# Pure Birth Process Question

I would appreciate any possible help for this question because I have no clue what to do! Thanks so much!

Consider a population made of a fixed number (N) of people. At time t=0 there is only one infected individual and N-1 susceptible people in the population. When you get infected, you remain in the infected state forever. In any short time interval that is h long, any given infected person will transmit the disease to any susceptible person with a probability of alpha * h + o(h) where o(h) is an error term and alpha is the individual infection rate. Let X(t) denote the number of infected individuals in the population at time t >= 0. So X(t) is a pure birth process on states 0, 1, ... N.

What are the birth parameters?

EDIT:

Some issues I am having are: 1) What exactly does the question mean by birth parameters? 2) How would I set up this problem in the form of a probability statement? 3) Any hints to help me out would be great!

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Hint: Suppose you have k infected people and N-k non-infected ones at the start of your very small interval of time h. Let $A_{ij}$ be the event that infected person #i transmits the disease to non-infected person #j in this time interval. The assumption should be that each $P(A_{ij}) = \alpha h + O(h^2)$ and the $A_{ij}$ are independent. So what about the probability (up to $O(h^2)$) of the union of all the $A_{ij}$?

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Hi! Thanks for the response. Would you kindly be able to set up the problem in mathematical notation (in terms of probability statements) because I am having difficulty following the hint. Also, what exactly are "birth parameters"? Is the question looking for me to find the value of lambda? – icobes Mar 19 '11 at 1:14
An additional question: will this require me solving differential equations? – icobes Mar 19 '11 at 1:28

The number of persons is $N$. there are two cases. these are; when it is in state zero ($0$) and when it is in a state greater than $0$.

case 1: when in state zero at time $t$ and at time $[t+h)$ it is still in zero. i.e the person not affected. is has a propability given as: $$p_0(t+h)=p_0(t)[1-\alpha*] +(0)h \quad \text{equ1}$$

case 2: when it is in state $n-1, n \geq 1$, at time $t$, and at $[t,t+h)$ it is in $n$. it means there is a transmission. so the probability is: $$p_{n-1}(t) \cdot \alpha \cdot h + (0)h,$$ and if in state $n, n \geq 1$ at time $t$, and in time $[t,t+h)$ it is in state $n$. the probability is: $$p_n(t) \cdot (1- \alpha) + (0)h.$$ so for case $2$ we have: $$p_n(t+h)= p_n(t)[1-\alpha] + p_{n-1}[\alpha] + (0)h \quad \text{equ 2}$$ when you solve equ 1 and 2 it will give you two differntial equations as: $$p_0'(t) = -\alpha \cdot p_0(t) \dots 1$$ $$p_n'(t) = -\alpha \cdot p_n(t) + \alpha \cdot p_{n-1}(t), \text{ for } n \geq 1 \dots 2$$

initial conditions: at state zero, only one person was affected, and at state $n, n \geq 1$, no other person was affected, so we have: $$p_0(0) = 1, \text{ and } p_n(0)=0.$$ when you solve those two equations using the initial conditions, you will have an expression for the birth parameter. although its ALPHA. BIRTH PROCESS IS A KIND OF STOCHASTIC PROCESS THAT ASSUMES A PROCESS TO HAVE ONE DIRECTION. I.E. NO DEATH., FROM STATE $N$ TO $N+1$. NO GOING BACK TO $N-1$. NOTE: I USED $p(t)$ INSTEAD OF $X(t)$

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This is unreadable. Can you please format better (including line breaks and whatnot)? You could use MathJax (LaTeX-like engine) to format the mathematics. – Asaf Karagila Mar 1 '13 at 23:32
Welcome to MSE, benjamin. I put this into mathjax, but some of your notation wasn't perfectly clear to me. Please have a look and change what, if anything, needs changing. Also, please avoid all caps language. If you would like to emphasize, you can put passages between *s (italics) or **s (bold face). You can click on 'edit' and make changes to your own post, and also if you'd like to compare how this was put into tex. – gnometorule Mar 1 '13 at 23:43