Density dependent jump Markov processes
In the present section we shall approximate certain jump Markov processes as a parameter n, interpreted as the population size, becomes large. The results will be presented in a form general enough for our purposes. More general results, as well as other extensions, may be found in Chapter 11 of Ethier and Kurtz (1986), which has served as our main source. With the aim to explain the intuition behind the theory we start with a simple example, a birth and death process with constant birth rate (`immigration’) and constant individual death rate. The results in Sections 5.3 and 5.4 are applied to this example, thus giving explicit solutions. In Section 5.5 we apply the results to the Markovian version of the epidemic model described in Section 2.3. It is shown that this process converges weakly to a certain Gaussian process but in this case it is not possible to obtain explicit solutions for the deterministic limit and the covariance function. It is worth mentioning that the techniques presented in this chapter may be applied to a wide range of problems such as more general epidemic models and models for chemical reactions and population genetics, as well as other population processes.
KeywordsPoisson Process Central Limit Theorem Epidemic Model Death Process Standard Brownian Motion
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