There are N individuals in a population, some of whom have a certain infection that spreads as follows. Contacts between two members of this population occur in accordance with a Poisson process having rate λ. When a contact occurs it is equally likely to involve any of the (N choose 2) pairs of individuals in the population. If a contact involves an infected and a noninfected individual, then the noninfected individual becomes infected with probability p. Once infected, and individual remains infected throughout. Let {X (t)} denote the number of infected members of the population at time t. i) {X (t)} is a pure birth process. Find the birth rates λi for each state i, and write down the Q matrix. ii) Starting with a single infected individual, ï¬nd an expression for the expected time until all individuals are infected. iii) Suppose now that each infected individual loses his or her infection independently at rate µ. (Or you can imagine that each individual once becoming infected, remains so for an exponential distributed length of time, independently of other infected members.) Assume that individuals do not gain immunity. Is this a birth and death chain, and if so, what are the birth and death rates? Solution www.tdx.cat/bitstream/10803/31943/2/tasc.pdf.txt?.

1. There are N individuals in a population, some of whom
have a certain infection that spreads as follows. Contacts between two members of this
population occur in accordance with a Poisson process having rate λ. When a contact occurs
it is equally likely to involve any of the (N choose 2)
pairs of individuals in the population. If a contact
involves an infected and a noninfected individual, then the noninfected individual becomes
infected with probability p. Once infected, and individual remains infected throughout. Let
{X (t)} denote the number of infected members of the population at time t.
i)
{X (t)} is a pure birth process. Find the birth rates λi for each state i, and write down
the Q matrix.
ii) Starting with a single infected individual, ï¬nd an expression for the expected time
until all individuals are infected.
iii) Suppose now that each infected individual loses his or her infection independently at
rate µ. (Or you can imagine that each individual once becoming infected, remains so for an
exponential distributed length of time, independently of other infected members.) Assume
that individuals do not gain immunity. Is this a birth and death chain, and if so, what are
the birth and death rates?
Solution
www.tdx.cat/bitstream/10803/31943/2/tasc.pdf.txt?