I am self studying continuous time Markov Chains. The book unfortunately does not contain solutions to the problems. I am writing the problems and my answers to them, and kindly tell me if they are correct.
Problem 1: Consider a population comprising a fixed number $N$ of individuals. Suppose that at time $t=0$, there is exactly one infected individual and $N-1$ susceptible individuals in the population. Once infected, an individual remains in that state forever. In any short time interval of length $h$, any given infected person will transmit the disease to any given susceptible person with probability $\alpha h+o(h)$.(The parameter $\alpha$ is the individual infection rate.) Let $X(t)$ denote the number of infected individuals in the population at time $t > 0$. Then, $X(t)$ is a pure birth process on the states $0,1,... , N$. Specify the birth parameters.
Answer: For $k\geq1$, $$P[X(t+h)-X(t)=1|X(t)=k]=k(N-k)(\alpha h + o(h))$$ since to increase the number of infected individuals by $1$ I have to choose $1$ infected individual (out of $k$ choices) and $1$ susceptible individual (out of $N-k$ choices) so the birth parameters are $\lambda_k=k(N-k)\alpha$ for $1\leq k\leq N-1$ and $\lambda_k=0$ otherwise.
But one can also argue in this way: if $X(t)=k$ (and hence number of susceptible individuals is $N-k$), it does not matter "who" gives the disease. It only maters that out of $N-k$ susceptible individuals, one has to be chosen as that person gets the disease. So $$P[X(t+h)-X(t)=1|X(t)=k]=(N-k)(\alpha h+o(h))$$Which of these two viewpoints is credible? I think the first one is credible because of the sentence "In any short time interval of length $h$, any given infected person will transmit the disease to any given susceptible person with probability $\alpha h+o(h)$", so it sort of fixes a "disease-spreader" and a "disease-catcher".
Problem 2: A new product (a "Home Helicopter" to solve the commuting problem) is being introduced. The sales are expected to be determined by both media (newspaper and television) advertising and word-of-mouth advertising, wherein satisfied customers tell others about the product. Assume that media advertising creates new customers according to a Poisson process of rate $\alpha = 1$ customer per month. For the word-of-mouth advertising, assume that each
purchaser of a Home Helicopter will generate sales to new customers at a rate of $\theta = 2$ customers per month. Let $X(t)$ be the total number of Home Helicopter customers up to time t. Model $X(t)$ as a pure birth process.
Answer: I think the answer will be, for $k\geq0$, $$P[X(t+h)-X(t)=1|X(t)=k]=(1+2k)h+o(h)$$ I cannot justify why I wrote this, but it seems similar to a Yule process with migration. Very non-rigorous. If this idea is correct, can there be a formal justification?
Thank you for your help!