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Research article

An efficient numerical method based on QSC for multi-term variable-order time fractional mobile-immobile diffusion equation with Neumann boundary condition

  • In this work, we aimed at a kind of multi-term variable-order time fractional mobile-immobile diffusion (TF-MID) equation satisfying the Neumann boundary condition, with fractional orders αm(t) for m=1,2,,P, and introduced a QSC-L1+ scheme by applying the quadratic spline collocation (QSC) method along the spatial direction and using the L1+ formula for the temporal direction. This new scheme was shown to be unconditionally stable and convergent with the accuracy O(τmin{3αα(0), 2}+Δx2+Δy2), where Δx, Δy, and τ denoted the space-time mesh sizes. α was the maximum of αm(t) over the time interval, and α(0) was the maximum of αm(0) in all values of m. The QSC-L1+ scheme, under certain appropriate conditions on αm(t), is capable of attaining a second order convergence in time, even on a uniform space-time grid. Additionally, we also implemented a fast computation approach which leveraged the exponential-sum-approximation technique to increase the computational efficiency. A numerical example with different fractional orders was attached to confirm the theoretical findings.

    Citation: Jun Liu, Yue Liu, Xiaoge Yu, Xiao Ye. An efficient numerical method based on QSC for multi-term variable-order time fractional mobile-immobile diffusion equation with Neumann boundary condition[J]. Electronic Research Archive, 2025, 33(2): 642-666. doi: 10.3934/era.2025030

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  • In this work, we aimed at a kind of multi-term variable-order time fractional mobile-immobile diffusion (TF-MID) equation satisfying the Neumann boundary condition, with fractional orders αm(t) for m=1,2,,P, and introduced a QSC-L1+ scheme by applying the quadratic spline collocation (QSC) method along the spatial direction and using the L1+ formula for the temporal direction. This new scheme was shown to be unconditionally stable and convergent with the accuracy O(τmin{3αα(0), 2}+Δx2+Δy2), where Δx, Δy, and τ denoted the space-time mesh sizes. α was the maximum of αm(t) over the time interval, and α(0) was the maximum of αm(0) in all values of m. The QSC-L1+ scheme, under certain appropriate conditions on αm(t), is capable of attaining a second order convergence in time, even on a uniform space-time grid. Additionally, we also implemented a fast computation approach which leveraged the exponential-sum-approximation technique to increase the computational efficiency. A numerical example with different fractional orders was attached to confirm the theoretical findings.



    Clemens von Pirquet originally defined the term Allergy as meaning the increased capacity of the body to react to a foreign substance. Today the term allergy means oversensitivity to a foreign substance that is normally harmless [1]. A major safety concern is food allergy [2]. Food allergy affects many millions of peoples and is responsible for substantial morbidity and reduced quality of life in patients, families and communities [3]. To date, the focus of the adult food allergy examination has been on a limited number of specific allergens [4]. In recent decades, it is generally accepted that the prevalence of food allergy has increased, especially in westernized countries [5].

    The incidence of food allergies is increasing globally [6]. Recent studies show that in the Middle East Region, allergic diseases are growing strongly [7]. The prevalence of clinically identified food allergy ranges widely from 1% to 13% [8]. Food allergy (FA) prevalence varies in different countries, as estimates are influenced by several factors; such as age, race, dietary intake frequency, and cooking method [9]. Food allergy, which now affects up to 8% of children and 5% of adults in Westernized countries, has become a public health priority for developing therapies for this potentially life-threatening condition [10].

    Food allergies are caused to exposure to certain life-threatening antigens by IgE- or cell-mediated humoral immune response. Such allergies are one of the main food safety issues in developed countries, affecting 1–10 percent of the global population, with a higher incidence in children [11]. Immunoglobulin E (IgE)-mediated food allergy is a leading cause of anaphylaxis, and so it is important to refer to an allergist for prompt and effective diagnosis and care [12]. Type I hypersensitivity responses caused by the cross-linking of IgE attached to the surface of mast cells and basophils underlie adverse food allergy reactions [13]. Important evidence indicates the key roles that mast cells, IgE and TH2 cytokines play in mediating food allergy [14]. Despite the vast number of foods that trigger immunoglobulin E (IgE)-mediated reactions, most prevalence studies have concentrated on the most common allergenic foods, i.e. cow's milk, hen's egg, peanut, tree nut, wheat, soy, fish and shellfish [15].

    A total of 170 blood samples were collected from patients suffering from allergy from an unknown source. All serum samples were subjected for determination of the presence of food allergy by varieties of foods. They visited the private clinical sectors of Erbil Province, Kurdistan Region, Iraq between 2018 to 2020. All blood samples collected inside the (10 mL) gel tube, it contains clotting activators, after clotting, centrifuged them for 15 min at 5000 round per minute (RPM). Blood serums have been subjected to food allergy analysis regarding the manufactures instructions of the test. Blood samples were sorted in the different aged groups (13–30 and 31–52 years) and also classifying them regarding to the genders in males and females.

    Clarifications of food allergies can carry out by using various food allergy profiles. Human IgE antibodies against the most frequent food allergens in serum, can determine semi-quantitatively or qualitatively based on the test system. Country-specific food allergy profiles are available which have been developed with about regional eating habits. The EUROLINE test kit provides semi-quantitative in vitro determination of allergen-specific (sIgE) in serum, contributing to the diagnosis of allergies. The test is multipara meter assay containing optimized combinations of relevant allergens, enabling the analysis of sIgE against these different allergens in one test. In this study, we used the country-specific food allergy profile “Food Iraq 1” (Catalog no: DP 3436-160-1 E, IVD-approved, and CE-certified EUROLINE immunoblot) test kit contains a test strip with 36 different allergens. All serum samples have been subjected for determinations of food allergies according to the manufacturer's instructions. Briefly, test strips coated with 36 food allergens: egg white (f1), egg yolk (f75), cow's milk (f2), nBos d8-casein (f78), wheat flour (f4), rye flour (f5), rice (f9), grain mix 2 (fs13), sesame (f10), peanut (f13), soybean (f14), hazelnut (f17), almond (f20), pistachio (f144), walnut (f256), gluten (f79), strawberry (f44), apple (f49), blue grape (f50), kiwi (f84), banana (f92), mango (f93), peach (f95), cherry (f97), honeydew melon (f87), citrus mix 2 (fs32), tomato (f25), potato (f35), bell pepper (f46), garlic (f47), celery (f85), eggplant (f262), chicken (f83), meat mix 5 (fs28), shrimp/prawn (f24) and seafood mix 3 (fs12). In addition to food allergens, strip coated with a cross-reactive carbohydrate determinant (CCD) and an indicator band. Test strips are first moistened with 1.0 mL of working strength universal buffer for 5 min, then aspirate off all liquid. Directly incubated with 400 µL of undiluted patient serum to bind s-IgE antibodies if present. If samples contain specific antibodies of class IgE, they will bind to the allergens coated on the strip. Subsequently, the bound s-IgE antibodies were detected using an enzyme-linked anti-human IgE catalyzing a color reaction. After stopping the reaction by using deionized or distilled water, place the incubated test strip onto the adhesive foil of the green work protocol (created in the EUROLINE scan program) using a pair of tweezers. The position of the test strips can be corrected while they are wet. As soon as all test strips have been placed onto the protocol, they should be pressed hard using filter paper and left to air dry. The drying process should take place without any direct light, in a room as dark as possible. After they have dried, the test strip will be stuck to the adhesive foil. Incubated strips that are still moist show a background coloring that disappears when they are completely dry. Therefore the evaluation of the strips only takes place after strips have completely dried. Class of antibodies can be classified into the following types depending on the concentration range and their explanation (Table 1).

    Table 1.  Class of antibodies with concentration range and explanations.
    Class Concentration (Ku/I) Explanation
    0 <0.35 No specific antibodies detected
    1 0.35 ≤ sIgE < 0.7 Very weak antibody detection
    2 0.7 ≤ sIgE < 3.5 Weak antibody detection
    3 3.5 ≤ sIgE < 17.5 Definite antibody detection
    4 17.5 ≤ sIgE < 50 Strong antibody detection
    5 50 ≤ sIgE < 100 Very high antibody titer
    6 ≥100 Very high antibody titer

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    Patients' rights and ethic approval statement explaining: Ethical approval for this research has not been obtained from the institutional review board or committee because all blood samples have been collected from a private clinical diagnostic laboratory, which officially has been recognized by the ministry of health Kurdistan Regional government of Iraq. Ministry of health instructions for Ethical criteria and patient's rights mandatory should take into consideration at all diagnostic laboratories in the Kurdistan Region of Iraq. Contentment for blood drawing orally earned from all patients after proceeding of all ethical instructions. Patients' and clients' information (name, age, and gender) electronically submitted to the lab database.

    The chi-square was applied to examine the relationship between the prevalence of food allergy and the types of antibodies detection in allergic patients from different genders and aged groups. P-values <0.05 were considered to be statistically significant.

    The present study illustrated that the food allergy prevalence (measured by specific IgE concentration) in males is 10% and 12.35% in females. Food allergy prevalence was 12.35% among individuals age 13–30 years and 10% among 31–52 years. Table 2 clarifying the prevalence of food allergy in between different aged groups with male and female.

    Table 2.  Distribution of positive and negative food allergy according to ages and gender.
    N (%) positive N (%) negative Total (%)
    Age groups (years) 13–30 21 (12.35) 55 (32.35) 76 (44.70)
    31–52 17 (10) 77 (45.29) 94 (55.29)
    Total 38 (22.35) 132 (77.64) 170 (100)
    Gender Male 17 (10) 77 (45.29) 94 (55.29)
    Female 21 (12.35) 55 (32.35) 76 (44.70)
    Total 38 (22.35) 132 (77.64) 170 (100)

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    Prevalence of positive results for 36 food allergens items in all 170 screened samples illustrated in Table 3, regarding to aged groups and genders.

    Table 3.  Prevalence of positive food allergy in all 170 screened samples according to gender and aged groups.
    Food allergy Number of screened
    Positive food allergy
    Number of screened
    Positive food allergy
    Male Female Male Female 13–30 years 31–52 years 13–30 years 31–52 years
    Egg white 76 94 4 (5.26) 2 (2.12) 76 94 4 (5.26) 2 (2.12)
    Egg yolk 76 94 2 (2.63) 2 (2.12) 76 94 3 (3.94) 1 (1.06)
    Cow's milk 76 94 1 (1.31) 2 (2.12) 76 94 2 (2.63) 1 (1.06)
    nBos d8-casein 76 94 1 (1.31) 2 (2.12) 76 94 2 (2.63) 1 (1.06)
    Wheat flour 76 94 2 (2.63) 2 (2.12) 76 94 2 (2.63) 2 (2.12)
    Rye flour 76 94 2 (2.63) 3 (3.19) 76 94 1 (1.31) 4 (4.25)
    Rice (f9) 76 94 3 (3.94) 3 (3.19) 76 94 3 (3.94) 3 (3.19)
    Grain mix 76 94 7 (9.21) 4 (4.25) 76 94 5 (6.57) 6 (6.38)
    Sesame 76 94 5 (6.57) 5 (5.31) 76 94 4 (5.26) 6 (6.38)
    Soybean 76 94 5 (6.57) 3 (3.19) 76 94 6 (7.89) 2 (2.12)
    Gluten 76 94 1 (1.31) 2 (2.12) 76 94 1 (1.31) 2 (2.12)
    Hazelnut 76 94 3 (3.94) 0 (0) 76 94 2 (2.63) 1 (1.06)
    Almond 76 94 2 (2.63) 3 (3.19) 76 94 4 (5.26) 1 (1.06)
    Walnut 76 94 2 (2.63) 2 (2.12) 76 94 3 (3.94) 3 (3.19)
    Pistachio 76 94 3 (3.94) 3 (3.19) 76 94 4 (5.26) 2 (2.12)
    Peanut 76 94 3 (3.94) 4 (4.25) 76 94 4 (5.26) 3 (3.19)
    Strawberry 76 94 2 (2.63) 2 (2.12) 76 94 1 (1.31) 3 (3.19)
    Apple 76 94 2 (2.63) 4 (4.25) 76 94 5 (6.57) 1 (1.06)
    Blue grape 76 94 1 (1.31) 0 (0) 76 94 1 (1.31) 0 (0)
    Kiwi 76 94 4 (5.26) 2 (2.12) 76 94 1 (1.31) 5 (5.31)
    Banana 76 94 2 (2.63) 0 (0) 76 94 2 (2.63) 0 (0)
    Mango 76 94 1 (1.31) 3 (3.19) 76 94 3 (3.94) 1 (1.06)
    Peach 76 94 2 (2.63) 4 (4.25) 76 94 4 (5.26) 2 (2.12)
    Cherry 76 94 1 (1.31) 3 (3.19) 76 94 4 (5.26) 0 (0)
    Honeydew melon 76 94 1 (1.31) 1 (1.06) 76 94 2 (2.63) 0 (0)
    Citrus mix 76 94 2 (2.63) 1 (1.06) 76 94 1 (1.31) 2 (2.12)
    Bell pepper 76 94 2 (2.63) 1 (1.06) 76 94 1 (1.31) 2 (2.12)
    Eggplant 76 94 2 (2.63) 1 (1.06) 76 94 1 (1.31) 2 (2.12)
    Tomato 76 94 2 (2.63) 2 (2.12) 76 94 2 (2.63) 2 (2.12)
    Potato 76 94 3 (3.94) 2 (2.12) 76 94 3 (3.94) 2 (2.12)
    Garlic 76 94 3 (3.94) 5 (5.31) 76 94 4 (5.26) 4 (4.25)
    Celery 76 94 3 (3.94) 0 (0) 76 94 2 (2.63) 1 (1.06)
    Chicken 76 94 4 (5.26) 7 (7.44) 76 94 5 (6.57) 6 (6.38)
    Meat mix 76 94 1 (1.31) 2 (2.12) 76 94 1 (1.31) 2 (2.12)
    Shrimp/prawn 76 94 3 (3.94) 4 (4.25) 76 94 2 (2.63) 5 (5.31)
    Seafood mix 76 94 10 (13.15) 12 (12.76) 76 94 11 (14.47) 11 (11.70)

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    Prevalence percentage for all allergens items have been checked in all positive cases, regarding to the aged groups and gender (Table 4).

    Table 4.  Prevalence (%) of food allergens in all 38 allergic patients according to aged groups and gender.
    Item number Food allergy Age groups
    Gender
    13–30 31–52 Male Female
    1 Egg white 4 (19.04) 2 (11.76) 4 (23.52) 2 (9.52)
    2 Egg yolk 3 (14.28) 1 (5.88) 2 (11.76) 2 (9.52)
    3 Cow's milk 2 (9.52) 1 (5.88) 1 (5.88) 2 (9.52)
    4 nBos d8-casein 2 (9.52) 1 (5.88) 1 (5.88) 2 (9.52)
    5 Wheat flour 2 (9.52) 2 (11.76) 2 (11.76) 2 (9.52)
    6 Rye flour 1 (4.76) 4 (23.52) 2 (11.76) 3 (14.28)
    7 Rice (f9) 3 (14.28) 3 (17.64) 3 (17.64) 3 (14.28)
    8 Grain mix 5 (23.80) 6 (35.29) 7 (41.17) 4 (19.04)
    9 Sesame 4 (19.04) 6 (35.29) 5 (29.41) 5 (23.80)
    10 Soybean 6 (28.57) 2 (11.76) 5 (29.41) 3 (14.28)
    11 Gluten 1 (4.76) 2 (11.76) 1 (5.88) 2 (9.52)
    12 Hazelnut 2 (9.52) 1 (5.88) 3 (17.64) 0 (0)
    13 Almond 4 (19.04) 1 (5.88) 2 (11.76) 3 (14.28)
    14 Walnut 3 (14.28) 3 (17.64) 2 (11.76) 2 (9.52)
    15 Pistachio 4 (19.04) 2 (11.76) 3 (17.64) 3 (14.28)
    16 Peanut 4 (19.04) 3 (17.64) 3 (17.64) 4 (19.04)
    17 Strawberry 1 (4.76) 3 (17.64) 2 (11.76) 2 (9.52)
    18 Apple 5 (23.80) 1 (5.88) 2 (11.76) 4 (19.04)
    19 Blue grape 1 (4.76) 0 (0) 1 (5.88) 0 (0)
    20 Kiwi 1 (4.76) 5 (29.41) 4 (23.52) 2 (9.52)
    21 Banana 2 (9.52) 0 (0) 2 (11.76) 0 (0)
    22 Mango 3 (14.28) 1 (5.88) 1 (5.88) 3 (14.28)
    23 Peach 4 (19.04) 2 (11.76) 2 (11.76) 4 (19.04)
    24 Cherry 4 (19.04) 0 (0) 1 (5.88) 3 (14.28)
    25 Honeydew melon 2 (9.52) 0 (0) 1 (5.88) 1 (4.76)
    26 Citrus mix 1 (4.76) 2 (11.76) 2 (11.76) 1 (4.76)
    27 Bell pepper 1 (4.76) 2 (11.76) 2 (11.76) 1 (4.76)
    28 Eggplant 1 (4.76) 2 (11.76) 2 (11.76) 1 (4.76)
    29 Tomato 2 (9.52) 2 (11.76) 2 (11.76) 2 (9.52)
    30 Potato 3 (14.28) 2 (11.76) 3 (17.64) 2 (9.52)
    31 Garlic 4 (19.04) 4 (23.52) 3 (17.64) 5 (23.80)
    32 Celery 2 (9.52) 1 (5.88) 3 (17.64) 0 (0)
    33 Chicken 5 (23.80) 6 (35.29) 4 (23.52) 7 (33.3)
    34 Meat mix 1 (4.76) 2 (11.76) 1 (5.88) 2 (9.52)
    35 Shrimp/prawn 2 (9.52) 5 (29.41) 3 (17.64) 4 (19.04)
    36 Seafood mix 11 (52.38) 11 (64.70) 10 (58.82) 12 (57.14)
    P value 0.08 0.0006 0.14 0.002

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    Allergic response class and response strength to all food allergen products have been tested in all positive cases. The prevalence percentage of food allergen products with class and severity of allergic reaction is shown in Table 5.

    Table 5.  Prevalence (%) of food allergens with class and intensity of antibodies detection in all 38 allergic patients.
    Number Food allergen No specific antibodie detected (class 0)
    Very weak antibody detection (class 1)
    Weak antibody detection (class 2)
    Definite antibody detection (class 3)
    Strong antibody detection (class 4)
    Very high antibody titer (class 5)
    Total
    N % N % N % N % N % N % N %
    1 Egg white 32 84.21 1 2.63 2 5.26 2 5.26 0 0 1 2.63 38 100
    2 Egg yolk 34 89.47 3 7.89 0 0 0 0 0 0 1 2.63 38 100
    3 Cow's milk 35 92.10 2 5.26 1 2.63 0 0 0 0 0 0 38 100
    4 nBos d8-casein 35 92.10 2 5.26 1 2.63 0 0 0 0 0 0 38 100
    5 Wheat flour 34 89.47 1 2.63 2 5.26 3 7.89 0 0 0 0 38 100
    6 Rye flour 33 86.84 4 10.52 1 2.63 0 0 0 0 0 0 38 100
    7 Rice 32 84.21 3 7.89 2 5.26 1 2.63 0 0 0 0 38 100
    8 Grain mix 27 71.05 6 15.78 4 10.52 1 2.63 0 0 0 0 38 100
    9 Sesame 28 73.68 6 15.78 2 5.26 2 5.26 0 0 0 0 38 100
    10 Soybean 30 78.94 3 7.89 3 7.89 2 5.26 0 0 0 0 38 100
    11 Gluten 35 92.10 3 7.89 0 0 0 0 0 0 0 0 38 100
    12 Hazelnut 35 92.10 2 5.26 1 2.63 0 0 0 0 0 0 38 100
    13 Almond 33 86.84 1 2.63 2 5.26 2 5.26 0 0 0 0 38 100
    14 Walnut 32 84.21 4 10.52 2 5.26 0 0 0 0 0 0 38 100
    15 Pistachio 32 84.21 2 5.26 2 5.26 2 5.26 0 0 0 0 38 100
    16 Peanut 31 81.57 5 13.15 1 2.63 1 2.63 0 0 0 0 38 100
    17 Strawberry 34 89.47 1 2.63 2 5.26 1 2.63 0 0 0 0 38 100
    18 Apple 32 84.21 2 5.26 2 5.26 1 2.63 1 2.63 0 0 38 100
    19 Blue grape 37 97.36 0 0 1 2.63 0 0 0 0 0 0 38 100
    20 Kiwi 32 84.21 4 10.52 2 5.26 0 0 0 0 0 0 38 100
    21 Banana 36 94.73 2 2.63 0 0 0 0 0 0 0 0 38 100
    22 Mango 34 89.47 3 7.89 1 2.63 0 0 0 0 0 0 38 100
    23 Peach 32 84.21 2 5.26 2 5.26 1 2.63 1 2.63 0 0 38 100
    24 Cherry 34 84.21 1 2.63 2 5.26 0 0 1 2.63 0 0 38 100
    25 Honeydew melon 36 94.73 1 2.63 1 2.63 0 0 0 0 0 0 38 100
    26 Citrus mix 35 92.10 2 5.26 1 2.63 0 0 0 0 0 0 38 100
    27 Bell pepper 35 92.10 2 5.26 1 2.63 0 0 0 0 0 0 38 100
    28 Eggplant 35 92.10 2 5.26 1 2.63 0 0 0 0 0 0 38 100
    29 Tomato 34 84.21 3 7.89 1 2.63 0 0 0 0 0 0 38 100
    30 Potato 33 86.84 2 5.26 2 5.26 1 2.63 0 0 0 0 38 100
    31 Garlic 30 78.94 2 5.26 4 10.52 1 2.63 1 2.63 0 0 38 100
    32 Celery 35 92.10 0 0 3 7.89 0 0 0 0 0 0 38 100
    33 Chicken 27 71.05 1 7.89 4 10.52 1 2.63 3 7.89 2 0 38 100
    34 Meat mix 35 92.10 2 5.26 0 0 1 2.63 0 0 0 0 38 100
    35 Shrimp/prawn 31 81.57 3 7.89 1 2.63 3 7.89 0 0 0 0 38 100
    36 Seafood mix 16 42.10 4 10.52 5 13.15 10 26.31 2 5.26 1 2.63 38 100
    P value 0.9 0.7 0.7 0.00001 0.006 0.1 - -

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    This research was carried out to determine the prevalence of food allergy in the Erbil Province Kurdistan Region of Iraq. We were unable to find any published data about prevalence food allergy in Erbil. Total 94 males (55.29%) and 76 females (44.70%) were studied and their ages range were in between 13 to 52 years old and mean of SD 32.57 ± 11.36 years. In the Middle East, studies about food allergy prevalence are scarce. A recent study done in the United Arab Emirates showed that the most common food allergens were seafood and nuts [16]. Another study showed that the highest frequency of food specific IgE is to hazelnuts and peanuts, with a marked increase in reactions to hazelnut [17].

    Most sensitivities to food allergens in males observed in these food items respectively, seafood: 10 (13.15%), grain mix: 7 (9.21%), sesame and soybean: 5 (6.57%) and followed by chicken, kiwi, and egg white: 4 (5.26%). But in females most sensitivity to food allergens were observed in seafood: 12 (12.76%), chicken: 7 (7.44%), garlic and sesame: 5 (5.31%) respectively. Also, negative results observed for allergic response among females for each one of the following items: almond, blue grape, banana, and celery. Among the two aged groups, seafood was the most prevalent food allergen: 11 (14.47%) in both aged groups respectively (Table 2). The study proved that the eight most common food allergens are eggs, milk, peanuts, tree nuts, soy, wheat, crustacean shellfish and fish, all of them are frequently consumed in the US [18].

    Prevalence of food allergy in asthmatic children under 18 years of age had significant association with gender, birth weight, history of other allergies, family history of allergy, type of coexistent allergy and age of initiation of symptoms, age of introduction of complementary feeding, consumption of cow milk before one year of age and also duration of breast feeding [19]. The prevalence of food allergy among adolescent age group has been confirmed to be comparatively low in Turkey. Peanuts and tree nuts were determined to be the most common causes of IgE-mediated food allergy [20].

    Recent findings are informing changes to population health guidelines in developed countries, which have the potential to halt or reverse the increase in food allergy prevalence. By contrast, food allergy in the developing world remains understudied [21]. There were no statistically significant differences in median age, sex, family history of allergy, or history of allergic diseases and food reactions [22]. We separated the food allergen in allergic patients in too many classes: egg, milk, grain/cereal, nut, fruits, vegetables/green and meats/seafood. Among these classes, the most sensitive response in allergic patients observed to seafood which has a higher percentage: 22 (57.89%), then chicken: 11 (28.94%), shrimp/prawn 7: (18.42%) and meat mix: 3 (7.89%) respectively. Among fruit classes, the apple with 6 (15.78%) was more prevalence in allergic patients, as well as peanuts with 7 (18.42%) showed the greatest sensitivity in allergic patients rather than other nuts. Cereals in grain mix 2 with 11 (28.94%) are more recorded allergen among grains. Cow's milk with 3 (7.89%) and egg white with 6 (15.78%) were the most common allergens in milk and in egg groups (Table 3).

    There are differences among allergic patients for types of food allergen according to genders and aged groups. The highest percentage of food allergens were in seafood mix (contains codfish, shrimp/prawn, blue mussel, tuna, salmon): 10 (58.82%), grain mix (contains wheat flour, oat flour, corn, sesame, buckwheat flour): 7 (41.17%), sesame and soybean: 5 (29.41%) in male groups. While in female group, the most food allergens were seafood mix, chicken, sesame, and garlic. Also statistically there is a significant relationship between food allergens in females and the p-value was 0.002, on the other hand statistically there are no significant relations for food allergens in males groups and the p-value was 0.14. Based on our results, in female group no positive results observed for the these items, hazelnut, garlic, and banana: 0 (0%) but in contrast for male group the was 3 (17.64%) for garlic and hazelnut respectively and 0 (0%) to the banana. There are many differences in the prevalence of food allergens in allergic patients based on the aged groups, for instance in allergic patients which their age were in between (31–52 years), no allergic responses have been detected against these items banana, cherry and honeydew melon. But some of these items mentioned above, give positive results 2 (9.52%), 4 (19.04%) and 2 (9.52%) respectively, in allergic patients which their ages range from (13–30 years). There are statically differences between these two groups, the p-value was 0.08 in 13–30 years which statically not significant, however, the p-value was 0.0006 in 31–52 years which statically significant (Table 4).

    Allergy with banana has been identified (0.04% to 1.2%) in different studies all over the world. Rarely, anaphylaxis has been reported with banana, oral-cutaneous allergy with it mostly observed. Symptoms of systemic reactions such as skin involvement, hypotension, angioedema, respiratory arrest have been reported in patients with anaphylaxis [23]. Betv 2-specific IgE antibodies have been shown to recognize profilins in apple, banana, carrot, celery, cherry, hazelnut, pear, pineapple, potato and tomato. Patients with birch pollen allergy who exhibit IgE reactivity to birch-related allergens in foods frequently ignore clinical symptoms when eating certain foods [24]. In the absence of a cure for food allergy, prevalence will increase steadily even if incidence remains stable. If incidence also increases, then the rates of increase in prevalence will likely accelerate exponentially [25]. In a meta-analysis of 51 studies, self-reported allergy to milk, egg, peanut and seafood ranged from 3% to 35% [26].

    In the serum of the majority of food allergy patients, specific IgE antibodies against nuts (f13, f17), soybean (f14), cows' milk (f2) and especially casein (f78) were dominant and reached to class (3–6), but the most often diagnosed allergic response was caused by potato (f35) (antibodies in 2–4 classes) [27]. Of all the different reactions induced by food allergens, the most frequent are hypersensitivity mediated by specific immunoglobulins E (IgE). In vitro biosensing of specific IgE levels is an alternative approach to invasive risky in vivo tests for diagnosing food allergy in humans [28]. Network analysis identified most associations between Early epitope-specific IgE with either early epitope-specific (IgEG4) or early epitope-specific (IgEIgD), indicating that IgE-secreting plasma cells could originate from either sequential isotype switch from antigen-experienced intermediate isotypes or directly from the IgD+ B cells [29].

    The classes of antibodies detection vary among 36 different food allergens: no specific antibodies detected, very weak antibody detection, weak antibody detection, definite antibody detection strong antibody detection, very high antibody titer. The highest frequency of very high antibody titer was detected only in egg white, egg yolk, and seafood mix (2.63%). Also, the p-value was 0.1 and statistically was not significant. Out of 36 food allergens, only in 6 of them strong antibody has been detected. The highest rate of strong antibody has been found in chicken (7.89%) rather than 5 remained food allergens. Statistically, there was a strong and significant relationship observed between food allergen and classes of antibody detection, the p-value was 0.006. Very weak antibody detection was observed in most food allergen. No allergic reactions recorded for only blue grape among all other allergens. Rye flour, grain mix, sesame, walnut, peanut, kiwi, shrimp/prawn, and seafood mix were showed the highest frequency of very weak antibodies and statistically not significant was observed, the p-value was 0.7. Also for most of the food allergens, weak antibodies were seen except for egg yolk, gluten, blue grape, banana, and meat mix, as well as there no relationship between food allergens and antibodies detection with class 2 and the value was 0.7. Definite antibodies were detected in certain food allergens and the p-value was 0.00001. The association between allergic patients was statistically strong with a food allergen. Among the 36 allergic foods, a larger number of antibodies are found in the seafood mix (Table 5).

    Our work represents the first prevalence and detection study for the 36 food allergens in different genders and aged groups in Erbil province, among allergic patients. The highest prevalence of food allergy observed in females in comparison with the male, as well as the highest frequency of food allergy seen among the 13–30 years allergic patients in comparison with 31–52 years.



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