Mô hình toán ứng dụng

#1

Em đang học môn hô hình toán ứng dụng và đang viết báo cáo về mô hình ung thu vú nhưng phần chứng minh toán học cho địng lý 3.2 rất khó hiểu, anh chị có thể giải thích giúp em với được không ạ. Em cảm ơn ạ.

https://www.sciencedirect.com/science/article/pii/S0378475422003718

Link drive: https://drive.google.com/file/d/1QCOomjQziUCcj_-N9UzW9vXASBMVhDHw/view?usp=sharing

#2

I would like to help you, but I have to “purchase the PDF” in order to know what the địng lý 3.2 is. Your link to this site is useless in this case. Sorry!

#3

Screenshot 2024-10-17 203529
Screenshot 2024-10-17 203529805×409 58.6 KB
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I don’t know why this issue occurred, but I accessed it for free by clicking the ‘view pdf’ button. Please help me try again

#5

I have uploaded the file to Drive

#6

Thanks for the last link on Google Drive. I got it and read through Theorem 3.2. Since my Vietnamese is limited and my computer doesn’t have a Vietnamese font, I don’t know how good your English is, so if I try to explain it in English, it might be a problem for you again. Two possibilities:

  1. You tell me the section you don’t quite understand and I’ll try to explain it.
  2. Or I try to explain it briefly like this:
    Theorem 3.2 states that under certain circumstances (or conditions), the healthy immune system can eliminate a small tumor with T(0) ≤ ¯T.
    Proof:
  • assuming td = in 2/a: T(t) T (t) ≤ 2 ˜T by t lies within 0…td and N ′ ≥ − f N − 2 ˜T p2 N
  • the consequence is then N (t) ≥ ¯N e−ηt ≥ ¯N e−ηtd for t ∈ [0, td ], where η = f + 2 p2 ˜T
  • Suppose ˜N = ¯N e−ηtd , η1 = a − p1 ˜N 2/(1 + 2α2 ˜T + β2 ˜N 2), η2 = kη1 with k > 1, ˜L = η2/( p6 − β6η2) + 2 ˜T α6η2/( p6 − β6η2), and ˜α5 = 1/(1 + α5).
  • assume that (i) ˜T and ˜L are small enough that ρP ˜LρL ϵ1(2 ˜T ) ≪ 1
  • so the equation 1 / (1 + ρP ˜LρL ϵ1(2 ˜T )/KY Q) is approximately 1 (21)
  • (ii) ˜L satisfies the condition 0 < ˜α5 p4 L N (1 − ˜L/KL ) − d ˜L. (22)
  • Equation (22) could (imply) L′(t) > 0 if L(t) < ˜L and T (t) ≥ 1
  • assuming η3 = ˜α5 p4 L N and η4 = η3/KL + d…
  • Result of equation (22) leads to L(t) ≥ L(0)e−η4t + η3(1 − e−η4t )/η4 while T (t) ≥ 1.
  • if there exists ˜t ∈ [0, td ) such that
    L(˜t) ≥ η2… (23)
  • then T ′(˜t) < (η1 − … (24)
    and L(t) ≥ min{L(˜t), ˜L} for t ≥ ˜t and T (t) ≥ 1
  • Note that L(0) satisfies equation (23) when L(0) ≥ ˜L. Thus
    T ′(t) < (1 − k)η1 < 0 for t ≥ ˜t and T (t) ≥ 1. This means that T (t) decreases to a size of less than one cell.
  • The healthy immune system is able to kill a small tumor T (0) ≤ ˜T when equation (21)–(23) holds.
    …
    The above analysis shows that the immune system is able to eliminate a small tumor and prevent tumor formation,
    even though the tumor-free equilibrium is locally unstable. This suggests the existence of a positive stable equilibrium,
    where the equilibrium tumor size is close to zero. A more detailed analysis or bifurcation analysis is required to prove this argument and unfold the dynamics of the system. This paper does not deal with a detailed mathematical analysis, but focuses on the biological interpretation of the model and its solutions.
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#7

Thank you, I’m quite confident in my English reading comprehension, so you can respond in English. However, my math skills are rather poor, and I really don’t understand how the author constructs equations (21)-(24) and their significance. I hope you can help me prove Theorem 3.2 in detail so that I can understand why those formulas exist and what they mean.

I understand that equation (23) sets a minimum limit for 𝐿 to be able to destroy the tumor, and equation (24) indicates that the tumor will be eliminated. However, I don’t understand the significance of equations (21) and (22).

Thank you for your help; I really appreciate it.

#8

Oh…

You need to dig deeper into his research on breast cancer. Usually a scientist likes to cite himself or refer to his other works (he assumes that the readers or research colleagues have already read them). In this case, he is probably referring to his paper “Wei2019 – Mathematical modeling of tumor growth of MCF-7 breast cancer cell line”. Mr. Wen tried to prove that tumors can be eliminated by a healthy immune system under certain conditions and supported his thesis with a mathematical model. I am neither a medical doctor nor a biologist, so I cannot say whether his mathematical proof is acceptable or not. And I do not have time to study all his given references. Sorry!

However, he assumes that under certain conditions the tumor cell population (T) can be derived within a period between 0…td of the active NK cell population (N) and the active CTL (or CD8+ T cell) population L (see equations 1…8). He assumes that if L and T are so small, equation 21 is equal to 1 and L is less than 0 (equation 22). This leads to equation 23 and results in equation 24: tumor cells are less than 0 – meaning: tumor population T is destroyed.

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#9

Bạn có nền Toán chưa? Đặc biệt là chuyên môn bên Phương trình đạo hàm riêng (Partial Differential Equations), hoặc rộng hơn là Giải tích số (Numerical Analysis).

Mình hỏi để biết cách trao đổi với bạn một cách dễ nhất thôi.

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#10

Em đã học mấy môn này, nhưng điểm số khá tệ =((

#11

Vậy mình nói thuần như một Mathematician đi, bắt đầu chứng minh Định lý, em hiểu đến khúc nào, và khúc nào em bị khó hiểu?

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#12

Em chỉ hiểu đến dòng dòng thứ 2, xác định giới hạn cho N ≥ ¯Ne^−ηtd. Bắt đầu từ đây em không hiểu cách tác giả lập luận toán học và xây dựng lên các công thức ˜L = η2/( p6 − β6η2) + 2 ˜T α6η2/( p6 − β6η2); L(t) ≥ L(0)e−η4t + η3(1 − e−η4t )/η4; (23) và (24) ạ

#13

Anh có thể giúp em giải thích chi tiết hơn về các công thức trên được không ạ, em cảm ơn

83% thành viên diễn đàn không hỏi bài tập, còn bạn thì sao?