Questions tagged [mathematical-modeling]

A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modelling.

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Trying to fit a model to “incomplete” data

Suppose the following dataset $\mathcal{D}=\{z_1,z_2,...,z_{80}\}$, was obtained by taking measurements on a natural phenomenon that follows the law: $z=f(x,y)=ax^2+by+c$, where $a,b,c \in \mathbb{R}$....
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Pandemic spread simulation learning subject [closed]

hello im a student from math department, i have a question about my project, pic above is covid 19 spread patern, i need to figuring out the math model of the spread to make a simulation what ...
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1answer
59 views

How to quantify fairness of exams?

So I was thinking about how to model exams. I assumed that teacher will give out $N$ problems to study. Due to the nature of exams, one can't have all $N$ problems on the actual exam, let's say ...
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What is meant with persistence time in stochastic models?

I've been reading about epidemiological modeling using stochastic models (discrete/continuous Markov chains and stochastic differential equations). I've come across the term persistence time multiple ...
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Is my average chance of death for a cohort correct

I am trying to model a human population. My simulation calculates at 1 day intervals. I want to track my population counts in cohorts of varying sizes (age 0, 1-14, 15-29, etc). The chance of death ...
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How can I find equationts for the EM field $E=(E_x,E_y,0)$ and $B=(0,0,B_z)$?

I am trying to find the motion's equation for a charged particle $q$ with mass $m$ in an EM field. First the 2 equations I have are: $$\frac{\mathrm dU_x}{\mathrm dt}= W_cU_y + \frac{W_cE_x}{B_z}\\ \...
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How to obtain the factor values in factor analysis?

I'm trying to understand factor analyisis. Let's say we observed $n$ $d$-dimensional values and want to have $p$ factors. Then after doing a factor analysis the j-th component of one observation can ...
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1answer
35 views

Double integral significance

If Integration $\int_{-1}^{1}\int_{-^{\sqrt{(1-x^{2})}}}^{^{\sqrt{(1+x^{2})}}}dy dx $ Gives area of a circle But what does this integral give? $$\int_{-1}^{1}\int_{-{\sqrt{(1-x^{2})}}}^{^{\sqrt{(1+...
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Plot a stiff system of three first order ODEs in Excel with sufficient accuracy

I want to plot a system of three stiff ODEs (Oregonator model - chemistry). That model describes a chemical oscillator. I do not have much experience with plotting ODEs, neither do I have experience ...
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dimensionless form of mathematic modeling [closed]

Given a set of equations : $\begin{aligned}&\frac{dX_1}{dt} = bN-PX_1-\mu hX_1-vX_1\\&\frac{dX_2}{dt}= PvX_1-(1-Pc)X_2-(Pc+\mu h)X2\\&\frac{dX_3}{dt}= (1-Pc)X_2-\mu hX3+vX1\\&\...
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Relation between Leslie dinamical system matrix and homogeneous 3rd degree recurrence relation

For a simple Leslie Matrix and asociated system we can get the value of a vector $x_0$ by using $X_{n+1}=AX_N$. For example with the following system $X_{N+1}=AX_N=\begin{pmatrix} 0 & 3 & 5\\ ...
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147 views

Method of characteristics with discontinuous source term

Let's say I have the system of equations $$ \frac{\mathrm{d}\rho}{\mathrm{d}t} = \left\{\begin{matrix} \beta, \hspace{3mm} x < 0 \\ 0, \hspace{3mm} x \geq 0 \end{matrix}\right., \\ \frac{\mathrm{...
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How to model a quasi-exponential function?

This is the graphic of official daily deaths caused by Covid-19 in Brazil. And this is the same graphic with an exponential model in red. We see the model don't fit the data. The misadjustment is ...
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Trouble Proving Invertability

I have this theorem. So if A is a nonnegative matrix, then the following conditions are equivalent. $(I-A)^{-1}$ exists and is non-negative There is a non-negative vector $\vec{x}$ so that $(I-A)\...
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Interpreting the reciprocal of the reproductive ratio, $R_0$

$R_0$ is the average number of secondary cases arising from a single infectious individual in a fully susceptible population. In many of the compartmental models for epidemiology, the parameter $\...
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QMLE and MLE differences [closed]

What is the difference in computation between the MLE and the QMLE?.I heard that the estimators are the same but the covariance is different. Thank you!
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real dynamical systems

I am a bit confused on how to solve this system or generally these kind of systems $$x(t+1)= x(t) -2y(t)$$ $$y(t+1) = 2x(t) +y(t)$$ My confusion derives from that fact that I am working with $(t+1)$ ...
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1answer
25 views

Tiles, dominos, black and white, even and odd [closed]

A cell is good if the board without this cell can be tiled by dominoes ($1\times 2$ tiles). What is the number of good cells? (In other words, you want to delete one cell in such a way that the rest ...
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Angles of a suspended lamina using centre of mass

A uniform rectangular lamina $ABCD$, where $AB$ is of length $a$ and $BC$ of length $2a$, has a mass $10m$. Further point masses $m$, $2m$, $3m$ and $4m$ are fixed to the points $A$, $B$, $C$ and $D$ ...
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How does the reproduction number depend on characteristics of the physical contact graph of a population?

Related question at Biology SE: How to model social structure in SIR models Preliminaries Let me start with the definition of the reproduction number of an airborne virus, e.g. as given here, only ...
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Help with this transfer function of a geared servo motor

I need to find the Transfer Function. I have these mathematical model equations (already with Laplace applied): 1) $U(s) = Kp\left( \theta I(s) - \theta_{o}(s) \right)$ 2) $U(s) = RI(s) + LsI(s) + ...
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The ”SIR” model is for investigating spread of a disease [closed]

The ”SIR” model is used to predict the spread of an infectious disease. Our aim with this question is to observe in detail how one can apply mathematics to investigate spread of a disease. • Let S(t) ...
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1answer
34 views

Leslie Matrix for Population of Plants and Seeds

I am asked to onsider a population of annual plants with the following life cycle: Seeds are produced at the end of summer, after which all adult plants die. A proportion of the seeds survives winter ...
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Modeling Via ODE System

My Problem José invests $\$2500$ of his $\$5000$ into an account with a $6\%$ annual interest rate, compounded continuously. If he continuously deposits an additional $3.5\%$ of his annual ...
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1answer
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Sketch a phase portrait for the non-linear system $\dot x =(1+x-2y)x$, $\dot y = (x-1)y$

Example 3.10 Sketch a phase portrait for the non-linear system \begin{aligned} \dot x &=(1+x-2y)x \\ \dot y &= (x-1)y \end{aligned} To do this we need to identity the critical points which ...
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Where does the Maximum Theorem break down with separate continuity?

If we have a function $f:X \times Y \rightarrow \mathbb{R}$, then Berge's Maximum Theorem states that if $f$ is jointly continuous on $X \times Y$, where $X$ and $Y$ are compact,then the argmax ...
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Stationary increment datasets

I have a stochastic counting process that has stationary increments and am struggling to find applicable datasets that I could use for modeling. (The process is not Poisson.) I know that I could ...
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1answer
55 views

Understanding the model of a fishery's effect on a fish population

The following differential equation is given to model the population of fish which is affected by a fishery: $\frac{\text{d}{U}}{\text{d}T}=rU\left(1-\frac{U}{K}\right)-\alpha\frac{U}{H_2+U}$ The ...
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1answer
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Modelling health using differential equations

Note: this is mathematics question but necessarily involves some non mathematical concepts. I hope that we can consider it to be a good fit for this site. If not, happy to close, and it will still ...
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Find a function

It's a cuttout from Rempe article "A mathematical model of the sleep/wake cycle" about a function which I don't understand how to describe: $h(t)$ decays exponentially while the system is asleep and ...
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a non linear curve that returns values between 0 and 1 but the x axis range is very long.

I am new to mathematical modelling and trying to understand how I can fit a non linear curve for my requirement below The curve should increase from 0 and finish at 1 within the given range of ...
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How does Elo rating scale with games played?

I only take two players starting out at 0 Elo and have them play against each other with one player winning all games. Also I consider a pure version of the Elo system without artificially introduced ...
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2answers
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Why is the recovery time for a logistic model $\frac{1}{r}$

What I mean by Logistic Model is the case that $\frac{dN}{dt}=f(N)=rN(1-\frac{N}{K})$. Also the steady states are $N$ when $f(N)=rN(1-\frac{N}{K})=0$ This model has two steady states, one at $0$ ...
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How do I find the unknown variables in a logistic equation?

I am trying to fit a function to the spread of a disease, in this case, the disease is Swine Flu. So, I eventually stumbled upon the logistic equation: $$y=\frac C{(1+a(e^Bx))}$$ I saw a video that ...
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1answer
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how to get $f(t,y)$ function of Verhulst differential equation

Take the Verhulst logistic differential equation down blew: $$I'(t) = \beta I(t)\left(1-\dfrac {I(t)}N \right)$$ with beta an N as constants. How can I write this equation as a f(t,y) equation int ...
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3answers
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SIR model - Why does an increase in infectious individuals cause the rate of infectious to recovered to increase?

When looking at dI/dt we have a term gamma*I, this is the rate in which infectious become recovered. Why is this dependent on I, the number of infectious people? Since if you have lots infectious ...
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2answers
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Express three-factor product in matrix form

For a small project of mine, I would like to model a phenomenon (impact of mood and opinion on a social network's member). The data I have is stored in square, same-sized matrices. I would like to ...
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MAPLE and equilibrium points

for example in this example Solving system of nonlinear differential equation in MAPLE I want evaluate the jacobian matrix of the system for each equilibrium point with mapel ?
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Why is “five-divide method” and “random-divide method” equal in “number guess game”?

set a range like [0,1000)digital, pick one number a like 733. "two-devide-method":First time you choose 500,and the host tells you it is smaller than a.Then you pick 750,next (500+750)/2.until pick ...
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Josephus Problem from 1 to N

can someone explain to me the recursive soultion of josephus problem I have found many solutions here and in other places but i can't understand it the recursive solution is ...
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139 views

Mathematics system components probability

For mathematics, I have to perform the following exersice. Consider a system consisting of 6 components: C1, C2, C3, C4, C5 and C6. The system works if at least 3 components work and at least 1 of ...
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Multicollinearity in logistic regression model

Suppose I want to fit a logistic regression model to my training data.However, I know that some of my regressors may be dependent on each other i.e. some of them are functions of common covariates for ...
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1answer
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Mathematical modeling for mixtures

A $200$ liter tank contains $100$ liters of pure water. Starting at time $t = 0$, alcohol is pumped into the tank at a rate of two liters per minute. At the same time the well-stirred mixture is ...
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1answer
20 views

Initial values for oscillations about a point problem (normal modes)

I have the Lagrangian $L=\frac{1}{2}(\dot{x}^2+\dot{y}^2) -7x^2+2xy-\frac{11}{2}y^2$. So the two equations of motion are $\ddot{x}+14x-2y=0$ and $\ddot{y}-2x+11y=0$. Hence the general solution (by ...
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Explain using mathematical terms

I have a list of job descriptions that require mathematical models to explain each of them, there are 8 tasks in number and would like explain the tasks to a non-mathematician. The tasks are as ...
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Simulation of Hard Core Model

i already learned about Hard core Model and i know the steps Using markov chain monte carlo (MCMC) to realise it.But i don't know how to make/do a Simulation of Hard code Model in practice. Can ...
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1answer
17 views

Linear Program for Hyperplane Sparation

GOAL: I want create a simple linear programming model that can be used to classify new data based off of a found hyperplane. In this particular case I have two classes and I'm given a data set in ...
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1answer
26 views

Different state space models derived from the same differential equation

I hace come accross the book "Robust Adaptive Control" from the authors "Petros A. Ioannou" & "Jing Sun". There is a free PDF version of the book at the webpage of one of the authors if anyone is ...
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1answer
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No penalty functions are perfect.

The quadratic penalty function was not great because we needed $ρ → ∞$ to get an optimal solution, and for finite values of $ρ$ we could get infeasible solutions. The exact penalty solved this issue, ...
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How do you do Monte Carlo without computer programs?

I am trying to approximate the area under the curve of $f(x) = \sqrt{x}$ over the integral $1/2 ≤ x ≤ 3/2$. I am so confused on how to use Monte Carlo without the use of a computer program to estimate ...

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