# Tagged Questions

Questions related to difference equations, which are discrete analogs of differential equations.

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### How to find the basic reproductive number of a discrete SIS epidemic model

I have been following a textbook called Mathematical Models in Population Biology and Epidemiology. The SIS model is given by the system \begin{aligned} S_{n+1} &= \Lambda + S_n e^{-\mu} ...
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### What are the main similarities between difference equations and differential equations, specifically in methods for solving them?

I know that that difference equations can be used to represent discrete dynamical systems and differential equations can be used to represent continuous dynamical systems. Therefore both are ...
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### A similar, but hopefully easier problem than Gilbreath's conjecture

Gilbreath's conjecture says that for every positive integer $n$, if we write out the first $n$ primes $2,3,5,7,11,13,\ldots,p_n$ take the differences between consecutive terms ...
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### Using Difference Equations to Solve Word problems

While I was studying about finite differences I noticed a question in difference equations.Does anyone knows how to solve this using difference equations? WORD PROBLEM Imagine you are to jump from ...
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### Let $g_n^k=p_{n+k}-p_n$, where $p_n$ is the $n$th prime. Does there exist $g_{k+1}^1=2$ such that $g_1^k,g_2^k,\ldots$ is a “Gilbreath sequence?”

Call $(S_i)_{i=1}^{\infty}$ a Gilbreath sequence if $1=\lvert S_2-S_1\rvert=\lvert \lvert S_3-S_2\rvert-\lvert S_2-S_1\rvert\rvert=\cdots$, i.e., if the sequence can be substituted for the primes in ...
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### Probability of a population increasing from size $N$ to size $N + 1$ in a time interval $(t, t + dt)$?

Let $\lambda$ be the birth rate and consider a time interval $(t, t + dt)$. If we have a population of size $N$ the probability of it increasing to size $N + 1$ within the interval $(t, t + dt)$ is ...
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### Solving $T(n)= 2T(n/2) + \sqrt{n}$ without master theorem (algebraically & recurrence tree)

$$T(n)= 2T(n/2) + \sqrt{n}$$ This recurrence was in a stackoverflow question, and I want to solve it without relying on the master method. The solution was given, but wolframAlpha gives a slightly ...
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### Difference equation of second order system with zero

I saw from lecture notes that difference equation of a first order system is like this: (1) (2) (3) (4) 1. What happens between (3) and (4)? It looks like inverse Z-transform but according ...