# Questions tagged [mathematical-modeling]

This tag is used to refer to mathematical/probabilistic/statistical modeling questions, usually this tag is used to ask about questions that are related with the mathematical formalism of the model instead of the correctness of a specific model in practice.

113
questions

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### How to force my differential equations give a bounded solution?

I have modeled the interaction of two physical quantities, $S$ and $B$, by the following differential equations (the second one is a delay differential equation):
$$S'(t) = 0.31 S(t) \Big( 1 - \frac{S(...

0
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1
answer

59
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### How to integrate an indicator function/constraint into the cost function of a linear program?

I have a mathematical model $P$ for which I optimize two cost functions say $F_1$ and $F_2$ subject to a set of constraints $C1$–$C10$.
In $F_2$, I want it to be included only when its expression ...

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### Gaussian white noise model in application

I am interested in applications (to data) of non-parametric statistics, and my question concerned the Gaussian white noise model defined by,
$$
X_{t_1, \ldots, t_d}=f\left(t_1, \ldots, t_d\right) d ...

0
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1
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185
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### Poisson Process x SIR model [closed]

Consider the simplest SIR model:
$$S'=-a SI$$
$$I'=a SI - b I$$
$$R'=b I$$
It is known that the waiting time of an infeccious person in the compartment $I$ follows an exponential behavior with rate $b$...

1
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0
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82
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### How to smoothly interpolate gravitational field between trajectories in high dimension?

I'm looking for the adequate numerical interpolation technique to solve the following problem. This is probably trivial for physicists who study gravitational fields, but I didn't find clear answers ...

4
votes

1
answer

185
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### Equation in epidemic SIR model with the influence of vaccinations

I am currently preparing a presentation on different modifications of the SIR model. In my sources about the use of vaccines, I came across the following model for a specific rate at which the ...

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0
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80
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### Advice on constructing a Non-structural Flood Mitigation Model [closed]

I am not sure if this is the right site to post this. But I seek some valuable suggestions, and I believe I can get them here.
At present, I am in the final semester of my BSMS Mathematics course. I ...

3
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### Mathematical formulation of beam: get stress/strain from forces and momentum

I'm working with static beams with Euler–Bernoulli model which ODE is
$$
\dfrac{d^2}{dx^2} \left(EI \cdot \dfrac{d^2w}{dx^2}\right) = q(x).
$$
With a beam along the $x$ axis, the solution consists of ...

1
vote

2
answers

453
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### Why should the logarithmic series distribution model the number of "Items" bought?

Suppose you're a shopkeeper in the business of selling Items. An "Item" is a thing whose only property is that the quantity that can be bought by a purchaser must be a positive integer; all ...

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69
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### Turing reaction diffusion equations and neural networks

Suppose you have a Turing-type reaction-diffusion system
$$
\begin{cases}
\partial_t \phi = & f(\phi, \psi) + D_\phi \nabla^2\phi \\
\partial_t \psi = & g(\phi, \psi) + D_\psi \nabla^2\psi
\...

2
votes

1
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### Mechanics: Model beam using differential vectorial formulation

At the Wikipedia there are the differential formulation for Euler-Bernoulli Beam \eqref{1} and Timoshenko Beam \eqref{2}
$$
\begin{align}
&\dfrac{d^2}{dx^2}\left(EI\dfrac{d^2w}{dx^2}\right) = q(x) ...

3
votes

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131
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### How to mix Lagrange mechanics + KKT conditions?

Question: How can I mix the concepts of Lagrange Mechanics and KKT conditions? I've learned that Lagrange Mechanics derivation comes from variational calculus, and in some formulations, we can add ...

2
votes

1
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327
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### Examples of ODEs with complex constant coefficients and applications to physics?

This question is asked on stackexchange: Are there examples for ODEs with complex coefficients with applications in physics?
but received no answers. I am reposting it here on the hope that it catches ...

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### What is the meaning of column integrated fluxes?

I am solving an equation where one term $\bar{P}$ is given and is called the integrated column flux. In the equation, the term $P$ is the precipitation. I am doing this on the discrete domain.
Anyone ...

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0
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### How to define Mock Hadley Cell in mathematical modeling?

I am computing a force term in which one component is $F_{ext}$. To define this, the following content given in the paper.
To capture the possible large-scale effects on precipitation clusters, we ...

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3
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495
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### Integer linear constraint(s) for y= x1 XOR x2 [closed]

Is there any way to convert $y=x_1~ \text{XOR} ~x_2$ to linear constraints? It means we write some linear relations with:
if $x_1=x_2 =0$ or $x_1=x_2=1$ $\to$ $y=0$,
else, $y=1$?

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vote

1
answer

81
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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

1
answer

198
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### Literature on reaction diffusion equations

My research area is age structure modelling, basically when applied to reaction diffusion equations. We mainly discuss the existence of travelling wave solutions; I want to work on the stability of ...

2
votes

0
answers

101
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### How to quantify the non-commutativity of human body motion? [closed]

Some years ago, there was that question on this forum:"How to quantify noncommutativity?".
I am asking that question in a context, human movement, which implies kinematic chains (like in ...

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73
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### Real life applications of distributions through models or simulations [closed]

What are the areas we can apply distributions in classical harmonic analysis? I don't mean probability distributions but distributions that are continuous linear functionals on the space of test ...

6
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1
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264
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### 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 ...

3
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0
answers

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### 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 ...

2
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1
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131
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### Reference request: probabilistic models on climate (change)

I am looking for probabilistic models to address climate change. Are they known in the existing literature?
I have found the post Math behind climate modeling. concerning PDE models.
Many thanks for ...

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70
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### Canonical representation of the a probability distribution for Hammersley Clifford Theorem

I'm reading the following paper
http://www2.stat.duke.edu/~scs/Courses/Stat376/Papers/GibbsFieldEst/BesagJRSSB1974.pdf
On page 7 they give the result that
$$Q(\textbf{x}) = \sum_{1 \leq i \leq n} ...

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0
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84
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### Discrete-time model for spread of information when the probability of information transfer between each pair is known

[This question is cross-posted from MSE.]
I'm interested in the behaviour of the following sort of system.
We are given:
a finite set $X$,
a subset $A_0 \subset X$, and
a function $f : X \times X \to [...

6
votes

1
answer

462
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### Graphs resembling the math genealogy graph must have concentration in a small number of families?

I was talking with a non-mathematician the other week at a workshop about the fact that many mathematicians, like myself, are indexed in the math genealogy database. We talked a little about how many ...

2
votes

1
answer

76
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### Introductory literature on the Voter Model

I am looking for a good introduction to the voter model appropriate for the Bachelor-Maths level (Europe). I need something that introduces the model on a low level, as a Glauber dynamics or similar.
...

2
votes

1
answer

337
views

### Logistic sequence convergence

1) How can we prove that the logistic sequence
$$x_{n+1}=rx_n(1-x_n),\ x_1=a\in (0,1)$$
converges to $\frac{r-1}{r}$, for $r\in [1,3]$?
2) Also I wonder how can we prove that the sequence $(x_n)_{n\in\...

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votes

1
answer

142
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### What is the ideal form of an h-curve?

This question concerns mathematical modelling of the citation curve, well-known in the sciencemetry.
The citation curve (or else the $h$-curve) of an individual researcher is the vector $(c_1,c_2,\...

3
votes

0
answers

127
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### Notions of "completeness" and "sufficiency" of a mathematical model

I'm modelling a real-world problem as having instances $i$ in a set $P$. As a very simple artificial example, consider the problem of choosing a meeting room given a certain number of people. Then $i =...

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votes

1
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585
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### Generalized linear models: What's the benefit of the underlying theory?

I was recently trying to understand generalized linear models (GLMs) and after investing quite a few days, it still hasn't dawned on me what the fundamental benefit of the framework is. Normally, I am ...

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0
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123
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### Next-generation matrix of infectious disease

If the population is classified into $\mathbf{S}$, $\mathbf{E}$, $\mathbf{I}$ and $\mathbf{R}$ compartments such that
\begin{equation}
\label{eq4}
\begin{aligned}
\mathbf{S} &=\dfrac{S_{1}N_{1}+...

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votes

2
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291
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### Searching for an early, highly theoretical, even philosophical, math paper on models or small-world networks

All I can remember is that it was very high-level / abstact and kind of philosophical, explaining (the discovery or interdependence of) small world networks. I assume that it was +50 years old and '...

6
votes

1
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256
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### 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 -...

4
votes

1
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### 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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votes

2
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### Approximated solutions of SEIR models

Numerical solutions of the SEIR equations (describing the spreading of an epidemic disease) – or variations thereof –
$\dot{S} = - N$
$\dot{E} = + N - E/\lambda$
$\dot{I} = + E/\lambda - I/\delta$
...

3
votes

0
answers

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### Image restoration quality general lower bounds

A typical image restoration model posits that, starting from a true image $f = f(x,y)$, we observe
$$
\tilde f = f \star h + n
$$
where $\star$ is convolution, $h$ is the point spread function (caused,...

2
votes

0
answers

67
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### SIR model constraint [closed]

During these past months, I've heard a lot about some pandemic modelling techniques, specially the so-called SIR model. Before I begin, I'd like to stress that my interest and question are just a ...

2
votes

1
answer

274
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### How many persons pass your 1.5 meter neighbourhood during 1 week ? If the distribution is power law what is the exponent?

Consider a graph with vertices being people (in some region), and make an edge if one person pass another closer than say 1.5 meter during say one week.
(Such a graph might be thought a kind of ...

12
votes

6
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### 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 ...

8
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4
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### Virus community spread mathematical modeling [closed]

What is the basic math behind the Virus community spread mathematical modeling,and how the time variable;(in these models),interacts with knowledge (data)?.
I am not asking about how the virus is ...

0
votes

1
answer

726
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### Reflecting Boundary conditions for advection-diffusion equations

I am trying to model the dynamics of phytoplankton in a water column using one-dimensional advection-diffusion partial differential equations.
$$\frac{\partial P}{\partial t}= D\frac{\partial^2 P}{\...

30
votes

7
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### 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 ...

29
votes

15
answers

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### Unconventional examples of mathematical modelling

I'll soon be teaching a (basic) course on mathematical control theory. The first part of the course will focus on mathematical modelling of dynamical systems. More precisely, I will present examples ...

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0
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69
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### A mathematical area capable of describing nonstationary game-like problem [closed]

Here is my definition of the problem that I am trying to model:
Let's have two agents and an environment. Each agent can do two types of actions. They are either supporting the environment or don't. ...

0
votes

0
answers

100
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### Global stability question for system with a unique locally-asymptotically-stable steady state

I have an ordinary differential system of dimension 3 that contains a locally-asymptotically-stable unique fixed point. Additionally, the system is strictly-positively invariant and bounded.
Now, ...

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1
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### 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\...

1
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0
answers

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### Mathematical modeling of multi-site reaction-diffusion

(I asked a similar question on Mathematics SE, but based on the Help section it might be better suited for this site, as it is focused on research-level mathematical modeling.)
I am wondering if ...

2
votes

0
answers

49
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### Finding a queuing model for waste accumulation

I've been tasked with modeling the accumulation of solid waste in an urban setting. In particular, the objective is to find the steady state distribution describing the amount of waste in a given ...

3
votes

1
answer

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### How to value the extent of separation or mixing of point sets in plane?

As the image presented below, the reddish point set is totally separated from the blueish one and the greenish one, while the blueish point set is quite mixed with the greenish one.
A number of ...