Research Article | Open Access
The Preventive Control of Zoonotic Visceral Leishmaniasis: Efficacy and Economic Evaluation
Zoonotic Visceral Leishmaniasis (ZVL) is one of the world’s deadliest and neglected infectious diseases, according to World Health Organization. This disease is one of major human and veterinary medical significance. The sandfly and the reservoir in urban areas remain among the major challenges for the control activities. In this paper, we evaluated five control strategies (positive dog elimination, insecticide impregnated dog collar, dog vaccination, dog treatment, and sandfly population control), considering disease control results and cost-effectiveness. We elaborated a mathematical model based on a set of differential equations in which three populations were represented (human, dog, and sandfly). Humans and dogs were divided into susceptible, latent, clinically ill, and recovery categories. Sandflies were divided into noninfected, infected, and infective. As the main conclusions, the insecticide impregnated dog collar was the strategy that presented the best combination between disease control and cost-effectiveness. But, depending on the population target, the control results and cost-effectiveness of each strategy may differ. More and detailed studies are needed, specially one which optimizes the control considering more than one strategy in activity.
Zoonotic Visceral Leishmaniasis (ZVL) is one of the world deadliest and neglected infectious diseases, according to World Health Organization. This disease is endemic in 80 countries worldwide, in which 90% of all cases occur in Bangladesh, Brazil, India, Nepal, and Sudan. Thus, about 360 million of people are exposed to risk of infection in the world [1–4]. The ZVL is a disease of major human and veterinary medical significance that involves a complex interplay between trypanosomatids protozoan from Leishmania complex, arthropod vectors (in Brazil, we find the female sandflies Lutzomyia longipalpis and Lutzomyia cruzi), environmental influence on vector distribution, small companion animal (dog) reservoir of infection, and susceptible human populations. In American continent, Leishmania infantum chagasi is the most important species from Leishmania complex.
From the last few years, ZVL has been emerging within nonendemic areas, mostly because of transportation of dogs from endemic areas and climatic changes with the expansion of the geographical range of the sandfly vector. Thus, the effective control will essentially involve interdisciplinary teams of microbiologists, parasitologists, entomologists, ecologists, epidemiologists, immunologists, veterinarians, public health officers, and human physicians .
Besides the publication of guidelines of ZVL control and the investments made in general surveillance activities, the sandfly and the reservoir in urban areas remain among the major challenges for the control activities. These challenges are due to the necessity to better understand the vector behavior in urban environment, the operational and logistic difficulties to carry out activities in sufficient time to obtain good results, and the high costs involved in these activities [2, 6].
Usually, health is not analyzed as an economical activity. However, economical analysis in health studies is important for comprehension of health polices dynamics and trends. From those results, it is possible to obtain arguments and support to organize and supervise health polices programs. In short, economic health expresses the universal desire of reaching the best investment, not only in terms of clinical effectiveness, but also in terms of approaching cost-effectiveness about healthcare procedures [7, 8].
Marinho et al.  observed that there are few studies that analyzed the economical impact on visceral leishmaniasis considering social and collective approach. In addition, there are several difficulties to develop economical analysis of visceral leishmaniasis transmission due to (I) the interval of time between the intervention and epidemiological impact or/and (II) the difficulty to relate the intervention activities to the resulting impact. Considering those difficulties and still open-questions about ZVL dynamics and impact, the use of mathematical models should become a very interesting alternative of analysis.
Some deterministic models have been published in literature and all of them analyze the dynamic of this disease and make any evaluation of strategies controls. In particular, since 1998 our research group has been working on ZVL modeling. Burattini et al.  worked on a mathematical model to visceral leishmaniasis where both humans and dogs were considered source of infection. Later, Ribas et al.  developed a model, based on Burattini et al. , which was restricted to LVZ and some preventive control strategies were also considered. Newly, an original article was published by Shimozako et al. , where they reviewed the model published by Burattini et al.  and not only updated some parameters but also provided a more complete mathematical analysis. In this most recent paper, we were able to fit the model to real data from Araçatuba/SP city (Brazil), carrying out a very robust model and results. And, besides those models published by our research team, we also have other researchers who published mathematical models for LVZ, as Zhao et al. , in which their model differs from ours by the adopted mathematical structure and the presence of backward bifurcation.
Even though the result from mathematical model indicates epidemiological availability for visceral leishmaniasis control, we should evaluate carefully the practical and economical viability. In this case, regarding public health, the disease control activities should work considering the best cost-effectiveness, since the available resources are limited. We also know that it is important to be aware of the time-response and applicability-practicality conditions. In other words, it is necessary to be careful with investment time and method application relationship and the respective expected result [5, 14–17].
In this work we propose an evaluation of five ZVL control strategies (positive dog elimination, insecticide impregnated dog collar, dog vaccination, dog treatment, and sandfly population control), by mathematical modeling. This mathematical model was based on the previous models published by Burattini et al.  and Ribas et al. . We studied the impact of those control strategies on human and dog population by approaching the epidemiological control and cost-effectiveness. Then, we discussed the most efficient control strategies and how they act on visceral leishmaniasis epidemiological chain.
2. The Model
We used a mathematical model that is an adaptation of the one proposed by Burattini et al. . In our model, we assume(1)a human and a dog population, with the biological vector transmitting the infection within and between the two populations;(2)those three populations (humans, dogs, and vectors) being constants;(3)both human (indexed as h) and dog (indexed as d) populations being divided into four categories: susceptible (xh and xd), infected but without noticeable disease (lh and ld) (i.e., “latent”), clinically ill (yh and yd), and recovering immunes (zh and zd). On the other hand, the vector population is divided into three categories: noninfected, infected but not infective, and infective individuals, denoted as , , and , respectively.The flowchart and compartment model (Figure 1) and the set of differential equations describing the model’s dynamics (system (1)) are presented as shown in Figure 1 and are as follows:The definition, biological meaning, and values of each of parameter are described in Table 1.
A brief description of system (1) should clarify their meaning.
Let be the total number of sandflies. The number of bites inflicted in the human host population in an infinitesimal time interval is , where ah is the biting rate on humans. The number of bites inflicted by infected flies is , where is the number of infected flies.
Let now be the total number of susceptible individuals in the human population. In an infinitesimal time interval , varies as follows:(i)The infected flies are able to bite on any category of human population. Thus, only a fraction of the infected bites are on uninfected individuals: , where is the fraction of uninfected humans. But, a fraction bh of becomes latent, so diminishes by .(ii)Simultaneously, individuals, latent and immune, revert to the susceptible condition, and die by natural causes other than the disease.(iii)We must add an entrance term, due to natality, which we choose to be , where is the disease-induced mortality rate, is the number of infected humans (clinically ill humans), and is the total number of humans needed to maintain a constant population (where , with as the number of latent humans and as the number of recovering humans).Thus we haveDividing this equation by and calling , we get the first equation of system (1).
Observe that is a time-dependent function: . This expression is the simplest way to simulate the changes on sandfly population size dynamics between 1999 and 2015.
We can apply the same process in order to obtain the equation for the dynamic of susceptible dogs (). However, observe from Table 1 that the sandfly : dog ratio depends on the sandfly : human ratio and on the human : dog ratio: . Although all the populations are constant, if we consider the real number of individuals, we expect more humans than dogs. Thus, if the sandfly population is constant, we have different values for and .
The last three equations of system (1) refer to the flies. When infected, a fly remains in a latent stage for a period of time . This time corresponds to the extrinsic incubation period of the parasite inside the vector fly. Numerically it lasts for about half the life expectancy of the flies.
Let be the number of susceptible flies. In an infinitesimal period of time , bites due to uninfected flies occur on latent and infected dogs (humans are not considered to be infective for flies; see Tesh ). A fraction, and , of the flies (that bites latent and clinically ill dogs, resp.) becomes latently infected as a result. Therefore, we haveDividing by and by , we get the equation for noninfected sandflies ().
Although this is a brief but detailed description of the noninfected categories equations (i.e., , , and ), we can note that each term of our system equation has a biological meaning. The meaning of each term depends on the respective parameters that set them (e.g., means the amount of latent dogs that develop immunity per day).
3. The Number of Clinically Ill Humans and Reported Cases
In Brazil, ZVL is a notifiable disease [17, 34]. Thus, we can assume the following:(i)An infected human should look for medical treatment when he/she will become clinically ill ().(ii)Only a fraction of those humans that are clinically ill will be reported to sanitary authorities. The remaining fraction (I) will not look for medical help, even if the clinical symptoms and signs appear or (II) will not be correctly reported in the hospitals.Now, let us see again the equation for in system (1): The term in (4) means the rate of latent humans who become clinically ill per day. Thus, in order to calculate the total of humans that become clinically ill along an interval of time, we havewhere is the total of humans that become clinically ill from an initial moment, , to a final one, .
Now, let us consider that, per day, the number of humans is eligible to look for medical help. However, only a fraction of those clinically ill humans will be correctly notified to sanitary authorities, where means the proportion of unreported cases . Therefore, the daily rate of reported human cases is defined byThe Centre of Epidemiological Surveillance of São Paulo State (CES-SP)  is the institution that administrates the data about ZVL in São Paulo State. In order to validate our model, we decided to use the data of human reported cases from the municipality of Araçatuba (São Paulo State, Brazil) as reference, because it is an endemic city for this disease. Those data are presented in Table 3 and are available on CES-SP website .
Note that we have the total of reported cases per year. Thus, since our time scale is day, we estimated an average of human reported cases per day for each year (dividing the total from each year by 365). Finally, we also have to consider that our model works with normalized population (all three populations are constant). Thus, as a last step, we have to divide each rate of human reported cases per day by the official population size of Araçatuba municipality. The population size of Araçatuba municipality is available on Brazilian Institute of Geography and Statistics website 
In order to fit and compare our results to real data, we also calculated a normalized average of reported cases per day from every 365 days of simulation. This simulation was run considering 60 years and the obtained curve was fitted by simple handling along the time-axis (e.g., we could assume the initial day as the first day of 1970 or 1980, depending on how best the simulated curve fits on the real data). Thus, we could obtain the yearly average of reported human cases per day and compare it to the real yearly average provided by CES-SP  (Table 3).
4. Fitting the Human : Sandflies Ratio
Among all used parameters for this work, the sandfly/human ratio is one of the most challenging to be estimated. Although we had found some field studies that tried to estimate sandfly population size and other demographic characteristics , we did not find any study regarding this ratio for Araçatuba city. Therefore, in our simulation, we decided to fit this ratio according to real data of human cases. Since we are studying visceral leishmaniasis dynamics, it is necessary that the disease is persistent in the population. Considering this condition, we assumed the condition and estimated the minimum value for (calculation is not shown, but we followed the method described by van den Driessche and Watmough ).
The real data provided in Table 3 suggests that the incidence was not constant along those years in which the data was collected (1999 to 2015). One reasonable hypothesis is the climate changes that have been occurring for the last years . Thus, since the sandfly population dynamics depend on climate and geographical conditions, we can include this idea in our model by fitting as time-function. It is not the scope of this paper to model the sandfly population dynamics according to climatic and geographic variations. Therefore, we will assume that a simple function for , which can fit the simulation data to the real data, should include those climatic and geographic variabilities.
Let us consider the following function for :The parameter values for (7) are in Table 4. Biologically, we can suppose that sandfly population reaches stability and oscillations decrease over time. Thus, note that for we have trending to .
5. Modeling the Dynamic of Control Strategies
System (1) models the disease dynamics over time, considering humans, dogs, and sandfly population. In order to evaluate the effect of preventive controls, we have to introduce new terms that indicate each of those methods. Since our focus is preventive control method, the target populations are dogs and sandflies.
In the following sections we present the inclusion of those new terms on system (1). We consider the parameters from Araçatuba municipality for simulation of those methods.
Each of the five control strategies considered in this work acts in a specific point of the ZVL dynamics. Because of this, it becomes clearer if we redescribe our model for each strategy separately. Therefore, we simulated 6 sceneries (one without control strategies and one for each strategy) and, for each evaluated strategy, we counted the number of individuals (dog or houses) that were controlled.
The estimation of control strategy rates is presented apart in the following sections.
5.1. Elimination of Positive Dogs
The elimination of positive dogs has already been indicated as in system (1), in the equations for dog population, and in Table 1. In this case, we suppose that this elimination rate is in accordance with the average produced by epidemiological surveillance system of Araçatuba . In other words, means the usual dog elimination rate (i.e., the dog elimination provided by health authorities in a common routine). In addition, since the official diagnosis method is serology, we assume any dog that is indicated as having antibody against Leishmania parasite as disease positive.
Note from Figure 1 that dog population is considered constant in our model. As a result, if the dog mortality is intensified due to elimination of positive ones (i.e., there is an extra/additional elimination rate by , e.g., if the health services receive better working conditions and if they are supplied by more materials), it induces an increase of dog population renewing. This renewing makes sense, since, ecologically, an eliminated dog allows a new one to replace it. In addition, as the official diagnostic techniques are based on serology, only susceptible dogs are not eligible to be eliminated. We adopted this idea because we considered the latent (, clinically ill (, and recovering () dogs had contracted the Leishmania antigen in any moment of its life. Therefore, they are eligible to be positive for diagnostic test.
5.2. Deltamethrin 4% Impregnated Dog Collar
Theoretically, the deltamethrin 4% impregnated dog collar could be applied in any dog. Therefore, we can assume that all of the four classes of dog in our model are eligible to use it and we adopted as the rate of dogs using collar per day. In this case, we indicated by the categories of dogs that use collar (susceptible dogs using collar , latent dogs using collar , clinically ill dogs using collar , and recovering dog using collar ) from those that do not use it. Basically, once a dog has this collar, it becomes protected from sandfly biting. If there is no contact between them, there will not be parasite transmission (either from infected dog to noninfected sandfly or from infective sandfly to susceptible dog).
Also, let us assume that those collars are available for inhabitants at local health centers. Thus, we suppose that owners would actively go to health center and acquire the collar for each dog they have. Since we consider that all preventive activities are supported by health policies, we can consider that the owner acquires the collar with no charge. If we imagine this simple hypothesis, we conclude that the only additional cost to the health policies is the purchasing of the collar.
Figure 2 refers to the flowchart considering the inclusion of deltamethrin 4% impregnated dog collar. Next, we have system (9), in which we included the collar-classes, and Table 5 where we describe the additional parameters for this control.
Note from Figure 2 that once the collar is fitted, there is a loss rate and a decrease of insecticide effect rate . Also, according to Halbig et al. , the efficacy of the collar is around 80%. Therefore, we considered that a proportion of those dogs using collar is protected.
5.3. Dog Vaccination
Biologically, the vaccination would be effective only in susceptible dogs , avoiding them to become infected by infective sandfly bites. Thus, if the vaccine distribution was only for susceptible dogs, it would be necessary to submit several dogs to diagnostic procedure. However, in practical terms, this is not feasible. Therefore, we suppose that all dogs are eligible to be vaccinated and this category of vaccinated dogs is indicated by (lowercase “”).
In our model, we considered that leishmaniasis vaccination would be offered together with rabies vaccine. In other words, we suppose that the rabies vaccination campaign would distribute not only rabies vaccines but also leishmaniasis vaccine. Since the rabies vaccination campaign has been already included in the annual municipality budget, the minimum additional cost to operation of vaccination as control strategy would be only the leishmaniasis vaccine purchasing. This is an idea similar to the one adopted to dog collar. However, in this model we are considering only the leishmaniasis vaccination rate (lowercase “ipsilon”) and its respective impact as control activity.
Figure 3 refers to the flowchart considering the inclusion of leishmaniasis vaccination. Next, we have system (10), in which we included the vaccinated dog compartment, and Table 6 where we describe the additional parameters for this control.
Note from Figure 3 that once the dog is vaccinated, there is a loss of immunity rate . Also, according to Fernandes et al. , the efficacy of leishmaniasis vaccination is around 96.4%. Therefore, we considered that a proportion of these vaccinated dogs against leishmaniasis is immunized.
5.4. Dog Treatment
In this control strategy, the objective is reducing the number of infected dogs, which works as source of infection. However, the probability of treating a latent dog is quite null, since this category of dog is visually healthy. Thus, we assume that only dogs that present clinical signs and/or symptoms are eligible to be treated and the dog treatment rate is indicated as .
We will consider the treatment protocol described by Miró et al. , which was composed by meglumine antimoniate plus allopurinol. In this work, the authors found a proportion of dogs that healed but still continued to be infected. In other words, once a dog is treated, there is a probability of a dog eliminating the parasitemia or not.
Furthermore, we also consider that the dog treatment would be offered by public health policies. Therefore, if the public health services have already included veterinarians in the staff, the minimum additional cost would be the acquisition of the medicine (meglumine antimoniate and allopurinol) and hospital material (e.g., syringes and needles).
Figure 4 refers to the flowchart considering the inclusion of dog treatment. Then, we have system (11), in which we included the treated dogs flux (from clinically ill to susceptible or to latent), and Table 7 where we describe the additional parameters for this control.
Note from Figure 4 that once the dog is treated, there is a probability to be recovered, but without parasitemia elimination. We adopt as a proportion of dogs that obtain only clinical recovery but are still infected . Also, once the treatment started, we assumed that any dog gives up on the treatment process over time (i.e., the proportion of dogs that receive the complete treatment is ).
5.5. Sandfly Population Control
The activities of sandfly population control focus on two approaches, both of them on the environment. First, according to Brazilian Ministry of Health , the sandfly population control includes a chemical control (spraying of insecticide on the houses) and a land clearing (that reduces the sandfly carry capacity). In order to simplify our study, we just considered that those both approaches included in the sandfly population control result in an increase of sandfly mortality rate, . On the other hand, it is unfeasible to organize a sandfly control considering the sandfly mortality rate, as “eliminated sandfly/day” (i.e., working in function of the amount ). Because of this, we considered as sandfly control rate the dimension of “treated houses/(sandfly × day)”: . Therefore, once the number of treated houses to be treated per day and per sandfly is determined, we can easily find the additional sandfly mortality rate:where means the average human/house and is the ratio sandfly/human.
In a proportional approach, note from Figure 5 that sandfly population is considered constant in our model (we remember that our three populations in our model are normalized). As a result, the proportional increase of its mortality rate induces an increase of population renewing at the same proportion. This acceleration of population renewing refers to the conception of carry capacity. Here, carry capacity means the maximum population size of biological species in an environment. Thus, whenever the sandfly population is under the carry capacity, it will tend to increase until it becomes fitted to it. Also, the opposite occurs if it is over the carry capacity (the population will decrease until its size fits the carry capacity). Finally, as our model considers the sandfly population proportionally constant, it means that when sandflies die, the population will decrease and it will be under the maximum size allowed by carry capacity. As a consequence, the population will increase by recruitment of new individuals (mathematically, this is the entrance term ). Therefore, in short, we conclude that if the sandfly mortality rate increases, the sandfly population renewing rate will also increase.
According to Burattini et al. , the acceleration of the sandfly population renewing (or, in other words, the decrease of life expectancy of sandfly population) affects directly the LVZ dynamics, since the infected sandfly is also eliminated in a shorter time. As a consequence, the parasite Leishmania will not have time enough to complete its development inside the sandfly and the proportion of infective will also naturally decrease.
6. The Estimated Costs and Calculation of Control Strategy Rates
It is very important to consider not only the result of the control strategy at the light of epidemiological approach but also the economical one too. Therefore, since the number of controlled elements is in dimension of elements/day, the estimation of the cost/elements would provide us with the estimated cost per day (i.e., cost/individual × individual/day = cost/day). Here we suppressed the cost calculation of each method, but we indicated the references from where we preceded our estimations. Table 8 summarizes those costs.
Usually, the operating of preventive control strategies is limited by logistic and financial resources. Therefore, in order to estimate the preventive control rates, firstly it is necessary to estimate those restrictions.
First, considering the data from Table 3. From “Human reported cases per year” column we estimated the year average, which is 20.18 human cases/year. Then, from Table 8, the estimated cost for human treatment is around 397.25 USD/human . Therefore, per year, the average expanses with human treatment are around 20.18 × 397.25 = 8015 USD/year. If we consider the costs per day, we have around 22 USD/day. For simplicity, we will consider that this value includes not only financial aspects but also logistic one.
Now, let us suppose that instead of this 22 USD/day that is invested on patient treatments, it would be invested on preventive control strategies. However, we should consider that this 22 USD/day is invested on the prevention of the whole dog population or houses. If we consider the human : dog ratio for Araçatuba/SP city of 10/1.8 human/dog  and the human : house ratio of 3 humans/house , the estimation of dog population and the number of houses for 2016 is around 34889 dogs and 64609 houses. Then, the invested cost per dog is estimated as 22/34889 = 6.29 × 10−4 USD/(dog × day) and, considering houses, 22/64609 = 3.40 × 10−4 USD/(house × day). Since we obtained the estimated costs for each control strategy (Table 8), it is possible to estimate the maximum rate of each control. As example, if the cost for elimination of one positive dogs is 170.71 USD/dog, the maximum rate of elimination dog per day would be 6.29 × 10−4/170.71 = 3.69 × 10−6/day. In the same way, we just repeated the calculation process and estimated the maximum rate of each control strategy, but in the case of vector control we used the estimated number of houses instead of estimated dog population. All estimated control rates are in Table 8.
It is important to present a special consideration about dimension. According to (12), dimension is “houses/(sandfly × day).” Since the dimension of estimated investment cost per house is “USD/(house × day),” we concluded that the cost estimated of sandfly population control presents the dimension “USD × sandfly/(house)2.” This dimension can be splitted as “(USD/house) × (sandfly/house).” Thus, we can observe that the cost of sandfly population control depends on density sandfly/house. The higher this density sandfly/house is, the more expensive the cost becomes. Therefore, we considered the sandfly population control average cost as 23.24 USD × sandfly/(house)2.
7. The Impact of Control Strategies on Total of Saved Humans
According to Table 2, we accessed official data of Araçatuba municipality from 1999 to 2015. Later, from those data, we were able to fit the model from system (1) by observing the resulting curve from reported human cases in (4) from fitting in (7).
Once we have the model from system (1) defined and calibrated, we are able to evaluate the dynamics of each control strategy and compare them with the no-control strategy scenery.
First, we considered the numerical simulation of system (1) from 1999 to 2025. Since we are interested in understanding the dynamics of the disease over time in an as real as possible behavior, we present Figure 6 with bars that indicate the official data. However, considering the prediction evaluation of the control strategies, we assumed in our simulation that those control strategies would start to be operated in 2018. Therefore, we observe in Figure 6 the numerical simulation and the prediction result if we consider the introduction of those strategies starting from 2018 (for a better view, see Figure 7).
Once we observed the control strategy dynamics in terms of reported human cases, it is very useful to estimate the quantity of people that avoided the infection. Just for simplification, in this text we refer to those people as “saved” human.
Since we developed a computational simulation, we had the control of sceneries. Therefore, in order to evaluate the impact of each control strategy, we compared the simulation results between introduced control strategy and no-control simulations. It is important to remember that in all simulations we computed the real total of clinically ill humans, according to (14). However, if we calculate the difference between the no-control simulation and introduced control simulation, we have the quantity of humans that were prevented to become clinically ill (the saved one).where is the total of saved humans until time and is the correspondent control strategy. Figure 8 represents the result of those totals of saved humans for each control strategy.
According to Figure 8, the dog treatment was the strategy that presented the lower number of saved people. It makes some sense, since the dog treatment does not eliminate the parasitemia status. Therefore even if an infected dog is treated, it may still continue being source of infection. On the other hand, the insecticide impregnated dog collar and dog vaccination were the strategies that most saved humans. Those two strategies reduce the amount of exposed susceptible individuals to infective sandfly biting. As a consequence, the proportion of infected humans decreases. However, although this interpretation is correct, we did not consider the restriction of resources, as financial, material, or human support. In the next section, we will include our observations about this.
8. Number of Controlled Elements and the Estimation of Total Cost
According to Table 8, each control strategy has a cost per controlled element (dog or house). Therefore, it is essential to understand how to find the equilibrium between the disease control and the available resources (material and/or financial).
In general idea, to count the controlled elements it is necessary to sum the amount of controlled elements per day over an interval of time: total of controlled elements = controlled elements/day × interval of time (days).
From total of controlled elements it is simple to estimate the invested total. Here, we are interested to compare the cost of the control strategies with the cost of human treatment. Therefore, if the cost of each strategy per element has already been normalized in terms of the human treatment cost, we are able to estimate the total cost as total cost = total of controlled elements × cost (normalized by human patient cost)/element.
Table 9 presents the expressions that calculate the total of controlled elements and the total cost of each strategy. Figures 9 and 10 present, respectively, the dynamics of total of controlled dogs or houses and the total cost normalized by human patient cost.
|In terms of patient cost.|
The index stands for the respective control strategy.
From Figures 9 and 10, it is possible to observe a similarity and correspondence between the curve responses. In terms of costs, vector control, dog vaccination, and dog collar are very close to each other. But, the difference is related to the number of controlled elements, in which we found that there were more dogs with collar than vaccinated dogs or treated houses. Also, although dog treatment and dog elimination presented reduced costs, they also controlled fewer elements too.
9. Calculation of as Function of Each Preventive Control and Evaluation of Dynamics
For each evaluation, we calculated the respective (Basic Reproduction Number) in function of the preventive control method. The Basic Reproduction Number indicates the quantity of infected individuals generated from one infective individual, when introduced in a population in disease-free equilibrium state . We assumed that is calculated when the time is high enough, where . As stated before, the full calculations are not described in this work, but we adopted the review published by van den Driessche and Watmough . Once is calculated, we calculated the respective (effective reproduction number) and investigated which one of the 5 control strategies (elimination of positive dogs , use of deltamethrin 4% impregnated dog collar , dog treatment with allopurinol and meglumine antimoniate , dog vaccination , and sandfly population control ) makes converge fastest to a value lower than 1.
Table 10 summarizes the expressions for each control strategy.