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.

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3 votes
1 answer
111 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(...
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1 vote
1 answer
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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 ...
2 votes
0 answers
54 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: \...
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19 votes
4 answers
2k 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 ...
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6 votes
1 answer
195 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 ...
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3 votes
0 answers
87 views

Epidemics: distribution of interarrival times

In models of disease transmission, after an individual is getting infected, he can generate a number of secondary infections. The number of secondary infections depends on the infectiousness of the ...
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8 votes
3 answers
600 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 ...
3 votes
1 answer
109 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 ...
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0 votes
0 answers
164 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)-...
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6 votes
1 answer
218 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 -...
3 votes
1 answer
1k 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 ...
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64 votes
3 answers
5k 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 ...
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4 votes
2 answers
226 views

Approximated solutions of SEIR models

Numerical solutions of the SEIR equations (describing the spreading of an epidemic disease) – or variations thereof – $\dot{S} = - N$ $\dot{E} = + N - E/\lambda$ $\dot{I} = + E/\lambda - I/\delta$ ...
0 votes
0 answers
79 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 ...
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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 ...
13 votes
6 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 ...
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0 votes
1 answer
147 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 ...
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5 votes
0 answers
458 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 $\mathbb P^3 \times \mathbb P^3 \times \mathbb P^3$ ...
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4 votes
1 answer
608 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 ...
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6 votes
1 answer
373 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 ...
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30 votes
7 answers
5k 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 ...
17 votes
4 answers
1k 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 ...
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3 votes
0 answers
73 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 ...
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7 votes
1 answer
373 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\...
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9 votes
1 answer
938 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 ...
2 votes
1 answer
51 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 ...
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1 vote
0 answers
293 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 ...
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....
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10 votes
2 answers
5k 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 ...
10 votes
3 answers
2k 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
7k 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 — ...
27 votes
16 answers
7k 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 ...
36 votes
16 answers
9k 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 ...