: Rising allergy and intolerance diagnosis rates A leading theory behind the rising allergy and intolerance diagnosis rates is the "hygiene hypothesis." This theory suggests that living conditions in much of the world might

In general, it seems that there has to be a slight distinction made between a diet during pregnancy and a mother's diet during breast feeding. Overall the available evidence is judged as 'not there' or 'too weak' to make any recommendation to avoid specific antigens.

Evidence is inadequate to advise women to avoid specific foods during pregnancy or breastfeeding to protect their children from allergic diseases like eczema and asthma.
We included five trials, involving 952 participants. Trials of mothers' avoidance of milk, eggs, and other potentially 'antigenic' foods during pregnancy or breastfeeding, or both, provide inadequate evidence about whether such avoidance helps prevent atopic eczema or asthma in the child. Women who avoided eating these foods gained significantly less weight during pregnancy in the one trial reporting on this outcome, raising the possibility of adverse nutritional effects on the mother or fetus. Finally, one small trial reported an inconclusive response of breastfed infants with atopic eczema when their mothers avoided consumption of cow milk and egg.
Prescription of an antigen avoidance diet to a high-risk woman during pregnancy is unlikely to reduce substantially her child's risk of atopic diseases, and such a diet may adversely affect maternal or fetal nutrition, or both. Prescription of an antigen avoidance diet to a high-risk woman during lactation may reduce her child's risk of developing atopic eczema, but better trials are needed.
Dietary antigen avoidance by lactating mothers of infants with atopic eczema may reduce the severity of the eczema, but larger trials are needed.
Kramer MS, Kakuma R.: "Maternal dietary antigen avoidance during pregnancy or lactation, or both, for preventing or treating atopic disease in the child." Cochrane Database of Systematic Reviews 2012, Issue 9. Art. No.: CD000133. DOI: 10.1002/14651858.CD000133.pub3

To the contrary, a few studies highlight a possiblly protective effect revealed by correlational outcomes in a prospective study:

Among mothers without P/TN (peanuts/tree nuts) allergy, higher peripregnancy consumption of P/TN was associated with lower risk of P/TN allergy in their offspring. Our study supports the hypothesis that early allergen exposure increases tolerance and lowers risk of childhood food allergy.
Frazier AL, Camargo CA Jr, Malspeis S, Willett WC, Young MC: "Prospective study of peripregnancy consumption of peanuts or tree nuts by mothers and the risk of peanut or tree nut allergy in their offspring.", JAMA Pediatr. 2014;168(2):156.

But that is not a universal:

Maternal CM avoidance was associated with lower levels of mucosal-specific IgA levels and the development of CMA in infants.
(CM = cow's milk; CMA = cow milk allergy)
Järvinen KM1, Westfall JE, Seppo MS, James AK, Tsuang AJ, Feustel PJ, Sampson HA, Berin C.: "Role of maternal elimination diets and human milk IgA in the development of cow's milk allergy in the infants.", Clin Exp Allergy. 2014 Jan;44(1):69-78. doi: 10.1111/cea.12228.

And if you then compare

High maternal consumption of milk products during pregnancy may protect children from developing CMA, especially in offspring of non-allergic mothers.
Tuokkola J et al.: "Maternal diet during pregnancy and lactation and cow's milk allergy in offspring.", Eur J Clin Nutr. 2016 May;70(5):554-9. doi: 10.1038/ejcn.2015.223. Epub 2016 Jan 13.

Then I conclude that we do not know anything certain about this –– but any avoidance scheme seems quite unfounded.
That is most unfortunate, as mothers were apparently quite shaken by 'scientific' and 'pseudo-scientific' in recent years:

In conclusion, all mothers in the present study restricted at least one type of food without scientific rationale while breastfeeding and more than a third of nursing mothers experienced difficulties with diet restriction. Nursing mothers should be educated on proper diet practices while being warned about unscientific approaches to diet restriction. In recent systematic review, education and emotional support by healthcare providers could enhance breastfeeding). We expect this study will give scientific basis for dietary recommendation to breastfeeding mothers and could promote a breastfeeding.
Goun Jeong et al.: "Maternal food restrictions during breastfeeding", Korean J Pediatr. 2017 Mar; 60(3): 70–76.
Published online 2017 Mar 27. doi: 10.3345/kjp.2017.60.3.70
PMCID: PMC5383635

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This answer would be for the Hygiene Hypothesis part of the question

Epidemiology studies in favour of Hypothesis

The geographical distribution of allergic and autoimmune diseases is a mirror image of the geographical distribution of various infectious diseases, including HAV, gastrointestinal infections and parasitic infections. 3

migration studies have shown that offspring of immigrants coming from a country with a low incidence acquire the same incidence as the host country, as rapidly as the first generation for T1D and MS.  3

Eg. :

This is well illustrated by the increasing frequency of diabetes in families of immigrants from Pakistan to the United Kingdom or the increased risk of MS in Asian immigrants moving to the United States 
The prevalence of systemic lupus erythematosus (SLE) is also much higher in African Americans compared to West Africans

In countries where good health standards do not exist, people are chronically infected by those various pathogens. In those countries, the prevalence of allergic diseases remains low. Interestingly, several countries that have eradicated those common infections see the emergence of allergic and autoimmune diseases.  3

Proof of principle of the casual relationship between the decline of
infectious diseases and an increase of immunological disorders.
The answer to this question comes from animal models of autoimmune and allergic diseases and, to a lesser degree, from clinical intervention studies.
The incidence of spontaneous T1D is directly correlated with the sanitary conditions of the animal facilities, for both the non-obese diabetic (NOD) mouse and the bio-breeding diabetes-prone (BB-DP) rat : the lower the infectious burden, the higher the disease incidence 3
Other examples -
infection of NOD mice with a wide variety of bacteria, virus and parasites protects completely (‘clean’ NOD mice) from diabetes 
mycobacteria (e.g. complete Freund's adjuvant) prevent induction of experimental autoimmune encephalomyelitis and ovalbumin-induced allergic asthma.

The decline is particularly clear for hepatitis A (HAV), childhood diarrhoea and perhaps even more spectacular for parasitic diseases such as filariasis, onchocercosis, schistosomiasis or other soil-transmitted helminthiasis. 3

Epidemiological data of cross-sectional studies revealed that Schistosoma infections have a strong protective effect against atopy, as reviewed recently. Hookworms such as Necator americanus also seem to protect from asthma
(https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3434398/)

Interestingly, it's been thought that, if parasitic infections if decline, then eosinophils which are involved in allergic reactions and parasitic reactions, may in fact have a role in increased allergy.
(Ref. Robbins and Cotran- Pathologic basis of diseases)

It has been shown that helminth eradication increases atopic skin
sensitization in Venezuela.
 3

Mechanism of Hygiene Hypothesis:
 3

Th1 to Th2 deviation -

Th1 T cells produce inflammatory cytokines such as IL-2, interferon
(IFN)-? and tumour necrosis factor (TNF)-? that are operational in
cell-mediated immunity (including autoimmune diabetes).
In contrast, Th2 T cells that produce IL-4, IL-5, IL-6 and IL-13
contribute to IgE production and allergic responses.
IL - 5 most potent eosinophil activating cytokine known. Upon activation, eosinophils liberate proteolytic enzymes as well as two unique proteins called major basic protein and eosinophil cationic protein which damage tissue.It is now believed that the late phase reaction is a major cause of symptoms in some type 1 hypersensitivity reactions such as allergic asthma.
(Ref. Robbins and Cotran- Pathologic basis of diseases)

IL-4 enhances Th2 differentiation; also promote class switching of B cell to IgE.
IL-13 enhances IgE production and acts on epithelial cells to stimulate mucus secretion.
(Ref. Robbins and Cotran- Pathologic basis of diseases)

IgE is involved in Type-1 Hypersensitivity, and hence it increases
will increase HST, like allergic asthma. (Ref. Robbins and Cotran- Pathologic basis of diseases)
Given the reciprocal down-regulation of Th1 and Th2 cells, some authors suggested initially that in developed countries the lack of microbial burden in early childhood, which normally favours a strong Th1-biased immunity, redirects the immune response towards a Th2 phenotype and therefore predisposes the host to allergic disorders.

2.Antigen Competition

The development of strong immune responses against antigens from infectious agents could inhibit responses to ‘weak’ antigens such as autoantigens and allergens. 

(https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2841828/)

Immunoregulation

Regulatory T cells which can suppress immune responses distinct from responses against the antigen in question, here antigens expressed by infectious agents (a phenomenon called bystander suppression)

4.Non Antigenic Ligand

A number of experiments indicate that infectious agents can promote protection from allergic diseases through mechanisms independent of their constitutive antigens, leading to stimulation of non-antigen specific receptors. This concept is well illustrated by the example of Toll-like receptors (TLRs)
It has also been observed that TLR stimulation could prevent the onset of spontaneous autoimmune diseases such as T1D in NOD.
Another mechanism of TLR4 is studied,
where a virus (RSV) which is linked with childhood asthma flips the 'switch' of TLR4. Now LPS a bacterial endotoxin also flips the switch but in a different way.

When this education is lacking or weak, the response to RSV by some critical cells in the immune system’s defence against infections—called “T-cells”—might inadvertently trigger asthma instead of protecting the infant and clearing the infection. How this happens is a mystery that we are trying to solve.
(Ref.- www.fda.gov/BiologicsBloodVaccines/ResourcesforYou/Consumers/ucm167471.htm)
Gene-Environment Interaction
Eg. NOD2 - cytoplasmic sensor of bacteria expressed in Paneth and other intestinal epithelial cells; May control resistance to gut commensal bacteria.

Correlation with Inflammatory Bowel Diseases. (Ref. Robbins and Cotran- Pathologic basis of diseases).

Hygiene during early childhood/pre-natal

Graham Rook and colleagues proposed a new explanation for the rise of immune disorders, which Rook called the “old friends” hypothesis. “We realized human beings coevolved with a whole host of organisms, and it was far more likely what was going on was that we were being deprived of organisms on which we are dependent.

Maternal microbes colonize the human gut while babies are in utero, and again as they pass through the birth canal and start breastfeeding. Young children continue amassing microbiota in every contact with family members, while playing outside in the dirt, getting licked by dogs, and sharing toys with friends. The developing immune system takes cues from all of these encounters.

Recent Thoughts

To the public, “hygiene” is interpreted as personal cleanliness: washing hands, keeping food clean and fresh, sanitizing the home.2
However, because the hypothesis has been largely uncoupled from infections, the idea that we need to be less hygienic is wrong.
2
Relaxing hygiene standards would not reverse the trend but only serve to increase the risks of infectious disease, says Bloomfield.
2
The second major concern among researchers is a lack of evidence demonstrating how to reduce rates of allergic and autoimmune diseases. 2
Although hygiene hypothesis proposes germ exposure but still diseases like Diabetes and Asthma are multifactorial in origin and hygiene may be playing only a part.1

Eg. Asthma

But preventing asthma isn't as simple as avoiding antibacterial soap, having a big family or spending time on the farm. For one thing, a number of microbes — such as a respiratory syncytial virus (RSV) — may cause asthma rather than prevent it. In addition, infections that might help prevent asthma can cause a number of other health problems. The type of germ isn't the only factor that plays a role, either. The severity of infection and when the infection occurs during childhood also appear to matter.
(Ref. www.mayoclinic.org/diseases-conditions/childhood-asthma/expert-answers/hygiene-hypothesis/faq-20058102)

More research is needed to understand exactly how childhood germ exposure might help prevent asthma. What we do know is that in children with asthma, exposure to germs is likely to do more harm than good.

Abbreviations

HAV - hepatitis A Virus
T1D - type 1 diabetes
MS- multiple sclerosis
NOD - non-obese diabetic
HST - hypersensitivity
TLR - toll-like receptors
LPS - lipopolysaccharide

References:

www.mayoclinic.org/diseases-conditions/childhood-asthma/expert-answers/hygiene-hypothesis/faq-20058102 https://www.fda.gov/BiologicsBloodVaccines/ResourcesforYou/Consumers/ucm167471.htm www.ncbi.nlm.nih.gov/pmc/articles/PMC2841828/ Robbins and Cotran- Pathologic basis of diseases

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