DOI: S0074-02762008000600004

Giovanini E Coelho

^{I}Programa Nacional de Controle da Dengue, Ministério da Saúde, Brasília, Brasil

^{II}Faculdade de Medicina, Universidade de São Paulo, R. Teodoro Sampaio 115, 05405-000 São Paulo, SP, Brasil

^{III}Instituto de Saúde Coletiva, Universidade Federal da Bahia, Salvador, BA, Brasil

^{IV}London School of Hygiene and Tropical Medicine, London, UK

Key words: dengue - larval index - basic reproduction number - force of infection

**ABSTRACT**

We analyzed dengue incidence in the period between October 2006-July 2007 of 146 cities around the country were Larval Index Rapid Assay (LIRA) surveillance was carried out in October 2006. Of these, we chosen 61 cities that had 500 or more cases reported during this period. We calculated the incidence coefficient, the force of infection (λ) and the basic reproduction number (R_{0}) of dengue in those 61 cities and correlated those variables with the LIRA. We concluded that λ and R_{0} are more associated with the number of cases than LIRA. In addition, the average R_{0} for the 2006/2007 dengue season was almost as high as that calculated for the 2001/2002 season, the worst in Brazilian history.

Dengue fever (DF), a Flaviridae infection transmitted by peridomestic *Aedes aegypti*(Gubler & Kuno 1997) re-appeared as a major urban epidemic in Brazil in 1986 (Marques et al. 1994, Massad et al. 2001, Luz et al. 2003) and is currently the most important arthropod-borne viral disease in Brazil (Luz et al. 2003). By the year 2000, all 26 states had reported DF cases. In the period 1986-2007, DF caused 4,559,812 officially reported cases (MS 2007a) with 493 deaths (MS 2007b).

The worst year in history was 2002, with the introduction of DEN-3, when 794,219 cases were reported. In the following years, DEN-3 spread itself throughout the country, causing several important outbreaks (MS 2007c). In 2006, 345,922 dengue cases were reported, with 682 dengue hemorrhagic fever (DHF) cases causing 76 deaths (MS 2007c). From January to July 2007, 438,949 classical DF cases were reported, with 926 cases of DHF, causing 98 deaths (MS 2007c). In this year DEN-3 virus caused 81% of the cases.

An important observation related to dengue transmission is that, in some subtropical areas like Brazil, the density of mosquitoes (Forattini et al. 1993a, b, 1995) and the number of DF cases starts to increase in number at the beginning of the rainy season (October) with the cases of dengue peaking more than three months later (Coutinho et al. 2006).

Long-term control of dengue and condition for outbreak occurrence has relied mainly on larval index (LI), that is, the proportion of houses with breeding places harboring aedes larvae which, paradoxically, has not demonstrated a good correlation with the number of DF cases (Gubler & Kuno 1997, Burattini et al. 2007). One of the few studies that indirectly show some correlation between the house index and the number of dengue cases is the work by Cunha et al. (1997), in which it is suggested, although it is not demonstrated, a relationship between low number of cases and low house index in a community of Rio de Janeiro. In Brazil, an alternative approach to the so-called container indexes (German et al. 1980, Bang et al. 1981) for larval surveillance was proposed in order to optimize the logistics of its operation. It consists of random sampling a relatively small number of dwellings in which surveillance of aedes-positive breeding places is carried out simultaneously in several cities around the country. This strategy has been called Larval Index Rapid Assay (LIRA).

In this paper we analyze the period October 2006-July 2007 of 146 Brazilian cities that carried out LIRA surveillance in October 2006. Of these, we choose those 61 cities that experienced 500 or more dengue cases in the period. We calculate the incidence coefficient (IC), the force of infection (l) and the basic reproduction number (R_{0}) of dengue in those 61 cities and cross-correlate those variables along with the LIRA, in order to estimate which parameter related to the intensity of transmission is the best predictor of the number of cases.

**MATERIALS AND METHODS**

*LIRA* - Consisted in the simultaneous visit of dwellings randomly chosen of 146 cities representing all the Brazilian regions (MS 2005). LIRA is a random sampling technique in which the sample unit corresponds to 9,000 to 12,000 dwellings, from which a maximum of 450 houses are randomly selected for inspection. The LIs obtained are the proportion of houses with *A. aegypti* larvae and the Breteau index. It is being used in Brazil since 2003.

*Calculation of* ? *and R _{0}* - Of the 146 cities visited for the LIRA surveillance, we selected 61 that had 500 or more reported cases and experienced a dengue outbreak in the period. By outbreak we understand the sudden exponential rise in the number of cases in the beginning of the transmission season. The list of those 61 cities can be seen in Table.

The calculation of R_{0} can be carried out from the initial exponential growing phase of the number of cases as detailed in Massad et al. (2001). However, we show in the appendix a summary of the derivation of R_{0} and ?.

From equations (5) and (6) of the appendix (Supplementary data) we obtain the relationship between R_{0} and ? for each one of the 61 cities, such as:

In this equation, ? is the growing rate of the number of dengue cases in the initial exponential phase, the incidence rate or ? of the disease in this phase. By fitting the proportion of new cases to an exponential curve, we can estimate R_{0}. The values of the parameters were µ = 0.156 weeks^{-1} and ? = 1 week^{-1} (Massad et al. 2001).

*Statistical analysis* - We fitted exponential curves by the method of the Minimum Squares to the number of cases in each week starting in October 2006 for each one of the 61 chosen cities. As mentioned above, the growing rate of the number of cases in the initial exponential phase, ? corresponds to the incidence rate or ? of the disease in this phase (equation 4 of the appendix).

The IC per 100,000 inhabitants, ? and R_{0} of dengue in those 61 cities were correlated with their respective LIRA. As a first approximation to estimate the correlations between the variables, we carried out a linear regression analysis that correlated the LIRA versus ?; LIRA versus R_{0}; LIRA versus IC; ? versus the IC; and R_{0} versus IC. The strength of the above correlations was estimated by the calculation of the Pearson's correlation coefficients. Of course other functional associations between the variables could serve the same purpose. However, we chosen a linear regression because the correlations found were poor for all tested function and so we chosen the linear for simplicity. All the statistical analyses were carried out with the software SPSS 15.0.

**RESULTS**

Table shows the 61 cities analyzed with the respective values of the LIRA, ?, R_{0}, the population size, IC and the goodness of fit of the exponential model.

In Fig. 1 we show an example of a city (Paranavaí) that resulted in a particularly accurate fitting of dengue cases. Note the exponential shape of the fitted curve and the resulting incidence of 0.32 per capita new cases per week.

In Figs 2 and 3 we show the correlation between LIRA and ? and LIRA and the R_{0}, respectively. Note that LIRA significantly correlates with both ? and R_{0}.

Finally, we tested the relation between LIRA, ? and R_{0} with dengue IC per 100,000 inhabitants for each city analyzed. Results are shown in Figs 4-6, respectively. It is noteworthy that LIRA correlates with both ? and R_{0} but none of them correlated with IC. Note also the presence of an outlier in Figs. However, its presence does not change significantly the correlation between the variables analyzed, and so we decided to keep it in the analysis.

**DISCUSSION**

The 2006/2007 dengue season in Brazil was the most severe in number of cases since the 2001/2002 season. This is reflected by the values of R_{0} available so far. If we compare the average values of R_{0} for the years of 1991 (2.03, Marques et al. 1994), 2001 (6.29, Massad et al. 2001) and 2007 (4.51, Burattini et al. 2007) we conclude that 2007 is closer to 2001 values than to 1991 as far as dengue intensity of transmission is concerned. We should stress here that the entire estimated LIRA used in this analysis refer to whole town averages, although dengue, as many of the vector-borne infections, have heterogeneous spatial distribution over the affected areas. This would theoretically underestimate the incidence, l and R_{0} estimates (Coutinho et al. 1999). However, our model is based on known cases and our equations system is scale-invariant, that is, it could be expressed in terms of proportions or absolute numbers in an approximately homogeneous area where the cases are happening. Moreover, dengue control measures as proposed by the Brazilian National Program of Dengue Control, in spite of being aimed to cover whole towns, should prioritize the focuses where transmission is likely to occur.

The regression analysis indicates that, despite the low values obtained for the correlation coefficients, they are significant for the associations between LIRA and ? and LIRA and R_{0}. Since LIRA is an entomologically derived variable and ? and R_{0} were estimated from human cases, it is noteworthy that there are weak, although significant correlations between them. To the best of our knowledge, this has not been demonstrated before. The correlations found between the entomological indexes and the estimators of the intensity of transmission are due to the fact that all indexes used are based on larval stages. These have to develop into the further stages until the adult mosquitoes that, in turn, have to bite infected individuals and the new susceptibles. It is, therefore, expected that the large number of unknown parameters governing each step of that chain, would vary from place to place and from moment to moment and, therefore, would weakened those correlation. In spite of this, LIRA could be used as a proxy for the velocity of the following dengue epidemic growth. In addition, those regions with the higher R_{0} are those likely to experience major impact of future outbreaks, especially when a new serotype starts to circulate.

However, when we try to calculate the association of these parameters with the incidence of dengue, none demonstrated to be significantly correlated with it. Nevertheless, when we compare the significance of the association between LIRA and incidence rate of dengue with the associations between ? and *R _{0}* and the incidence rate we observe that both ? and R

We would like to emphasize the importance of the intensity of transmission parameters proposed in this work, namely ? and R_{0}, both of which are more closely associated with the total number of cases than the LIs, although far from being reliable predictors of incidence. However, they are clear indicators of the intensity of outbreaks and relatively easy to calculate in the particular situation in which the number of cases start to increase after a relatively long period with very small number of cases.

As mentioned above, the current predominant strain in Brazil is DEN-3. It would be interesting to include a comment on potential differences in the outcomes now that DEN-2 started to predominate in several states of the country. Certainly, the fact that a great cohort of susceptibles to DEN-2 is available would increase all the estimators of intensity of transmission analyzed in this study.

Finally, it is noteworthy that LIRA performed in nearly October has some predictive power on the velocity of epidemic growth, as demonstrated by the association found between LIRA, ? and R_{0}. Nevertheless, LIRA is not reliable as a predictor of dengue incidence, a fact already known for other LIs. However, the relation between LIRA and the rate of growth of dengue epidemic demonstrated herein is a new result and it is one of the most important results of this paper, since the results of LIRA obtained almost two months before the beginning of dengue outbreaks could serve as a guide for early intervention.

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Received 28 February 2008

Accepted 21 July 2008

Financial support: CNPq, FAPESP, LIM01 HC-FMUSP

+ Corresponding author: edmassad@usp.br

*Appendix* - Let *S _{H}* and

In the first equation of system (1) *a* is the average number of bites a mosquito inflicts in the human population, so that *aN _{m}* is the total number of bites per unit of time and

In the second equation of system (1) the extrinsic incubation period is represented by ?. Therefore, *a*S* _{m}*(

Dividing the first equation of system (1) by *N _{H}* (total human population) and the second by

where *m = N _{m}/N_{H}* and the lower case letters represent proportions.

At the beginning of an outbreak, we can assume that S* _{H}* ? 1 and S

whose general solution is

Taking the derivative of equations (4) and substituting the result in equations (3) we get:

Remembering that R_{0} is

Therefore, from equations (5) and (6) we obtain (Massad et al. 2001):

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DOI: 10.1590/S0074-02762009000600013

Marcos Amaku

^{I}Departamento de Medicina Veterinária

^{II}Departamento de Patologia LIM 01, Hospital das Clínicas, Faculdade de Medicina Preventiva e Saúde Animal, Faculdade de Medicina Veterinária e Zootecnia, Universidade de São Paulo, Av. Prof. Dr. Orlando Marques de Paiva 87, 05508-270 São Paulo, SP, Brasil

^{III}Instituto de Estudos Avançados, São José dos Campos, SP, Brasil

Key words: epidemiology - seroprevalence - force of infection - viral diseases - average age - Monte Carlo method

**ABSTRACT**

Epidemiological parameters, such as age-dependent force of infection and average age at infection () were estimated for rubella, varicella, rotavirus A, respiratory syncytial virus, hepatitis A and parvovirus B19 infections for a non-immunized Brazilian community, using the same sera samples. The for the aforementioned diseases were 8.45 years (yr) [95% CI: (7.23, 9.48) yr], 3.90 yr [95% CI: (3.51, 4.28) yr], 1.03 yr [95% CI: (0.96, 1.09) yr], 1.58 yr [95% CI: (1.39, 1.79) yr], 7.17 yr [95% CI: (6.48, 7.80) yr] and 7.43 yr [95% CI: (5.68, 9.59) yr], respectively. The differences between average ages could be explained by factors such as differences in the effectiveness of the protection conferred to newborns by maternally derived antibodies, competition between virus species and age-dependent host susceptibility. Our seroprevalence data may illustrate a case of the above-mentioned mechanisms working together within the same population.

Certain viral infections are more likely to occur around a given age. For example, rotavirus and respiratory syncytial virus (RSV) typically occur in children under three years of age, while rubella affects 5-9-year-olds (Knipe & Howley 2007).

Utilizing seroprevalence data, it is possible to estimate epidemiological parameters and subsequently analyse the dynamics of an infectious disease in different populations. Examples of this approach include analyses of measles, mumps and rubella seroprevalence in different European countries (Edmunds et al. 2000) and in countries in Europe and Latin America (Amaku et al. 2003). A complementary approach is to analyse the dynamics of different infectious diseases within the same community.

Information about the average age at infection () will help us characterize the dynamics of an infection in a population. A higher incidence of cases in people of younger or older ages may be associated with several factors, such as sanitary conditions, population density (Almeida et al. 2002), age at which children enter the school system (Yu et al. 2001) and other social and economic variables. For instance, improvements in sanitation as well as the introduction of a vaccination programme may lead to changes in the , which demonstrates the importance of this parameter in the assessment of both the effects of an environmental control programme and the efficacy of a vaccination strategy.

To characterise the dynamics of different infectious diseases in the same population, we have estimated the force of infection (? (*a*)) and the , based on data from a seroprevalence survey carried out in a Brazilian community for antibodies against the following viruses: rubella, varicella, group A rotavirus (rotaA), human RSV, hepatitis A (HAV) and human parvovirus B19 (parvoB19). We have also estimated confidence intervals for using Monte Carlo simulations.

**SUBJECTS, MATERIALS AND METHODS**

Data consisted of seroprevalence surveys for IgG antibodies against rubella (Azevedo Neto et al. 1994), varicella (Valentim et al. 2008), rotaA (Cox et al. 1998b), RSV (Cox et al. 1998a), HAV (unpublished data) and parvoB19 (Huatuco et al. 2008) carried out in the community of Caieiras, a small town in the state of São Paulo (SP), Brazil. The population of Caieiras was about 30,000 inhabitants in 1990, most of them inhabiting an urban area occupying 104 km^{2}. Samples were collected from 476 individuals randomly selected by cluster sampling from November 1990-January 1991. The seroprevalence study was carried out in dwellings from 30 of the 60 administrative regions to obtain a sample representative of the population living in Caieiras, including newborns to adults 40 years of age. More details are published elsewhere (Azevedo Neto et al. 1994).

Let *S*^{+}(*a*) be the proportion of seropositive individuals with age *a*. An estimate of the function *S*^{+}(*a*) resulted from fitting the serological data to each disease (Farrington 1990, Amaku et al. 2003)

where *k*_{1} and *k*_{2} are fitting parameters, estimated by either the Maximum Likelihood or the Least Squares technique.

In our model, the seropositive individuals correspond to those who are either infected or immune. The ? (*a*) was estimated from the seroprevalence data for the so-called catalytic approach (Griffiths 1974):

.

Equation (2), expressed in terms of equation (1), is

From the ? (*a*), we can define the average age at which susceptibles acquire infection:

We have taken the highest age observed in the seroepidemiological survey (*L* = 40 years) as the upper integration limit of the integrals of equation (4).

In this paper, we are assuming that the immune response is not lost over time for the diseases studied.

*Monte Carlo simulations* - From the fitting procedure, we have estimated the variance-covariance matrix V for the parameters of the seroprevalence function [equation (1)]

where *S _{k}*

The correlation coefficient between *k*_{1} and *k*_{2} is defined as

and it always lies in the interval -1 < ? < 1. If ? =0, *k*_{1} and *k*_{2} are said to be uncorrelated, if ? > 0, they are positively correlated and, if ? < 0, negatively correlated.

As we have an estimate of the variance-covariance matrix V, we can randomly generate pairs of values (*k*_{1}, *k*_{2}) and calculate the .

The Monte Carlo algorithm we have adopted is based on the generation of random numbers for *k*_{1} and *k*_{2} and on the substitution of these values in equation (4) for the . We have supposed that *k*_{1} and *k*_{2} are normally distributed with the means and and the standard deviations *S _{k}*

where *g*_{1}() and *g*_{2}() are two different normal random generators with mean zero and variance 1. The computational routines we have used to generate random numbers are those described by Press et al. (1992).

**RESULTS**

Fig. 1 shows the fitted seroprevalence curves (Fig. 1A) for the six infectious diseases considered and the respective forces of infections (Fig. 1B).

The estimates for the fitting parameters are shown in Table. The and the 95% confidence intervals estimated using the Monte Carlo method for rubella, varicella, rotavirus A, human RSV, HAV and parvoB19 were 8.45 years (yr) [95% CI: (7.23, 9.48) yr], 3.90 yr [95% CI: (3.51, 4.28) yr], 1.03 yr [95% CI: (0.96, 1.09) yr], 1.58 yr [95% CI: (1.39, 1.79) yr], 7.17 yr [95% CI: (6.48, 7.80) yr] and 7.43 yr [95% CI: (5.68, 9.59) yr], respectively.

The highest values for the ? (*a*) were observed for rotavirus A and RSV, followed by varicella, all of which had an of less than five years. The ? (*a*) curve for rotavirus A showed a sharp peak around one year of age, followed by a steep decrease and vanished after five years of age. The sharp peak for RSV occurred around two years and was followed by a decrease slower than that observed for rotavirus A, vanishing only after the age of 10 years.

The ? (*a*) curve for parvo B19 showed the smallest intensity among the infections studied.

The ? (*a*) curves for rubella and HAV indicated intermediate intensity, being higher than that of parvo B19, but not reaching the intensity of the RSV and rotavirus A curves. The average ages observed for rubella and HAV were 8.45 yr and 7.17 yr, respectively.

By visual inspection, taking into account the confidence intervals, differences between values for the infections studied are noticeable, with the exception of parvo B19 when compared to HAV and rubella (Fig. 2).

**DISCUSSION**

We analyzed seroepidemiological data for six viral infectious diseases in the same population. Our results for rotavirus A are in agreement with the prevalence observed by Candeias et al. (1989) in the city of São Paulo from data obtained in outpatient departments of hospitals, showing this disease occurs early during infancy. The same is true for RSV, as shown by Vieira et al. (2007).

In the case of varicella, two different samples from Caieiras (Yu et al. 2001, Valentim et al. 2008) showed similar seroprofile patterns. However, the seroprevalence for Caieiras is different than that of city of São Paulo (Yu et al. 2001) due to differences in the children's social behaviour.

The HAV seroprevalence found in our analysis is very similar to the profile observed in the Rio de Janeiro metropolitan area (Almeida et al. 2002) and is typical of urban areas.

As for parvoB19, serological data published for Rio de Janeiro (Nascimento et al. 1990) showed a higher prevalence in young adults (80%) compared with the data presented here for Caieiras.

The rubella seroprevalence described here for Caieiras is similar to data published by Souza et al. (1994) for the city of São Paulo from a survey carried out on sera collected in 1987, which suggests that the Caieiras population is representative of the São Paulo metropolitan region population.

Based on seroprevalence data, we estimated the age-dependent ? (*a*) and the average age of first infection. These epidemiological parameters were used in previous publications to verify the effect of different vaccination strategies (Amaku et al. 2003).

The average age at first infection is an important reference for determining the optimum age for vaccine introduction. It was demonstrated that immunization for rubella is not effective if proposed for ages around or above the (Amaku et al. 2003). Although this is a relative simple parameter, as shown by our results, it can support health authorities' decisions around establishing a vaccination schedule.

The application of the parameters estimated in this paper proved to be useful when the immunization campaign against rubella was designed for SP in 1992 (Massad et al. 1994, 1995). This campaign was successful at reducing the incidence of rubella.

The differences observed among the epidemiological parameters estimated for these six infectious diseases may not be solely due to the contact pattern among individuals because the individuals sampled are the same. Therefore, other factors should be considered as influential to the epidemiology of each disease by mathematical modelling.

If we consider that all viruses are circulating in the population, why don't we observe constant proportions of population infected across all age groups?

Depending on the infection, factors such as protection by maternal derived antibodies, transmission route, age-dependent host susceptibility and competition among different virus species may also help to explain the observed age-dependent pattern of disease transmission.

The decline of maternal derived antibodies from birth onwards may interfere with the acquisition of an infection in the first year of life, altering the average age at first infection. RSV, for example, requires a high titer of maternal derived antibodies to be neutralized, making infants susceptible after two months of age (Collins & Crowe 2007). This fact may explain the rapid spread of RSV infection in the first two years of life. It may also be the explanation for other diseases that spread among infants, such as rotavirus A infection.

According to our results, the route of transmission did not seem to make a difference in terms of when an infection would occur. For instance, rotavirus A and HAV are both mainly transmitted by a faecal-oral route, but the former has an average age of first infection of one year and the latter seven years.

Infectivity is another characteristic that varies among pathogens and it may be important in determining the success of rotavirus and RSV acquisition so early in an infant's life as compared to other illnesses such as rubella or parvoB19, for example.

Regarding competition among viruses, it should be considered that nonspecific immune responses produced by one viral infection may inhibit a concomitant infection by another virus. Although simultaneous virus infection in the same host was demonstrated for RSV, adenovirus, metapneumovirus, rhinovirus, enterovirus, influenza and parainfluenza in different combinations (Calvo et al. 2008), concurrent infection is less likely to occur, making competition a good hypothesis to explain why some diseases like rubella and parvoB19 appear less frequently in infants and toddlers.

We have observed that viral infections like rotavirus A and RSV have a lower average age of first infection as compared with varicella, which, in turn, is lower than HAV, parvoB19 and rubella. Our seroprevalence data may illustrate the above-mentioned mechanisms working together within the same population.

If it is true that there is an interaction among virus and human populations that drives the observed age-dependent incidences, it is plausible that a change in the seroprevalence pattern of a disease may be instigated by the introduction of vaccine against another disease.

Further investigation of the seroprevalence profiles of different diseases within the same population should be carried out to elucidate whether such an interaction occurs after the introduction of immunization.

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Massad E, Burattini MN, Azevedo Neto RS, Yang HM, Coutinho FAB, Zanetta DMT 1994. A model-based design of a vaccination strategy against rubella in a non-immunized community of São Paulo state, Brazil. *Epidemiol Infect 112*: 579-594.

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Valentim J, Sartori AM, de Soárez PC, Amaku M, Azevedo RS, Novaes HM 2008. Cost-effectiveness analysis of universal childhood vaccination against varicella. *Vaccine 26*: 6281-6291.

Vieira SE, Gilio AE, Durigon EL, Ejzenberg B 2007. Lower respiratory tract infection caused by respiratory syncytial virus in infants: the role played by specific antibodies. *Clinics 62*: 709-716.

Yu AL, Amaku M, Burattini MN, Massad E, Azevedo RS 2001. Varicella transmission in two samples of children with different social behaviour in the state of São Paulo, Brazil. *Epidemiol Infect 127*: 493-500.

Received 10 March 2009

Accepted 9 July 2009

Financial support: CNPq, FAPESP

+ Corresponding author: amaku@usp.br

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DOI: 10.1590/S0074-02762012000400022

Eduardo Massad

^{I}LIM 01-Hospital de Clínicas, Faculdade de Medicina, Universidade de São Paulo, São Paulo, SP, Brasil

^{II}London School of Hygiene and Tropical Medicine, London University, London, UK

Key words: vectorial capacity - sporozoite rate - force of infection - vectorborne infections - basic reproduction rate - dimensional analysis

**ABSTRACT**

A dimensional analysis of the classical equations related to the dynamics of vector-borne infections is presented. It is provided a formal notation to complete the expressions for the Ross' Threshold Theorem, the Macdonald's basic reproduction "rate" and sporozoite "rate", Garret-Jones' vectorial capacity and Dietz-Molineaux-Thomas' force of infection. The analysis was intended to provide a formal notation that complete the classical equations proposed by these authors.

It is now accepted that infectious diseases will contribute a proportionately smaller burden of disease over the next two decades as non-communicable diseases emerge as public health problems. Infectious diseases, however, will continue to contribute proportionately more in the poorest quintile of the population (Molineaux 2003).

Currently the burden of vector-born infections poses a heavy toll on the major fraction of the world. Dengue represents a threat to the integrity of almost half the world's population and malaria still kills a child every 35 sec (Healthy Environments for Children Alliance) (available from: who.int/heca/infomaterials/vector-borne.pdf), not to mention several hundred thousand new yearly cases of leishmaniasis, African trypanosomiasis, among other infections transmitted by the bites of insects. In Asia, Japanese encephalitis puts three billion people at risk every year. Malaria and dengue are also major public health problems in south-east Asia. While 8% of the world population lives in the eastern Mediterranean Region, almost 11% of the global burden of vector-borne diseases is found there. Malaria is among the most prevalent causes of death and illness in Africa, where 90% of the globally reported cases of malaria take place. An estimated one million deaths from malaria occur each year, most of them in children under five years old. In Latin America and the Caribbean, dengue haemorrhagic fever has infested all major cities. Nearly 95 million people in the western Pacific are at risk of contracting malaria. To make things worse, the burden of many of these diseases is borne largely by developing countries (Hill et al. 2005).

Since the seminal work by Ronald Ross, mathematical models have provided a great deal of theoretical support for understanding the complex dynamics of those infections, in addition to the important role those models have played in designing and assessing control strategies. Key concepts like the basic reproduction number, vectorial capacity and the force of infection derived from the theoretical works on vector-borne infections are currently central to the quantification of transmission, as well as to the proposal of public health measures to control them. Some of the original mathematical expressions of these concepts suffer from minor (although important) formal inadequacies. These dimensionality inadequacies are related to incomplete notation.

In this note we revisit some of the central tenets of vector-borne infections theory from the perspective of dimensional analysis. As mentioned above, we point to some small (but important) formal inadequacies in some of the theoretical developments of some important quantifiers of risk and transmission of vector-borne infections.

*Dimensional analysis -* At the heart of dimensional analysis is the concept of similarity, due to Newton (1686). In physical terms, similarity refers to some equivalence between two things or phenomena that are actually different (Sonin 2012).

Essentially, the arguments of transcendental functions such as exponential, trigonometric and logarithmic functions, or of inhomogeneous polynomials, must be dimensionless quantities (Gibbings 2011). Therefore, an expression like log(30*km*) is meaningless because the argument has an explicit dimension (*km*).

More generally, dimensional analysis is, or should be, a crucial step in any mathematical approach to the real world, but surprisingly dimensional methods are so little understood and applied in the biological sciences (Stahl 1961).

*Ross' Threshold Theorem -* Ronald Ross, the father of the theory behind the dynamics of vector-borne infections, in a set of publications (Ross 1911), proposed what he called "pathometry", a term he later defined as "the quantitative study of disease" (Ross 1923, Bailey 1982). From a set of simple assumptions, Ross derived a system of equations for the incidence of malaria, beginning with two difference equations (Fine 1975). Let *a _{t}* be the number of uninfected and

Where, according to Fine (1975), "*h* is the infection rate *H* is the recovery rate, *v*, *V* refer to births, deaths, immigrations and emigrations of affected and unaffected individuals (respectively)". From the dimensional point of view, it is difficult to understand system (1). For example, *h* and *H* are rates and, therefore, have dimension *T* ^{-1} . Therefore expressions like (1-*h*) and (1-*H*) are, as written, dimensionally incorrect. Below we show how to circumvent these difficulties.

The so-called Threshold Theorem refers to the critical density of mosquitoes, *a*, below which the disease would disappear from the human population. Ross proposed that this could be calculated by

where *r* is the recovery rate, *b* is the proportion of uninfected mosquitoes which feed on man, *i* is the proportion of infected individuals who are infectious and *s* is the proportion of mosquitoes which survive through the extrinsic incubation period. Note that according to this equation *a* has dimension *T* ^{-1} which is more like a rate than a mosquito density.

*Macdonald's basic reproduction "rate" and sporozoite "rate" -* The year of 1952 is a landmark in the history of mathematical epidemiology. It was when George Macdonald proposed his expression for what he defined as the "basic reproduction rate" (*R*_{0}) of malaria. In his words, it is defined as the number of secondary cases produced by a single infected individual (index case) along his/her infectious period in an entirely susceptible population (Macdonald 1952a, Dietz 1993, Massad et al. 1994, 2010, Heesterbeek 2002, Lopez et al. 2002, Burattini et al. 2008). His derivation of *R*_{0} is as follows. Let us begin by assuming that the index case is a human host. The question to be answered is how many human secondary infections this index case produces in his/her entire infectiousness period.

Let *N _{m}* be the number of female mosquitoes. Let

or calling *N _{m}*/

Similarly, if we begin with an infective mosquito as an index case and compute the number of infected mosquitoes this index case produces in the first generation we get the same expression. Note that in his original formulation Macdonald did not consider the parameter *c.*

In equation (4), it is possible to identify the dimensional problem in the term 1/-1n(*p*). If, on the one hand, the parameter *p* has a dimension (per day), then 1n(*p*) is ill defined. On the other hand, if *p* is dimensionless, then *R*_{0} ends up with dimension *T* ^{-1}, being, therefore, like a rate. We shall see below how to overcome these difficulties.

In the same year, Macdonald (1952b) proposed another useful quantity for the epidemiology of malaria, namely the "sporozoite rate" (*S*), which he defined as the proportion of mosquitoes with sporozoites in the salivary glands (therefore it cannot be a "rate"). His definition of *S* was the lifespan of the mosquitoes with sporozoites in their salivary glands and hence it can be deduced as follows. Let *y* be the proportion of human hosts infected (a prevalence). If the mosquitoes bite an average *a* bites per day then the average number of infective bites (to the mosquito) per day is *ay* and the probability of biting at least once a day is (1-*e*^{-ay}) (here the problem begins: if *a* is a rate it cannot be the argument of the exponential). Macdonald continues his argument by assuming that the probability of no infective bites is *e*^{-ay}. Now *p* was defined as the probability of mosquitoes survival for one day hence the probability of survival for one day in a non-infected state is *pe*^{-ay}. Therefore, the mosquitoes' life expectancy in a non-infected state is

but we are interested only in the life expectancy of the mosquitoes after the extrinsic incubation period *n* days of the parasite, that is:

So, the sporozoite "rate" can be expressed as

which simplifies to

Again, if the parameter *p* has a dimension (per day), then 1n(*p*) is ill defined; if *p* is dimensionless, then the denominator of equation (8) is meaningless.

*Garret-Jones' vectorial capacity -* In 1964, Garret-Jones (1964a, b) (Garret-Jones & Shidrawi 1969, Dye 1990, 1992) proposed a new parameter to estimate the risk of malaria introduction which he called the "vectorial capacity" denoted *C*. This parameter is a measure which is essentially independent of the prevalence of *Plasmodium* infection. Strictly defined, the vectorial capacity is the daily rate at which future inoculations arise from a currently infective case, provided that all females mosquitoes biting that case become infected (Dye 1986). According to Garret-Jones, *C* could be obtained by the product of the number of bites per person per day [man-biting rate (*ma*)], the number of bites per day by each female mosquito [man-biting habit (*a*)] and the expectation of a mosquitoe's infective life [*p ^{n}*/-1n(

Here the problem is even more serious. If *p* has a dimension (per day), then 1n(*p*) is ill defined, but if *p* is dimensionless, then expression (9) ends up with dimension *T* ^{-2}!

Again, this can be easily corrected as we show below.

*Dietz-Molineaux-Thomas' force of infection -* In what can be considered the most important work on modelling malaria Dietz et al. (1974, 1980) extended the concept of vectorial capacity for a multi strains mosquitoes' populations such that if there are *J* vectors populations, with time-dependent densities [*m _{j}*(

They then defined a time-dependent inoculation rate *h*(*t*) (the force of infection) as

which, for stable situations reduces to *h* = *g*[1-exp(-*Cy*)], where *g* is equivalent to *b* in Macdonald's expressions (the authors call *g* the hosts' susceptibility but we prefer to call it vector competence). At first sight the latter expression is dimensionally incorrect because *C* has dimension *T* ^{-1}. However, it is possible to understand the author's intention and in the next section we propose a notational completion to make things clearer.

*Completing the notation to solve some dimensional problems -* Let us begin by trying to complete equations (1). For this we define a time interval *T*. Let *a _{t+T}* be the number of uninfected individuals at time

Now, the way to make equation (2) dimensionally correct is to multiply the recovery rate *r* by a time unit *T*, which is made equal to 1.

Now, the dimensional inadequacies of the equations related to the basic reproduction number (*R*_{0}), the sporozoite rate (*S*) and the vectorial capacity (*C*) can be entirely solved when we use the modern notation. To the best of our knowledge the introduction of the terms *µ* for the mortality rate of the mosquitoes and *e ^{-µ}*

From equation (12) it is possible to write equation (4) by identify *p* = *e*^{-µ} and therefore 1n(*p*) = *µ.* This, however, is dimensionally incorrect. However, it is important to note that p = *e*^{-µT} , where T = 1, such that = -1n(*p*)/*T*. Therefore, equations (4), (6) and (9) should have their term -1n(*p*) replaced by , which makes they dimensionally correct.

The assumption behind this deduction is an exponential distribution for the mosquitoes' survival curve such that if we suppose that a mosquito female is infected as soon as she emerges as adult, it is possible to calculate what is her life expectancy in the infective condition, that is, after the extrinsic incubation period is elapsed:

which is the term that substitutes *p ^{n}*/-1n(

Now, equation (11) for the time-dependent force of infection has a hidden time interval in the sense that what is computed is the number of new infections that occurred in a give interval ?*t*, such that in that specific interval a certain number of potentially infective bites is *C*?*t*, which makes equation (11) dimensionally correct. So, equation (11) should read

which reduces to (11) if ?*t* = 1. We should not, however, forget that ?*t* = 1 is one unit time (e.g. 1 day).

This note was intended exclusively to provide complete notational formality to the classical equations in the field of vector-borne infections. It would be preposterous to propose a "correction" to these equations since we are pretty sure that all the authors mentioned in this note knew exactly what they were doing. It is just that they took for granted that the hidden time dimension (*T*=1) would be readily noticed by the readers of their paper. However, perhaps due to the fact that we are both physicists, we decided that we could contribute to readers interested in understanding in dept the theory behind the rich dynamics of vector-borne infections. Therefore, we hope that the analysis presented above can have a didactical role for the new generations of researchers in this fascinating area.

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Bailey NTJ 1982. *The biomathematics of malaria*, Charles Griffin & Company, London/Hugh Wycombe, 210 pp.

Burattini MN, Chen M, Chow A, Coutinho FAB, Goh KT, Lopez LF, Ma S, Massad E 2008. Modelling the control strategies against dengue in Singapore. *Epidemiol Infect 136*: 309-319.

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Massad E, Coutinho FAB, Burattini MN, Amaku M 2010. Estimation of *R*_{0} from the initial phase of an outbreak of a vector-born infection. *Trop Med Int Health 15*: 120-126.

Massad E, Coutinho FAB, Yang HM, De Carvalho HB, Mesquita F, Burattini MN 1994. The basic reproduction ratio of HIV among intravenous-drug-users. *Math Biosc 123*: 227-247.

Molyneaux DH 2003. Common themes in changing vector-borne disease scenarios. *Trans R Soc Trop Med Hyg 97*: 129-132.

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Ross R 1911. *The Prevention of malaria*, 2nd ed., John Murray, London, 772 pp.

Ross R 1923. *Memoirs: whit full account of the great malaria problem and its solution*, John Murray, London, 547 pp.

Sonin AA 2012. [Assessed 2 May 2012]. The physical basis of dimensional analysis, 2nd ed. Available from: web.mit.edu/2.25/www/pdf/DA_unified.pdf.

Stahl WR 1961. Dimensional analysis in mathematical biology. *Bull Math Biophys 23*: 355-376.

Received 7 April 2012

Accepted 7 May 2012

Financial support: European Union's Seventh Framework Programme (FP7/2007-2013) (282589), LIM01 HCFMUSP, CNPq

+ Corresponding author: edmassad@usp.br

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DOI:

Jorge Diego Marco, Angel Marcelo Padilla, Patricio Diosque, Marisa Mariel Fernández*, Emilio Luis Malchiodi*, Miguel Angel Basombrío

Laboratorio de Patología Experimental, Facultad de Ciencias de la Salud, Universidad Nacional de Salta, Calle Buenos Aires 177, 4.400 Salta, Argentina

*Instituto de Estudios de Inmunidad Humoral, Cátedra de Inmunología, Facultad de Farmacia y Bioquímica, Universidad de Buenos Aires, Buenos Aires, Argentina

Key words: Argentina - dog - force of infection - followup - tegumentary leishmaniasis

**ABSTRACT**

A clinical-serological follow-up was carried out in a canine population in endemic foci of *Leishmania braziliensis *spread in northwestern Argentina. Each dog was studied in at least two visits, 309±15 days (X±SE) apart. Some initially healthy dogs (n=52) developed seroconversion or lesions. The clinical evolution of the disease in dogs resembles in many aspects the human disease. Similarities include the long duration of most ulcers with occasional healing or appearance of new ones and the late appearance of erosive snout lesions in some animals. Yearly incidence rates of 22.7% for seroconversion and of 13.5% for disease were calculated as indicators of the force of infection by this parasite upon the canine population.

Dogs are highly susceptible to the development of leishmaniotic ulcerative lesions in areas where *Leishmania (Viannia) braziliensis* is transmitted among human populations. The possible role of these animals as reservoirs and their availability as targets for field studies on vaccination and therapy, add a medical and epidemiological dimension to the veterinary problem. Many "cross sectional" surveys and very few "longitudinal", follow-up studies have been performed on canine tegumentary leishmaniasis (CTL). Using the former approach, several authors (Reithinger & Davies 1999) have determined the pathologic, serologic and parasitologic features of CTL. Alternatively, follow-up studies of canine populations determining the risk of acquiring infection in non-infected animals, the evolution of recent infection, and the outcome of incipient or advanced disease, are quite relevant for vaccination or therapy and have seldom been addressed in CTL. With this purpose, we have undertaken a "longitudinal" follow-up study of non-infected, recently infected and chronically infected dogs of a high-transmission region.

The work was undertaken at the departments of Orán and San Martín, province of Salta, Argentina. Humid forests abound in this area, where intensive agriculture has expanded in the last three decades. Many human cases of leishmaniotic cutaneous or mucocutaneous ulcers have been recorded in recent years. Most lesions occur in rural personnel working at the edge of deforestation fronts. Urban outbreaks in the cities of Tartagal and Pichanal where recorded in 1985 and 1994 (Sosa Estani et al. 1998).

The dogs studied lived in rural or periurban dwellings, selected for having had human patients in either the same house or in the close vicinity. Although we have so far been unable to characterize the species of *Leishmania* isolated from dogs, at least seven out of eight human *Leishmania* isolates from the same area were classified by different methods as belonging to the subgenus *Viannia*, braziliensis complex (Campanini et al. 1993, Cuba et al. 1996, Sinagra et al. 1997). Groups which have succeeded in the isolation of dog parasites in other areas of human transmission of *L. (V.) braziliensis* could show that most dog isolates were homologous to the human ones (Cuba et al. 1985, Aguilar et al. 1987).

A serologic, clinical and parasitologic follow-up was carried out in three groups of animals: Group 1: apparently healthy dogs, confirmed as seronegative, n=52; Group 2: seropositive dogs, bearing ulcerative lesions (n=28) and Group 3: seropositive dogs without clinically detectable lesions (n=3). Every dog was visited, examined and analyzed at least twice. The interval between the first and the last rounds of visits averaged 309 days (standard error: 15 days; range: 95-625 days). The skin of each animal was examined in detail, paying special attention to ears and genitals.

Blood was drawn by venous puncture and the serum was kept frozen until tested with an ELISA assay (Malchiodi et al. 1994). A panel of antigens, including F45 of *L. mexicana*, and Ag163B6 of *Trypanosoma cruzi* allowed the distinction between *Leishmania*-infected (F45+, Ag163B6-) and *T. cruzi* infected dogs (Ag163B6+; Chiaramonte et al. 1999).

The criteria used to define lesions as "compatible with leishmaniasis" were: ulcerative character, long duration, and rounded, raised and indured edges. Lesions probably induced by trauma were not considered as due to *Leishmania*infection.

Material for Giemsa-stained smears was obtained mainly by either scratching the ulcer margin with a toothpick or by touch-printing fresh biopsied skin tissue.

*Group 1 - *Clinically, these dogs showed no lesions and had been completely seronegative in the first round of visits. In the subsequent rounds, 6/52 (11.6%) had developed typical ulcerative lesions, characteristic of leishmaniasis and 10/52 (19.2%) had seroconverted. Coincidence between appearance of lesions and seroconversion occurred in three dogs, seroconversion without apparent lesions occurred in seven and lesions without seroconversion in three.

*Group 2 -* This group had 28 dogs bearing lesions, all of them with positive serology in the first round. Half of these animals maintained their lesions without noticeable cure or aggravation during all the period of observation. In four animals (14.3%) bearing multiple lesions in the first round, it was observed that the ear or snout lesions progressed while other lesions, mostly on furred skin, underwent cure (Table, "divergent evolution"). Seven dogs (25%) of this group showed evident aggravation of their lesions, which increased either in size or number. Finally, in the remaining three dogs, (10.7%) complete cure was observed (Table).

*Group 3 -* Of the three seropositive dogs without lesions in the first round, two developed no lesions and the remaining one developed a large lesion in the ear.

The interval between the first and last observation of each dog (309 ± 15 days) was only a fraction of the lifetime of these animals. However, our observation period was enough to witness several primary infections, as attested by 10/52 cases of seroconversion, the appearance of new lesions in 7/58 animals, the aggravation or persistence of lesions in 25/28 animals and apparent cures in a small proportion of them (3/28). The "divergent evolution" of multiple lesions, where some lesions in snout and ears aggravated and others (mostly in furred skin) improved, seems to be consistent with two stages in this disease. The first would consist of lesions occurring at the site of inoculation (Kirkpatrick et al. 1987). The second would involve the metastatic, progressive, erosive mucocutaneous nasal lesions, equivalent to the so-called "espundia" or "uta" in humans (WHO 1990). The fact that either lesions in the nasal or auricular cartilage were mostly found in elderly dogs, further supports this assumption. The average age of dogs with erosive snout or ear lesions (n=10) was 5.4 ± 1.7 years whereas dogs with lesions in other sites (n= 48) were 3.6 ± 2.2 years old (p<0.014).

Given the rates of seroconversion/lesions and the average intervals between observations in our samples, a yearly incidence rate of 22.7% for seroconversion and of 13.5% for disease can be expected for normal dogs exposed under the same epidemiological conditions. Studies on the force of infection for canine visceral leishmaniasis in northern Brazil (Quinnell et al. 1997), based on estimates of *per capita* incidence rates and basic reproduction number, have shown an even stronger risk of infection, pointing to the high susceptibility of dogs to both types of leishmaniasis.

The serological and clinical survey of dogs owned by highly exposed human communities might thus provide a measurable and consistent parameter to evaluate therapeutic or preventive measures, including vaccination.

Both parasite load and time of exposure to vectors are main determinants of the ability of canine populations to disseminate the infection. In a previous report (Padilla et al. 1999) we have described the scarcity of parasites in dog lesions. Only 14 of 25 smears were positive and the load was much scantier than in human lesions of the same area. This raised doubts about the role of dogs as reservoirs. However, the present observations point again to the importance of dogs in this respect. Sand fly-mediated propagation of CTL is highly dependent on whether the insects feed on lesions as opposed to normal skin (Vexenat et al. 1986). The long periods during which dogs maintain open, exposed lesions, might counterbalance the scarcity of parasites to secure an efficient vectorial transmission.

**ACKNOWLEDGEMENTS**

To Maria Celia Mora and Alejandro Uncos for technical support. Drs Alberto Marinconz, Liliana Canini and Néstor Taranto helped with their guidance in the endemic area.

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+Corresponding author. Fax: 54-387-425.5333. E-mail: basombri@ciunsa.edu.ar

Received 31 July 2000

Accepted 13 December 2000

This work received financial support from Consejo Nacional de Investigaciones Científicas y Técnicas, Consejo de Investigación of the University of Salta, Universidad de Buenos Aires and Agencia Nacional de Promoción Científica y Tecnológica.

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