This study was designed to use a mathematical model to estimate the risk of acquiring falciparum malaria for travelers to the endemic regions of Brazil. The model assumes that the population of humans is subdivided into three classes and the population of mosquitoes is similarly divided into three compartments summarized in table S1, Additional file 1.
It was separated from the human general population individuals that are in the area a cohort [ 7 ], denoted by primes and named "probe", which represent a cohort of travelers, followed through their entire exposure in the region, to calculate the risk of malaria acquisition. A deterministic version precisely determined through a known relationship of the model was used to describe the malaria dynamics in the resident population level and a stochastic version using a ranges of variable values providing a probability. On the equations analyzing the probes to describe the risk probability of contracting malaria of a single individual traveler visiting the region.
This is based on the assumption that, since the probe is a small number of individuals, the biting rate will randomly fluctuate and the probability of infection is unpredictable. The model parameters are shown in table S2, Additional file 2. The seasonality parameters c S and d S where chosen to represent the observed seasonal variation in the Amazon region described by Tadei [ 10 ], who described a 30 fold difference in mosquito number between summer and winter.
Using the parameters in table S1, Additional file 1 the model calculates that around ,, falciparum malaria cases will occur annually, a number similar to WHO estimates for Brazil in [ 2 ].
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The probability of infection in this self-limiting outbreak is then given by the following expression:. In the above equation, S ' H t and N ' H t are respectively the number of susceptible hosts and the total population of the cohort used as a probe, and h mal t is the force of infection of malaria, defined as the per capita number of new cases per time unit [ 7 ] and expressed as.
This is done by setting the limits of integration in equation 1 as:. The average risk for a traveler who arrived in the Amazonian region at four different time periods was calculated, namely, in the dry season winter in the spring, in the wet season summer and in the fall.
The model produces a result of , cases falciparum malaria per year. This number is very dependent on a number of other variables and parameters. In the sensitivity analysis below all the parameters are varied and as a consequence the yearly number of cases varies.
- Modeling the risk of malaria for travelers to areas with stable malaria transmission.
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The result for the risk calculation is shown in figure 1. Stochastic risk of catching falciparum malaria for travelers as a function of the period of the year they arrive and the time remaining in the area the figure shows 20 iterations of the simulated. In this section we analyze the sensitivity of the model to the parameters. This is done in two steps: a deterministic analysis at the populational level, which describes the sensitivity of the model to measurement variance in the parameters; and a stochastic analysis at the individual level, which determines the variation in the model's outcomes due to intrinsic stochasticity in some of the parameters.
The result of this analysis is given in table S3, Additional file 3. The sensitivity of the model is significantly influenced by with the season of the year. Biting rate and mosquito mortality are well recognized by entomologists as important parameters as they describe vectorial capacity a quadratic component and mortality expressed exponentially in the equation.
Of all the parameters described in table S1, Additional file 1 the biting rate a single individual is subject, a' , and the probability that an infectious biting is infective to the individual, b' , for an occasional traveler are obviously stochastic variables. As mentioned above, it was assumed a Poisson distribution for the parameter a' and a Gamma distribution for the parameter b' with a small variance. The result is shown in table S4, Additional file 4 which shows the risk of malaria acquisition for a traveler who arrives at different moments of the year and remains 30 days in the area.
The highest risk of malaria acquisition occurs for individuals arriving in autumn around one case for every visitors as the infected mosquitoes population is close to its peak and the proportion infected high see figure 2.
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In the figure we show in the x-axis time in days. It starts arbitrarily at day and illustrates seasonal variation of a mosquito population susceptible and infected. The arrows define the relative season and seasonal impact on the susceptible mosquitoes populations thin line over the year. The figure also shows the infected mosquitoes thick line amplified 20 times. Where an individual remains for a 1 year their risk is approximately 1. This rate closely correlates with the incidence observed in Amazon residents of 1. PAHO [ 12 ] estimates that this incidence, a maximum of 75 malaria cases per inhabitants annually will occur.
We are estimating only the cases of falciparum malaria, which represent about one third of the total malaria cases, hereby total predicted P falciaprum malaria cases for this region is of the order of 33 cases per inhabitants per year. Mathematical models for estimating risks, as described in this paper, should be considered as auxiliary tools for decision-makers.
Some caveats, however, are necessary; the model's outcomes are determined by assumptions in the dynamics of the system modeled and on the values given to the parameters. In our model, the most critical assumptions relate to homogeneity. For example, the Amazon region is very large and therefore, it is likely that some parameters will vary from region to region, such as the densities of vectors and human hosts determined by the respective carrying capacities.
The seasonal variations assumed in the model are simple and are only an approximation of the actual climatic variations that occur in the Amazon region. Notwithstanding the oversimplifications of our model, we believe that our results are a good approximation in the sense that the actual risk of malaria lies within the estimated confidence intervals calculated by the model.
Previous studies have attempted to determine the cumulative risk of acquiring malaria in travelers [ 6 , 13 , 14 ], but the estimated incidence rates were not generalizable to all travelers at all times, as malaria incidence varies greatly from year to year [ 15 ]. Mathematical modeling is well suited to adjust for seasonality and annual variations. In a previous analysis we modeled the risk of dengue and yellow fever, with similar approach to the one described here [ 16 — 18 ].
This is the first time that travelers' malaria risk estimates have been calculated using mathematical modeling. Our models are robust and have been tested extensively on Amazonian data [ 19 ]. Risk for malaria risk for endemic populations has also been estimated using modeling by Okell et.
A traveler arriving in summer Dec-Feb exposed for days has at least a ten-fold higher risk of infection than a traveler who arrives in the winter June-Aug for a visit of similar duration. It is shown that the risk increases nonlinearly with time, but this again varies by season of exposure. National and international recommendations for long term travelers; particularly those traveling through regions of varying transmission and with different malaria species have been very crude all or nothing and have a very limited evidence base [ 21 ]. The model can be used for highlighting the malaria risk in a way that many advisors and their clients can interpret.
An individual arriving during the summer Jan-Mar , that is, rainy season, has a probability of 0. In fact, Behrens et al.
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