Questions tagged [mathematical-biology]

For questions that lie between the intersection of significant mathematical problems and fundamental questions in biology.

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How to "linearize" a term of the form $\frac{\partial}{\partial x} \left( m \frac{\partial c}{\partial x} \right)$ in a nonlinear PDE?

I am currently studying the following nonlinear PDE model for cells migrating under the influence of diffusion and chemotaxis: \begin{align} \hspace{2cm} \frac{\partial m}{\partial t} & = \...
Leonidas's user avatar
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Calculating pitch and roll from XYZ accelerometer data collected from a wildlife collar

I work with wildlife collars (lions, elephants) which also collect XYZ accelerometer data. From this, I can visualize pitch and roll of the animals in the collar manufacturers web application, but ...
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Genomic and sum of geometric random variables

In their paper The Maximum of independent Geometric Random Variables as the Time for Genomic Evolutionthe authors noted that if to consider the genomic word of L letters, than the measure of the time ...
user124297's user avatar
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Problem in performing sensitivity analysis in prey-predator model

I tried to create a general prey predator model to simulate sensitivity analysis using PRCC and Latin hypercube sampling. But I can a bar plots which are empty. What is wrong in my approach? The code: ...
LOVEMATH's user avatar
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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 ...
Enforce's user avatar
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Nondimensionalization of given equation

I have the following equation, which is a logistic system with a death term, $$\frac{dN}{dt}=rN(1-N/k)-p(N),$$ where $p(N)=\frac{BN^2}{A^2+N^2},$ for unknown constans $A,B$. I want to ...
Gonzalo de Ulloa's user avatar
6 votes
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96 views

A combinatorial model for multi-sexual reproduction

I was thinking about the following question: why do most creatures on earth reproduce asexually or bisexually, but not trisexually? Looking on the internet, I read an interesting perspective https://...
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2 votes
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How to slightly modify my ODE system in order to capture bump in the data?

I have the following two sets of data, which show the dependencies of two quantities, namely, $S$ and $B$, on time ($0$ h, $3$ h, $6$ h, $9$ h, $15$ h, $18$ h, $21$ h, and $24$ h): ...
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How to measure local density of A's in a sequence of A's and B's?

Let me preface this by stating that I am not a mathmagician. I have a DNA sequencing problem, but it boils down to a math problem. But I don't have the math skills to adequately describe the issue, ...
NAMEGOESHERE's user avatar
3 votes
1 answer
122 views

Stability and Asymptotical behavior of a nonlinear system

A simple mathematical model to describe how the HIV/AIDS virus infects healthy cells is given by the following equations: $$ \begin{align} \frac{dT}{dt} &= s - dT - \beta Tv \\ \frac{dT^*}{dt} &...
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Basic reproduction number for complicated diseases model

For an epidemic model, the basic reproduction number is defined as the average number of new infections (e.g. infectious individuals) generated by one infectious individual in an otherwise completely ...
N00BMaster's user avatar
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Solving the PDE $\frac{\partial n}{\partial t} = -v \frac{\partial n}{\partial \alpha} - \mu n $ using a given ansatz

I'm working on exercise 25 of Chapter 10 in Mathematical Models in Biology by Edelstein-Keshet. In the exercise we analyze the following chemotherapy model which accounts for the process of cell aging/...
Leonidas's user avatar
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Help clarifying a proof: next generation matrix in mathematical epidemiology

For the past few years I have been using the Next Generation Matrix to calculate the basic reproductive number $R_0$ in many epidemiological applications. Lately, I've tried to understand the proof of ...
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Choosing correct parameters in a model with hamiltonian equation

I am working on the hamiltonian of a system related to the extension of the Potts Model which is Cellular Potts Model. The total hamiltonian of the system is: $$ H = H_1 + H_2 $$ $$ H_1 = - J \sum_{\...
wallevic's user avatar
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Analytical solution of quadratic recurrence equation with limited capacity

Consider the famous equation associated to the logistic map: $$x_{n+1}=r x_n\left(1-x_n\right),\quad r>1 $$ It is well known that there is no closed formula for $x_n$ as a function of $n$. However, ...
Matt's user avatar
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Book recommendation about Modeling on Mathematical Biology

I would like to ask if you have a good book to suggest about Mathematical Models for the Dynamics of DNA and Proteins.
Athanasios Paraskevopoulos's user avatar
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How to approximate $1-\frac{1}{2\lambda}(\lambda+\mu+\beta-\sqrt{(\lambda+\mu+\beta)^2-4\lambda\mu})$?

Problem: I'm trying to verify a claim in this paper where the expression $$P^\infty_1=1-\frac{1}{2\lambda}(\lambda+\mu+\beta-\sqrt{(\lambda+\mu+\beta)^2-4\lambda\mu})$$ is approximated under three ...
Diplodokus's user avatar
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How do I interpret a low Jaccard similarity and a statistically significant Chi-Squared Value?

There are two genes I am interested in. They both have many samples in which they have no expression. So I decided to see if there is a relationship between when they are expressed. I was looking for ...
Ben Oppenheimer's user avatar
1 vote
1 answer
134 views

Reaction Diffusion Equation general solution

I have been struggling to find the general solution of the following BVP of reaction-diffusion equation: $$\frac{\partial N}{\partial t}=\frac{\partial^2 N}{\partial x^2}+N(1-N)-\sigma N$$ $$N(0,x)=...
Peachy April's user avatar
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Adjusting death rate based on known half-life

Most health applications use exponential decay for estimating the death rate. This assumes at every age, there is an equal chance of dying. Is there a simple way to slightly improve this assumption by ...
Paichu's user avatar
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4 votes
1 answer
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Help Finding a Traveling Wave Solution

I am looking for traveling wave solutions of \begin{align} \frac{\partial U}{\partial t} &= AU\left(1-\frac{U}{K}\right)-BUV+D_{1}\nabla^{2}U \\ \frac{\partial V}{\partial t} &= CUV-DV+D_{2}\...
Iesha Patterson's user avatar
5 votes
2 answers
103 views

Traveling-wave solution for McKendrick age-structure model with finite lifespan

The model was described many times before, so I keep the details concise. We have $$ \frac{\partial}{\partial t} \rho(t,a) + \frac{\partial}{\partial a} \rho(t,a) = -\delta(a) \rho(t,a) $$ where $\...
Durden's user avatar
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3 species Lotka–Volterra model. Limit cycle

Good day, I have 3 species Lotka–Volterra model. My goal is to determine if there is a limit cycle in the system $$ \left\{ \begin{array}{l} \frac{d c}{d t}=r_c c(1-c)-\frac{c h}{c+\theta_1} \\ \frac{...
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Manual calculation of the interaction p-value using effect estimates and 95% CIs of two (or more) subgroups.

I was wondering if there was a way to calculate the interaction p-value by hand using estimates from a published article that does not report the interaction p-value. For instance, if the hazard ratio ...
Spencer K's user avatar
0 votes
1 answer
126 views

Recovery Time for Logistic model equation with harvesting

We are given a logistic growth model with constant harvesting as: $\frac{dN}{dt} = rN(1-\frac{N}{K})-Y_0$ We are asked to show that the recovery time for harvesting a yield $Y_0$, $T_R(Y_0)$, ...
Green Ideology's user avatar
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Understanding the Role of Matrices in Capturing Disease Dynamics on Networks

I am currently reading an article related to Network epidemic, Markov chain and the relationship with automorphism graph, I'am confusing onp age 483: Here the link to the article What are the ...
Zbigniew's user avatar
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persistence of SEIR epidemic model

We have the following stochastic SEIR model $dS=\Lambda - \beta SI - \mu S - \sigma SI dB(t)$ $dE=\beta SI - (\lambda +\mu) E+\sigma SI dB(t)$ $dI=\lambda E-(\gamma +\alpha +\mu) I$ $dR=\gamma I-\mu R$...
Markiii's user avatar
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How do I calculate the equilibria, rate of production of infected cells, average lifespan of infected cells and basic reproduction number?

I have the following infection system: $ \frac{dx}{dt}=\lambda-d\cdot x(t)-\beta\cdot v(t)\cdot x(t) $ $\rightarrow$ Susceptible cells $ \frac{dy}{dt}=\beta\cdot v(t)\cdot x(t)-(a+d)\cdot y(t) $ $\...
williantafsilva's user avatar
1 vote
0 answers
55 views

Showing an endemic steady state is stable

I need to show that the steady state of this non-dimensional model is stable using minimal algebra however I am not sure how to approach this without long lines of working. The model is: $$\frac{dS}{...
user00134857693's user avatar
1 vote
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An alternative to the popular Hutchinson population model

As an alternative to the popular Hutchinson population model, which introduces a delay in the per capita growth rate, one can introduce a delay solely in the growth contribution and consider a ...
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2 votes
1 answer
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Epidemiology SEI disease model

I have the following model for simple endemic with susceptible (S), exposed (E), and infective (I), $$\frac{dS}{dt}=-\beta SI,$$ $$\frac{dE}{dt}=\beta SI-\delta E,$$ $$\frac{dI}{dt}=\delta E.$$ I have ...
user00134857693's user avatar
3 votes
0 answers
491 views

Some good books for ODE and Dynamical system.

I am trying to shift my research area from Pure math to math bio for various reasons. So whatever time I invested in my algebra is not of much use plus I have to make the basics of ODE and the ...
Ri-Li's user avatar
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How to find the basic reproduction number?

I have just begun studying stability of dynamic systems. I came across this model system where I have no information about the coefficients, but I assume that they are strictly positive: \begin{...
CHOSM's user avatar
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Can protein folding be used to solve SAT problems?

Given a sequence of amino acids, the protein folding problem is to find a geometric structure of the amino acids that minimizes energy. Given that this problem is NP-Hard, one should be able to do the ...
Kai Arakawa's user avatar
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Stability Analysis of DFE in terms of R0 Using Ruth-Hurwitz Criterion

I am working with an epidemiological model in the form of a dynamical system of seven ODEs and I am trying to show that the DFE is stable if R0<1. When applying the Routh-Hurwitz criterion, I see ...
Hew123's user avatar
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1 vote
1 answer
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Linearized Michaelis-Menten equation for LP

I am attempting to include the Michaelis-Menten equation in (continuous) variables $S$ and $v$ $$v=\frac{V_{\max} S}{K_m + S}$$ where $V_{\max}$ and $K_m$ are given constants, as a linear programming ...
Freiburgermsu's user avatar
4 votes
1 answer
220 views

How do you calculate how long until a drug reaches its steady state?

I encountered a mathematically intriguing conundrum, in that it's related to medicine but is centered around mathematics. Suppose drug A has a half-life in the body of 30 hours. The patient takes 40mg ...
askquestions4's user avatar
2 votes
0 answers
157 views

Calculation of the “basic reproduction number” (R0) for “reaction-diffusion-advection” type epidemic models

I would like to calculate the “basic reproduction number” (R0) for a “reaction-diffusion-advection” type epidemic model. I calculated R0 for the ordinary differential equations of my model using the « ...
Nell's user avatar
  • 53
2 votes
1 answer
75 views

Where to learn about whether a travelling wave solution to the reaction diffusion equation is a pushed or pulled wave?

I'm trying to understand pushed and pulled waves as seen in many biology articles such as: Gene Surfing in Expanding Populations by Hallatschek Spatial gene drives and pushed genetic waves by Tanaka ...
Yous's user avatar
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0 answers
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Stability of two consecutive equilibrium points

I was studying the existence of two species in a ecosystem. I was thinking if there could be two consecutive stable equilibrium points. I don't have a valid proof in this regard. But if geometrically ...
Manjoy Das's user avatar
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0 votes
2 answers
132 views

Why life expectancy calculate as 1/μ in SIR model?

In the SIR model, when the death rate is μ, life expectancy is calculated as 1/μ. Can anyone explain it intuitively?
Kaz's user avatar
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1 answer
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Finding the steady state solutions of a model.

Problem: Find the steady states and check for stability of the model \begin{align*} X_{t+1} &= rX_te^{r(1-X/k)-aY_t} \\ Y_{t+1} &= X_t(1-e^{-aY_t}) \end{align*} Attempt: Calculating ...
Tokita Ohma's user avatar
1 vote
0 answers
55 views

Finding nullclines and linearization about the steady state

Here's my system: $$ \left\{ \begin{aligned} \frac{du}{dt} &= u(1-u^2)-w \\[5pt] \frac{dw}{dt} &= u \end{aligned} \right. $$ I'm coming to the conclusion that the only steady state ...
user5896534's user avatar
2 votes
1 answer
86 views

For the given delay-differential equation, $x'(t)=x(t-T)e^{3-x(t-T)}-x(t)$, how do you find stability for given equilibria?

The delay differential equation given by, $x'(t)=x(t-T)e^{3-x(t-T)}-x(t)$, how to find stability conditions? So here's my attempt: to find equilibria we have, $x^*e^{3-x^*}-x^*=0$ which leads us to $x^...
user5896534's user avatar
1 vote
1 answer
69 views

For the difference equation $x_{n+1}=ax_n\exp(-x_n)$, find values of $a$ that lead to a period-doubling bifurcation and values that lead to extinction

For the difference equation, $x_{n+1}=ax_n\exp(-x_n)$, find values of $a\in(-1,1)\cup(1,e^2)$ that lead to a period-doubling bifurcation, and values that lead to extinction So for $x_{n+1}=ax_n\exp(-...
user5896534's user avatar
1 vote
1 answer
77 views

For given system of ODE's, in a region $\Omega$ in the $uw$-plane, show it's a trapping region for large R

Need help figuring out what this question wants me to do. No need to do entire problem for me, just need a push in the right direction on how to solve Here's my system: \begin{gather*} \frac{du}{...
user5896534's user avatar
0 votes
1 answer
73 views

Performing linear stability analysis for nonlinear discrete system by approximating function for large values of the varying bifurcation parameter

Here's my system, \begin{gather*} N_{t+2}=N_t\exp{[r(1-\frac{N_t}{K})]}\frac{1-e^{-aP_t}}{aP_t} \\ P_{t+1}=N_t[1-\frac{1-e^{-aP_t}}{aP_t}] \end{gather*} In the research paper, it states that ...
user5896534's user avatar
0 votes
1 answer
172 views

How to determine $R_0$, the basic reproduction number, for following system?

Consider a population of size $N$ with per capita birth rate $b(N)$ and death rate $d(N)$. Assume that it reaches stable steady state $N^*$ in absence of disease, with $b(N^*)=d(N^*)$. Find $R_0$ when ...
user5896534's user avatar
0 votes
0 answers
43 views

Where did the r value go for this ODE?

Question: Answer: So I got the same answer except mine has e^rt instead of e^t so can someone explain why the r just disappeared and if my answer is still correct?
Balkaran Mali's user avatar
0 votes
0 answers
33 views

How to determine bifurcation diagram for following system of difference equations?

So given this system, $\begin{gather*}N_{t+2}=N_t\exp[r(1-\frac{N_t}{K})]\frac{1-e^{-aP_t}}{aP_t}\\ P_{t+1}=N_t[1-\frac{1-e^{-aP_t}}{aP_t}]\end{gather*}$, how do I determine the bifurcation diagram? ...
user5896534's user avatar