as quite interchangeable, from the point of view of a equal-finality parametrization. The variables are the degree-day index, the accumulated entropy change, and the accumulated enthalpy change. Therefore, we have considered:

• Atmospheric degree-day index―Lets to be the basal temperature for mosquitoes metabolic performance, assumed equal to the mean temperature of July 1^st, actualized each year. So we can write

where represents the degree-day index, measured here in energy units by mass unity (J∙kg⁻¹), and defined by the difference between the actual () and the basal temperature (), multiplied by the specific heat at constant pressure of the air,. The associated accumulated diurnal variation is repre-

sented by, with initial value reset to zero at the initial time (each July, 1^st). The integer n indicates the number of observations since July, 1^st. The integer index i is the observation counter along the data series.

• Atmospheric entropy change―The diurnal variation of enthalpy, entropy and degree-day index are com- puted. The accumulated diurnal variation of the entropy () can be obtained as

where is a typical scale of the air temperature, usually assumed equal to 300 (K), is the latent heat capacity of the air at constant pressure,. is the specific volume of air (i.e., the inverse of air density) and is the median value of, here estimated by, where is the gas (air) constant. In the r.h.s. of the equation, the second term (due to pressure variation), is so smaller than the first term (due to the temperature variation) that will be not considered in the present calculation. The start date to account the values is July, 1^st for each year in the simulation period. and are the successive values of the air temperature, separated by the time step in units of (s). The is obtained in units of.

• Atmospheric enthaphy change―Formally the enthaply (H) is the sum of the internal energy (U) plus the work (W), i.e., or indeed, , where is the latent heat capacity of the air at constant volume, V is the air volume, and P the air pressure. For a ideal gas, yields

since the state relation can be written as, and the specific heat relationship is. This equation states simply the increase of enthalpy is equal to the heat entering into the system for a ideal gas. The atmosphere can be usually considered as a ideal gas.

Inspecting Equations (1)-(3), the relationship between, and yields,

With this relationships at hand, it is possible to verify the relation between the air environment thermody- namics and the performance of the kinetic rate of enzymatic reactions, the duration of the maturation-emergency period for the different phase of the mosquito life.

In the MARJ, the variation of enthalpy (numerically equal to the variation of accumulated degree day) at the end of a winter-to-summer period of 180 days is approximately equal to or, considering the molar mass of the air given approximately by, or indeed, since. The corresponding entropy variation was estimated close to.

These estimations were obtained by integration of a cos-sin function representing the first harmonic of mean air temperature in Rio de Janeiro city, along 180 days (i.e., radians), starting in July, 1^st (winter), and ending at January, 31^th (summer), given by, in ˚C, where is the rank of the day in the interval, and assuming and. The present estimation of is in agreement with the optimum indicated in Table 1 from the literature.

Without considering the details of the diurnal variation, we can estimate a value of same order to the inter-seasonal variation of enthalpy, i.e.,. For that we can assume a mean daily variation of air temperature in Rio de Janeiro city near to 10 (˚C×day⁻¹) applied to Equation (3). The result is an estimation of the mean daily enthalpy variation, given by. The obtained value

is comparable to the average value of variation computed from observations to Rio de Janeiro city (Figure 7). This diurnal evaluation can be then multiplied by 180 days (i.e., approximately half year) in order to obtain the estimation of the inter-seasonal variation, i.e., considering the period of 6 months, started in.

There are some investigations on the epidemics precursors. For instance, a comprehensible one was provided by Descloux et al. (2012) [28] which have investigated the epidemic dynamics of dengue in Noumea, that is essentially driven by the climate of previous forty years. In accord to theses authors the maximal temperature and relative humidity thresholds and their persistence were determinant in outbreak occurrences. For epidemics in Noumea, the best explicative variables were the number of days with maximal temperature exceeding 32˚C during Jan-Feb-Mar and the number of days with maximal relative humidity exceeding 95% during Jan. The best predictive variables were the maximal temperature in Dec and maximal relative humidity during Oct-Nov- Dec of the previous year.

In the present work, the enthalpy change and seasonal accumulation were used to define a predictive metabolic performance factor controlling the vector development and maturation in a tropical metropolis. Additionally, the water content into breeding sites is very important for the reproduction of vectors. The complete parametrization will be describe in the following sections, as also the model skill in explaining the Dengue outbreaks.

References: # This work, e Otero et al. (2006) [23] , * Christophers (1960) [12] . Other consulted reference works are: ! Yang et al. (2009) [45] , Ä Watts et al. (1987) [43] , Å Rueda et al. (1990) [57] , $ Focks et al. (1993) [34] [35] , and % Focks et al. (1995) [36] .

The observational data were employed both to run and to verify the model results. First, atmospheric data observed at surface at the International Airport of Rio de Janeiro (i.e., Tom Jobin airport) are assimilated in the model, providing the necessary meteorological variables (i.e., air temperature, pressure, humidity, wind speed and direction) to the calculation of the terms of the Surface Energy Budget (SEB). Precipitation flux data (i.e., rainfall) are also obtained of local mesonet in MARJ, and assimilated by nudging, for calculations of the Surface Water Balance (SWB). A Newtonian relaxation (i.e., nudging) was used to assimilate all the atmospheric data in the model, considering a sequence of assimilation cycles of 1 hour, along all the integration time. According to Kister (1974) [29] , and Haltiner and Williams (1980) [30] , nudging is a dynamic method to bring the simulated variables close to the observations, without introduces shocks in the model dynamics (i.e., by filtering very short-wave along assimilation cycles). Second, Dengue fever notifications, actually available from the Brazilian public health system, were used in comparison with modelled variables.

The employed data are:

• Notification of Dengue―Used to verify the skill of hte model against observations. The data are obtained directly from the Ministério da Saúde (MS) of Brazil (SVS, 2006) [9] , Secretaria de Vigilância em Saúde (SVS) and Sistema Único de Saúde (SUS, 2006) [31] , and downloaded from Sistema de Informação de Agravos de Notificação (SINAN) at URL http://dtr2004.saude.gov.br/sinanweb/.

• Atmospheric data―The surface meteorological data are assimilated using cycles of 1 hour in the model. The data are obtained from a weather station installed at the International Airport of Rio de Janeiro (i.e., Tom Jobim airport). The station is maintained by the Brazilian Aeronautic Command (COMAER). The original coded data (i.e., METAR) was decode, reformatted and checked statistically (i.e., comparing each variable decoded with its mean ± 2 standard deviation threshold). Additionally, a continuous time coordinates (i.e., decimal year) was computed and added to the data. This continuous coordinates (e.g., 2015.103269290) was employed in the assimilation numerical procedure to control the flux of data in relation the internal integration time (i.e., also computed as a decimal year coordinate). The data preparation was accomplished in a Fortran-90 program created for this task. The model request some atmospheric variables which are not directly available in the observational dataset, such as the density flux of solar radiance (W×m⁻²) at surface and its components (i.e., direct, diffuse and global). To supply these variables, we have parametrized them following Flores R. et al. (2015) [32] that optimally have fit the Iqbal solar irradiance model (IQC) (Iqbal, 1983) [21] to the observed conditions in MARJ. Indeed, some necessary thermodynamic variables (e.g., specific and relative humidity) are computed following Bolton (1980) [33] .

• Hydrometeorological data-Data of precipitation sampled to each 15 minutes were also assimilated in the model. The data origin is automatic rain-gage mesoscale network (i.e., the AlertaRio system), composed by 32 weather stations dispersed on the Rio de Janeiro’s municipality area, and available from URL http://alertario.rio.rj.gov.br/.

Presently, there is a set of Dengue infection dynamics models for human and mosquito population as prognostic variables, or indeed stochastic models with probability of infection as the unknown, written using different levels of complexity. Below we are listing some of them:

Focks et al. (1993, 1995) [34] - [36] proposed a mathematical treatment of the epidemiology of dengue with a view to control based in the concepts of 1) mass action principal, i.e., the course of an epidemic is dependent on the rate of contact between susceptible hosts and infectious vectors, and 2) threshold theory, i.e., the introduction of a few infectious individuals into a community of susceptible individuals will not give rise to an outbreak unless the density of vectors exceeds a certain critical level. These result in two numerical weather-driven models (CIMSiM, the Container-Inhabiting Mosquito Simulation Model, and DENSiM, the Dengue Simulation Model) written in function of the site-specific observed air temperature and precipitation. In accord with Focks and Barrera (2006) [37] , whereas CIMSiM is essentially an accounting program of vector dynamics, DENSiM performes the dynamics of human population and virus transmission between hosts and vectors. Both being weather-driven stochastic models, with a daily time step, that is usually parametrized by the correlations between observed monthly cases and lagged cases, monthly average temperature, monthly rainfall, length of the gonotrophic cycle, in days, and the extrinsic incubation period (EIP), also in days, for the site-specific Dengue [37] .

Otero et al. (2006) [23] described a dry weather model for Dengue dynamics, with temperature-dependent effects, without considering rainfall effects, directly.

Morin et al. (2010) [38] described the Dynamic Mosquito Simulation Model (DMSM), with rainfall and temperature-dependent effects, applied to a Central America tropical city.

Luz et al. (2003) [39] discussed theoretically some uncertainties to the development of Dengue models for Rio de Janeiro-RJ, Brazil, concluding that the two most important entomological parameters to be estimated in the field to compare with a numerical model of Dengue epidemics are mortality rate (of humans) and the extrinsic incubation period.

Sporleder et al. (2009) [40] proposed the Insect Life Cycle Modeling (ILCYM), a software for developing precipitation and temperature-based insect phenology models with applications for regional and global pest risk assessments and mapping, and

Lana et al. (2014) [41] with base in statistic tests with a set of prototype vector models, reccomending that future dengue models should investigate the heterogeneous levels A. aegypit of eggs through the seasons of successive years, as it may affect virus development along the years and also the occurrence of further epidemic outbreaks.

In view of the present state of development of modeling of Dengue epidemics, the following characteristics were chosen for the numerical model described in this work:

• assimilation of both variables, precipitation and air temperature, to compute the entomological parameters. In the same way, other meteorological surface variables such as the incoming solar radiation flux, the air humidity and pressure were assimilated by the Newtonian relaxation method (i.e., nudging), throughout the integration time;

• metabolic factor in function of accumulated thermodynamic variables (i.e., enthalpy, entropy, degree-day index);

• an human reinfection reservoir to implement a positive feedback in the epidemic simulation;

• four virus serotypes reservoirs and associated viremia effecting reinfection parameters and the human mortality rates due to Dengue infection; and

• free software (written in f90) and free distribution.

Consistently in the proposed model, the human mortality rate was estimated from the reinfection rate, so that this last variable can be compared with the notifications of Dengue observed. Also the metabolic factor can be activated by level of the accumulated enthalpy along the months, triggering the reproductive behaviour of the mosquitoes.

A epidemic reinfection model in function of available enthalpy

In this work a model for a infectious disease, based in a system of non-linear differential partial equations, correspondent to the temporal evolution of the dynamics of population during epidemic cycles of Dengue was implemented. The numerical implementation was done in the Fortran-90. Additionally, the assimilation of observed atmospheric data, the role of precipitation, and the effects of feedback associated with reinfections were taken in account.

The atmospheric data (air temperature, precipitation rate, specific humidity, direct and diffuse shortwave radiation flux and air pressure) were prepared and formatted as XYZ.DAT files for a 4D assimilation, imple- mented by Newtonian nudging.

The equations system was divided in three population reservoirs, which in turn were subdivided in boxes. The forecasting variables are relative fractions of the total populations, therefore the unity here represents the pandemic that has spread through the whole population and area.

Dengue virus dynamics equations

The virus population dynamics was based in the theory of exponential growing of Malthus (1798) and also in logistic growing described by Verhulst (1845) [42] . The former reference was mainly applied in population dynamics over effect of different viruses, and the later one, supposes that temporal variation of the number of infected individuals is proportional to the amount of infections, with the variation rate proportional to the fraction of non-infected population, i.e., in accord with a time scale in function of the incubation period.

The model consider four serotypes of Dengue virus, i.e., DENV-1, DENV-2, DENV-3, and DENV-4, indicated by the suffix ()_i in the following equation. The time of entry of each serotype also was considered in the model, set up to MARJ, in the years 1986, 1990, 2001, and 2011, respectively. In the model, the dynamics of virus population is controlled by the following proposed equation,

where and are consistency weights representing the increasing immunization after the time period after the introduction of serotype ()_i. is assumed equal to 700 days.

The first term in the r.h.s of Equation (5) represents the logistic development of the virus in the nervous system, intestine and glanders of mosquitoes, associated to the extrinsic incubation period (EIP). The second term was proposed to represent the increase of viremia associated to the infection and re-infection in the intermediate and final hosts; and, finally, the third and forth terms were introduced to represent a small effect associated with random mutation from a serotype into other.

The parameter controlling mutation from serotype j to i that is expressed by where is a random number with uniform distribution in the range.

The air temperature was used to control the rate of incubation of the virus in the mosquitoes. In accord with the laboratory experiments of Watts et al. (1987) [43] the DENV-2 virus can be transmitted to monkeys only by Aedes aegypti was retained at 30˚C or more for 25 days. Indeed, the extrinsic incubation period was 12 days for mosquitoes at 30˚C, and was reduced to 7 days for mosquitoes incubated at 32˚C and 35˚C. Consequently, to represent the temperature-dependence of the extrinsic incubation in the present work, the parameter () was considered as a function of the incubation temperature of the mosquitoes, given by the following expression

where is the environment temperature, in (˚C).

The virus quantification in laboratory involves counting the number of viruses in a specific volume of blood to determine the virus concentration. In accord with Focks and Barrera (2006) [37] , studies comparing the consequences of viral titre (i.e., virus quantification) on the dynamics of endemic dengue suggest that titre, through the agency of probability of dissemination and EIP, does indeed play a role (by hypothesis).

Focks et al. (1995) [36] evaluated virus titres between 10⁵ (i.e., a low value with partial impact on the population) and 10⁶ (i.e., a high value with very broad impact), in relation the median infective dose ().

To update of the virus population in the numerical solution, we apply the Euler’s method in finite differences (Mesinger and Arakawa, 1976) [44] , written as

where is the indicator function of unit that is equal to 1 if the DENV-i is present, and 0 otherwise.

Mosquitoes Population Dynamics Equations

The population dynamics of mosquitoes is described from the differential equations system for the relative population dynamics of the Aedes aegypti in the aquatic and winged phases. This system allows to represent the interrelations between the different phases, in which the metamorphosis stages are related to each other throughout the different phases (i.e., the discriminant compartments of the mathematical model).

The representation in form of differential partial equations system considers different population compart- ments, that represent the successive stages of metamorphosis during the mosquito life cycle.

The equations depend on the entomological parameters, as well as environmental parameters (e.g., Yang et al., 2009 [45] ). In this work, the entomological parameters were established in function of the temperature, as usual, but also in function of the estimated water availability into the breeding sites, from observed rainfall.

In the proposed model, the mosquitoes population dynamics equation are given by

where:

(E) is the relative population of eggs, i.e., the number of eggs divided by the maximum number of eggs supported by the environmental condition.

The dynamic of the eggs population depends of the oviposition rate, , which is associated with the gonostrophic cycle, fertility of female mosquitoes, and the opportune rainfall occurrences.

is the rate of eggs eclosion, from eggs to larvae The number of eggs decreases in function of a specific mortality rate, , in usual units of (day⁻¹).

(L) represents the relative number of larvae, written in function of the eclosion rate, , a parameter controlling the pupation (i.e., the transformation of larvae into pupae), and, the larva mortality rate, in usual units of (day⁻¹).

(P) represents the relative population of pupae, which is function of the pupation rate, the rate of metamorphosis of pupa into winged mosquitoes (i.e.,), and, the pupae mortality rate, in usual units of (day⁻¹).

represents the relative number of non-infected mosquitoes, in usual units of (day⁻¹).

represents the relative number of exposed mosquitoes, of non-infectious Aedes aegypti, in usual units of day⁻¹.

represents the relative number of winged infectious mosquitoes, after the extrinsic incubation period of exposed wings to the Dengue virus, in usual units of day⁻¹.

is the relative number of mosquitoes Aedes aegypti, i.e.,.

Additionally, the following parameters are employed.

is the rate of infection of succeptible mosquitoes by biting infectant humans, in usual units of day⁻¹.

, the rate of transformation of into, i.e., the inverse of the extrinsic incubation period, in usual units of day⁻¹.

is the rate of mortality of adult mosquitoes, in usual units of day⁻¹.

The larval behavior is controlled in Equation (8) by parameter, that depends of both the available enthalpy of the environment and of the rainfall distribution (i.e., its intensity and frequency). As proposition of this work, is parametrized by the following

The first factor in the r.h.s. of Equation (9) represents the effect of the capacity of the breeding site, considered the minimum necessary volume of water to support larvae development. The variable, in (m), is the height of the water into breeding site, obtained from the integration of the balance of water for the linear reservoir of surface, given the atmospheric precipitation. The scale is the optimum value of before spill. The second term represents the basal metabolism control by the environment change of enthalpy, here proposed by the following expression

where the parameter was tuned out to the period of late summer in Rio de Janeiro. The first

factor in r.h.s. of the BM equation is used to represent the effect of the energy of environment that is favourable to the fertility of the females of Aedes aegypti. The second effect is used to account of the oviposition of the previous year, associated with the intensity of the accumulated DG. When the oviposition was smaller in previous year, the possibility of a Dengue surge in the present year is decreased. The third factor is used to express the accumulation of energy in joules per kg and time step.

The development of the larval breeding depends on two parameters: the volume of nutrient water in the breeding and the total number of larvae present. Thus, an appropriate variable to larval development estimated rate is the concentration of larvae per unit volume of water in units (larvae×m⁻³), or its reciprocal, which is the specific volume of nutrient water divided by the number of larvae present, that is, the volume available to the liquid nutritional development of each larva in (m³×larva⁻¹).

In the Equation (9), represents the height of the water accumulated in the reservoir of a typical larva breeding site, and is its optimum value, generally achieved before spill, which occurs when the rainfall implicates. In a linear reservoir representation the ratio of water heights is equal to the ratio of present and optimum volume of water in the breeding, i.e., since, with is the mean cross section area of the breeding site, constant by simplicity.

The specific volumes suitable for larvae breeding can be written as

where is the total number of breeding sites in the city, and is the optimum population of larvae (integer number). The normalized difference between the larvae specific volume (), in (m³×larva⁻¹), and this optimum value (), i.e., that can be shown to be equal the Equation (9), (dimensionless).

The available water volume and feeding reservoir can constrain the larvae development. The minimal water volume for supporting the maintenance of larvae in the breeding site is approximately,

associated with a concentration between 46 to 60 (larvae per litre), as the optimum. The effect of the water volume in simulated breeding sites is the following: if there is too much larva (higher them 46 to 60 larva per litre), the value of becomes too small (close to zero). In this case, the larva cannot survive the competition for food and space. Thus, the interruption of the cycle of the Aedes aegypti can be emulated by the simulation. Consequently, the maximum number of larvae into a breeding has a upper limit given by the maximum volume of the reservoir.

Epidemic model on human population

The Epidemic model on human population, Equation (13), comprehends five compartments of individuals in different moments of the infection by the Dengue virus (i.e., in a similar way the exemplified models by White et al., 2010). Here, we propose the coexistence of five classe: exposed population, infected population, infected population with transmission capacity, and recover population. Additionally, in the present model, we have a compartment for the re-infected fraction of population, , generally re-infected by a different serotype.

The equations are inter-related and time dependent. The progression of the infection can be controled by the following model parameters:, the frequency of recuperation after a primary infection;, the period of intrinsic incubation of the virus in the human population; and (), the mean rate of mortality of population for non-epidemic years. Additionally, the parameter establishes the rate of recuperation after a re-infection by a new Dengue serotype.

The dynamic of population under dengue epidemics can be described by

The prognostic model variables for human population, in usual units of (day⁻¹), are:

is the relative fraction of population suceptible to a primary infection by Dengue;

is the relative fraction of population exposed to the infection;

is the relative fraction of population composed by infectious individuous;

is the relative fraction of population reinfected by a second type serotype;

is the relative fraction of population recuperated from Dengue infection;

is the category of individuals that died due to Dengue fever infection.

The parameters of Equation (13), in usual units of day⁻¹, are

is the rate at which susceptible individuals become exposed;

is the rate at which exposed individuals become infected;

is the rate at which infected individuals become recuperated;

is the susceptibility rate after infection, reinfection or recuperation;

is the mortality rate of susceptible individuals,;

is the mortality rate of exposed individuals,;

is the mortality rate of the primarily infected individuals;

is the mostality rate of reinfected individuous;

is the rate of mortality of recovered individuals,;

The immunization effect after a epidemic is parametrized by

The summation of the equations in the system (13) implies the cancellation of many opposite terms. The obtained equation can be expressed in terms of the total population (T), as follows

This equation states the physical consistency associated to conservation of population. In the other hand, is the condition of a closed population model. Since that, the system can be closed for obtaining s as a residue.

Metabolism parametrizations

Yang et al. (2009) [45] he has presented polynomial fit curves of third and sixth order to represent rates of natality, metamorphosis and mortality of Aedes aegypti as a function of the temperature of the breeding site. Unhappily, we cannot use directly these curves for numerical modelling purpose because they diverge from expected values under hypo and hyperthermia, i.e., outside the range of considered observations (»8.5˚C to »35.0˚C). As in the MARJ, the observed maximal air temperature is frequently above 35˚C [46] [47] , due to the Urban Heat Island and the anticiclonic weather condition, the practical alternative is extending the limit range of application of the curves in proper way (i.e., considering the prior technical knowledge on hypothermic and hyper- thermic inactivation effects [12] ), as have been suggested by Morin (2012) [48] and Morin et al. (2013) [49] .

Among others, Otero (2006) [23] has considered the metabolic rates of the mosquitoes in function of enzymatic activity, that is, depending on the poikilothermic enzymatic activity, in units of day⁻¹, following the equation proposed by Sharpe and DeMichele (1977) [50] and Schoolfield et al. (1981) [51] .

The numerical realizations considered in the present work have tested both parametrizations represented by polynomial fits [45] and Sharpe-Schoolfield’s performance equation [50] [51] . The parametrization of poikilo- thermic enzymatic activity seem more appropriate for the simulations since can be really used outside from the optimum range of temperatures, and the model approximates more of the observations.

Sharpe-Schoolfield’s equation

This subsection describes the parametrization used to represent the relative performance of ectotherms in function of the environmental temperature of breeding sites. The temperature of breeding sites is supposed to be a non-linear function of the external air temperature. The breeding sites are considered as a opened cavity to the environment, able to exchange the metabolic gases, thermal energy and water vapor by conduction and convec- tion, and acting as a reservoir for a fraction of the intercepted rainwater. The enzymatic activation/deactivation of maturation and other basal metabolisms can be associated to the relative performance of the ectotherms by equations of reaction kinetics rates (), in a similar way to the proposition of Sharpe and DeMichele (1977) [50] for poikilotherm development, of Van der Have and De Jong (1977) [23] for the ectotherms in general, and also applied by Otero et al. (2006) [52] for a numerical investigation of the Aedes aegypti’s ovoposition, for a metropolis of subtropical climate.

The basal reactions kinetics rate, in units of (day⁻¹), associated with the optimum temperature for activation/sativation reactions, in (K). is the low level thermodynamic enthalpy () associated with a low temperature for half inativation of reations ();is the high level thermodynamic enthalpy for the high temperature for half inactivation reactions ().is the enthalpy level under optimum conditions ().

The Sharpe-Schoolfield’s equation (Schoolfield et al., 1981) [51] can be rewritten using the formalism of a Bayesian posterior, such as the posterior distribution of the parameters (e.g., as those of kinetic reactions) can be obtained by modifying the prior distribution () by multiplication with the likelihood distribution of parameters (evidence). Therefore we write

where

Generally, and. Considering the rule, i.e., for each 10˚C should the performance increases by twice (or three times), and given the values of, and, then we can rule estimate the enthalpy change due to the lowering the performance by half of this optimum by the following expression

It is common in the thermal vicinity to approximate Equation (19) by (Salon et al., 1987) [53] , specially when the linearity of rule can be supposed valid beyond the lower extreme of the range of temperatures (e.g., under reduced survivor-ship and development rates for the eggs and larval stages, peeked until favourable environmental conditions take place), or, indeed, for some asymmetrical distributions to high values, , e.g., for pupa to wing development time (Otero et al., 2006) [23] .

The enthalpy in (J×mol⁻¹) refer to the heat released or absorbed per unit of mole during a metabolic reaction, completed under thermal optimum conditions. However, the enthalpies and occur under less favourable conditions to the metabolic support. Therefore, the enthalpy of the enzyme kinetic reactions of the animal specimen refers to the transfer of energy that accompanying a change of state, from a reference state to a further metabolic state.

The values of enthalpy and temperatures applied here are shown in Table 1.

Bayesian statistics have been used to build robust framework for parameter estimation and uncertainty analysis in dynamical models for epidemics forecasting (Coelho et al., 2008, 2011) [54] [55] . The quality of Bayesian statistics increases with of number of samples (i.e., observations) used to defined the Likelihood, and, by product with the prior, to achieve the posterior distribution of parameters.

The Sharpe-Schoolfield’s equation [51] can be interpreted in terms of the Bayesian model statistics (i.e., as a theoretical expression of Bayes’s equation). In this case, the prior distribution can indicate the simply fact of metabolic reactions tend to increase with the temperature (e.g., as in Arrenhius’s law). The likelihood informs about the kind of distribution of the model parameters around the thermal optimum. In the Sharpe-Schoolfield’s equation, the likelihood is due to the empirical evidence, the performance have typically a Normal distribution around the optimum. The posterior distribution in function of the temperature is proportional to the product of the prior by the likelihood distribution of the parameter, resulting that Sharpe-Schoolfield’s equation follows an adjunct distribution of the exponential family, associated with a weak informative prior. Consistently, the asymmetry between the low and high temperature of a half deactivation/activation of enzyme reactions, for each site of the optimum, would result in some (positive or negative) asymmetric distribution of performance.

On the other hand, Rall et al. (2012) [56] have suggested that the temperature dependence of functional response parameters is more complex than simples Arrhenius terms, which is basically a expression of tempera- ture dependence. Thus, the biological rates depend not only on temperature, as all chemical reactions, but also on body mass. In Metabolic Theory of Ecology the Arrhenius equation is given by

where (eV) is the activation energy, k (eV∙K⁻¹) the Boltzmann constant, T (K) is the absolute temperature, (K) sets the temperature for metabolic activation, and m is the body mass.

In the case of interaction between the dengue virus, the insect vector and human host, a minimal virus critical mass seems necessary to trigger an outbreak, since the mass of an isolated virus is extremely small in com- parison to the insect vector masses and the human host. Thus, the insertion capacity and virus proliferation in the human host will likely depend on extrinsic incubation the virus vector, which in turn, is a function of ambient temperature. This effect is implicitly considered in Equation (6).

Mortality rate for Aedes aegypti

In this work, mortality rates for different insect life stages Aedes aegypti were parameterized following the proposition Hosfall (1955), Bar-Zeev (1958) and Rueda et al. (1990). Table 2 shows the mean values of the parameters used in the model to the conditions found in the MARJ.

Open source model

The idea is that the model can receive user community contributions in a dynamic and participatory process of continuous development. Therefore, the model code is proposed as open source, available in the Internet through the Internet site of the Laboratorio de Hidrometeorologia Experimental, located in the Institute of Geosciences of the Federal University of Rio de Janeiro (IGEO-UFRJ).

Basic reproduction number

The basic reproduction number, , is a measure of the outbreak potential, being defined by the number of secondary infections produced after the introduction of an single index case (i.e. a single serotype) as observed

at the end of a extrinsic incubation period of the vector, in a large crowded conurbation (Massad et al., 2001 [58] ; Luz et al., 2003 [39] ).

There are some different estimations of in the literature. For instance, Coelho et al. [54] have suggested the treatment of model uncertainties to obtaining a Bayesian posterior estimation of (), given the observa- tional evidence; Khan et al. (2014) [59] estimate for a single-strain Dengue Fever epidemics, using the model evidences (i.e., prognostic variables); and Luz et al. (2003) [39] also proposed a method to estimate for Rio de Janeiro based in a Markov chain analyse and quite uninformative prior distributions.

Interestingly, Luz et al. (2003) [39] considered the risk of dengue transmission in humans proportional to density of infected vectors into the epidemic areas of lower l and high h densities. Hence, () is computed starting an infectious period in the areas of low and high vector densities, called l or h. During a full incubation period, a fraction of population commutes between the two areas with daily probability. The expected number of secondary cases produced during the entire infectious period () can be estimated by a Markov chain method as the summation on the areas of the expected daily number of secondary cases produced times the time spent in each area by infected people,. This yields,.

On the other hand, considering a model of statistical correlations, Luz et al. (2003, 2008) [39] [60] have applied box-plots in comparisons of the simulated parameter performance that lead to prediction of Dengue fever epidemics outbreak (when) and disease extinction (), for viremia in humans, mosquito mortality, and mosquito biting rate.

In the present work, we estimated simply by counting the increasing number of infections in a fortnight.

Breeding sites modeled by linear reservoirs

Christophers (1960) [12] characterizes the cycle of development of the Aedes aegypti mosquito, vector of the yellow fever and of the Dengue, as well as discusses qualitatively the factors of its survival, among them temperature and precipitation. These factors are classified as associated to, first, the daily cumulative develop- ment (CD), which explicitly modify the Aedes aegypti’s metamorphosis in function of the air temperature and surface precipitation, and second, the maintenance factors during the different phases of the insect life. The metamorphosis depends on the presence of water.

In particular, the oviposition and eclosion of the eggs, as well as the survival of the larva, which, after going through the pupation process achieve the stage of maturation, where the metamorphosis into pupa occurs. In all the three stages of metamorphosis mentioned above, it is known that the minimum water may be accumulated from precipitation, irrigation or accumulation for human consumption in the urban reservoirs and must be 10 (mm) or higher in height.

The Surface Energy Balance (SEB) is fundamental to the determination of the temporal evolution of atmospheric variables at surfaces (Oke, 1988) [18] , particularly in cities, where urban surface topography and land cover is able to define different partitions of energy, defining different urban microclimates (e.g., as observed in down- towns with high buildings, green parks, concrete surfaces of parking, roads and even microclimates within urban dwellings). Part of net flux of energy is stored by the urban materials and buildings, providing a 24 hours cycle of hysteresis between temperature and the net flux of radiation (Ferreira et al., 2013) [61] .

According to Piovezan (2009) [62] , the predominant breeding sites for the development of the aquatic phase of the mosquito are presented in three types: dishes and pots for plants, cans, pots and flasks, and other types of removable receptacles. Additionally, it is interesting to include in this list: open water tanks, abandoned pools, big cans, flower pots in cemeteries [23] , refuse dumps etc.

Hypothetically, places considered prone to mosquito aquatic phase development are those subject to recept a partial flux of incoming water during precipitation vents (i.e., a smoothed flux), with thermal conditions around to the most favourable thermal conditions (i.e., close to the optimum), obtained with the Mosquito’s oviposition preference by partially shadowed sites. With that in mind, the breeding sites in the MARJ most propitious to the development of the aquatic phase of Aedes aegypti are those under partial shadowing and with only partial interception of rainfall.

Once the reservoirs of water of the urban breeding sites are essentials to the aquatic phase of the mosquito behavior, the data analysis of precipitation and relative humidity were accomplished in detail. The goal was quantifying the model sensibility in relation to these variables. As result, we observed that a large number of weak to moderate rainfall events, alternated with hot sunny periods, can be most favourable to aquatic develop- ment of larvae than the occurrence of some few strong rainfall events of equivalent height of water. By hypothesis, with rain interspersed with periods of sun the breeding sites can lose water due to evaporation, but not in the potential rate due to the partial shadowing conditions inside them.

A Linear Reservoir Model (LRM) was used to represent the average dynamics of breeding sites. To supply the water for breeding reservoirs the atmospheric precipitation was assimilated by nudging. The lost of water due to evapotranspiration and leaks was also represented in the modelled breeding sites. The loss is parametrized by the sum of the estimated Potential Evapotranspiration (PET) plus a smaller amount reservoir leaking, in the form of a simplified Surface Water Balance (SWB) (Grimmond et al., 2002 [63] ; Karam et al., 2010 [47] ).

The Potential Evapotranspiration (PET) was obtained with the known Penman’s Combination Method (Penman, 1948 [64] ; Thom e Oliver, 1977 [65] , and Monteith and Unsworth, 2007 [66] ). The calculation of the PET depends on the data assimilation of the net irradiance flux ().

The LRM used in this work requires a small number of parameters, scale of time of the response time of the reservoir, the ratio of average loss due to leakages, horizontal surface area exposed to rain, and maximum reservoir capacity. A comprehensive introduction about the Surface Energy Budget (SEB) and WSB can be found in Oke (1988) [18] . The specificities of the SEB of different cities have been discussed (e.g., Karam and Pereira Filho (2006) [67] , Lemonsu et al. (2007) [68] and Masson et al. (2014) [69] ). In hydrology, the application of LRM is well known (e.g., Nasch, 1959 [70] ; Beven, 2001 [71] ; and Brutsaert, 2005 [19] ).

In this work, for representing the urban breeding sites we use the LRM and the PET, provided modelled data of the net irradiance flux (Flores Rojas et al., 2015) [32] . In a way similar to doing in simulations of WSB to natural surfaces, for the urban breeding sites the variation of the water volume along the time was given in function of the observed influx and lose of water. The modelled recharge occurs during atmospheric precipi- tation events due to the interception, modelled by a small fraction of the total rainfall, estimated near to 1% for Rio de Janeiro metropolis. Due to the shadowing condition of the breeding sites, it is also considered a fraction of the PET calculated (i.e., equal to 60%) to simulate the evaporative conditions found in the breeding sites.

The idea of using linear reservoirs to simulate the breeding site is well known in the literature. For instance, a application of SWB and SEB for modelling breeding sites for larvae and pupae development, in function of the observed temperature and precipitation, was comprehensively described by Sporleder et al. (2009) [40] . In this case, the researchers obtained the spatial and temporal distributions of the vector population on Australia territory, in a GIS platform. Differently, no rainfall was considered in Otero’s temperature-dependent model for model the summer surges of Dengue fever, for Buenos Aires city, in Argentina, where the climate is subtropical, with cold and rainy weather in winters (Otero et al., 2006) [23] . In the Otero’s model, the oviposition presents hysteresis behaviour to the seasonal variation of the air temperature. That is not the expected case of a tropical city as the Rio de Janeiro’s metropolis, where the diurnal variation of temperature is commonly larger than the seasonal variation (i.e., as typical of a tropical climate city).

Goodness-to-fit statistics

Lana et al. (2014) [41] have applied several goodness-to-fit statistics to verify their model performance, e.g., statistics based on the log-likelihood functions for transition probabilities of fitted Markov chains, e.g., as the Akaike Information Criterion (AIC).

In this work, a set of statistical estimators were applied to verify the model performance and tuning the model parameters. These scalar accuracy measures are the Mean Absolute Error (MAE), the Mean Squared Error (MSE), the Root Mean Squared Error (RMSE), the Coefficient of Determination R², and the Nash-Sutcliffe Efficiency (NSE), as recommended by Nash and Sutcliffe (1970) [72] , Singh et al., (2005) [73] , Moriasi et al. (2000) [74] , and Wilks (2006) [75] .