^{1}

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In this work, we present results of an investigation of environmental precursors of infectious epidemic of dengue fever in the Metropolitan Area of Rio de Janeiro, RJ, Brazil, obtained by a numerical model with representation of infection and reinfection of the population. The period considered extend between 2000 and 2011, in which it was possible to pair meteorological data and the reporting of dengue patients worsening. These data should also be considered in the numerical model, by assimilation, to obtain simulations of Dengue epidemics. The model contains compartments for the human population, for the vector
*Aedes aegypti* and four virus serotypes. The results provide consistent evidence that worsening infection and disease outbreaks are due to the occurrence of environmental precursors, as the dynamics of the accumulation of water in the breeding and energy availability in the form of metabolic activation enthalpy during pre-epidemic periods.

Dengue fever is an acute viral disease of fast dissemination, featured by an infirmity that is caused by the Dengue fever virus, an arbovirus of the family Flaviviridae, genus Flavivirus, which comprises 4 immunological types (i.e. serotypes) which are activelly circulating among humans (DENV-1, DENV-2, DENV-3 and DENV-4) [

In Brazil, Dengue fever infections have been reported since 1986 [

The primary infection caused by any one of the serotypes can provide permanent protection against reinfec- tion by the same serotype and temporary protection against the remaining serotype [

According to the 2005 technical relatory from the Brazilian Wealth Surveilance Service (Serviço de Vigilân- cia em Saúde) [

The viremia period on humans is of seven days, being shorter on other primates [

The Dengue vector is the Aedes aegypti mosquito that has two distinct phases in its life cycle: aquatic, which comprehends the stages of egg, larva and pupa; and winged, also known as adult phase [

When the mosquito transmits the disease to a human, through the inoculation of the virus, an incubation period called viremia starts in the human organism. After the contamination, the virus remains in the human blood from the day before the fever to the sixth day of disease. In this period, a non-infected female Aedes aegypti that bites the infected human, also contaminates itself, initiating the extrinsic incubation process [

The life cycle of the Dengue vector is well known. For a extensive description see Christophers (1960) [

In a summarized form, only the virus infected female Aedes aegypti mosquitoes are able to transmit the disease to men. The specificity of females as a vector of the virus is due to the haematophagous nutrition necessary to mature the eggs [

According to Brigdman and Oliver (2006) [

In the present work, a numerical model was proposed in the form of a set of coupled non-linear partial differential equations used to simulate the epidemic dynamics in the human population, vectors and virus. The model equations were designed to permit representation of Dengue fever infections, as also the assimilation of atmospheric observed data, including air temperature, pressure, umidade, precipitation, shortwave irradiance, etc. The simulated period was between Jan./2000 and Dec./2011, when Dengue fever epidemics were observed on the Metropolitan Area of Rio de Janeiro (MARJ).

The distribution and intensity of the precipitation events in a tropical location are important during the dengue epidemics development. A complex relationship exists between the rainfall occurrences and the Dengue vector development such as exemplyfied by Campos (2009) [^{−1}) are able to remove by washing the eggs from the walls of the breeding, hence- forth limiting the epidemic occurrences.

The dataset used in this work were obtained from surface weather stations located at the international airport of Rio, for 12 years (2000-2011). Also data from the public health service of Brazil (called SUS) were used. The SUS data are records of notifications of patients who were diagnosed with dengue fever. The frequency of the dengue notifications by SUS was of fortnight in the used period. Additionally, atmospheric data such as the density of the flux of global solar radiation (i.e., the shortwave irradiance) has been considered to estimate the potential evapotranspiration [

The efficiency of a prognostic model for Dengue epidemic dynamics can be evaluated by comparing the predicted number of re-infections with the number of reported notifications by SUS.

The results obtained seem suggest the atmospheric thermodynamic variables such as the rate of variation and accumulation of the environmental enthalpy can be assimilated in a epidemic model of Dengue, as a way to make estimations about the population of vectors, in such a way, that epidemics outbreak can be predicted seasonally for a tropical metropole, as MARJ in Brazil.

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) [

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) [

According to Murphy and O’Neill (1997) [

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) [

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 ^{st}, actualized each year. So we can write

where ^{−1}), and defined by the difference between the actual (

sented by^{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 (

where ^{st} for each year in the simulation period.

• Atmospheric enthaphy change―Formally the enthaply (H) is the sum of the internal energy (U) plus the work (W), i.e.,

since the state relation can be written as

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

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

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., ^{st} (winter), and ending at January, 31^{th} (summer), given by

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.,^{−1}) applied to Equation (3). The result is an estimation of the mean daily enthalpy variation, given by

is comparable to the average value of variation computed from observations to Rio de Janeiro city (

There are some investigations on the epidemics precursors. For instance, a comprehensible one was provided by Descloux et al. (2012) [

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.

Develop. cycle | ||||||
---|---|---|---|---|---|---|

Egg hatching | 0.2400 e | 10798 e | 100000 e | 288.15 * | 311.15 * | |

Larval develop. | 0.2088 e | 26018 e | 55990 e | 284.15 # | 304.6 e | |

Pupal develop. | 0.3840 e | 14931 e | 472379 # | 284.15 # | 308.4 # | |

Gonostrophic cycle | 0.3720 e | 15725 e | 175648 # | 284.15 # | 314.2 # |

References: # This work, e Otero et al. (2006) [

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) [

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) [

• 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) [

• 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) [

Otero et al. (2006) [

Morin et al. (2010) [

Luz et al. (2003) [

Sporleder et al. (2009) [

Lana et al. (2014) [

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) [

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 _{i}.

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

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) [

where

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) [

Focks et al. (1995) [^{5} (i.e., a low value with partial impact on the population) and 10^{6} (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) [

where

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 [

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,

^{−1}).

(L) represents the relative number of larvae, written in function of the eclosion rate^{−1}).

(P) represents the relative population of pupae, which is function of the pupation rate^{−1}).

^{−1}).

^{−1}.

^{−1}.

Additionally, the following parameters are employed.

^{−1}.

^{−1}.

^{−1}.

The larval behavior is controlled in Equation (8) by parameter

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

where the parameter

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 ^{−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 (m^{3}×larva^{−1}).

In the Equation (9),

The specific volumes suitable for larvae breeding can be written as

where ^{3}×larva^{−1}), and this optimum value (

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

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:

The equations are inter-related and time dependent. The progression of the infection can be controled by the following model parameters:

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:

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

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,

Metabolism parametrizations

Yang et al. (2009) [

Among others, Otero (2006) [^{−1}, following the equation proposed by Sharpe and DeMichele (1977) [

The numerical realizations considered in the present work have tested both parametrizations represented by polynomial fits [

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 (

The basal reactions kinetics rate^{−1}), associated with the optimum temperature

The Sharpe-Schoolfield’s equation (Schoolfield et al., 1981) [

where

Generally,

It is common in the thermal vicinity

The enthalpy ^{−1}) refer to the heat released or absorbed per unit of mole during a metabolic reaction, completed under thermal optimum conditions. However, the enthalpies

The values of enthalpy and temperatures applied here are shown in

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) [

The Sharpe-Schoolfield’s equation [

On the other hand, Rall et al. (2012) [

where ^{−1}) the Boltzmann constant, T (K) is the absolute temperature,

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).

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,

cte. | Name | Value or reference | Units |
---|---|---|---|

1 | Yes | - | |

2 | (s^{−1}) | ||

3 | 1.1574e−8 | (s^{−1}) | |

4 | 5.787e−8 | (s^{−1}) | |

5 | 0.000000012 | (s^{−1}) | |

6 | 0.000007720 | (s^{−1}) | |

7 | 0.000002512 | (s^{−1}) | |

8 | Area of breeding | 0.050 | (m^{2}) |

9 | Breeding height (max.) | 0.050 | (m) |

10 | Leak timescale for breeding | 2592000. | (s) |

11 | Intercepted rainfall by breedings | 1.0 | (%) |

12 | Total breeding sites number | 2500 | - |

13 | Human life expectancy | 2075068800.0 | (s) |

14 | 2.0 | - | |

15 | 4.977e−6 | (s^{−1}) | |

16 | 4.24e−8 | (s^{−1}) | |

17 | 9.260e−8 | (s^{−1}) | |

18 | 1.1574e−14 | (s^{−1}) | |

19 | 1.1574e−12 | (s^{−1}) | |

20 | 1.0000e−7 | (s^{−1}) | |

21 | DENV-1 (arrival year) | 1986.0 | - |

22 | DENV-2 (arrival year) | 1990.0 | - |

23 | DENV-3 (arrival year) | 2001.5 | - |

24 | DENV-4 (arrival year) | 2010.0 | - |

at the end of a extrinsic incubation period of the vector, in a large crowded conurbation (Massad et al., 2001 [

There are some different estimations of

Interestingly, Luz et al. (2003) [

On the other hand, considering a model of statistical correlations, Luz et al. (2003, 2008) [

In the present work, we estimated

Breeding sites modeled by linear reservoirs

Christophers (1960) [

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) [

According to Piovezan (2009) [

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 [

The Potential Evapotranspiration (PET) was obtained with the known Penman’s Combination Method (Penman, 1948 [

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) [

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) [

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) [

Goodness-to-fit statistics

Lana et al. (2014) [

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 R^{2}, and the Nash-Sutcliffe Efficiency (NSE), as recommended by Nash and Sutcliffe (1970) [

The analysis of atmospheric observations and Dengue fever notifications were performed, as well as, simula- tions and parameter tuning.

First, a simple statistical analysis was accomplished on dengue notification data from SUS.

The major epidemic events in the analysis period are shown in

The threshold number for establishing the beginning of epidemics seems to be very small. The variability of notifications (

The three bigger epidemics of Dengue in the period 2000-2011, occurred in the years of 2002, 2008 and 2011 (

Based in absolute values of number of Dengue notifications, the month of March is the month with biggest number of cases, followed by February, April and January. Although the month of March presents a larger absolute value in the notified Dengue cases, the month of April shows a larger epidemic volume, occurring a decline in the number of cases from April on.

Dengue shows a seasonal aspect, with a cyclic tendency of the epidemic varying from August to April, suggesting that the monthly distribution of the cases varies in function of the seasons of the year, with a probable chronological delay addressed to the period comprehended between the mosquito life cycle and the appearance of clinical characteristics in human organisms, this delay lasts an average of 40 to 45 days.

The shape of the envelope shown in

It is noteworthy the favorable conditions for dengue surge in years 2014 and 2015 in the Brazilian Southeast, associated to a very hot and relatively dry summer in 2014. In association to the recovering of precipitations in summer of 2015 a strong increase of Dengue was verified in State of São Paulo. In the last years, the epidemic character of the Dengue infectious disease seems to be changed to endemic in many areas of Southeast Brazil.

Summer is the season of the year with the largest number of registered Dengue cases, since the increased temperatures, as well as the distribution and major frequency of accumulated rainfall water, that generally occurs in this period, can favor the development of the mosquito biological cycle and the consequent pro- liferation of the mosquito.

Whereas in summer the egg takes 10 days to reach the adult stage, in the winter this time increases to 30 days, due to the major frequency of cold and dry weather.

A low number of occurrences were notified in the years 2004 and 2005, while, differently, in 2002 and 2008, when a higher number was notified, resulting in a larger accumulated epidemic volume. The increase also occurred in 2011, and also in 2015. In this later year, the epidemics followed a very hot and extremely dry year in part of Southeast Brazil (i.e., during 2014), with particularly severe effects in the hydrologic budget of State of São Paulo.

There is a seasonal tendency of the epidemics for all the administrative subdivisions of MARJ, with the start of the epidemic cycle in the month of September and the ending in March or April. One can observe a signifi- cant gradual decrease of the number of cases during the following months, present along the microclimates of MARJ.

Two fundamental factors for the epidemic seasonality are associated with the annual cycle of weather conditions in MARJ, i.e., hot temperature in the summer and a series of rainfall events to start the aquatic phase of A. aegypti. The temperature presented as a function of MARJ seasons, with a maximum occurrence during Dec/Feb/Mar, frequently.

The results of the simulation is presented ahead. The time step (i.e., time resolution) of the model was 15 minutes. The time integration was accomplished between January-2000 and December-2011.

The model time resolution was set in 15 minutes, considerably greater than the input data resolution (i.e., from dengue notification from SUS, of 30 days). To permit a peer comparison, a simulation output for each 30 days period was also provided (not shown here).

The results for the vector population dynamics are shown in

The simulations generated for the linear hydrological reservoir together with the available precipitable water data were important for the analysis of the model results, because we want to investigate the correlation between Dengue infection data and the distribution of urban rainfall, along the seasons and years.

In the simulation of the virus relative population is presented in

In the simulation, the virus population is dynamically variable, meanwhile the serotypes can maintain a degree of interrelationship, in function of the virus population and incubation temperature.

There is considerable variation in the virus population number (as it appears in the simulation), suggesting that their number can be modulated by the population dynamics of mosquitoes, which in turn is controlled by the enthalpy available as a result of environmental conditions.

A numerical essay was realized to investigate the role of the rainfall intensity. In this essay, the assimilation of only half of observed rainfall is enough for decreasing of the number of re-infected individuals. The results indicated that the distribution of rainfall along the year is also important especially after a long hot spell.

The simulation of the human population dynamics shows two periods in the epidemic cycle, in which the compartment rates shows connections between themselves (in the beginning of 2002 and 2011). Such instants seem to be influenced by the force of the reinfection, as of the introduction of a determined serotype of the virus in the population (i.e.,

Associated with the annual cycle of air temperature on RMRJ, the annual variation of enthalpy can be estimated. The available energy in the environment can be used to maintain and control the mosquitoes phy- siology behaviour. The more favorable condition is applied in reproduction. The environmental accumulation of energy was estimated in function of the accumulated enthalpy (i.e., in a similar way, we can consider the DG or entropy variation).

The estimations from observations indicate accumulated DG values up to ^{−1}∙K^{−1}) for enthalpy change during winter-to-summer period. These values likely can be associated with the occurrence of a definite sequence of hot days, resulting in very favourable conditions for enzymatic kinetic performance and vector reproduction. In this case, the air temperature fre- quently reaches values near or above to 35˚C.

It is noticeable that the preceding months of the years of reduced notification (i.e., 2004, 2009 and 2010) show a reduction in the value of accumulated DG (or enthalpy), avoiding or restriting the epidemic develop- ment.

Therefore it is suggested that years with small accumulated DG, due to a long period of negative perturbation of temperature, is able to prevent development of dengue epidemics, at least under similar tropical conditions as found in the Rio de Janeiro city.

If the enthalpy decreases also decreases the probability of dengue epidemics. This appears have occurred in the years 2001, 2003, 2004 and 2005, when the number of notifications were very small compared to epidemic years.

In the graphic is noticeable the occurrence of high maximum preceding the epidemics of 2002 and 2008, hence it is suggested that the energetic organization driven by enthalpy can be a favourable factor to some epidemic surges (i.e., controlling the gnostrophic cycle of the vector).

The effectiveness of the rainwater catchment by breeding sites was estimated in this model in the form of a parameter (

General discussion

This work started with the investigatory process of the notified cases of hospitalization and death by Dengue, obtained from SINAN-Net and SVS-RJ.

The atmospheric variables were obtained from NOAA-NCDC, referring to observational data of surface meteorological stations in the airports of the RMRJ maintained by the COMAER, and also from the network of mesoscale precipitation observation in the RMRJ, maintained by GEORIO netwrok of rain gauges.

The observational atmospheric data and Dengue notification data were analysed separately and, after that, assimilate in the model for simulations.

The occurrence of correlation between epidemic cycles and atmospheric data was verified.

The simulation allowed a detailed analysis of the population dynamics of living organisms involved in events of the infectious disease Dengue, considering the populations of the virus, mosquito and humans.

The numerical results show the amplitude and phase (lag) in the rates of infection, through the variation of minimums and maximums of the fractions of the population infected (i) and reinfected (ir), during epidemic cycles.

Variables derived from a-biotic observations were also computed for the period of investigation. Among these a-biotic variables are enthalpy, number of rainfall events, water accumulated in reservoirs, etc. Abiotic is not the proper category for the general description of these variables, which are directly related to animal and vegetable physiology, for example, raw primary production and the level hydric stress which plants and animals consider in different environments or moments of their development.

With the introduction of different serotypes in one region, considering the combined effects of reinfections in the population becomes necessary. The viral action of a determined serotype is enlarged by the introduction of other serotype in the previously infected population, as observed in the simulations from the model.

The high temperature is a determining factor in the rise in the number of notified cases of Dengue. The water also contributes to this rise. However the infections are not only modulated by the form of the distribution and intensity of the rains in the Aedes aegypti life cycle, but they were modulated by the accumulation of water in hydrological reservoirs. Thus it is concluded that water is a main factor for the infection, as a maintainer of the minimal quantity for the eggs eclosion, development and survival of the different stages of life of the mosquito.

Regarding the enthalpy and epidemic cycles, it is concluded that energetic biological transformations involv- ing epidemiological cycles follow the laws of thermodynamics and the living organisms involved are thermo- dynamic systems in vivo, therefore demanding constant energy supply; as well as using a large part of this energy in biochemical processes to obtain energy.

It was observed that although the population dynamics of the involved organisms with the Dengue vary concurrently with the temporal-spatial variation of entropy and degree day, being those the determining factors for the biochemical processes of the involved organisms, as the population dynamics presented a good return to available energy in the cycles. The negative rate of accumulated degree days holds high return in the reduction of epidemiological cycles, as well as their maximum amplitudes alongside the epidemic.

Regarding the cycles, it was observed that there is a lag between the simulated and the observed, principally regarding the accumulation of water in the reservoirs of the mosquito breeding sites. This is probably due to the period preceding the infection, as the mosquito eggs can only hold in latency for 450 days.

The prognostic results for reinfection was compared with the observed data of Dengue fever worsening (

The analysis of the temporal evolution of

Comparison between observation and simulation

The model skill was computed and presented in Figures 9-11. A statistical hypothesis

Moriasi et al. (2007) [

efficiency (NSE), percent bias (PBIAS), and ratio of the root mean square error to the standard deviation of measured data (RSR), being that the following ranges are recommended for satisfactory modelling,

In

Observation | Simulation | Correlation | Simulation | Correlation |
---|---|---|---|---|

Notification | anor | 0.021 | w3 | 0.196 |

Notification | anoi | 0.046 | wt | 0.152 |

Notification | mesi | -0.290 | s | −0.279 |

Notification | virus1 | 0.178 | e | 0.239 |

Notification | virus2 | 0.178 | i | 0.277 |

Notification | virus3 | 0.335 | ir | 0.619 |

Notification | virus4 | 0.019 | r | 0.200 |

Notification | o | 0.078 | m | 0.033 |

Notification | l | 0.083 | tot | −0.026 |

Notification | p | 0.097 | dmdt15 | 0.417 |

Notification | w1 | 0.114 | r0 | 0.291 |

Notification | w2 | 0.175 | rn | −0.342 |

Baldwin and Kain (2006) [

build up the contingency table used to obtain the parameters

quently, the model can be addressed more to determine the change of magnitude order than to discriminate a specific maximum. At least, it can be useful as a skilled tool to update the tendency of epidemic growth. Therefore, the model performance seem to be conditionally acceptable, even considering the many uncertainties and assumed hypothesis.

The main conclusions in this research are:

• The accumulation of energy (since July 1^{st} of each year) associated with the degree-day index variation allows us to evaluate the degree of maturation of females of the mosquito Aedes aegypti, thereby determining whether an epidemic period is more or less likely. The accumulated variation of enthalpy or entropy also can be used similarly.

• If the entomological parameters can be parametrized in function of air temperature and accumulation of energy from the environment (through the concepts of degree-day index or the accumulated enthalpy change), a numerical model for the population dynamics under dengue epidemic can be developed considering this relationship.

• The numerical model was developed and the equations were presented in this work. The model was run along 12 years (from Jan. 2000 to Dec. 2012), assimilating continuously some meteorological data available in the area about the Dengue epidemics in the MARJ-Brazil.

• The model design considered the perspective of: a) high portability, based in use of free software and compiler, that is ideal but not absolutely necessary to run the model; b) use of dummy variables in the subroutine, to easy coupling into some advanced atmospheric mesoscale models (a present proposition), c) simply customised, considering preparation of input data, and d) automatized graphical output, accessed by a single line command (job), written in shell script and gnuplot.

• The epidemic model can be used in an externalized way (e.g., as done in this work), or still integrated/ coupled with a atmospheric numerical forecasting model in high resolution. The prediction of Dengue outbreaks is the most suitable for tropical cities characterized by high temperatures and rainfall during the summers. On the other hand, this kind of prediction can also be very useful in the evaluation of advanced risk scenarios for subtropical cities, which are likely to have increased their average temperatures due to climate change over the twenty-first century.

• The model skill was verified with the Student’s t-test and other statistics such as MBE and RMSE, describing the uncertainties of the model along the simulation period. The Nash-Sutcliffe efficiency also was employed to verify the model skill above of the statistical persistence (i.e., daily cycle repetitions).

• The numerical results appear indicating that a greater number of weak and moderate rainfall events, during the year, are more favourable to larvae breeding, than a small number of intense rainfall events. Very intense precipitations are not favourable too due to the washing out effect depleting the egg and larvae population remaining in the breeding.

• The simulated results have supported the existence of correlation between the seasonal variation of degree- day/entropy/enthalpy (as precursor variables) and the outbreaks of epidemic Dengue fever (as a predicted event characterized by the dengue fever worsening notifications).

• According to the simulation results, the seasonal variation of degree day, or the equivalent in terms of entropy, or indeed, in terms of enthalpy, can be associated with the basal metabolism of the insect vector.

• In association and for this reason the amount of eggs deposited in the previous year may constrain (or limit) the epidemic onset in the following summer.

• The oviposition can be established by the state of metabolism and maturation of Aedes aegypti females that are consequence of the net entropy/enthalpy seasonal change, mainly accumulated along the preceding months (i.e., before oviposition and gonostrophic cycle establishment).

• By their turn, the consequence of a reduced oviposition is the decrease in the chance of a epidemic surge, and this is so even though in the months following oviposition; we can observe environmental conditions very favourable to the development of Aedes aegypti, both in their water as aerial phase.

• Furthermore, in the case in what the amount of eggs is not considered a limitation, the net accumulation of enthalpy, entropy or degree-day to above of a threshold of approximately 2 million (J∙kg^{−1}) appears to be a necessary condition for the outbreak of Dengue fever and for the establishment of the epidemic.

• The statistical analysis on model results enlightened the need of a long data series, typically 10 years, in way to tune the parameters and use the epidemic model operationally. This is in accord with the proposition of Coelho et al. (2008, 2011) [

• Finally, we must emphasize the importance of reinfection process for the computational modelling of epidemics of acute Dengue fever, without which the dynamics of infected populations cannot be achieved satisfactorily.

• Due to the autocorrelation of hydrometeorological variables through successive years (e.g., Yue et al., 2002) [

As next steps of the research we can investigate the effect of climate change on the occurrence of outbreaks of dengue fever. To this end, the numerical model described in this paper can be very useful to assimilate atmospheric conditions from different global warming scenarios, which are expected for the next 50 years. For example, a scenario in which the global temperature may be some degrees above the current average, as described by the Intergovernmental Panel on Climate Change (IPCC).

This work was an effort to investigate and to develop prognostic tools able to predict dengue outbreaks months in advance in the metropolitan tropical areas such as the MARJ in Brazil. Hopefully, the biophysical relationships tested there may be useful for predictions Dengue epidemics also in other tropical cities.

The first author thanks to Luiz Carlos Ciafrino Neto that kindly read the preliminary version of this work.

HAK and JCBS proposed the presented numerical model and prepared input dataset. HAK wrote the numerical code in Fortran-90 and performed larger amount of numerics and parameters tuning. HAK and JCBS run the simulations. The analysis of results and discussions received contributions of all authors (HAK, JCBS, AUPF and JLFR). All authors meet the ICMJE criteria for authorship read and met, and agree with the manuscript’s results and conclusions.

Hugo AbiKaram,Julio Cesar Barreto daSilva,Augusto José PereiraFilho,José Luis FloresRojas, (2016) Dynamic Modelling of Dengue Epidemics in Function of Available Enthalpy and Rainfall. Open Journal of Epidemiology,06,50-79. doi: 10.4236/ojepi.2016.61007