This paper proposed an approach for the identification of mutation mechanisms of breast cancer in women in four member countries of the Middle East Cancer Consortium i.e. Egypt, Jordan, Cyprus and Israel (Arabs and Jews). We set up multistage models including both gene mutation and the clonal expansion of intermediate cells. We fit the data-set related to the incidence of female breast cancer in the four member countries. Our simulation results show that the maximum number of driver mutations of breast epithelium stem cells of Egyptian women is 13, whereas there are 14 driver mutations in the genome of stem cells of female patients in Jordan, Cyprus and Israel (Arabs and Jews). In addition, the 3, 10, 5, 5 and 4 stage models are the optimal ones for the tumorigenesis of females in Egypt, Jordan, Cyprus, Israel (Arabs) and Israel (Jews), respectively. The genomic instability is caused by first three driver mutations.

#### Keywords

- breast cancer
- multi-stage model
- incidence rate
- chi square test

Cancer is the uncontrolled growth of abnormal cells that exist in a human body. This disease has become one of the major cause of death all over the world. Breast Cancer is the most common type of cancer that affect women. It constitutes 10% of all new cancer cases worldwide and 1–2% of mortality among women populations throughout the world [1, 2]. Recently, studies on female breast cancer have attracted the attention of biologists and geneticists [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. Breast cancer emerges when an unregulated growth of abnormal cells begins in different parts of tissue. A tissue is a group of similar cells which have an identical general function. This may develop in milk ducts and glands of the breast. Every year, over one million new cases are diagnosed with breast cancer. However, recently many breast cancer deaths have been prevented. This is due to improvements in treatment and early detection through mammography. It is globally estimated that at least 400,000 patients are dying of this disease every year [18,19,20,21].

Breast cancer is considered a major public health problem in both developing and developed countries. The incidence of breast cancer differs in various regions. In developing countries, it is lower than in developed countries. They range between 22 and 71/100,000 of women population, but mortality rates are considerably higher [1, 2, 22,23,24,25,26]. Nevertheless, breast cancer is the most commonly diagnosed cancer among women in the Middle East, because many women do not seek medical care promptly [22,23,24, 27, 28]. Therefore, the increase in the rate of incidences slows down in this region. The average age of women affected by breast cancer in the Middle East is around a decade earlier than the average of that in the west. Comprehensive studies show that the incidence rates of breast cancer are increasing among four countries of the Middle East Cancer Consortium (MECC).

The Middle East Cancer Consortium (MECC) is an example of regional cooperation in cancer registry. The comparison of the countries in the region helps to identify common risk factors and sharing cancer prevention and control strategies. Najla et al. [2] showed age-standardized and age-specific incidence rates of breast cancer among Lebanese women and compared them with those of the regional and western countries. Sherko et al. [29] demonstrated that the incidence of breast cancer among Iraqi Kurdish women is lower than some other countries of Middle East and the west. Hanhan et al. [30] suggested eight biological capabilities that should be acquired during the multistep development of human tumors, namely, self-sufficiency in growth signals, insensitivity to growth-inhibitory signals, resistance of programmed cell death (apoptosis), limitless replicative potential, sustained angiogenesis, tissue invasion and metastasis, reprogramming of energy metabolism, and evading immune destruction.

The process of carcinogenesis is considered as a result of the degeneration of a cell from the normal to a malignant state through a finite number of intermediate stages. The deterioration of a normal cell leads to the proliferation of transformed cells whose descendants generate a tumor. A normal cell may generate a tumor after it has gone through two mutations. How many driver mutations are needed for human breast cancer which are necessary for tumor diagnosis and therapy? Wood et al. [31] obtained an important result that less than 15 (< 15) driver mutations were likely to be responsible for genomic landscapes of human breast and colorectal cancers. The earliest two-stage model with clonal expansion of intermediate cells were developed by Moolgavkar et al. in age-specific incidence rates of breast and colorectal cancers [32,33,34]. Zhang and Simon in 2005 [35] extended the two-stage model to six-stage model with clonal expansion of intermediate cells in each compartment. The two to six stage models with clonal expansion of intermediate cells fit the age-specific incidence rates of breast cancer very well. Zhang et al. [36] in 2014 used the statistical inference (Chi-square test) to test both the two to six stage models with clonal expansion of intermediate cells and the selection between mutation and clonal expansion of intermediate cells. By using the method of statistical inference, Li et al. [37] obtained the maximum number of driver mutations for the breast tumorigenesis of females in United States of America.

The data pattern for the age-specific incidence rates of breast cancer in the USA [38] is different from that in the Middle East countries. In this work, our main objective is to investigate the main differences in the mutation mechanism of breast cancer of women among four member countries (Egypt, Jordan, Cyprus and Israel) of the Middle East Cancer Consortium (MECC) and compare them with the results obtained by Li et al. [37] for the United States of America (USA). In section 2, we briefly describe the data resources. In section 3, we present the mathematical model. Section 4 shows the inference method. Section 5 provides the simulation results. Finally, section 6 describes brief discussion of the results.

The data used in this work are the age-specific incidence rates of breast cancer for female patients in four member countries of the Middle East Cancer Consortium (MECC), i.e., Egypt, Jordan, Cyprus and Israel (Arabs and Jews) [1]. The source of Egypt data is the National Cancer Registry Program (NCRP) during the period 2008–2011 [39]. The Jordan Cancer Registry covers the population from 1996–2001 [1]. The Cyprus National Cancer Registry reports the cyprus data from 1998 to 2001 [1]. Israel data for Arabs and Jews are extracted from the Israel National Cancer Registry from the year 1996 to the year 2001 [1]. The collection of data by the MECC registries is guided by the Manual of Standards for Cancer Registration [40]. For our analysis, we used the incidence of breast cancer rates by age at diagnosis, date of birth and sex. Incidence rates are expressed as cases per 100,000 females.

Figure 1 shows the age-specific incidence rates of female breast cancer in Egypt, Jordan, Cyprus, Israel (Arabs and Jews) and the USA. Most women diagnosed with breast cancer are over 45. Female breast cancer is rare under the age of 25. In the Middle East countries, the lowest age-specific rate is from age groups of 0–4 and 20–24, then increases gradually till it reaches the highest rate and then decreases. Whereas, in the USA, incidence rate increases gradually till it reaches the highest rate without decrease. The majority of female breast cancer cases were diagnosed in the USA and Israel (Jews). The data pattern for the age-specific incidence of female breast cancer is not similar in Middle East countries, and it is different from that in the USA. The peak of the incidence rate is at the age group of 60–64 especially for Egypt, Jordan and Israel (Arabs) and 70–74 for Cyprus, and Israel (Jews). However, the peak of the incidence rates is at the age group of 75–79 for the USA data.

The oncology processes should be a sequence of genetic events according to the findings of Hanahan et al. [30]. Figure 2 shows the schematic representation of multi-stage model with clonal expansion in each compartment of intermediate cells. We supposed that the growth of normal stem cells can be denoted by a logistic curve from age 0 to age 20. Our assumptions are according to ref. [35, 36]. There are ten cells in each breast tissue at birth or at age 0 and 10^{7} cells at the age of 20. There are 10^{6} stem cells during age 80. Since 10^{7} × ^{−0.0667(80−45)} = 10^{6}, in this work, it is assumed that the number of stem cells decreases after age 45 at a rate −0.0667 per cell per year because of the expression levels of hormone and menopause.

For a k-stage model (_{1}(_{i}_{k}_{+1}(_{i}_{1},_{2}, …, _{k}_{+1};

Then, the Kolmogorov forward differential equation [33, 36, 43] for Ψ is defined by:
_{i}_{i}_{i}_{i}

The hazard function (incidence function) is the instantaneous rate of the appearance of malignant tumor in a previously tumor free tissue [33, 35]. Let

From equation (2),

The hazard and probability are related by the given equation [33, 35]

According to ref [44],

Then hazard function or the incidence function should be written in terms of Ψ and Ψ^{′} as follows:
^{′}(1,1, …,1,0;

Furthermore the conditional expectation is given by

By substituting from equations (7) and (8) in (6), we obtained

Since cancer is a rare disease, the probability P (t) of the presence of a malignant cell at time t is close to zero. Thus,

Then, we have the approximation for one breast tissue:

In this section, we proposed the multi-stage model of ordinary differential equations [35, 36] that we used in our calculations, in order to estimate the expectation _{i}_{i}

For _{i}

At _{i}^{7}, _{1} = 1.0074, and
_{i}_{i}

In order to study some biological mutation mechanisms of female breast cancer such as impact of genetic mutation and clonal expansion of cells on cancer evolution, we have to hypothesize two of the following:

Hypothesis one: There is a mutator phenotype in the progression of the tumor of breast cancer for multistage models, that is, the mutation rates per cell per year in _{1} ≤ _{2} ≤ … ≤ _{k}

Hypothesis two: There is no mutator phenotype in the tumorigenesis of breast cancer for multi-stage models, that is, the mutation rates per cell per year in the _{1} = _{2} = …= _{k}

To study the mutation mechanism of female breast cancer, we use the statistical inference to test whether the optimal fittings for the 2–15 stage model are available for each country. In this work, Chi-square test is employed to test those optimal fittings. For these countries, there are insufficient data available for the age of 79 years. However, there are only 16 values for the age-specific incidence rates of women breast cancer patients in Egypt, Jordan, Cyprus and Israel. For the 15-stage model, there are 43 unknown parameters. Therefore, it is impossible for us to examine the 15-stage model because the amount of data is small. To produce enough data to implement the Chi-square test, we use the previous formula (4) to convert the data set of incidence data into data set for probability of female breast cancer. Since the probability values are always small, we handle this problem by amplifying ^{3}); we get 49, 47, 48, 46 and 49 data points for Egypt, Jordan, Cyprus, Israel (Arabs) and Israel (Jews), respectively, which are greater than the one which is adequate for performing the Chi-Square test. The goodness of fit is measured by Chi-square test which is applied within this relation
_{i}

The key question in this study is how many driver mutations are needed in the oncology processes for women patients in the four countries of the MECC, i.e., Egypt, Jordan, Cyprus and Israel. A useful tool to answer this question is the use of statistical inference. The mortality may decrease if the breast cancer is diagnosed at early stages, by the implementation of proper awareness and screening programmes. In this paper, Chi-square test is employed to test whether the 2–15 stage models are available. The calculations were obtained using the numerical optimization routine, fminsearch in MATLAB. Table 1 exhibits the results of Chi-square test for the four countries calculated using equation (14), and comparing them with the values at 5% significant level for the 2–15 stage models under the first hypothesis. The fitting of models more than 15 stages given worse results than the 15-stage model.

Minimum Chi-square values for multi-stage models in the four member countries (Egypt, Jordan, Cyprus and Israel) of the Middle East Cancer Consortium (MECC); Chi2 denotes the value of Chi-square; d

Hypothesis One | ||||||
---|---|---|---|---|---|---|

Multi-stage model | Chi-square test | Egypt | Jordan | Cyprus | Israel(Arabs) | Israel(Jews) |

Stage 2 | Chi2 | 1.5901 | 1.9497 | 3.9351 | 1.4681 | 2.9320 |

df | 44 | 42 | 43 | 41 | 44 | |

S.5% | 60.481 | 58.124 | 59.304 | 56.942 | 60.481 | |

Stage 3 | Chi2 | 0.2860 | 1.0191 | 1.0778 | 0.9077 | 0.9929 |

df | 41 | 39 | 40 | 38 | 41 | |

S.5% | 56.942 | 54.572 | 55.758 | 53.384 | 56.942 | |

Stage 4 | Chi2 | 3.1342 | 1.2497 | 0.8470 | 5.3882 | 0.0289 |

df | 38 | 36 | 37 | 35 | 38 | |

S.5% | 53.384 | 50.998 | 52.192 | 49.802 | 53.384 | |

Stage 5 | Chi2 | 0.4743 | 25.2820 | 0.0609 | 0.2932 | 0.3452 |

df | 35 | 33 | 34 | 32 | 35 | |

S.5% | 49.802 | 47.400 | 48.602 | 46.194 | 49.802 | |

Stage 6 | Chi2 | 2.8163 | 0.7947 | 0.2816 | 0.8875 | 0.6357 |

df | 32 | 30 | 31 | 29 | 32 | |

S.5% | 46.194 | 43.773 | 44.985 | 42.557 | 46.194 | |

Stage 7 | Chi2 | 1.3874 | 0.3559 | 0.1831 | 1.1421 | 1.4832 |

df | 29 | 27 | 28 | 26 | 29 | |

S.5% | 42.557 | 40.113 | 41.337 | 38.885 | 42.557 | |

Stage 8 | Chi2 | 2.1740 | 0.4034 | 0.3379 | 1.1119 | 0.8344 |

df | 26 | 24 | 25 | 23 | 26 | |

S.5% | 38.885 | 36.415 | 37.652 | 35.172 | 38.885 | |

Stage 9 | Chi2 | 1.1726 | 0.4277 | 0.6023 | 1.4395 | 1.0687 |

df | 23 | 21 | 22 | 20 | 23 | |

S.5% | 35.172 | 32.671 | 33.924 | 31.410 | 35.172 | |

Stage 10 | Chi2 | 2.3850 | 0.3292 | 3.4140 | 1.6899 | 1.3919 |

df | 20 | 18 | 19 | 17 | 20 | |

S.5% | 31.410 | 28.869 | 30.144 | 27.587 | 31.410 | |

Stage 11 | Chi2 | 4.7264 | 1.0679 | 1.3535 | 3.0451 | 2.3307 |

df | 17 | 15 | 16 | 14 | 17 | |

S.5% | 27.587 | 24.996 | 26.296 | 23.685 | 27.587 | |

Stage 12 | Chi2 | 8.5654 | 2.7151 | 1.4607 | 3.9033 | 4.6111 |

df | 14 | 12 | 13 | 11 | 14 | |

S.5% | 23.685 | 21.026 | 22.362 | 19.675 | 23.685 | |

Stage 13 | Chi2 | 10.1917 | 4.5112 | 1.7721 | 5.8185 | 5.1914 |

df | 11 | 9 | 10 | 8 | 11 | |

S.5% | 19.675 | 16.919 | 18.307 | 15.507 | 19.675 | |

Stage 14 | Chi2 | 63.2724 | 6.8347 | 6.8223 | 6.3965 | 11.1831 |

df | 8 | 6 | 7 | 5 | 8 | |

S.5% | 15.507 | 12.592 | 14.067 | 11.070 | 15.507 | |

Stage 15 | Chi2 | 92.6729 | 71.2838 | 53.7725 | 91.2415 | 82.2308 |

df | 5 | 3 | 4 | 2 | 5 | |

S.5% | 11.070 | 7.815 | 9.488 | 5.991 | 11.070 |

In Egypt data, the Chi-square values of the multi-stage models starting from 2 to 13 are less than the values at 5% significant level, whereas the 14 and 15 stage models give Chi-square values which are greater than the values at 5% significant level. Therefore, we reject 14 and 15 stage models and accept 2–13 stage models. However, we reject only the 15 stage model and accept 2–14 stage models for Jordan, Cyprus, and Israel, which are similar to that of the USA [37]. It is clear that the 2–14 stage models fit the data very well and these models provide a very small error. The error is the difference between real data and simulated data. It is obvious from the optimal parameters for the net proliferation rates that the net proliferation rates of premalignant cells of the 2–14 stage models before age 45 are larger than those after 45 years of age, which are similar to that in the USA [37]. Net proliferation rates of intermediate cells with one mutation approach zero. This confirms the fact that normal stem cells need two mutations in tumor suppressor genes to become the intermediate cells that expand clonally to destroy the homeostasis of the cells, or the first mutation occurs in genetic instability genes. By the optimal parameters for the mutation rates, it is found that only the first three mutation rates are small; however, the mutation rates approach to one after three stages. Therefore, the genomic instability is caused by the first three driver mutations (see supplementary materials). Our results are consistent with results obtained by Li et al. for USA [37]. They have used stochastic multi-stage models to determine how many mutations are needed for the breast cancer processes. In addition, 3, 10, 5, 5 and 4-stage models are optimal models for Egypt, Jordan, Cyprus, Israel (Arabs) and Israel (Jews), respectively. This means that for tumorigenesis, only Egypt has the three-stage model which is the optimal stage for USA.

We have given the optimal fittings for the 2–15 stage models in Figures 3–7. It is clear that models from stage 2 to 14 can fit all data very well except the data from Egypt. The models which are more than 13-stage cannot fit the data from Egypt. Figure 3 displays the optimal fitting for the 2–15 stage models of Egypt for the period from 2008 to 2011. We can see that 14 and 15 stage models cannot fit the data, so we can only accept the mathematical models with 2–13 mutations at the 5% significant level. The optimal fittings for the 2–15 stage models in Jordan from 1996 to 2001 are shown in Figure 4. We accept the 2–14 stage models at the 5% significant level and reject the 15 stage model at the 5% significant level. The age-specific incidence rates per 100,000 females for breast cancer in Cyprus during 1998–2001 are given in Figure 5. Two to fourteen stage models fit the data very well, while the 15 stage model provides a bad fitting with a large error. We plot the age-specific incidence rates for Israel (Arabs) during 1996–2001 in Figure 6. The 15 stage has led to a very large error, so we reject it and accept the 2–14 stage models. Figure 7 shows the age-specific breast cancer incidence for Israel (Jews) in 1996–2001. The 2–14 stage models fit the data very well, and thus we reject the 15 stage model.

We applied hypothesis two of constant mutation rate per cell per year to check the genetic instability. This hypothesis means that there is no mutator phenotype but only the clonal expansion of intermediate cells in the process of breast carcinogenesis. Mutator phenotype refers to the increase in mutation rates of breast cancer cells [45]. Table 2 shows the minimum Chi-square values of the goodness fit of the data for the 2–6 stage models. The fitting of the models in more than 6 stages given worse results than the 6-stage model. We found that models with two to four stages can fit Egypt, Jordan and Cyprus data and reject all other models that are more than 4-stage of the selection of clonal expansion of intermediate cells in the process of breast carcinogenesis, because it gives a large error. Models with two to five stages can fit the data from Israel. However, models with more than three stages cannot fit the SEER data [37]. We conclude that the first two or three stages lead to genetic instability before generating a cancer cell. We plot the probability under hypothesis two for models with two to six stages for the four countries as shown in Figures 8–12. In figures (8–10), it can be observed that two, three, and four stage models provided good simulations with data, while five and six stage models cannot fit the data well. Therefore, we should accept the two-four stage models and reject the models with more than four stages. In Figures 11 and 12, the 2–5 stage models fit the data very well. So, we reject all other models of the selection of clonal expansion of intermediate cells in the process of breast carcinogenesis. This hypothesis gives the best results for the data from Israel (Arabs and Jews) because two to five stage models give us good simulation for the data from Israel.

Chi-square values for the biological hypothesis two.

Hypothesis Two | ||||||
---|---|---|---|---|---|---|

Multi-stage model | Chi-square test | Egypt | Jordan | Cyprus | Israel(Arabs) | Israel(Jews) |

Stage 2 | Chi2 | 1.5984 | 1.9152 | 3.9383 | 53.384 | 2.9285 |

df | 45 | 43 | 44 | 42 | 45 | |

S.5% | 61.656 | 59.304 | 60.481 | 58.124 | 61.656 | |

Stage 3 | Chi2 | 0.3994 | 0.5781 | 1.1131 | 0.2319 | 0.8121 |

df | 43 | 41 | 42 | 40 | 43 | |

S.5% | 59.304 | 56.942 | 58.124 | 55.758 | 59.304 | |

Stage 4 | Chi2 | 6.9790 | 6.5856 | 3.1302 | 3.3066 | 4.3960 |

df | 41 | 39 | 40 | 38 | 41 | |

S.5% | 56.942 | 54.572 | 55.758 | 53.384 | 56.942 | |

Stage 5 | Chi2 | 59.9711 | 74.4047 | 55.3319 | 10.6929 | 51.2783 |

df | 39 | 37 | 38 | 36 | 39 | |

S.5% | 54.572 | 52.192 | 53.384 | 50.998 | 54.572 | |

Stage 6 | Chi2 | 72.6603 | 164.2708 | 143.3481 | 82.7561 | 159.2855 |

df | 37 | 35 | 36 | 34 | 37 | |

S.5% | 52.192 | 49.802 | 50.998 | 48.602 | 52.192 |

The purpose of this work is to investigate the main differences in the mutation mechanism of breast cancer of women in the four member countries (Egypt, Jordan, Cyprus and Israel) of the Middle East Cancer Consortium (MECC) and compare them with the results obtained by Li et al. [37] for the United States of America (USA). Li et al. [37] have used the stochastic multistage models to fit the data set from America on age-specific incidence rates of breast cancer by using several coupled ordinary differential equations derived from the Kolmogorov backward equations. We used different mathematical models to estimate the age-specific incidence rates of breast cancer for women in four member countries (Egypt, Jordan, Cyprus and Israel) of the Middle East Cancer Consortium (MECC). In addition, we have used statistical inference to test the number of mutation stages that are required for female breast cancer. We estimated the optimal parameters by using the numerical optimization routine, fminsearch in MATLAB, to calculate the minimum Chi-square value. We made comparisons among four countries to know the differences in the process of breast carcinogenesis. The regional comparisons can shed light on the health care factors and population, to identify common risk factors and share cancer prevention and control strategies. The differences between countries reflect the variation in lifestyle, genetic influences, environmental exposure, and the availability of mammography screening.

The maximum number of driver mutations is 13 in Egypt. However, by goodness fit of data to the probabilities of female breast cancer occurrence in Jordan, Cyprus, and Israel (Arabs and Jews), the maximum number of driver mutations for three countries is found to be 14. These results support the results of Wood et al. [31] that are less than 15 mutations required to develop breast cancer and colorectal cancer. The optimal models are 3, 10, 5, 5 and 4-stage for Egypt, Jordan, Cyprus, Israel (Arabs) and Israel (Jews), respectively. The maximum number of driver mutations is 14 in the USA, and the three-stage model is the optimal model [37]. Therefore, Egypt has the same optimal model as United States of America (USA), but differs in the maximum number of driver mutations for breast cancer. Genomic instability is a characteristic of almost all human cancers. By optimal parameter values for mutation rates of multi-stage models, it is found that only the first three mutation rates are small. Therefore, the mutation of instability gene occurred due to the first three mutations (see supplementary materials). This confirms the fact that normal stem cells need two mutations to become intermediate cells that expand clonally to destroy the homeostasis of the cells, or the first mutation occurs in genetic instability genes. The mutations rates and clonal expansion of intermediate cells depend on menopause. Therefore, hormone therapy such as tamoxifen has a good effect on female patients before menopause. By testing hypothesis two, we have checked the selection between the mutator phenotype and the clonal expansion of intermediate cells. We conclude in the case of three countries (Egypt, Jordan and Cyprus), that if the driver mutations are less than five stages, we cannot reject the selection of both the mutator phenotype and the clonal expansion of intermediate cells. If the driver mutations are larger than four stages, we should reject the selection of the clonal expansion of intermediate cells. As a matter of fact, if the driver mutations are larger than four stages, carcinogenesis does not need the mutator phenotype because all mutation rates after four stages approach to one. However, in Israel (Arabs and Jews), we should accept the two-five stage models at the 5% significant level and reject models that are more than 5-stage. According to our results, hypothesis one gives the best results for the four countries compared to hypothesis two because 2 to 13 or 14 stage models are available and give us a good simulation and a small error.

#### Minimum Chi-square values for multi-stage models in the four member countries (Egypt, Jordan, Cyprus and Israel) of the Middle East Cancer Consortium (MECC); Chi2 denotes the value of Chi-square; df (=number of data-number of parameters-1) denotes the degrees of freedom; and S.5% denotes the value at 5% significant level.

Hypothesis One | ||||||
---|---|---|---|---|---|---|

Multi-stage model | Chi-square test | Egypt | Jordan | Cyprus | Israel(Arabs) | Israel(Jews) |

Stage 2 | Chi2 | 1.5901 | 1.9497 | 3.9351 | 1.4681 | 2.9320 |

df | 44 | 42 | 43 | 41 | 44 | |

S.5% | 60.481 | 58.124 | 59.304 | 56.942 | 60.481 | |

Stage 3 | Chi2 | 0.2860 | 1.0191 | 1.0778 | 0.9077 | 0.9929 |

df | 41 | 39 | 40 | 38 | 41 | |

S.5% | 56.942 | 54.572 | 55.758 | 53.384 | 56.942 | |

Stage 4 | Chi2 | 3.1342 | 1.2497 | 0.8470 | 5.3882 | 0.0289 |

df | 38 | 36 | 37 | 35 | 38 | |

S.5% | 53.384 | 50.998 | 52.192 | 49.802 | 53.384 | |

Stage 5 | Chi2 | 0.4743 | 25.2820 | 0.0609 | 0.2932 | 0.3452 |

df | 35 | 33 | 34 | 32 | 35 | |

S.5% | 49.802 | 47.400 | 48.602 | 46.194 | 49.802 | |

Stage 6 | Chi2 | 2.8163 | 0.7947 | 0.2816 | 0.8875 | 0.6357 |

df | 32 | 30 | 31 | 29 | 32 | |

S.5% | 46.194 | 43.773 | 44.985 | 42.557 | 46.194 | |

Stage 7 | Chi2 | 1.3874 | 0.3559 | 0.1831 | 1.1421 | 1.4832 |

df | 29 | 27 | 28 | 26 | 29 | |

S.5% | 42.557 | 40.113 | 41.337 | 38.885 | 42.557 | |

Stage 8 | Chi2 | 2.1740 | 0.4034 | 0.3379 | 1.1119 | 0.8344 |

df | 26 | 24 | 25 | 23 | 26 | |

S.5% | 38.885 | 36.415 | 37.652 | 35.172 | 38.885 | |

Stage 9 | Chi2 | 1.1726 | 0.4277 | 0.6023 | 1.4395 | 1.0687 |

df | 23 | 21 | 22 | 20 | 23 | |

S.5% | 35.172 | 32.671 | 33.924 | 31.410 | 35.172 | |

Stage 10 | Chi2 | 2.3850 | 0.3292 | 3.4140 | 1.6899 | 1.3919 |

df | 20 | 18 | 19 | 17 | 20 | |

S.5% | 31.410 | 28.869 | 30.144 | 27.587 | 31.410 | |

Stage 11 | Chi2 | 4.7264 | 1.0679 | 1.3535 | 3.0451 | 2.3307 |

df | 17 | 15 | 16 | 14 | 17 | |

S.5% | 27.587 | 24.996 | 26.296 | 23.685 | 27.587 | |

Stage 12 | Chi2 | 8.5654 | 2.7151 | 1.4607 | 3.9033 | 4.6111 |

df | 14 | 12 | 13 | 11 | 14 | |

S.5% | 23.685 | 21.026 | 22.362 | 19.675 | 23.685 | |

Stage 13 | Chi2 | 10.1917 | 4.5112 | 1.7721 | 5.8185 | 5.1914 |

df | 11 | 9 | 10 | 8 | 11 | |

S.5% | 19.675 | 16.919 | 18.307 | 15.507 | 19.675 | |

Stage 14 | Chi2 | 63.2724 | 6.8347 | 6.8223 | 6.3965 | 11.1831 |

df | 8 | 6 | 7 | 5 | 8 | |

S.5% | 15.507 | 12.592 | 14.067 | 11.070 | 15.507 | |

Stage 15 | Chi2 | 92.6729 | 71.2838 | 53.7725 | 91.2415 | 82.2308 |

df | 5 | 3 | 4 | 2 | 5 | |

S.5% | 11.070 | 7.815 | 9.488 | 5.991 | 11.070 |

#### Chi-square values for the biological hypothesis two.

Hypothesis Two | ||||||
---|---|---|---|---|---|---|

Multi-stage model | Chi-square test | Egypt | Jordan | Cyprus | Israel(Arabs) | Israel(Jews) |

Stage 2 | Chi2 | 1.5984 | 1.9152 | 3.9383 | 53.384 | 2.9285 |

df | 45 | 43 | 44 | 42 | 45 | |

S.5% | 61.656 | 59.304 | 60.481 | 58.124 | 61.656 | |

Stage 3 | Chi2 | 0.3994 | 0.5781 | 1.1131 | 0.2319 | 0.8121 |

df | 43 | 41 | 42 | 40 | 43 | |

S.5% | 59.304 | 56.942 | 58.124 | 55.758 | 59.304 | |

Stage 4 | Chi2 | 6.9790 | 6.5856 | 3.1302 | 3.3066 | 4.3960 |

df | 41 | 39 | 40 | 38 | 41 | |

S.5% | 56.942 | 54.572 | 55.758 | 53.384 | 56.942 | |

Stage 5 | Chi2 | 59.9711 | 74.4047 | 55.3319 | 10.6929 | 51.2783 |

df | 39 | 37 | 38 | 36 | 39 | |

S.5% | 54.572 | 52.192 | 53.384 | 50.998 | 54.572 | |

Stage 6 | Chi2 | 72.6603 | 164.2708 | 143.3481 | 82.7561 | 159.2855 |

df | 37 | 35 | 36 | 34 | 37 | |

S.5% | 52.192 | 49.802 | 50.998 | 48.602 | 52.192 |

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action images based on graph difference function Application of Higher Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Application of Forced Modulation Function Mathematical Model in the Characteristic Research of Reflective Intensity Fibre Sensors Radioactive source search problem and optimisation model based on meta-heuristic algorithm Research on a method of completeness index based on complex model Fake online review recognition algorithm and optimisation research based on deep learning Research on the sustainable development and renewal of Macao inner harbour under the background of digitisation Support design of main retracement passage in fully mechanised coal mining face based on numerical simulation Study on the crushing mechanism and parameters of the two-flow crusher Interaction design of financial insurance products under the Era of AIoT Modeling the pathway of breast cancer in the Middle East Corporate social responsibility fulfilment, product-market competition and debt risk: Evidence from China ARMA analysis of the green innovation technology of core enterprises under the ecosystem – Time series data Reconstruction of multimodal aesthetic critical discourse analysis framework Image design and interaction technology based on Fourier inverse transform What does students’ experience of e-portfolios suggest Research on China interregional industrial transformation slowdown and influencing factors of industrial transformation based on numerical simulation The medical health venture capital network community structure, information dissemination and the cognitive proximity Data mining of Chain convenience stores location The optimal model of employment and entrepreneurship models in colleges and universities based on probability theory and statistics A generative design method of building layout generated by path Parameter Id of Metal Hi-pressure State Equation Analysis of the causes of the influence of the industrial economy on the social economy based on multiple linear regression equation Research of neural network for weld penetration control P-Matrix Reasoning and Information Intelligent Mining Intelligent Recommendation System for English Vocabulary Learning – Based on Crowdsensing Regarding new wave distributions of the non-linear integro-partial Ito differential and fifth-order integrable equations Research on predictive control of students’ performance in PE classes based on the mathematical model of multiple linear regression equation Beam control method for multi-array antennas based on improved genetic algorithm The influence of X fuzzy mathematical method on basketball tactics scoring Application of regression function model based on panel data in bank resource allocation financial risk management Research on aerobics training posture motion capture based on mathematical similarity matching statistical analysis Application of Sobolev-Volterra projection and finite element numerical analysis of integral differential equations in modern art design Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Research on motion capture of dance training pose based on statistical analysis of mathematical similarity matching Application of data mining in basketball statistics Application of B-theory for numerical method of functional differential equations in the analysis of fair value in financial accounting Badminton players’ trajectory under numerical calculation method Research on the influence of fuzzy mathematics simulation model in the development of Wushu market Study on audio-visual family restoration of children with mental disorders based on the mathematical model of fuzzy comprehensive evaluation of differential equation Difference-in-differences test for micro effect of technological finance cooperation pilot in China Application of multi-attribute decision-making methods based on normal random variables in supply chain risk management Exploration on the collaborative relationship between government, industry, and university from the perspective of collaborative innovation The impact of financial repression on manufacturing upgrade based on fractional Fourier transform and probability AtanK-A New SVM Kernel for Classification Validity and reliability analysis of the Chinese version of planned happenstance career inventory based on mathematical statistics Visual positioning system for marine industrial robot assembly based on complex variable function Mechanical behaviour of continuous girder bridge with corrugated steel webs constructed by RW Research on the influencing factors of agricultural product purchase willingness in social e-commerce situation Study of a linear-physical-programming-based approach for web service selection under uncertain service quality A mathematical model of plasmid-carried antibiotic resistance transmission in two types of cells Fractional Differential Equations in the Exploration of Geological and Mineral Construction Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China The Law of Large Numbers in Children's Education Data structure simulation for the reform of the teaching process of university computer courses Calculating university education model based on finite element fractional differential equations and macro-control analysis Educational research on mathematics differential equation to simulate the model of children's mental health prevention and control system Analysis of enterprise management technology and innovation based on multilinear regression model Verifying the validity of the whole person model of mental health education activities in colleges based on differential equation RETRACTION NOTE Innovations to Attribute Reduction of Covering Decision System Based on Conditional Information Entropy Research on the mining of ideological and political knowledge elements in college courses based on the combination of LDA model and Apriori algorithm Adoption of deep learning Markov model combined with copula function in portfolio risk measurement Good congruences on weakly U-abundant semigroups Research on the processing method of multi-source heterogeneous data in the intelligent agriculture cloud platform Mathematical simulation analysis of optimal detection of shot-putters’ best path Internal control index and enterprise growth: An empirical study of Chinese listed-companies in the automobile manufacturing industry Determination of the minimum distance between vibration source and fibre under existing optical vibration signals: a study Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets Nonlinear Differential Equations in the Teaching Model of Educational Informatisation Fed-UserPro: A user profile construction method based on federated learning Smart Communities to Reduce Earthquake Damage: A Case Study in Xinheyuan, China Response Model of Teachers’ Psychological Education in Colleges and Universities Based on Nonlinear Finite Element Equations Institutional investor company social responsibility report and company performance Mathematical analysis of China's birth rate and research on the urgency of deepening the reform of art education Precision Machining Technology of Jewelry on CNC Machine Tool Based on Mathematical Modeling First-principles calculations of magnetic and mechanical properties of Fe-based nanocrystalline alloy Fe _{80}Si_{10}Nb_{6}B_{2}Cu_{2}Computer Vision Communication Technology in Mathematical Modeling The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics Has the belt and road initiative boosted the resident consumption in cities along the domestic route? – evidence from credit card consumption MCM of Student’s Physical Health Based on Mathematical Cone Attitude control for the rigid spacecraft with the improved extended state observer Sports health quantification method and system implementation based on multiple thermal physiology simulation Research on visual optimization design of machine–machine interface for mechanical industrial equipment based on nonlinear partial equations Research on identifying psychological health problems of college students by logistic regression model based on data mining Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching A mathematical model of PCNN for image fusion with non-sampled contourlet transform Nonlinear Differential Equations in Computer-Aided Modeling of Big Data Technology The Uniqueness of Solutions of Fractional Differential Equations in University Mathematics Teaching Based on the Principle of Compression Mapping Cognitive Computational Model Using Machine Learning Algorithm in Artificial Intelligence Environment Application of Higher-Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Recognition of Electrical Control System of Flexible Manipulator Based on Transfer Function Estimation Method Automatic Knowledge Integration Method of English Translation Corpus Based on Kmeans Algorithm Real Estate Economic Development Based on Logarithmic Growth Function Model Informatisation of educational reform based on fractional differential equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis Research on the control of quantitative economic management variables under the numerical method based on stochastic ordinary differential equations Network monitoring and processing accuracy of big data acquisition based on mathematical model of fractional differential equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with Diamond-Square Algorithm The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Cross-border E-commerce Platform Operation Based on Fractional Differential Equations Research on Stability of Time-delay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of Air-Conditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression System dynamics model of output of ball mill Optimisation of Modelling of Finite Element Differential Equations with Modern Art Design Theory Mathematical function data model analysis and synthesis system based on short-term human movement Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Human gait modelling and tracking based on motion functionalisation Analysis and synthesis of function data of human movement The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Financial Accounting Measurement Model Based on Numerical Analysis of Rigid Normal Differential Equation and Rigid Functional Equation Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Design of Morlet wavelet neural network to solve the non-linear influenza disease system Nonlinear Differential Equations in Cross-border E-commerce Controlling Return Rate Differential equation model of financial market stability based on Internet big data 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Children’s cognitive function and mental health based on finite element nonlinear mathematical model Motion about equilibrium points in the Jupiter-Europa system with oblateness Fractional Differential Equations in Electronic Information Models Badminton players’ trajectory under numerical calculation method BIM Engineering Management Oriented to Curve Equation Model Optimal preview repetitive control for impulse-free continuous-time descriptor systems Development of main functional modules for MVB and its application in rail transit Study on the impact of forest fire prevention policy on the health of forest resources Mathematical Method to Construct the Linear Programming of Football Training The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Value Creation of Real Estate Company Spin-off Property Service Company Listing Selection by differential mortality rates Digital model creation and image meticulous processing based on variational partial differential equation Dichotomy model based on the finite element differential equation in the educational informatisation teaching reform model Nonlinear Dissipative System Mathematical Equations in the Multi-regression Model of Information-based Teaching The modelling and implementation of the virtual 3D animation scene based on the geometric centre-of-mass algorithm The policy efficiency evaluation of the Beijing–Tianjin–Hebei regional government guidance fund based on the entropy method The transfer of stylised artistic images in eye movement experiments based on fuzzy differential equations Research on behavioural differences in the processing of tenant listing information: An eye-movement experiment A review of the treatment techniques of VOC Some classes of complete permutation polynomials in the form of ( x ^{pm}−x +δ )^{s}+ax ^{pm}+bx overF _{p2m}The consistency method of linguistic information and other four preference information in group decision-making Research on the willingness of Forest Land’s Management Rights transfer under the Beijing Forestry Development A mathematical model of the fractional differential method for structural design dynamics simulation of lower limb force movement step structure based on Sanda movement Fractal structure of magnetic island in tokamak plasma Numerical calculation and study of differential equations of muscle movement velocity based on martial articulation body ligament tension Study on the maximum value of flight distance based on the fractional differential equation for calculating the best path of shot put Sports intensity and energy consumption based on fractional linear regression equation Analysis of the properties of matrix rank and the relationship between matrix rank and matrix operations Study on Establishment and Improvement Strategy of Aviation Equipment Research on Financial Risk Early Warning of Listed Companies Based on Stochastic Effect Mode Characteristics of Mathematical Statistics Model of Student Emotion in College Physical Education Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Nonlinear strategic human resource management based on organisational mathematical model Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Fractional Differential Equations in Sports Training in Universities Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model