Grupo de Tecnología Bioquímica. Departamento de Bioquímica y Biología Molecular. Universidad de La Laguna. 38206. San Cristóbal de La Laguna. Tenerife. Spain

Instituto Universitario de Enfermedades Tropicales y Salud Pública de Canarias. Universidad de La Laguna. 38206. San Cristóbal de La Laguna. Tenerife. Spain

Grupo de Tecnología Bioquímica. Departamento de Estadística, I.O. y Ciencias de la Computación. Universidad de La Laguna. 38206. San Cristóbal de La Laguna. Tenerife. Spain

Abstract

Background

The WHO considers leishmaniasis as one of the six most important tropical diseases worldwide. It is caused by parasites of the genus

Results

Four biologically significant variables were chosen to develop a differential equation model based on the GMA power-law formalism. Parameters were determined to minimize error in the model dynamics and time series experimental data. Subsequently, the model robustness was tested and the model predictions were verified by comparing them with experimental observations made in different experimental conditions. The model obtained helps to quantify relationships between the selected variables, leads to a better understanding of disease progression, and aids in the identification of crucial points for introducing therapeutic methods.

Conclusions

Our model can be used to identify the biological factors that must be changed to minimize parasite load in the host body, and contributes to the design of effective therapies.

Background

The WHO considers leishmaniasis as one of the six most important tropical diseases worldwide

Mathematical modeling of the processes involved in parasite-host interactions has become a necessary tool in the study of diseases, leishmaniasis being no exception. A significant part of the modeling work in this field is epidemiological

There are many studies regarding the biology, epidemiology and immunology of leishmaniasis

In the present work we adopted a systems biology approach for understanding disease evolution, host-pathogen interactions, and immune response function. We performed this task by using experimental time series measurements in BALB/c mice infected with

Results

Mathematical Model

Experimental measurements obtained in BALB/c mice were used to fit the parameters of the mathematical model shown in Figure

Leishmaniasis progression model

**Leishmaniasis progression model**. Solid arrows represent synthesis (input arrows) or degradation (output arrows) fluxes (each flux number notated by its corresponding γ_{i}); dashed arrows are the signals among processes variables which are quantified by the corresponding g_{i }value. Positive and negative signs denote activation and inhibition of the corresponding fluxes respectively. See text for the nomenclature.

In the above expression, X_{i}, σ_{ij}, γ_{j }and g_{jk }represent the normalized variable set, the stoichiometric matrix, the rate constants, and the kinetic orders, respectively. The variables lymphocytes proliferation (X_{2}), IgG1 (X_{3}) and IgG2a (X_{4}) were normalized with respect to the respective value in the control group of mice. Because the control group is parasite-free, the same approach could not be used to normalize parasite load. In this case the variable was normalized with respect to its own mean value. This standardization reduces the range of variation of the parameters and computation time, and also exploits various properties of the GMA-PL formalism on the behavior of variables and parameters. The specific numerical values for the parameters _{ij}_{j }and g_{jk }are determined using prior biological knowledge, information about the basal steady-states of the system

Figure _{1}, ... , g_{14 }stand for kinetic orders representing influences on the creation or degradation fluxes (V_{i}) of the four variables.

The total parasite load in the host (X_{1}) stimulates its immune response. The parasites multiply in macrophages by binary division. The parasite load growth (V_{1}) has a nonlinear dependence on the parasite load through the kinetic order g_{1}. Increased parasite load leads to a decrease in the proliferation rate of lymphocytes (V_{4}); this interaction is represented by the kinetic order g_{7 }_{2}) occurs when naïve T cells are activated by antigens of the pathogen (g_{5}) and then differentiated into effector cells (Th1 or Th2) and memory cells. The activation of lymphocytes is an essential event in the production of specific immune responses (both humoral and cellular) against pathogens. Proliferation was measured following the protocol by Monks et al. _{3 }is also stimulated by X_{2}, through g_{6}. Cell mediated effectors enhance X_{1 }decay (V_{2}); this effect is represented by the positive kinetic order g_{3 }

The host immune system produces IgG1 (X_{3}) and IgG2a (X_{4}) antibodies which could be linked to the Th2 and Th1 mechanisms respectively _{2 }on the rate synthesis of IgG1 (V_{5}) through g_{9}, and on the rate synthesis of IgG2a (V_{7}) through g_{12}. These two immunoglobulins are antagonistic, so each of them has a negative influence on the generation rate of the other, namely X_{4 }on V_{5 }and X_{3 }on V_{7}. These effects are represented by the kinetic orders g_{10 }and g_{13}, respectively _{1 }rate decay, V_{2}. This interaction is represented in our model by the positive kinetic order g_{4}. It is assumed that the transformation rates V_{2}, V_{4}, V_{5 }and V_{8 }are proportional to X_{1}, X_{2}, X_{3 }and X_{4}. These dependences are represented in the model by the positive kinetic orders g_{2}, g_{8}, g_{11 }and g_{14}, respectively.

Given that every variable has an influx and an outflow, the stoichiometric coefficients are 1 and -1 for the synthesis and transformation processes respectively. Model parameters were determined by fitting the model to experimental data from mice using a genetic algorithm as described in the methods section.

Accordingly, the power law model derived from the above scheme is given by:

Figure

Data fit and predicted model dynamics of the four model variables

**Data fit and predicted model dynamics of the four model variables**. The panels under Model Data Fitting shows the data fit for the time series data of the four model variables. Panels under Predicted Model dynamics show the comparison of predicted and measured system variable dynamics for an initial parasite load of 10^{6}. The continuous lines indicate the estimated dynamics while the dotted lines indicate the experimental data. Error bars indicate the standard deviation among mice.

Model validation

We validated the model by using it to make predictions about the way the system would behave under initial parasite loads that were different from those used to calculate parameter models (106 as compared to 103, see Methods for details). We then performed the corresponding experiments in vivo (see Methods), measuring the four variables described by the model and comparing their observed behavior to model predictions. This initial number of parasites (which mimics a severe leishmaniasis condition) was chosen to check the model capacity to correctly describe the behavior in extreme and differing conditions of initial parasite load. Since the model's main purpose is for the design of therapeutic strategies, a model able to describe a wide range of parasite load dynamics is of foremost interest. Figure ^{6}. Since the model describes the evolution over the first 20 weeks after infection, the observed discrepancies in the two experimental conditions considered (model fitting and validation) can be deemed as reasonable in light of the associated experimental error. In this regard, we want to stress the fact that in the experimental data used for model verification, other elements of the immune system may be playing a significant role not addressed by the model, but which could be relevant in conditions of massive infection.

Sensitivity Analysis

Figure _{i}) and rate constants (γ_{j}). The System Parameter Dynamic Sensitivity is noted as S_{pk}^{Xi}; ^{-5 }(S(g_{10}; X_{2})) and 1.83 (S(γ_{1}; X_{1})). The lower value corresponds to the influence of g_{10 }on the lymphocyte proliferation, and the higher value measures the influence of the rate constant associated to the parasite multiplication rate on the parasite load. This range of values, together with the observation that the median of the absolute values of sensitivities is 0.067, indicates a robust model.

Absolute value of the system dynamic sensitivities S_{pk}^{Xi }(X_{j }p_{k})

**Absolute value of the system dynamic sensitivities S_{pk} ^{Xi }(X_{j }p_{k})**.

In general, the higher sensitivities correspond to the variable indicated by the arrow, except for the parameters directly influencing lymphocyte proliferation (γ_{3}, γ_{4}, g_{5}, g_{6}, g_{7 }and g_{8}). For the parasite load, sensitivities with absolute values greater than 1 are S_{γ1}^{X1 }and S_{g1}^{X}. This implies that the generation rate of parasites, and the effect of parasites on their own generation, strongly influence parasite load. Sensitivity S_{γ2}^{X1 }is much lower than S_{γ1}^{X1}. All the other parameters yield sensitivities with absolute values of less than 0.1 for parasite load. The sensitivity of IgG1 and IgG2a to most of the parameters is higher than the sensitivity of lymphocyte proliferation or parasite load to the same parameters. This could be a consequence of the fact that most of the values for parameters directly influencing the parasite load (γ_{1}, γ_{2}, g_{1}, g_{2}, g_{3}, and g_{4}) are small (< 1), as opposed to those directly influencing lymphocyte proliferation.

Systematic search of parameter profiles for the minimization of parasite load

In order to apply the model for therapeutic purposes, we carried out a systematic search of parameter values that minimize the parasite load. The aim was to discover the set of parameter values (kinetic constants, g_{i }and rate constants, γ_{i}) that yields a reduced, minimum value of the parasite load, both during the time of infection evolution and at the final, 24-week stage.

The search program was organized in two phases. In the first phase, we changed only one of the parameters at a time (g_{i }or γ_{i}), with the others maintaining their original values. In this case the value of the candidate parameter is initially set to 10% of the model estimated parameter, the following to 20%, 30% and so on, until the parameter reaches the upper-bound range that was assumed feasible and physiologically relevant. Then, for each changing factor, the model solutions were calculated. In order to evaluate the effectiveness of the change in parameter value, the mean, maximal and final parasite loads were calculated. The mean parasite load reflects the average severity of the disease, the maximal value accounts for the maximal number of parasites along the infection dynamics (which has to be lower than the maximum number of parasites the organism can bear), and the final parasite load represents the final outcome of the disease.

Single-parameter search for identification of optimum parameter values

_{i }

After a systematic search among all kinetic constants, we found that g_{1 }and g_{6 }were the most suitable parameters to be changed for reducing parasite load. g_{1}, describes the influence of parasites on their own proliferation and is the most significant in this regard, since changes in its value causes the greatest reduction. This is achieved by increasing g_{1 }value from 0.53 to 3. Figure _{1 }ranging from 0.01 to 3. Our model suggests that increases in g_{1 }from its initial value (0.53) have a therapeutic effect, because they lead to a decrease in parasite load and therefore to healing. It is important to note that for values of g_{1 }< 0.5, final parasite load is proportional to g_{1}. However, for values of g_{1 }> 0.5, final parasite load becomes inversely proportional to the value of this parameter. This latter region includes the actual g_{1 }value. This implies that therapeutic strategies should aim to increase g_{1}. If decreases are sought, such decreases must be well below 0.5 in order to have a similar effect.

Evolution of the parasite load over time for different values of g_{1}

**Evolution of the parasite load over time for different values of g _{1}**. A. Final parasite load during a time period of 24 weeks for different values of g

Figure

At first glance this result could appear paradoxical and certainly counterintuitive. But, in fact, the sensitivity of g_{1 }with respect to the parasite load (X_{1}), S_{g1}^{X1}, is negative (see Figure _{1 }should lead to a decrease in parasite load. This prediction holds true for the mean parasite load over a time period of 24 weeks. However, it has been observed that both the maximum and the final value of parasite load increase in the beginning until they reach a threshold value and decrease from that point on. Again, such a decrease in parasite load indicates that increasing g_{1 }produces a therapeutic effect. The same result has recently been observed by Dancik et al. _{1 }in our model, which is influenced by g_{1}) impairs the pathogen load in certain stages of the disease. Since the modeling strategy used by Dancik et al. _{1 }as the mechanism that increases parasite growth rate. They reported the same evolution pattern in the parasite load that we found: a higher parasite growth rate yields a higher increase in pathogen load in the beginning but also a higher decrease afterwards in such a way that, as a whole, parasites are eliminated earlier. They observed that infection was cleared after eight weeks versus the 17-20 weeks in our model, but this can be attributed to the different

Another g_{i }parameter with similarly minimizing effects on the parasite load is g_{6}. g_{6 }stands for the influence of lymphocytes on their own production. It has been observed that the parasite load (final, maximal and mean) can be reduced by increasing g_{6 }from its original value of 0.02 to 1.02 (Figure _{6 }could be related to system immune response enhancement. Lymphocytes need some time to identify the pathogens, thus there is a time lag between the start of immune response, the identification of parasites, and their elimination. Accordingly, the parasite load augments until reaching a point where it suddenly decreases.

Evolution of the final parasite load over time for different values of g_{6}

**Evolution of the final parasite load over time for different values of g _{6}**. A. Final parasite load during a time period of 24 weeks for different values of g

_{i }

We also carried out a systematic search among all rate constants. We observed that changes in all _{i }_{2 }_{2}_{2}

Evolution of the parasite load over time for different values of γ_{2}

**Evolution of the parasite load over time for different values of γ _{2}**. A. Final parasite load during a time period of 24 weeks for different values of γ

Combined two-parameter searches for identification of optimum parameter values

We carried out a systematic scanning of all the combinations of two parameters that yielded the minimum final parasite load. The rationale is that a combination of drugs makes it possible to reduce the parasite load in greater quantity, more quickly, and with lighter dosage than using only one drug. The search was limited to smaller parameter changes in the range of 60% - 180% of a parameter's original estimated value.

We found that a total reduction of the observed final (as well as the maximal and mean) parasite load can be attained by simultaneously increasing g_{1 }from its original value by a factor of 1.6 (approximately), and by changing any other of the remaining 21 parameters by different factors (Table _{1 }and g_{2 }are also good choices for the minimization of parasite load. The remaining combinations also produce a reduction of parasite load, but to a lesser degree, and are considered to be of minor interest.

Parameter change factors for the two parameter combinations involving g_{1.}

**Parameter combination**

**FC(g _{1})/FCp_{k}**

**Parameter combination**

**FC(g _{1})/FCp_{k}**

g_{1}/γ_{1}

1.73/0,67

g_{1}/g_{5}

1.77/0,81

g_{1}/γ_{2}

1.78/1,60

g_{1}/g_{6}

1.78/1,74

g_{1}/γ_{3}

1.79/1,20

g_{1}/g_{7}

1.78/1,49

g_{1}/γ_{4}

1.79/0,79

g_{1}/g_{8}

1.76/1,20

g_{1}/γ_{5}

1.78/1,26

g_{1}/g_{9}

1.77/1,18

g_{1}/γ_{6}

1.79/0,94

g_{1}/g_{10}

1.77/1,76

g_{1}/γ_{7}

1.79/1,20

g_{1}/g_{11}

1.79/1,58

g_{1}/γ_{8}

1.79/1,53

g_{1}/g_{12}

1.77/1,65

g_{1}/g_{2}

1.77/0,83

g_{1}/g_{13}

1.79/1,54

g_{1}/g_{3}

1.79/0,95

g_{1}/g_{14}

1.74/0,92

g_{1}/g_{4}

1.79/1,21

The Parameter combination columns stand for the combination of parameter while the FC(g_{1})/FCp_{k }columns indicate the ratio between the factor change (FC) of the original value or g1 over the FC of the corresponding parameter (p_{k})

Optimized final parasite load obtained for each possible combination of two parameters of the system

**Optimized final parasite load obtained for each possible combination of two parameters of the system**. The parasite load of the fitted system was 5.5.

By way of illustration, Table _{1 }(black column and file in Figure _{1}, γ_{4}, γ_{6}, g_{2}, g_{3}, g_{5}, and g_{14}. Their values are reduced by factors ranging from 5 to 85% (see Table

Discussion

The standard leishmaniasis treatments are chemotherapy based, though some new treatments are based on the use of immunotherapy. In our model, the chemotherapeutical agents are those that target parasite destruction (g_{2}) or inhibit proliferation (g_{1}), whereas immunotherapeutic treatment implies changing parameters g_{3}, ... g_{8 }and g_{3}, g_{4}, g_{6}, g_{8}, g_{9}, ...g_{12}. In most cases the exact interaction mechanism of the drug is not yet known, though it is possible to associate them to the corresponding parameters that are being influenced. It is important to mention that if a given therapeutic agent has an influence that is not represented by any of our model's parameters but corresponds with the in- or outflux of a model variable, the effect of this agent can be translated in our model by a change in the respective rate constant γ_{i}.

Regarding drug therapy, we have found three parameters which cause parasite load reduction: g_{1}, which describes the influence of parasites on their own proliferation; g_{6}, which represents the influence of lymphocytes on their own proliferation; and γ_{2}, the rate constant for parasite degradation.

Examination of the standard drugs used for leishmaniasis treatment shows that most are aimed at parasite destruction. In our model that translates a an increase in γ_{2}, the rate of parasite destruction _{2}. These observations constitute a pragmatic,

Most of the therapeutic drugs used also seem to inhibit, albeit through different mechanisms, parasite proliferation: aminoglycosides alter parasite messenger RNA, pentamidine inhibits polyamine and DNA synthesis in the parasite, imidazole and itraconazole inhibit demethylation of membrane, and pyrazolopyrimidines block protein synthesis and destroy parasite RNA. All these effects can be interpreted, in terms of our model, as a decrease in g_{1}. The discrepancy in our model's predictions can be explained by several facts. First, in all cases where a decrease in g_{1 }could be assumed, there is also the concomitant effect of increasing γ_{2}, as noted above. Thus, a trade-off of these two actions should be previously evaluated in order to have an accurate account of the whole drug effect. Second, it should be taken into account that the effect of a g_{1 }modulation could be different depending on the stage of the disease. It has also been shown that if parasites replicate quickly, the immune system is able to recognize them more easily

The factor that increases the influence of parasites on their own proliferation (g1) is crucial according to our model's results; and currently, no pharmaceuticals that increase g1 have been tested against Leishmania. Insuline-like growth factor 1, interferon, and possibly TNF-α cytikine could be considered as potential targets for stimulating parasite replication inside macrophages, and it would be of great interest to test their anti-leishmanial effectiveness. Insuline-like growth factor also increases the number of parasites (γ_{1}) and reduces parasite-toxic production of nitric oxide (γ_{2}).

Furthermore, no existing drug is known to have an effect on g_{6}, which, in our analysis, is also seen as a possible effective pharmaceutical target. This clearly points to the new, potential application of existing and current therapeutic strategies.

The approach used for detecting key processes that must be regulated in order to reduce parasite load also allowed us to identify combinations of two drugs that would eventually be more effective than a single drug treatment. As is showed in Figure _{1 }or g_{2 }and simultaneously change any other parameter, or, alternatively, combinations of drugs that decrease g_{1 }together with the change in another parameter, would cause significant reduction in the final parasite load. These findings greatly amplify the number of therapeutic options available, although they still remain to be tested. By way of illustration, we could suggest the combination of any of the available drugs that increase g_{2 }(amphotericin B, aminoglycosides, antimonials, pentamidine imidazole, itraconazole, and pyrazolopyrimidines) together with any of the following: interleukin-5,6,13 and MHC class II molecules (both increasing g_{3}), rLmSTI1 (increase in g_{4}), and chemokines (that increase g_{3 }and g_{2 }simultaneously). In the same mouse model we will test the effects on the variables of different drug combinations to verify the model's predictions and to eventually refine and extend the model by including new variables and mechanisms.

A limitation of the present approach is that our model is a simplification and does not include a detailed description of all the factors involved in the interaction mechanism of the drug in the body. However, given that these mechanisms are often not known, the modeling approach constitutes an approximation to the understanding of a complex dynamic system based on available information and informed hypothesis.

Conclusions

In the present work we have illustrated a novel approach for the design of effective therapeutic strategies for leishmaniasis treatment. The approach is based on the integration of experimentally available information on infection development in an animal model using a mathematical model that describes the system dynamics observed. Many of the predictions concur with the standard practice, while others remain to be explored. We are confident that this rational, model-based approach is of great interest given that it overcomes the limitations of a trial and error strategy, and provides an extra layer of rationality in the search for new therapeutic formulas. This approach is also readily applicable to other parasitic-related illnesses.

Methods

Mice

BALB/c mice, 6-8 weeks old, were obtained from the animal breeding facilities of the Universidad de La Laguna. The experimental protocols used were approved by the Animal Care and Use Committee of the University of La Laguna (Approval ID number 132).

Parasites and experimental infection of mice

Amastigotes of MPRO/BR/77/LTB0016 strain of

In order to follow up the evolution of the infection, 10^{3 }(10^{6 }in case of the verification experiments) stationary phase

Parasite quantitation

Estimates of the parasite number present in infected organs were done as described in Buffet et al.

Serial threefold dilutions ranging from 1 to 1/3 × 10^{-6 }were prepared twice for each homogenate in wells of 96-well plates, containing 200 μl of Schneider's insect medium, pH 7.2, supplemented with 10% heat-inactivated fetal bovine serum, 200 U/ml penicillin and 200 μg/ml streptomycin. After 7 and 10 days at 26°C, each well was examined and defined as positive or negative based on the presence or absence of viable promastigotes. A limiting dilution analysis was applied to the data to estimate the number of viable

Antigen

The soluble antigen of the parasite (LaAgS) used for enzyme-linked immunosorbent assay (ELISA) determination and splenocytes proliferation assay was obtained from stationary phase cultures of MHOM/BR/77/LTB0016 strain promastigotes of

Antibodies

The specific antibody response levels of IgG1 and IgG2a against LaAgS were determined by indirect ELISA in serum of BALB/c mice. ELISA assays were carried out using standard conditions. Microtiter plates were coated with 0.8 μg per well of the antigen. The sera from the mice were assayed at 1:80 dilutions. As secondary antibody the HRPO-conjugated goat anti-mouse IgG1 and IgG2a were used to 1:8000 and 1:1000, respectively. We used two groups of 8 mice, one experimental and one control.

Lymphoproliferation

_{4}Cl, 10 mM KHCO_{3}, 1 mM EDTA, pH 7.4) and remaining cells were finally resuspended to a density of 2.5 × 10^{6 }cells/ml in DMEM containing 10% FCS, 2 mM L-glutamine, 0.05 mM 2-mercaptoethanol, 12 mM HEPES, pH 7.1, 100 IU/ml penicillin and 100 μg/ml streptomycin. Lymphocytes were divided into 100 μl aliquots (2.5 × 10^{5 }cells) in 96-well plates and they were allowed to proliferate for 3 days at 37°C in an atmosphere containing 5% CO_{2 }and 95% humidity in the presence or absence of LaAgS (final concentration 40 μg/ml).

Proliferation was measured by SRB assay, following the protocol by Monks et al.

where _{Tw }_{Cw }

Parameter estimation

In the power-law model used, the parameters of the model, kinetic orders (g_{i}) and rate constants (γ_{i}) were estimated from experimental data using a genetic algorithm adapted for power-law models

where _{var }_{tp }_{j}(t_{i}) is the predicted value for the ^{th }^{th }_{j}^{exp}(t_{i}) is the value of the ^{th }^{th }

Parameter estimation was executed from the previously obtained time series of experimental data after normalization _{i }were permitted to vary within the range [0,10] and the g_{i }within the range [0,3], except the parameters g_{2}, g_{8}, g_{11 }and g_{14 }which were set to 1 since they relate variables with their own outflow (V_{2}, V_{4}, V_{6 }and V_{8}, respectively). This model hypothesis is a biologically plausible one that permits reduction in terms of the number of parameters. These ranges come from physiological as well as kinetic considerations. Processes in which kinetic orders are greater than 3 are rarely observed since this corresponds to processes or reactions with extreme sensitivity to changes in one of the reactants. A similar reasoning applies to higher γ_{i }values, although in this case the admissible range is wider. However, in the optimization search we used greater ranges. The rationale is that in the optimization search we can assume possible wider ranges as physiologically acceptable because the optimized system would correspond with altered kinetic behavior with the use of drugs or other therapeutic agents.

Parameter estimation using 1000 iterations which required a computation time of 96 hours. The objective function value finally reached was 0.0517, with a maximal absolute error between interpolated data and model of 0.5828. In accordance with the objective function definition, this measures the average distance between the experimental data and the model. Since the standardized experimental variation values range between 0 and 2.5, an objective function value of 0.0517 means an average relative error lower than 5%, which is less than the experimental error and therefore enough to ensure that the model represents the experimental data.

All variables in the model are normalized. Parasite load is normalized with respect to its proper mean and the remaining variables are normalized with respect to the control group of mice. This standardization reduces the range of variation of the parameters and computation time and also exploits various properties of the GMA-PL formalism concerning the behavior of variables and parameters. Initial values of parasite load between 10^{-10 }and 0.01 were implemented, comparing them in terms of the objective function. The smaller the minimum of the objective function, the better the model (with the respective initial value) fits the experimental data. Of the initial values for which simulations meet the criterion f_{obj }< 0.2, we found that the best fit and lowest objective function was attained with 10^{-6 }as the initial value for parasite load in the group infected with 10^{3 }parasites. Therefore 10^{-6 }was used as the initial value in that group.

Model selection strategies

The topology of the model is assumed to be as it appears in Figure

Dynamic Sensitivities

Sensitivity analysis is a tool useful for model robustness evaluation and system dynamics characterization. Since our model studies the system dynamics, this tool enables us to identify the parameters with major influence on the transient dynamics. We have used the System Parameter Dynamic Sensitivity, S_{pk}^{Xi}, defined in equation 6:

In the above expression, _{pk}^{Xi }value corresponds to the variation of the area under the variable time course after perturbation in parameter space. For each model variable the absolute values of the area _{k }_{k2 }_{k }

Authors' contributions

BV and NVT proposed the elaboration of a mathematical model based on the GMA power-law formalism of the progression of the disease and with the potential to be used for the design of effective therapies. They planned the work and coordinated the study. CP-B carried out parasite, antigen, antibodies and lymphoproliferation quantization. BML implemented the GMA-PL model.

BML, BW and CG-A performed the optimum parameter searches. NVT, BV, BML and CP-B and CG-A defined the model and obtained the numerical parameters used in the paper. All authors read and approved the final manuscript.

Acknowledgements

The authors gratefully acknowledge Dr Carlos González-Alcón and Guido Santos and for their helpful observations. This work was funded by research Grants from Spanish MICINN (Ref. BIO2011-29233-C02-02, PI081984 and RD06/0021/0005) and from Agencia Canaria de Investigación, Innovación y Sociedad de la Información (Ref. Project PIL2070901 and PIL2071001).