Questions tagged [mathematical-biology]
Mathematical biology is an interdisciplinary science that uses mathematical methods to study problems arising from biology, the science studying living organisms and ecosystems.
39 questions
5
votes
1
answer
481
views
Examples of "Cell divisions" in mathematics?
I stumbled upon a mathematical structure, which I would describe as a cell division from biology, while researching prime factorization trees:
The image show a cell division:
blue = Growth of classes,...
- 3,761
1
vote
0
answers
284
views
problem with a calculation in Biomathematics paper
I'm working on this paper but really struggling to understand how the derived equation (10). It does seems like something is missing or its abscense it's not well justified. Can someone help me?
link ...
- 191
1
vote
1
answer
172
views
Identifying player strategies in repeated games, based on payoffs
Background
In evolutionary game theory, one can what kinds of different strategies yield the most payoff to players that play the same game repeatedly. Consider, for instance, the iterated Prisoner's ...
- 5,065
3
votes
2
answers
212
views
What is the expected remaining life duration of a cell in the $t\to\infty$ limit?
Consider the following population model: We start with a population of a single cell at time $t=0$. Each cell divides into $k$ new cells at random times $T$ distributed according to a probability ...
- 304
4
votes
1
answer
242
views
The stability of the equilibria of a non-linear ODE system
I have the following coupled non-linear ODE system, which describes a biological system:
$$
\begin{cases}
\dfrac{dp}{dt} = -\gamma p f,\\
\\
\dfrac{df}{dt} = -c f + \gamma p f,\\
\\
\dfrac{dT}{dt} = \...
user524961
0
votes
0
answers
119
views
Texts on coalescent theory/probability methods for DNA evolution
I am starting a PhD on mitochondrial evolution modelling with a focus on probabilistic methods and coalescent theory. For this purpose, I am looking for
advanced textbooks on probability methods for ...
- 203
2
votes
0
answers
95
views
Where can I find resources for a paper "Stability analysis of a novel DDE of HIV CD4+ T-cells"?
I am currently working on a the paper [NND]:
Question:
On page 4, equation 6 introduces a concept related to the infection rate within the context of the HIV model. Unfortunately, the paper does not ...
- 248
3
votes
1
answer
330
views
Epidemic modelling: expectation of time of infection given the distribution of transmission and recovery
Can I express the expected value of
\begin{equation}
\langle \tau\rangle_\text{total}=\int_0^\infty \tau \psi_\text{inf}(\tau)\Psi_\text{rec}(\tau)\mathrm{d}\tau
\end{equation}
in terms of the moment(...
- 304
1
vote
1
answer
119
views
Resources/Reading Materials on PASA (optimal control theory)
I am currently working on my undergraduate thesis, and my adviser suggested that I look into a Polyhedral Active Set Algorithm (PASA) for my paper. I have been trying to find resources/materials on it ...
- 149
2
votes
0
answers
83
views
Turn a growing 1D lattice of size $N(t)$ into a constant size lattice by change of variables
I am studying some stochastic process (a voter model for example) on a finite size lattice of 1 dimension and of size $N_t$. However $N_t$ grows at rate $\lambda$ and can be represented as follows:
\...
- 117
19
votes
4
answers
3k
views
Applications of complex exponential
In calculus we learn about many applications of real exponentials like $e^x$ for bacteria growth, radioactive decay, compound interest, etc. These are very simple and direct applications. My question ...
- 199
6
votes
1
answer
388
views
Current status on Richardson's model (growth model)
A very simple stochastic growth model on a lattice is the Richardson's model (Actually originally defined by Murray Eden in the 60s).
Each point of the lattice can be either occupied or vacant, once ...
- 304
8
votes
3
answers
664
views
Explain seemingly non-random figures which arise from random Poisson points with normalization
Context Working with some biological datasets it was puzzling to see the patterns like Figure 2 (right) below. The first feeling was, that it corresponds to some biological effects like correlations ...
- 26.3k
4
votes
1
answer
178
views
The mean value of the reconstruction complexity of a random sequence
This problem is motivated by the problem of reconstructing a genome from the family of its short subwords.
Given a word $w$ and a positive integer $k$, let $M_k(w)$ be the family of all subwords of ...
- 44.4k
1
vote
1
answer
361
views
Proving positive invariance
I need to prove that set $D$(A picture for Set $D$) given by
$$D=\{(x,y):0\leq x\leq L_0,~0\leq y\leq X_0,~0\leq x+y \leq R_0\}\subseteq \mathbb{R}_+^2$$ of the system:
$$\dot{x}=k_1(R_0-x-y)(L_0-x)-...
- 11
6
votes
1
answer
366
views
Time of peak of an SIR epidemic
I've learned some classical results on the peak and the attack rate of an idealized epidemic which evolves according to a SIR model
$\dot{s} = -\beta\cdot i \cdot s$
$\dot{i} = +\beta\cdot i \cdot s -...
- 9,934
4
votes
1
answer
3k
views
How to mathematically characterize a feedback loop in ODEs?
I have a biological system that exhibits a feedback type of behavior. The diagram is a schematic of the system of ODEs. In this system, the total amount of $x_1, x_2, x_3$ is conserved; however, there ...
- 513
76
votes
3
answers
6k
views
Should water at the scale of a cell feel more like tar?
The Navier-Stokes equations are as follows,
$$\dot{u}+(u\cdot \nabla ) u +\nu \nabla^2 u =\nabla p$$
where $u$ is the velocity field, $\nu$ is the viscosity, and $p$ is the pressure.
Some elementary ...
- 1,151
4
votes
2
answers
373
views
Approximated solutions of SEIR models
Numerical solutions of the SEIR equations (describing the spreading of an epidemic disease) – or variations thereof –
$\dot{S} = - N$
$\dot{E} = + N - E/\lambda$
$\dot{I} = + E/\lambda - I/\delta$
...
- 9,934
0
votes
0
answers
120
views
Genes mirror geography on a torus?
Disclaimer: this is an open-ended, imprecise question, asking for speculation in a topic that I know relatively little about (random matrix theory and principal component analysis). I originally asked ...
- 1,377
12
votes
6
answers
1k
views
Suggestions for reducing the transmission rate?
What are suggestions for reducing the transmission rate of the current epidemics?
In summary, my best one so far is (once we are down to the stay home rule) to discretize time, i.e., to introduce the ...
17
votes
7
answers
3k
views
Mathematical physics without partial derivatives
Remark: All the answers so far have been very insightful and on point but after receiving public and private feedback from other mathematicians on the MathOverflow I decided to clarify a few notions ...
- 4,041
0
votes
1
answer
200
views
Conditions to determine sign of real roots
From a delay system, I obtain the following as part of a characteristic equation:
$$f(\lambda) = \lambda - a + be^{-c\lambda},$$
where $a, b,$ and $c$ are positive number and $a<b, ac<1$. My ...
- 513
6
votes
0
answers
648
views
Status of the Salmon Conjecture
The set-theoretic version of the Salmon Conjecture (that is, finding the equations that cut out the fourth secant variety of the Segre embedding of $\mathbb P^3 \times \mathbb P^3 \times \mathbb P^3$ ...
- 3,464
4
votes
1
answer
766
views
Proof that dynamical systems with bounded Kolmogorov complexity can't emulate all Turing machines
Motivation:
During a discussion with neuroscientists the question arose as to whether the human brain may emulate any Turing machine. If we assume that animal brains may be modelled as deterministic ...
- 4,041
6
votes
1
answer
465
views
Sphere packing processes during biological development
Within the context of mathematical biology, a sphere packing problem occurred to me. I must note that unlike the typical sphere packing problems, the variant I consider involves minimising the average ...
- 4,041
31
votes
7
answers
6k
views
Applications of mathematics in clinical setting
What are some examples of successful mathematical attempts in clinical setting, specifically at the patient-disease-drug level?
To clarify, by patient-disease-drug level, I mean the mathematical work ...
18
votes
4
answers
2k
views
Differential geometry applied to biology
This was originally a question posted here on MathSE. But I'll ask again here to see if I can get some different answers.
I'm looking for current areas of research which apply techniques from ...
- 181
3
votes
0
answers
104
views
conditions for asymptotic comparison to hold
I have the following simple dynamical system:
\begin{align}
x_1' &= a - f(x_2)x_1\\
x_2' &= bx_1 - cx_2,
\end{align}
where all parameters and initial conditions are positive. $f(x_2)$ is a ...
- 513
7
votes
1
answer
525
views
How to study the global stability for this 3D system?
I am studying a biological system (HIV) and arrived at this simplified dynamical system:
\begin{align}
x_1' &= a_1 + a_2x_2 - a_1x_2 - a_4x_1 - a_5\frac{1+a_6x_3}{1+a_7x_3}x_1\\
x_2' &= a_5\...
- 513
10
votes
1
answer
1k
views
Why did Voevodsky abandon his work on "singletons"?
In an interview (I link the Google translation), Voevodsky talks about how, in the late 2000s, he worked on the problem of "restoring the history of populations according to their modern genetic ...
- 251
2
votes
1
answer
78
views
Purpose of using a saturable logistic like term
I would like to know what is the purpose of using the term $P\over (k+P)$ in the following. I found it when reading the article found here but it was commonly used in few other related articles .
Is ...
- 131
2
votes
0
answers
327
views
Life. Intermediate stages
My question is pure mathematics when restricted to the cellular automata theory.
John von Neumann got the grasp of and defined life. Many years later biologists supported von Neumann's definition of ...
- 7,600
27
votes
4
answers
3k
views
Algebra and cancer research
Let me start by acknowledging the existence of this thread: Mathematics and cancer research
It is well-known that mathematical modeling and computational biology are effective tools in cancer research....
- 1,432
9
votes
2
answers
6k
views
Applications of algebraic geometry/commutative algebra to biology/pharmacology
Are there applications of algebraic geometry/commutative algebra to biology/pharmacology?
It might be that some Gröbner basis technique is used somewhere? I know there are some applications to ...
11
votes
3
answers
3k
views
Applications of knot theory to biology/pharmacology
What are the applications of knot theory to biology/pharmacology?
I guess there should be some, since proteins are quite long and some of their properties are probably related to whether they are ...
26
votes
7
answers
9k
views
Applications of group theory to mathematical biology (pharmacology)
Are there applications of group theory — broadly, say, representation theory, Lie algebras, $q$-groups, etc — to mathematical biology?
In particular, I am interested in applications to pharmacology — ...
29
votes
20
answers
8k
views
Mathematics and cancer research
What are applications of mathematics in cancer research?
Unfortunately, I heard quite little about applications of mathematics, but I heard something about applications of physics, and let me put this ...
40
votes
17
answers
10k
views
Interesting mathematical topics arising from biology
I've heard that there's a relatively new field of science called mathematical biology.
It will certainly apply well known and less known mathematical techniques to the understanding of some biological ...