The Assessment of the Arising of Food Allergy among Antiacid Users Using Mathematical Model

The first case of a new food allergy, an anaphylactic reaction to Manioc (Manihot esculenta or Manihot utilissima), also called cassava and tapioca, was described in 2001. Cassava is a tuber widely consumed in Brazil, which has been consumed by native Indians, i.e., more than 500 years ago, so why are the first cases just emerging now? We address this question by developing a mathematical model considering the fact that proton pump inhibitors (PPIs) for digestive disorders maintain the cassava allergen intact due to the elevation in pH of gastric juice, thereby facilitating its presentation to the immune system. The model assumed the mass action law including saturation to describe the recruitment of PPIs users, and Hill function to describe the sensitization of immune system by the allergens and the acquisition of full food allergy. Analytical results were obtained from the model, and numerical simulations were done. The estimated period of time elapsed since the introduction of antiacids before the diagnosis of food allergy was 15 years. The food allergy may become a public health problem, if PPIs are being used indiscriminately and irregularly. The results obtained from the analysis of the model suggest that the use of nonprescription antiacids, like PPIs, may be recommended or approved by the physician in order to avoid the rising of food allergy.


Introduction
In 2001, Chiron et al. [1] described the first case of a new food allergy, an anaphylactic reaction to Manioc (Manihot esculenta or Manihot utilissima), also called cassava and tapioca.In 2004, Galvão et al. [2] described, nearly at the same time as a group of researchers from Portugal [3], the first case of cross-reactivity between cassava and natural rubber latex.Another case was reported by Ibero et al. [4].Cassava is a tuber, rich source of carbohydrate widely consumed in Brazil and in more than 80 countries, mainly in South America, Asia and Africa.In Brazil, this food has been consumed since before the discovery of the country, by native Indians, i.e., more than 500 years ago, so why are the first cases just emerging now?A simple and superficial explanation would be a deficient diagnostics.However, this explanation is at least scarce giving that technical conditions and specific knowledge for a detailed diagnostics in allergy is available since years.Diagnostics in allergy evolved a lot in the last decades especially after description of IgE [5].Furthermore, even earlier than development of specialized techniques medical literature already discussed clinical reports and therefore new allergies could be detected and diagnosed by clinicians.Accordingly, it seems that allergy to cassava is indeed a recent problem and considering the long period of consumption and population exposure to this food we can conclude that presumably one or more factors contributed to sensitization process and progression to allergy.In attempt to elucidate the problem our group investigated new cassava allergens from clinics to molecular biology, including the identification of a new allergen (the Man e 5) with cross reactivity to Hev b 5 from natural rubber latex [6].
Back to the fundamental question, why only now is this allergy been detected and the cases are increasing?One hypothesis that has been considered is the increasing use of proton pump inhibitors (PPIs) for digestive disorders [7].This therapeutic class was introduced in 1988 and today they are among the most used drugs worldwide, including Brazil.Studies carried out by the group of HC-USP with this new allergen have demonstrated that this protein is part of a very special group of proteins called "natively unfolded or intrinsically unstructured" [8] that have been shown experimentally or are predicted to lack ordered structure [9].These characteristics make it extremely labile to the action of the pepsin in the low pH of gastric juice, leading to its degradation in seconds (unpublished data).Antacid medications (PPIs) cause leaky gut syndrome, where food is absorbed before it is completely digested.This can result in both food allergies and immune exhaustion [10,11].In this way, the elevation of the pH caused by PPIs results in maintenance of this cassava allergen intact, thereby facilitating its presentation to the immune system.
All the facts and evidences presented above lead us to a second question: why was allergy to cassava found in Brazil around 2002, if the PPIs were introduced around 14 years earlier in 1988?To answer this question we must consider some factors.From the beginning of the marketing of a new drug until the exponential growth in its use there is an elapse of time.In addition to this, new drugs with similar actions emerge.Moreover, we must consider that only part of the population uses the drug indiscriminately and irregularly, and that only part of it is genetically predisposed to sensitization.We must also take into consideration the time of sensitization to a new allergen and the period of time between the onset of symptoms and diagnosis.
The use of antiacids is triggered by a variety of conditions, ranging from stomach upset, heartburn, and nonulcer dyspepsia to verified gastric and duodenal ulcers.Antiacids have been found to be effective in the management of heartburn and ulcer dyspepsia and are also effective in the maintenance treatment of duodenal ulcer disease [12].Drugs such as the H2-receptor antagonists, sucralfate, pirenzepine, and omeprazole, which have been introduced during the last 15 -20 years, have to a great extent replaced antiacids in the treatment of peptic ulcer.The introduction during recent years of new treatment strategies that aim at the eradication of Helicobacter pylori may also have influenced the use of antiacids.However, the use of antiacids in less-severe forms of dyspepsia is very common, and antiacids still play a major role in the treatment of acid-related disorders [13].Most dyspeptic symptoms are dealt with by patients, without seeking medical advice [14].Corder et al. [15] have shown that the majority (80%) of those with dyspeptic complaints who did not consult a physician used simple antiacids.Furu and Straume [16] reported the overall prevalence of antiacid use about 10%, which is much higher than reported in a survey from Finland, in which only 3.7% reported use of antiacids [17].
Health surveys may collect information on nonprescription medications and also assess health status, risk factors, and sociodemographic variables at the individual level.Dyspepsia and related symptoms obviously repre-sent the most common reason for the use of antiacids.However, there is reason to believe that additional factors such as general health status, lifestyle, and utilization of health services are associated with the use of antiacids, but no studies have explicitly tried to assess these associations [16].Aiming to assess the risk of appearance of food allergy due to the increasing trend in the antiacid usage, we develop a mathematical model.
The paper is structured as follows.A simple mathematical model is developed aiming the assessment of the prevalence of food allergy among antiacid users (Section 2).We determine the steady states of the model, and the stability of the equilibrium points (Section 2).Also numerical results are presented with discussions (Section 3).Finally we present conclusion (Section 4).

Model
Our objective is the development of a simple mathematical model to describe the prevalence of food (cassava allergen) allergy in a population since the introducetion of antiacid by pharmaceutical industries.One of the effects of the antiacid is the elevation of pH in the stomach.As a consequence macro-proteins are not completely cleavage to aminoacids, and micro-proteins are absorbed by the surrounding epithelial cells and circulating immune cells (macrophages, dendritic cells, etc.) can recognize epitopes of these micro-proteins.In normal immune response, however, these epitopes are considered as self, and no immune reaction is promoted.Nevertheless, in some persons, the immune cells can be sensitized by epitopes, and, in conjunction with genetic predisposetion, environmental factors, etc., plasma cells are promoted to release antibodies (a comprehensive biological basis of food allergy can be found in [18]).
Our aim is to assess the prevalence of food allergy among persons in a community, rather than the individual development of the allergy [19].A mathematical model is developed by assuming the following hypotheses.
Before the introduction of antiacid by pharmaceutical industries, we assume that there are not food (cassava allergen) allergic persons, and they are considered naive persons.The number of naive persons at time t is denoted by P. In a given time, antiacid medication is introduced in a completely naive population.Depending on the amount of budget applied by industries to promote the medicine, and on the efficacy of the antiacids, the number of antiacid users at time t, denoted by U, increases with time, but attains a saturation after a period of time due to, for instance, regulation of the medicine usage, less adherence of naive population, and even the limited capacity of production by pharmaceutical industries.We translate both features supposing that the incidence of new users is proportional to the numbers of naive persons and antiacid users, that is, . This kind of interacting function was used to describe the uptake of drugs by cancer cells [20].The adherence to antiacid usage depends on the parameter  , named recruitment rate, and takes into account the marketing towards naive persons (P) and the efficacy measured by users (U).The recruitment of antiacid users follows mass action law for small number of antiacid users ( , for ) [21], but at increased number of users, the recruitment is saturated with respect to the users ( ).The parameter 1 U  P K is a scaling factor of recruitment: for lower values, the effect of saturation occurs very quickly, and the mass action law is practically absent (see Appendix A).
Due to the residuals of micro-proteins (food allergens) in stomach which are absorbed by epithelial cells, immune system of antiacid users can be sensitized.This sensitization is strongly dependent on the amount of food allergens [22] that are presented to antigenic presenting cells (APC).In general there is a threshold of food allergens concentration that can effectively sensitize immune cells.These micro-biological behavior are translated to the populational sensitization by the factor The parameter  is the sensitization rate of antiacid users, which transfers these users to a group of sensitized persons (S, the number of sensitized persons at time t).The occurrence of sensitization is low for small number of antiacid users ( ), but at increased number of users, the occurrence is saturated ( ).This saturation is due to the threshold in the amount of food allergens in a person.The parameter 1 U  U K is a scaling factor of sensitization.Among sensitized persons, due to genetic predisposetions, environment changes, etc., some individuals are actually driven to acquire immune response against food allergens.This biological event occurs less frequently than sensitization, hence we assume that acquisition of food allergy obeys . The parameter  is the allergic reaction acquisition rate of sensitized persons, which transfers these individuals to a group of food allergic persons, denoted by A, the number of food allergic persons at time t.The occurrence of food allergy acquisition is very low for small number of sensitized persons ( ), but at increased number of sensitized persons, the occurrence is saturated ( ).The parameter 1 S  S K is a scaling factor of food allergy acquisition.
Both sensitization and food allergy acquisition were described by Hill function of order n [23] (see Appendix A).In biochemistry, the binding of a ligand to a macromolecule is often enhanced if there are already other ligands present on the same macromolecule.In other words, once a ligand is bonded to the enzyme, its affinity for other ligand molecules increases.By using higher order for the allergy acquisition (n = 3) than sensitization of naive person (n = 2), we are assuming that there are much more factors needed to trigger allergic reaction than sensitization.
Besides the allergy acquisition parameters, we assume that all persons are under the same mortality rate  , with 1   being the life expectancy.We also assume that antiacid users return to naive group at a rate  of abandonment of the medicine.We also assume that there is an effective treatment (when available) that cures allergic persons, and they return to sensitized group at a rate  of treatment.
. Dynamical trajectories are obtained using the following initial conditions (at which represents introduction of antiacid medication at 0 t  in a community previously free of this medicine.The initial value 0 represents the proportion of persons who were coopted by intense marketing of pharmaceutical industries just before the introduction of antiacid.

u
In Table 1 we present the summary of variables and parameters of the model.
Let us determine the steady states of the system of Equation (1), and assess the stability of these equilibrium points.
The steady states, denoted by  = , , , Q p u s a, can be determined by letting the derivatives equal to zero in the first members of the system (1).One of the steady states is the trivial equilibrium point 0 Q given by which corresponds to the population free of antiacids, hence free of food allergic persons.
To establish the stability of the trivial equilibrium point, we evaluate the eigenvalues of the characteristic equation associated to the matrix obtained by linearizing the system (1) around this equilibrium point.All eigenvalues will have negative real part if the Routh-Hurwitz criteria are satisfied [21].Hence, , where th  is the threshold of  , and is given by From the first two equations of (1) in the steady state, we obtain with u being the positive solution of the equation where the rational functions   f u and   From the last two equations of (1) in the steady state, we obtain with s being the positive solution of the equation where the rational functions and are, in terms of the previous solution In Appendix B we present the analysis of Equations ( 6) and ( 8) with respect to the number of solutions: we have up to three positive solutions.
We next present numerical analysis regarded to the non-trivial equilibrium points.Also the system of Equations (1) is solved numerically in order to asses the time delay to occur the exponential growth phase of food allergy in a population.

Results and Discussion
The prevalence of antiacid users in 1996-1997 was higher than in 1987-1988 adjusted for age, gender and heartburn, and the proportion of antiacid users increased among those with dyspeptic complaints and also among those reporting no dyspeptic symptoms [24].In this section we assess the prevalence of food allergy arising among antiacid users.First, we study the steady states; and, then, we determine numerically the dynamical trajectories of Equation (1).Discussions are also provided.
Numerical simulations are performed taking into account the values of the model's parameters given in Ta- ble 2. The threshold of  assumes .The units given in Table 2 are not shown hereafter.days 

Results Regarding to the Steady States
Using values given in Table 2,  Considering the values given in Table 2, and by solving Equation (B.1), we have:   I) given by Equation (3), which is unique.
Table 2, the st  DDD/1000 inhabitants/day in 1997, respectively.The unit of measurement in wholesale statistics is usually the Defined Daily Dose (DDD), which is defined as the assumed average maintenance dose per day for a drug use on its main indication in adults [24].Health surveys may, however, collect information on nonprescription medications and also assess health status, risk factors, and sociodemographic variables at the individual level [16].In    this way we assessed the hazard of food allergy originating from self-care use of antiacids.How could nonprescription medications be managed after proving dangerous (in different levels) side effects?Let us drawn the log-scale bifurcation diagram using values given in Table 2, except higher ).Next we present so arios regarding to the control of the use of nonprescription medications.The discussions can be promptly applied to antiacids.
Figure 4 shows an obvious finding: the rate at which naive person becomes nonprescription medications user me scen  is the main control parameter.When the use is spread throughout the community for diversity of reasons, which case corresponds to the case 2 >   (region labelled as IV in Figure 4), the prevalence of users situates at very high level ( 3 u ).Let us suppose that a campaign is carried on in order to avoid self-care use of medications, which decreases the recruitment of users (  decreases).First, if this rate of the recruitment is still elevated ( 2    , region IV), a decreasing in the amount of users due to health campaign is temporary, and as time goes on, the previous prevalence is attained.In other words, a complete rejection of medications is only possible with the abandonment of medications by all users ( 0 u  ).Second, when the recruitment is lowered substantially ( 1 2      , region III), if the educational campaign reduces the users below the critical proportion ( 2 u ), then, as time goes on, the prevalence is decreased dramatically reaching low value 1 u (order of ), a perfect campaign redu the recruitment rate below its threshold, and as time goes on, all individuals abandon the use of medications.
With respect to the nonprescription antiacids, which can induce food allergy, public health authorities must design a campaign that reduces the recr ced uitment rate below 1  , without the necessity of reaching all persons If this goal is unreachable, a campaign must reduce the recruitment rate below 2  , and additional efforts must be spent to reduce the antiacid users below the critical value ( 2 u , coordinate of the stable equilibrium 2 Q  ).The reason is due to the fact that the preva-   Q  ) is so low that avoids the appearance of food allergic perns.
states analysis furnished qualitative behavior of erical solutions of this quations were obtained d the last tw so

Dynamical Trajectories
Steady the system of Equations (1).Num system of ordinary differential e by the 4th order Runge-Kutta method [25].In appropriate ranges of values of the model's parameters, we can observe the hysteresis behavior.Now let us determine the period of time spent in order to appear cases of cassava allergic persons in a measurable prevalence.
The system of Equation ( 1) is divided into two subsystems: the first two equations describing the recruitment of persons to become antiacid users, an o equations, which is feed by the first sub-system, describing the onset of food allergy among antiacid users.The parameters regarded to the first sub-system determine the qualitative behavior of the dynamical system: Q  is called break point [28].Notice that both 0 u are lower than 2 u ; because we used   in the dynamics system) [27].The hysteresis reports a social behavior in which a critical number of persons (in our example, the use of an r an intense collective response.

Conclusions
We developed a simple phe tiacids) is needed to trigge nomenological model to dece of food allergy (cassava allergen) sers of antiacids.Our argument was scribe the appearan in persons who are u that the increasing use of proton pump inhibitors for digestive disorders resulted in the elevation of the pH, and, consequently, the cassava allergen remains intact, thereby facilitating its presentation to the immune system.Adtionally, only part of it uses the drug indiscri di minately and regularly, and that only part of the population is ge-ir netically predisposed to be sensitized and acquire food allergy.We expect however an elapse in the time from the introduction of antiacids (specially PPIs) until the sensitization to a new allergen, the onset of symptoms and diagnosis of food allergy.
The model considered Hill function to describe the following events: 1) the recruitment of naive persons to become antiacid users ( ).The reason behind the choice of Hill type function was basically of three types: 1) no medicines (like antiacids) usage can be thought of or (collective response depends on a critical number of adhered persons); 2) the sensitization is strongly dependent on a threshold in the amount of food allergens; and 3) the acquisition of allergy is an event depending on many factors, such as genetic predisposition, modern life style including social behavior and food intake, and stress.
Due to the mass action law, there was a threshold of the recruitment rate th Q  depending on the initial conditions supplied to t dynam al system (Figure 6).When the rec itment rate situates between  6).In the situation as reported by Figure 6 (low value of recruitment rate), the pharmaceutical industry must spent a lot of money to convince peop tiacids.If this goal is successful, then the possibility of appearing food allergy is increased dramatically.
Population-based information showed that the consumption of antiacids, which are nonprescription drugs, mainly is explained by the prevalence of dyspept le to use an ic symptoms.Nonprescription antiacid medication, in conjunction with the report of some degree of relief in the treatment of acid-related disorders by users, can increase the recruitment rate  .A larger number of medical vis- its in the year 1987-1988 predicted an increased likelyhood of being an antiacid user, and this findings persisted throughout the period 16].When [  increases and sur- passes the critical value 2  , that is, 2    , the dynamical trajectories are driven to the big equilibrium 3 Q  , regardless the initial conditions supplied to the dynamical system.Therefore, food allergy originated from cassava allergen can be detectable if the recruitment rate  is situated above the critical value 2  , rather than the threshold th  (Figure 5).
From the dynamical trajectories of the model (Figure 5), we conclude that detectable numb of patients should be arose 15  able es a population level to be compared with the model's prediction.The food allergy may become a public health problem, if PPIs are being used indiscriminately and irregularly.The results obtained from the analysis of the model suggest that the use of nonprescription antiacids, like PPIs, may be recommended or approved by the physician in order to avoid the rising of food allergy.
Finally, as new biological findings occur, improved mathematical model can be developed and fitted against observed data.Notwithstanding, t veloped and analyzed here showed an elapse of 15 years from the introduction of antiacid medicine before the onset of food allergic persons, and the model is forecasting an exponential growth in the number of food allergic persons currently.

Appendix A. Hill Function
Let us consider the Hill function [23] of order n, written as The first derivative of the Hill function is , and the second derivative is . Notice that for , the first derivative is always positive, and The first order Hill function (n = 1) is increasing function without change of concavity.However, for , the increasing n-th order Hill function changes the concavity at , and the value of to a, while the latter from 1/4 to 1/2. Figure A1 illustrates the Hill function for 1 n  , 2 and 3, letting an arbitrary value .At 1

Appendix B. Analysis of the Model
We deal with the steady states of the system (1), and we assess the stability of the equilibrium points.

Steady States
The trivial equilibrium point 0 Q is given by Equation (3).The non-trivial equilibrium point   , , , Q p u s a   has the coordinates given by: Equation (5) for p , solution (s) of Equation ( 6) for u , solution of Equation ( 8) for s , and Equation ( 7) for a .Hence, let us analyze Equations ( 6) and (8).
From Equation ( 6), the function     ), Equation (6) has again a unique positive solution, such that > 3 u u k .We stress the fact that 0 u  is always a solution whatever be the value of  .Similar behavior occurs when we fix  and vary  .
The passage from one solution to three positive solutions, or vice-versa, as  varies, occurs at values of  at which Equation ( 6) has two positive solutions.Two positive solutions occur when there exists the collapse of pre-existing two solutions in one, that is, the curve of the function   , and instead of Equation ( 8).This second order polynomial has one positive solution (other is negative).The qualitative dynamics of the system also does not change.Hence, due to the less sensitive changes in the dynamics promoted by any order of the Hill function, we can revise the model's assumption about the less frequent occurrence of allergic reaction in sensitized persons, if necessary.Summarizing, the non-trivial equilibrium point Q  always exists for where u and s are the positive solutions of, respectively, Equations ( 6) and (8).We recall that the expressions for p and a are given by Equations ( 5) and ( 7), respectively.When there are three equilibrium points, for each 1 u , 2 u and 3 u , we calculate other coordinates and we have three equilibrium points 1 Q  , 2 Q  and 3 Q  .

Stability Analysis
We can also evaluate the discriminant of the matrix Q  with elevate number of antiacid users is locally asptotically stable.In other cases, the stability is determined numerically.
In the case of the trivial equilibrium point is the trivial equilibrium point.Then, from La-Salle Lyapunov Theorem [30], the equilibrium point ng be seen 0 Q corresponding to the absence of antiacid users is g ally asymptotically stable for th lob    .

(
see Appendix B, where the global stability using Lyapunov function is also shown).If the rate of recruit-ment of naive individuals (  ) is lower than the antiacid users are going to extinction.As a pharmacological marketing point of view, the efficacy of a medicine must be confirmed by users who maintain the use.When > th   , the trivial equilibrium becomes un-stable, and the anti-acid users are sustained by the recruitment of naive persons.Let us determine the non-trivial equilibrium point Q  .

, 2 u and 3 u 2 Q
we show the bifurcation diagram in Figure 1: the positive solutions 1 u (labelled, respectively, as 1 u , 2 and 3 u ) as a function of u  .These values are positive solutions of Equation (6), which were numerically obtained by the bisection (to find zeros of polynomials) method [25].For each 1 u , 2 u and 3 u , we calculate other coordinates of the equilibrium points 1 Q ,   and 3 Q respectively, using Equation (B.2).The vertical axis begins at -0.1, because the small positive solution  , 1 u assumes very lower values.The difference of magnitudes between 1 u and 3 u of some orders, which requires the use of log-scale (see figure below).In Figure 1 we show the coordinates at which we have two equilibrium points are

3 u 2 u 3 uFigure 1 . 2 u
Figure 1.Bifurcation diagram obtained using values give n in Table 2.The equilibrium points u 1 , u 2 and u 3 are labelled, respectively, as u 1 , u 2 and u 3 .Considering the critical values α 1 and α 2 , and the threshold α th , which is slightly smaller than α 1 , the attracting basins are labelled as I (all dynamical trajectories are attracted to the trivial 0 Q ), II (all dynamical trajectories are attracted to 1 u ), and III (all trajectories are attracted to 3 u ).In α 1 < α α 2 , the unstable equilibrium value <

Figure 2
Figure 2 shows, using values given in ability of the equilibrium points 1 u , 2 u and 3 u .According to Routh-Hurwitz criteria r a second egree polynomial, if fo d

Figure 2 u
) and big ( 3 u ) po iv re stable, while the intermediate root ( sit e solutions a 2 u ) is unstable.Notice that we have always   1 0 tr J   Figure 2(b), showing that the sta-( bility is e term that does not depend on given by th  [27]).Also, from ma J are real numbers, and dynamical trajectories do not have damped oscilla-, this according to Equation that ies near the equilibrium points are node (if stable, the model's parameters in 1 im .The ge where three positive solutions occur increases with increasing in 0 t es ran  and p k , while with decreasing in  and u k , the r ge increases.Hence, there are critical value c an s  , c p k , c  and c u k such that the hysteresis effects disappear (and, con uently, there is a unique positive solution for > th seq   ).At the critical values c  , (B.1) ution does not have positive sol in  , and hysteresis is absent.Notice that the range of  which the hysteresis effects occurs is less sensitive n hysteresis effect, in which case we have the forward bifurcation (only one stable non-trivial equilibrium point Q  for > th   ). Figure 3(a) shows the forward bifurcation, which was drawn using values given in Table 2, for many years, and still commonly used self-prescribed drugs.However, their importance has diminished since the development of histamine-2-receptor antagonists (H 2 RA) and more recently proton pump inhibitors.Based on the drug wholesale statistics in Norway, the sale of antiacids decreased from 7.6 DDD/1000 inhabitants/day in 1988 to 4.3 DDD/1000 inhabitants/day in 1997.The H 2 RA and PPI have in the same period increased from 4.0 DDD/ 1000 inhabitants/day (only H 2 RA) in 1988 to 5.2 and 7.8

Figure 2 .
Figure 2. Determinant (a), trace (b) and discriminant (c) of the matrix J 1 to assess the stability of the equilibrium values 1 u , 2 u and 3 u labelled, respectively, as u 1 , u 2 and u 3 .

Figure 3 .
Figure 3. Forward bifurcation diagram (a) obtained using values given in Table 2, except β = 1.0 × 10 −9 .Determinan trace and discriminant of J corresponding to the unique t,

Figure 4
the equilibrium points (as a function of  ) 1 u , 2 u and 3 u are labelled, respectively, as 1 u , 2 u and 3 u .We also show the critical values 1  and 2  , and the threshold th  , which are lower than those obtained n Figure 1.The arrows indicate the attracting quilibrium values (IV).The arrows indicate the attracting equilibrium values (stable equilibrium point Q 

s ri 1 u
(coordinate of the stable equilibrium 1

6 4 3 Q  , with coordinates obtained using 3 u
k and u k .The remaining parameters  ,  and s k can be managed to assess the food allergy manifestation in a community.Both sub-systems are up arameters co led by the p  and u k .The fo wing numerical results are obtained changing the values of the parameters llo those used to draw the bifurcation diagram given in Fig- ure 4. The threshold value be mes Brazil, the onditions, given by Equation (2), supplied to the dynamical system (1) are latter initial conditions describe, due to the absence of PPIs, the absence of sensitized persons rgy.F mical trajectories: long-term Figure 5(a) and short-term Figure 5(b).The year 0 in the simulations can be linked to the year 1988, when PPIs were introduced.Fifteen years later since the introduction of antiacid in a community, we have 8 cases of food allergy in a population of 200 million persons, and asymptotically attains 92 cases per 1 million persons (equilibrium value neither persons who shows the dyna acquired food alle igure 5 a ).With respect to sensitized persons, we find 48,000 cases after fifteen years in a population of 200 million persons, and asymptotically attains 1200 cases per 1 miln persons (equilibrium value lio s ).If we increase the initial conditions to of food allergy in a population of 200 million persons are observ d a e fter an elapsing 14.57 d 14.52 years, respectively.For decreased initial conof food allergy in a population of 200 million persons after elapsing 16.4 years.The asymptotic (equilibrium) proportions of the nai n 1.0 u  ve persons p and PPIs users u are near, respectively, 0.8 and 0.2 (Figure5(a)).The value assigned to  is much higher than 2  , hence we have only one equilibrium point Q .In this case, the i itial conditions only affect the delay in the time of arising of the exponential growth, and further hievement of asymptotic equilibrium.Now, let us illustrate the dependency of the dynamical system with the initial conditions, letting to  .The unstable equilibrium point 2
2) the sensitization of the immune system among antiacid users to the cassava allergen remained intact due to the elevation of pH ( antiacid users cannot be maintained.The use of Hill function to describe the flow between different status of persons in a community arose the teresis behavior, which appears when

6 8
if the initial amount of antiacid users (u 0 ) does not achieve a critical value (regarded to the coordinates of the unstable 2 Q  ), this comnity is lativel onset of allergic reactions in persons.However, the figure is completely different if u 0 surpasses the critical value: the numb of antiacid users increases around 345 times for

Figure A1 .
Figure A1.Hill function for the first three degrees, with H n (K x ) = 0.5 at x = K x = 1.

2  , with 1 2 <
f u is tangent (intercepts in one point) with the curve of the function   g u .The other positive solution always exists.This behavior occurs at two values of , named 1  and   .At these values we must have behaviors of the functions   h s and   r s , we conclude that the equation     r s  h s has only one positive solution s , with 0 s s  , for a given positive value of u .Notice that, if we consider Hill function of any order for the populational acquisition of the food allergy, that is, equation of the system (1), a similar conclusion with respect to unicity of the solution can be made.For instance, for , equilibrium points is deter-The local stability of the mined by the eigenvalues of the characteristic equation the2 2   null matrix, the diagonal matrices are consequently, 0 p  ), the coefficients of haracteristic Equation (B ) are positive, and both eigenvalues have negative real part.Therefore, the non-trivial equilibrium point the c .4 system of Equation (1), it can that the maximal invariant set contained in = 0 V 

Table 1 . Summary of the variables and parameters of the model.
s Food allergy acquisition scaling factor

Table 2
Journal of Allergy andas in 1988.Hence, in the year 2003 first cases of food allergy from antiacid users should be diagnosed.After 23 years, in the calendar year of 2011, we can estimate the incidence of sensitized and food allergic persons in 3500 and 2 cases per 10 million persons, respectively.These estimates were obtained using the values given in For small values of α, but greater than the threshold th 2 >